Three-Dimensional Motion Control and Dynamic Force

Sensing of a Magnetically Propelled Micro Particle Using a

Hexapole Magnetic Actuator

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy

in the Graduate School of The Ohio State University

By

Fei Long, M.S.

Graduate Program in Mechanical Engineering

The Ohio State University

2016

Dissertation Committee:

Chia-Hsiang Menq, Advisor

Manoj Srinivasan

Rama Yedavalli

Vadim I. Utkin

Copyright by

Fei Long

2016

ii

Abstract

This dissertation presents the development of a hexapole 3D magnetic actuator that

can be used as a probing system by actively controlling a magnetic bead in three

dimensional space. The magnetic force, which is a noncontact force, is an ideal force for

biological applications due to its biocompatibility and magnetic susceptibility. This

magnetic actuator can achieve magnetic bead stabilization, trajectory tracking, accurate

force modeling and dynamic force sensing. These capabilities will transform the

magnetic actuator from the traditional force applier to a three-dimensional scanning

probe system, which is not achieved in other magnetic actuator systems.

An over-actuated Hexapole magnetic actuator is employed to realize 3D motion

control of the magnetic bead. A lumped parameter magnetic force model is derived to

characterize the nonlinear relationship from the input current to the output magnetic

force. This electromagnetic actuating system achieves significantly greater force

generation capability compared with existing magnetic actuators [1, 2]. These

improvements are accomplished through enhanced design and optimization of the current

allocation of the over-actuated system. A magnetic bead can be stably controlled and

steered and the magnetic force model is experimentally validated.

The fundamental issues in this over-actuated multi-pole actuator are caused by the

following four characteristics of the magnetic force: a) redundancy and coupling, b)

instability, c) nonlinearity, and d) position dependency. An optimal inverse model of the

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over-actuated hexapole electromagnetic actuating system over the 3-D workspace is

derived to minimize 2-norm of the six input currents when applied to produce the desired

3-D magnetic force on the magnetic bead. This inverse model greatly facilitate the

feedback linearization in the feedback control. Due to the compact form of the optimal

inverse model, it can be implemented in high speed real-time control to achieve stable

magnetic bead trapping and precise motion control.

Another challenge in electromagnetic actuation system is the hysteresis effect. The

existing current-based magnetic force model relies on the assumption that the magnetic

flux generation is proportional to the input current, which is not valid under hysteresis

effect. The hysteresis effect will greatly degrade the magnetic force model accuracy. As

soft magnetic material is used in this magnetic actuator, the hysteresis effect is not only

nonlinear but also rate-dependent. Moreover, the magnetic coupling among six magnetic

poles will make the hysteresis issue beyond the capability of modeling. To solve this

problem, Hall sensors are introduced to directly measure the magnetic field in each pole

and a so called Hall-Sensor-based magnetic force model is proposed. Owning to the 3-D

motion control capability, the Hall-Sensor-based and current-based magnetic force

models can be experimentally calibrated and compared by steering the magnetic bead

wherein the viscous force can serve as the reference force. It is clearly seen that the Hall-

Sensor-based magnetic force model greatly outperforms the current-based magnetic force

model in term of force modeling accuracy. When the magnetic field becomes larger, the

magnetization saturation of the magnetic bead begin to emerge and a more accurate Hall-

iv

sensor based model is proposed in which the nonlinear magnetization effect of the

magnetic bead is modeled.

With accurate magnetic force model, a dynamic force sensing estimator can be

developed to achieve real-time dynamic force sensing and parameter estimation

simultaneously. With the measurement information of the magnetic bead and the Hall-

sensor based magnetic force model, the bead-sample interaction force can be dynamically

estimated. Moreover, the drag coefficient can be also estimated, which is an indication of

the environment change such as the wall effect, fluid property change and etc. The

Kalman filter algorithm is used to estimate the state variables since the dynamics subject

to random thermal force and measurement noises. Combined with motion control

capability, this magnetic actuator can achieve force control application and automatic

scanning of an unknown environment.

v

To my grandfather

vi

Acknowledgments

In the past one fifth of my life, I was very lucky to spend a meaningful time in the

Precision Measurement and Control Lab (PMCL). I would like to sincerely thank my

advisor Dr. Chia-Hsiang Menq, who continuously encourages me, challenges me and

advises me with his enthusiasm. His inquisitiveness and logical thinking inspired me all

the time. I am also very thankful for Dr. Manoj Srinivasan, Dr. Rama Yedavalli, and Dr.

Vadim Utkin for agreeing to be my committee members and giving me valuable advices.

I am very grateful to dear PMCL friends, with who I shared scores of achievements

and frustrations. I want to thank Dr. Peng Cheng for his strong support in all aspects,

especially the image processing work, which might be the most beautiful real-time visual

sensing system in this world. I sincerely thank my colleagues, Zhen Liu and Yanhai Ren,

with who I spent countless days and nights. They are always willing to give their

intelligent suggestions and kind encouragement. I want to thank my seniors Dr. Zhipeng

Zhang and Dr. Yanan Huang, who gave me a lot of constructive opinions in research and

life.

I want to thank Dr. Daisuke Matsuura, who finished the mechanical design part of

this project and put in all his effort to facilitate my work at the beginning of this project.

Most of all, I am so grateful to my beloved parents and grandparents. Especially, I

want to thank my grandparents. I am not able to become who I am if I were not brought

up by them. I cannot imagine how lucky I was to be influenced by their noble characters.

Finally, I am deeply indebted to my grandfather, whose interest in mathematics inspired

me to become a researcher.

vii

Vita

Feb, 1987 ....................................... Born – Dandong, Liaoning, China

2005 ............................................... Huainan No.2 Middle School, Huainan, Anhui,China

2009 ............................................... B.S. Mechanical Engineering, Shanghai Jiao Tong

University, Shanghai, China

2009 – 2010 .................................. University Fellowship, The Ohio State University

2010 - Present ............................... Graduate Research Assistant, PMCL, The Ohio State

University

Publications

1. Z. Zhang, F. Long, and C. H. Menq,“Three-dimensional visual servo control of a

magnetically propelled microscopic bead,” IEEE/ASME Trans. Robot. 29, 373–

382 (2013)

2. F. Long, D. Matsuura, and C. H. Menq,“ Actively Controlled Hexapole

Electromagnetic Actuating System Enabling 3-D Force Manipulation in Aqueous

Solutions,” IEEE/ASME Trans. Mechatronics., (accepted)

3. F. Long, P. Cheng, and C. H. Menq, “Optimal Inverse Modeling and Control of

Hexapole Electromagnetic Actuation”, (in process)

4. F. Long, P. Cheng and C. H. Menq, “A Hall Sensors Based 3-D Magnetic Force

Modeling and Experiment of an Actively Controlled Hexapole Electromagnetic

Actuator”, (in process)

5. F.Long, P. Cheng and C. H. Menq, “Accurate 3D Magnetic Force Modeling and

Experiments of Superparamagentic Magnetic Bead Propelled By Hexapole

Electromagnetic Actuator”, (in process)

Fields of Study

Major Field: Mechanical Engineering

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Table of Contents

Abstract ............................................................................................................................... ii

Acknowledgments.............................................................................................................. vi

Vita .................................................................................................................................... vii

List of Figures .................................................................................................................. xiii

Chapter 1: Introduction ................................................................................................... 1

1.1. Background and Motivation ............................................................................. 1

1.2. Objectives and specific aims .............................................................................. 5

1.3. Dissertation Overview ........................................................................................ 8

Chapter 2: design, force modeling and inverse modeling of the hexapole magnetic

actuator ............................................................................................................................ 10

2.1 Introduction ........................................................................................................... 10

2.2 Design of Hexapole Electromagnetic Actuator ................................................... 12

2.2.1 Design, synthesis, and fabrication .................................................................... 12

2.2.2 Finite element analysis of the magnetic field ................................................... 15

2.2.3 Hexapole magnetic field model ........................................................................ 17

2.3 Force Model and Inverse Modeling ..................................................................... 21

2.3.1 General magnetic force model .......................................................................... 21

ix

2.3.2 Current-Based Magnetic Force Model ............................................................. 22

2.3.3 Inverse Model Based on Constant Constraints ................................................. 24

2.3.4 Optimal inverse model at the center of the workspace ..................................... 25

2.4. Experimental Verification of the Optimal Inverse Model ................................ 28

2.5. Calibration and Validation of the Force Model ................................................ 31

2.6. Force Generation Capability ............................................................................... 38

2.7. Conclusion ............................................................................................................. 41

Chapter 3: The Optimal Inverse Model and Control of Hexapole Electromagnetic

Actuation .......................................................................................................................... 43

3.1 Introduction ........................................................................................................... 43

3.2 Hardware Implementation of High Speed Control ............................................ 44

3.3 The inverse model.................................................................................................. 46

3.3.1 The position dependent inverse model based on constant constraints ............. 46

3.3.2 The Optimal Inverse Model in the Entire Workspace ...................................... 47

3.4 Active feedback control: stabilization, Brownian motion control and tracking

control ........................................................................................................................... 53

3.4.1 Stabilization and Brownian motion control ...................................................... 53

3.4.2 Trajectory tracking ........................................................................................... 57

3.5 Comparison of High Speed Control in FPGA and Low Speed Control in PC 60

x

3.6. Conclusion ............................................................................................................. 63

Chapter 4: Hall Sensors Based 3-D Magnetic Force Modeling and Experiment ..... 65

4.1 Introduction ........................................................................................................... 65

4.2. The Magnetic force models.................................................................................. 66

4.2.1 The Hall-sensor-based magnetic force model .................................................. 66

4.2.2 The current-based magnetic force model ......................................................... 68

4.3 Hardware Integration and Validation of Hall Sensors based Hexapole

Magnetic Actuator....................................................................................................... 70

4.3.1 Hall sensors integrated with magnetic actuator ................................................ 70

4.3.2 Validation of the Hall Sensor measurement ..................................................... 71

4.4 Hall Sensor Measurement in Magnetic Bead Control and Calibration Of

Magnetic Force Models ............................................................................................... 73

4.4.1 Hall Sensor measurement in bead trapping ...................................................... 73

4.4.2 Force model calibration .................................................................................... 74

4.5 Application of Hall-Sensor-Based Model and application of Hall Sensors ..... 82

4.5.1 Force Prediction ................................................................................................ 82

4.5.2 Magnetic Field Feedback Control Using Hall-Sensor inner loop control ........ 86

4.6 Accurate Hall sensor based Magnetic Force Modeling for large magnetic force

generation ..................................................................................................................... 90

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4.6.1 The modeling of the magnetic bead magnetization .......................................... 90

4.6.2 The Modeling of The Accurate Hall Sensor Based Magnetic Force ................ 93

4.6.3 Calibration of the Magnetic Force Model ........................................................ 95

4.6.4 Application of the Accurate Hall-Sensor-Based Model ................................. 101

4.6.4.1 Magnetic Force Prediction Using the Accurate Hall-Sensor-Base Force

Model .................................................................................................................. 101

4.6.4.2 Parameter calibration and magnetization study ...................................... 105

4.6.4.3 Accurate Hall-sensor-based optimal inverse modeling .......................... 106

4.7 Conclusion ............................................................................................................ 109

Chapter 5: Dynamic force sensing and parameter estimation ................................. 111

5.1. Introduction ........................................................................................................ 111

5.2 Joint State-Parameter Estimator Algorithm .................................................... 113

5.2.1 Drag coefficient estimation based on the magnetic force model.................... 113

5.2.2 Drag coefficient estimation based on the thermal variance measurement ..... 118

5.3 Simulation result the joint state-parameter estimator ..................................... 120

5.3.1 The simulation of the drag coefficient estimation in water ............................ 120

5.3.2 The simulation of the drag coefficient estimation in Glycerol ....................... 121

5.3.3 The simulation of the simultaneous estimation of the drag coefficient and the

external force ........................................................................................................... 123

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5.4 Experiment results .............................................................................................. 125

5.5 Conclusion ............................................................................................................ 129

Chapter 6: Conclusion and future works ................................................................... 131

6.1 Conclusion ............................................................................................................ 131

6.2 Future Works....................................................................................................... 134

BIBLIOGRAPHY ......................................................................................................... 136

xiii

List of Figures

Fig. 2.1. (a) CAD model of motion stage and lower poles. (b) Assemble yoke ring and

three upper poles. .............................................................................................................. 12

Fig. 2.2. (a) Fabricated prototype, integrated on an inverted microscope. (P2, P4, P5)

forms the upper layer and (P1, P3, P6) forms the lower layer. Each pole is associated with

an actuation coil (for flux generation) and a measurement coil (for flux measurement). P2

→P1, P4→P3 and P6→P5 are +x, +y and +z directions of the actuation coordinate

system. (b) CAD model and meshing of the hexapole actuator ....................................... 12

Fig. 2.3. (a) The top view (measurement coordinate) of the vector plot of the magnetic

flux density distribution (unit: Tesla). (b) The magnetic flux density vectors near the

workspace center (actuation coordinate). (c) The magnetic field vectors near the tip of

pole 1. ................................................................................................................................ 16

Fig. 2.4. Magnitude of the magnetic flux density, i.e., |B|, associated with the

measurement coordinate system. (a) |B| in the horizontal plane (top view). (b) |B| in the

vertical plane (side view). ................................................................................................. 17

Fig. 2.5. (a) Hexapole magnetic actuator integrated on an inverted microscope. (P2, P4,

P5) forms the upper layer and (P1, P3, P6) forms the lower layer. Each pole is associated

with an actuation coil (for magnetic flux generation). P2→P1, P4→P3 and P6→P5 are

+x, +y and +z directions of the actuation coordinate system. (b) CAD model of six tips of

the hexapole magnetic actuator and the associated Measurement Coordinate {O;x ,y ,z }m m m

xiv

and Actuation Coordinate {O;x ,y ,z }a a a. (c) FEM analysis in ANSYS about magnetic

charge model of a sharp-tipped magnetic pole. (d) The sketch of the magnetic bead and

six magnetic charges. ........................................................................................................ 17

Fig. 2.6. Validation of the hexapole magnetic field model: (a) comparison of magnetic

induction vectors, and (b) normalized norms of error vectors. (c) the definition of the

fitting error. ....................................................................................................................... 21

Fig. 2.7. Illustration of the desired force in the spherical coordinate system. .................. 26

Fig. 2.8. Three orientation-dependent optimal constraints compared with their

corresponding constant constraints. .................................................................................. 27

Fig. 2.9. Optimal current allocation compared with that obtained using constant

constraints. ........................................................................................................................ 28

Fig. 2.10. Block diagram of the feedback motion control (dP is the desired position,

mP is

the measurement position, e is the error signal,dF is the desired force calculated by the

controller, ' (t) (t )d d a F F is caused by the actuator delay, MTF is the magnetic force, F is

the modeling error, TF is the thermal force,

a and m are the delay in actuator and

measurement) .................................................................................................................... 28

Fig. 2.11. Six input currents of the hexapole actuator ...................................................... 30

Fig. 2.12. Stabilization of the magnetic particle at the center of the workspace .............. 31

Fig. 2.13. Actuation current applied to coil 1 and voltage readings from the six

measurement coils. ............................................................................................................ 32

Fig. 2.14. Twelve linear trajectories on the horizontal plane passing the center of the

workspace (in measurement coordinate) .......................................................................... 34

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Fig. 2.15. Particle motion and viscous force displayed with reference to the actuation

coordinate, wherein black dashed lines are target motion and color solid lines are

measured motion (plotted in time sequences). .................................................................. 35

Fig. 2.16. Six actuation currents associated with motion control ..................................... 35

Fig. 2.17. The best-fitted results when using the nominal flux distribution matrix to

calibrate the force gain vector. .......................................................................................... 37

Fig. 2.18. The best-fitted results when using the measured flux distribution matrix to

calibrate the force gain vector ........................................................................................... 37

Fig. 2.19. The best-fitted results when using the modified flux distribution matrix to

calibrate the scaling factor and the force gain vector simultaneously. ............................. 38

Fig. 2.20. Testing the linear range of electromagnetic actuation: P2 (left) and P3 (right) . 39

Fig. 2.21. Comparing three force envelopes calculated using nominal force models: 3-D

actuator using current allocation based on constraint constraints (blue), newly developed

actuating system using current allocation based on constraint constraints (red), and newly

developed actuating system using optimal current allocation (grey)................................ 40

Fig. 2.22. Comparing three force envelopes calculated using calibrated force models: 3-D

actuator using current allocation based on constraint constraints (blue), newly developed

actuating system using current allocation based on constraint constraints (red), and newly

developed actuating system using optimal current allocation (grey)................................ 41

Fig. 3.1. (a) Magnetic setup integrate with the inverted microscope (b) High speed

embedded control system using FPGA (c) real-time display of the reference bead and the

xvi

control bead (the reference sticks to the cover glass surface to provide the information

such as drift, vibration and etc.) ........................................................................................ 44

Fig. 3.2. Inverse model based on constant constraints. (a) The percent force error at

(20,0,0)um, desired force ˆdF =1; (b) The percent force error at (20,0,0)um, desired force

ˆdF =10; (c) The percent force error at (40,0,0)um, desired force ˆ

dF =1; (d) The percent

force error at (40,0,0)um, desired force ˆdF =10 ................................................................. 48

Fig. 3.3. Optimal inverse model. (a) The percent force error at (20, 0, 0) um; (b) The

percent force error at (40, 0, 0) um ................................................................................... 48

Fig. 3.4. Locations where the optimal inverse model are obtained .................................. 50

Fig. 3.5. Comparison of percentage force error between Taylor expansion and least-

square fitting ..................................................................................................................... 52

Fig. 3.6. (a) Force generation envelops at (0,0,0)um. (b) Force generation envelops at

(20,0,0)um. (c) Force generation envelops (40,0,0)um .................................................... 53

Fig. 3.7. Block diagram of the feedback motion control using proportional controller (dP

is the desired position, mP is the measurement position, e is the error signal, Kp is the

proportional gain, dF is the desired force calculated by the controller, ' (t) (t )d d a F F is

caused by the actuator delay, a and

m are the delay in actuator and measurement, MTF is

the magnetic force, F is the modeling error, TF is the thermal force) ............................. 53

Fig. 3.8. 3D plot in bead stabilization experiment (the result in from The Optimal inverse

model) ............................................................................................................................... 55

xvii

Fig. 3.9. Actuation effort using optimal inverse model and the inverse model based on the

constant constraints. (For the inverse model based on constant constraints, I5 and I6 are

negative with much larger absolute value than optimal invers model) ............................ 56

Fig. 3.10. The positioning standard deviation in x,y and z axis using the optimal inverse

model and constant constraints ......................................................................................... 56

Fig. 3.11. The positioning performance from Proportional controller, using optimal

inverse model and constant constraints............................................................................. 56

Fig. 3.12. Trajectories in 3D tracking control. .................................................................. 58

Fig. 3.13. Actuation effort in 3D tracking control. (For the inverse model based on

constant constraints, I5 and I6 are negative with much larger absolute value than optimal

invers model) .................................................................................................................... 59

Fig. 3.14. Tracking error using the optimal inverse model and inverse model based on

constant constraints. .......................................................................................................... 59

Fig. 3.15. The measurement PSD curves and calibrated PSD curves (The peak at 71.37Hz

is due to structure vibration) ............................................................................................. 61

Fig. 3.16. The positioning result using the optimal trapping stiffness in 200Hz case and

1606Hz case ...................................................................................................................... 62

Fig. 3.17. Trapping stiffness vs. standard deviation of Brownian motion, 200Hz control

and 1606Hz control ........................................................................................................... 63

Fig. 4.1. Magnetic field around the tip of the magnetic pole and a suppositional Hall

Sensor. ............................................................................................................................... 67

xviii

Fig. 4.2. (a). Hall sensors integrated with Hexaple magnetic actuator. (b) Zoom-in plot of

Hall sensors associated with upper poles. (c) Zoom-in plot of Hall sensors associated with

lower poles. ....................................................................................................................... 70

Fig. 4.3. (a). The setup for studying the relationship between the surface-mount Hall

Sensor measurement and the Hall Sensor measurement at the tip. P1 and P2 are actuated

by current-driven coils. (b). Positions and dimensions of hall elements. ......................... 71

Fig. 4.4. Input-output plot between Vtip and Vsurface in two cases: 1). P1 is self-actuated, 2)

P2 is actuated, P1 is magnetized by the magnetic coupling between P1 and P2............... 72

Fig. 4.5. input-output plot between current and Hall Sensor voltage. .............................. 74

Fig. 4.6. Motion control result in trajectory tracking ........................................................ 75

Fig. 4.7. The input-output of six poles in two tracking experiments, the second

experiment is conducted by reversing the sign of the actuation currents. (To make the

input-output plot orientated with positive slope, the signs of some plots are changed

accordingly) ...................................................................................................................... 76

Fig. 4.8. Control loop for magnetic force analysis; (dP is the desired position,

mP is the

measurement position, pK is the proportional control gain, dF is the desired force

calculated by the controller,a and

a are the delay in actuator and measurement, MTF is the

magnetic force, TF is the thermal force, N is measurement noise) ................................... 77

Fig. 4.9. PSD analysis without low-pass-filter ................................................................. 78

Fig. 4.10. PSD analysis with low-pass-filter..................................................................... 78

Fig. 4.11. Hall-sensor-based magnetic force model along circles in xy plane (as Fig.4.6

(a)) ..................................................................................................................................... 80

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Fig. 4.12. Hall-sensor-based magnetic force model along circles in yz and xz planes (as in

Fig. 4.6 (b) and Fig.4.6 (c)) ............................................................................................... 80

Fig. 4.13. Current-based magnetic force model along circles in xy plane (as Fig.4.6 (a)) 81

Fig. 4.14. Current-based magnetic force model along circles in yz and xz planes (as in

Fig.4.6 (b) and Fig.4.6 (c)) ................................................................................................ 81

Fig. 4.15. Histogram of modeling error in force calibration (Hall-sensor-based model vs.

Current-based model; the black histogram are from Fig.4.11 and Fig.4.12, the green

histogram are from Fig.4.13 and Fig.4.14) ....................................................................... 82

Fig. 4.16. Bead motion trajectory for force prediction ..................................................... 83

Fig. 4.17. Force prediction using Hall-Sensor-based model (motion along circles in xy

plane)................................................................................................................................. 83

Fig. 4.18. Force prediction using Hall-Sensor-based model (motion along straight line in

xy plane)............................................................................................................................ 84

Fig. 4.19. Force prediction using Current-based model (motion along circles in xy plane)

........................................................................................................................................... 84

Fig. 4.20. Force prediction using Current-based model (motion along straight lines in xy

plane)................................................................................................................................. 84

Fig. 4.21. Histogram of modeling error in force prediction (Hall-sensor-based model vs.

Current-based model; the black histogram are from Fig.4.17, the green histogram are

from Fig.4.19) ................................................................................................................... 85

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Fig. 4.22. Histogram of modeling error in force prediction (Hall-sensor-based model vs.

Current-based model; the black histogram are from Fig.4.18, the green histogram are

from Fig.4.20) ................................................................................................................... 85

Fig. 4.23. Magnetic bead position control integrated with Hall-sensor inner loop

control. dP and

mP are the desired and measurement position, e is the positioning error, dF

is the desired force calculated from the motion controller. dV and

mV are the desired and

measured Hall-Sensor voltage. ......................................................................................... 86

Fig. 4.24. Normalized Response of different desired voltage dV . ...................................... 87

Fig. 4.25. Step response for different voltage inputs. ....................................................... 87

Fig. 4.26. Positioning performance using Hall-Sensor inner loop control. ...................... 89

Fig. 4.27. dV is the desired voltage and mV is the measurement voltage) ............................ 89

Fig. 4.28. Step Response of Hall-sensor feedback control. .............................................. 89

Fig. 4.29. Magnetization of the M450 superparamagnetic bead....................................... 91

Fig. 4.30. The plot of the Langevin Function (assume the saturation limit sm =1) ............ 92

Fig. 4.31. Motion control result in trajectory tracking ...................................................... 95

Fig. 4.32. The input-output of six poles in two tracking experiments, the second

experiment is conducted by reversing the sign of the actuation currents. (To make the

input-output plots orientated with positive slope, the signs of some plots are changed

accordingly) ...................................................................................................................... 97

Fig. 4.33. Calibration result of Hall-Sensor-based force model using Langevin function.

(a) Force calibration along circles in xy plane (as in Fig.4.31) (b) zoom-in plot of force

calibration ......................................................................................................................... 99

xxi

Fig. 4.34. Calibration result of Hall-Sensor-based model without using Langevin-

function. (a) Force calibration along circles in xy plane (as in Fig.4.31) (b) zoom-in plot

of force calibration ............................................................................................................ 99

Fig. 4.35. Calibration result of Hall-Sensor-based force model using Langevin function.

(a) Force calibration along circles in yz and xy planes (as in Fig.4.31) (b) zoom-in plot of

force calibration .............................................................................................................. 100

Fig. 4.36. Calibration result of Hall-Sensor-based force model without using Langevin

function. (a) Force calibration along circles in yz and xy planes (as in Fig. 4.31) (b)

zoom-in plot of force calibration .................................................................................... 100

Fig. 4.37. Histogram of the force calibration error for Hall-sensor-based model with

Langevin-function and without Langevin-function. (The red histograms are from the

modeling errors of Fig.4.33 and Fig.4.35, the blue histograms are from the modeling

errors of Fig.4.34 and Fig.4.36) ...................................................................................... 101

Fig. 4.38. Bead motion trajectory for force prediction ................................................... 101

Fig. 4.39. Force prediction of Hall-Sensor-based force model using Langevin function.

(a) Force calibration along circles in xy plane (as in Fig.4.38) (b) zoom-in plot of force

calibration ....................................................................................................................... 102

Fig. 4.40. Force prediction of Hall-Sensor-based force model without using Langevin

function. (a) Force calibration along circles in xy plane (as in Fig.4.38) (b) zoom-in plot

of force calibration .......................................................................................................... 103

xxii

Fig. 4.41. Force prediction of Hall-Sensor-based force model using Langevin function.

(a) Force calibration along line trajectories in xy plane (as in Fig.4.38) (b) zoom-in plot

of force calibration .......................................................................................................... 103

Fig. 4.42. Force prediction of Hall-Sensor-based force model without using Langevin

function. (a) Force calibration along line trajectories in xy plane (as in Fig.4.38) (b)

zoom-in plot of force calibration .................................................................................... 104

Fig. 4.43. Histogram of the force calibration error for Hall-sensor-based model with

Langevin-function and without Langevin-function. (The red histograms are from the

modeling errors of Fig.4.39, the blue histograms are from the modeling errors of

Fig.4.40) .......................................................................................................................... 104

Fig. 4.44. Histogram of the force calibration error for Hall-sensor-based model with

Langevin-function and without Langevin-function. (The red histograms are from the

modeling errors of Fig.4.41, the blue histograms are from the modeling errors of

Fig.4.42) .......................................................................................................................... 104

Fig. 4.45. relationship between external magnetic field ||B|| (Gauss) and the magnetic

moment ||m||. ................................................................................................................... 106

Fig. 4.46. Plots of || ||B , || ||V and || || / || ||B V .................................................................... 107

Fig. 4.47. (a). The plot of 2ˆ 20 || || || ||Lf a V V and ˆ 20 || ||Lf a V . (b). the plot of ˆ 20 || ||Lf a V .

(c). Zoom-in plot of 2ˆ 20 || || || ||Lf a V V when || ||V is small. (d). the plot of

2ˆ 20 || || || ||Lf a V V and the inverse function 2ˆ( 20 || || || || )Linverse f a V V ............................ 109

xxiii

Fig. 5.1. Joint state-parameter estimator (using the Hall sensor based force model)

running simultaneously with feedback controller. (dP is the desired position,

mP is the

measurement position, e is the error signal, a and

m are the delay in actuator and

measurement, I is the current from the nominal inverse model,dF is the desired force

calculated by the controller, ' (t) (t )d d a F F is caused by the actuator delay, F is the

modeling error, MTF is the magnetic force,

TF is the thermal force, EF is the bead-sample

interaction force, B is the magnetic flex density, mV is the Hall sensor measurement

voltage, ˆMTF is the modeling magnetic force, is the estimated drag coefficient, ˆ

EF is the

estimated external force) ................................................................................................. 113

Fig. 5. 2. Estimation result in water using the magnetic force model to estimate the drag

coefficient ....................................................................................................................... 121

Fig. 5. 3. Estimation result in water using the thermal variance to estimate the drag

coefficient ....................................................................................................................... 121

Fig. 5. 4. Estimation result in glycerol using the magnetic force model to estimate the

drag coefficient ............................................................................................................... 122

Fig. 5. 5. Estimation result in glycerol using the thermal variance to estimate the drag

coefficient ....................................................................................................................... 123

Fig. 5. 6. Estimation result in glycerol using the thermal variance to estimate the drag

coefficient (no measurement noise) ................................................................................ 123

Fig. 5. 7. Estimation result using the magnetic force model when the drag coefficient and

external force are changing simultaneously. ................................................................... 124

xxiv

Fig. 5.8. Joint state-parameter estimation using Hall sensor based model (a) estimation

result in x direction (b) estimation result in y direction (c) estimation result in z direction

(d) trajectories used to test joint state-parameter estimation algorithm. ......................... 125

Fig. 5.9. Joint state-parameter estimator (using current-based magnetic force model)

running simultaneously with feedback controller. (The meaning of all parameters is the

same as that in Fig.5.1, the only difference is that the modeling force ˆMTF is from current-

based model instead of Hall-sensor based model). ......................................................... 127

Fig. 5.10. Joint state-parameter estimation, Hall sensor based force model vs. Current-

based force model. (a) Joint state-parameter estimation result in x direction. (b) Joint

state-parameter estimation result in y direction. (c) Joint state-parameter estimation result

in z direction. (d) Trajectories used to test joint state-parameter estimation algorithm. 128

1

Chapter 1: Introduction

1.1. Background and Motivation

Probing biological samples and manipulating biological processes have become

important techniques in the study of cell mechanics and mechanobiology [3], since it is

widely discovered that the interaction force between cells/biomolecules plays an

important role in many physiological processes [4-6]. Especially, a dream of using

modern instruments to probe biomolecules in live cells has been shared by many

researchers in the field.

In the scanning probe microscopy (SPM) family [7], Atomic Force Microscopy

(AFM) offers high spatial resolution and enables force probing and scanning [8]. AFM,

however, remains as a 2-D surface tool [9-12] due to its kinematic constraints and the

restriction of mechanical connections employed [13]. It has been applied to study

biomolecules from isolated model systems [14, 15] rather than from their native and fully

active environment.

Optical trapping is another modern technique that is useful for the study of biological

systems under physiological conditions [16-22]. It is well suited for quasi-static force

measurement [23] and dynamic force sensing [24]. Whereas it has been theoretically

derived and experimentally verified that trapping smaller objects can be achieved by

increasing the power of the trapping laser, heating is a main issue that needs to be

2

resolved [25]. One feasible approach is to implement real-time Brownian motion control

to reduce heating [26, 27]. However, due to its underlying working principle, unwanted

trapping of debris may easily mix with the measurement probe. Lack of specificity in

optical trapping is, therefore, an important limitation that must be examined before it is

employed to probe biological samples [19].

Magnetic tweezers use magnets and/or electromagnets to generate magnetic field to

propel microscopic magnetic particles, which serve as measurement probes and exert

force on biological samples. They have several advantages over optical tweezers.

Magnetic field is intangible and safe to most biological materials. It is specific to

magnetic particles and there is no heat generated in the process. It is, however, necessary

to employ feedback control to achieve stable magnetic trapping since the magnetic force

field is inherently unstable [28]. Therefore, without active control, the magnetic

particles/probes are often functionalized and anchored to target bio samples to avoid

instability. Many magnetic actuators are, thus, actually simple force appliers, wherein the

force is adjusted according to a pre-calibrated force function [29]. They were used in a

wide range of applications, ranging from manipulating biological macromolecules [30-

34], probing cell membranes [35-37], to characterizing intracellular properties [38-41].

Among various magnetic tweezers, some could only apply forces in a single

direction, using only one coil-actuated pole [29, 42] or two poles facing each other [30,

39], some were 2-D systems with multiple poles [40, 43], and a few were able to generate

forces in 3-D space [34]. These magnetic tweezers were employed as simple force

appliers, without motion control, wherein magnetic forces were applied and the induced

3

motions of magnetic particles were recorded by appropriate measurement systems. One

development of magnetic tweezers did use motion control to achieve stabilization of a

4.5µm magnetic particle suspended in water [44]. It employed a six-pole magnetic

apparatus to generate an upward magnetic force, whereas the downward force was caused

by the gravity. Swimming robots propelled in liquid using external magnetic fields have

also been successfully demonstrated [45-48]. Moreover, an electromagnetic system was

designed to control intraocular micro-robots for delicate retinal procedures [49-51]. An

error less than 0.5 mm was achieved when controlling the micro-robot in a visual-

servoing framework [51].

Accurate force modeling and inverse modeling of electromagnetic actuation are

essential for effective force manipulation and for enabling stable magnetic trapping via

feedback control. The development of a 2-D quadrupole magnetic tweezers was reported

in [52, 53], wherein four tip-shaped electromagnetic poles were employed to control 2-D

magnetic force to propel a microscopic magnetic particle, serving as a force-sensing

probe. It was applied to characterize the mechanical property of live cells using a

functionalized probe [54]. A lumped-parameter analytical magnetic force model was

derived to characterize the nonlinearity of the magnetic force with respect to the input

currents and its position dependency in the workspace. Its extension to the realization of a

3-D hexapole magnetic actuator was detailed in [1, 2]. Employing a visual particle

tracking system [55], a visual servo control system was developed to enable precise

motion control of the magnetically propelled particle in the 3-D workspace [1], wherein a

4

simple inverse model was derived and used to implement a nonlinear feedback control

law to realize stable magnetic trapping.

These two multi-pole electromagnetic actuating systems had electromagnetic poles

made of thin (approximately 100μm thick) high-permeability nickel-iron magnetic alloy

(permalloy) film. Whereas the design of these systems had advantages, such as easy in

fabrication and assembly, it had several limitations. First, it was necessary to use

specially made sample chambers, therefore, standard culture dishes commonly used for

live cell experiments could not be employed due to space limitations. Second, small

cross-sectional area of the thin film resulted in large magnetic reluctance, and thus

yielded small magnetic flux and saturation flux density.

Other than hardware design, inverse modeling is also important for multi-pole

electromagnetic actuating systems, which are usually over-actuated systems. The issue

raised by redundancy was solved through applying constant constraints in [1, 52]. But the

use of constant constraints resulted in excessive actuation effort, severely limiting the

force generation capability. A micro-robot system [50] implemented a suboptimal inverse

model from pseudo inverse, which cannot be applied in our system due to the nonlinear

nature of the force model. A multi-pole magnetic actuator [56, 57] could steer a ferrofluid

drop by optimal control effort. But the sampling rate was limited to 15Hz due to the

cumbersome calculation. Moreover, 3-D realization has not been achieved.

Besides the position control capabilities, the magnetic force model accuracy and

dynamic force sensing capabilities are also crucial for biological applications since many

biological processes happen in a short time and the interaction forces between biological

5

samples can be as small as pN scale. Most researchers did not directly verify the accuracy

of the magnetic force models. The stability of the model-based feedback control [1, 2, 52]

is far less than enough to verify the accuracy of the magnetic force model since the

feedback control can stabilize the bead even when the modeling error exists. For the force

application, however, the inaccuracy of the magnetic force model will lead to biased

force estimation between the magnetic bead and biological samples. A common issue for

soft magnetic material is the hysteresis problem [58], which is usually rate-dependent

[59, 60] in soft magnetic material. Moreover, the poles in the hexapole magnetic actuator

are magnetically coupled together, which made the hysteresis effect very difficult to

model. Some researchers estimated the magnetic force from Brownian motion

fluctuations through image measurement microscopy [61, 62], which is actually a quasi-

static force estimation. New method/model need to be proposed to solve the complicated

hysteresis effect and achieve dynamic force estimation to capture the transient bead-

sample interaction force.

This research project aims to enable a haxapole magnetic actuator to perform active

scanning tasks, wherein a magnetic bead can be actively controlled and the force can be

accurately sensed/controlled dynamically.

1.2. Objectives and specific aims

The objective of this research project is to develop an over-actuated actively

controlled magnetic actuator that can achieve stabilization, motion control, and accurate

6

force sensing and force control. According the objectives, four specific aims are

identified.

(1). Hardware development, magnetic force modeling and calibration

A hexapole magnetic actuator was employed for 3D motion control and force

application. Six sharp-tipped magnetic poles, each of which is actuated by a coil, are

alignment in three orthogonal directions to control the magnetic bead in 3D space. The

actuator is over-actuated since the magnetic poles can only generate attractive force.

Besides the redundancy, the magnetic force on the magnetic bead is nonlinearly

dependent on the input current and is position dependent. A lumped parameter magnetic

force model is employed to characterize the relationship between the input current and

the resulting magnetic force. Due to the motion capability, the magnetic force model can

be experimentally calibrated and verified.

(2). Optimal inverse model and control: stabilization, Brownian motion control and

trajectory tracking

Inverse model is important for feedback linearization and feedback control. The

magnetic force generation capability also greatly depend on the current allocation. Since

the magnetic force is redundant, infinite many solutions exist for a specific desired force.

Constant constraints are used in [1, 2, 52] to remove the redundancy. However, the force

generation capability is greatly sacrificed under such constraints. Moreover, the

constraints result in excessive large current allocation, which will lead to larger modeling

7

error. An optimal inverse model based on the minimal squared norm criterion will be

developed to achieve improved feedback linearization. Stabilization, Brownian motion

control and trajectory tracking can be achieved and the performance will be compared

with that from constant constraints.

(3). Hall Sensor Based 3-D magnetic force modeling and feedback control

Compared with the motion control capability, accurate force modeling is a more

challenging task. With feedback control, the bead can be steered stably even when

modeling error exists. The force application, however, completely relies on the magnetic

force model. Due to the complicated nature of the hysteresis effect and the magnetic

coupling, hysteresis modeling is a very difficult task for the hexapole magnetic actuator.

The Hall sensors can be introduced to directly measure the magnetic field from each pole.

The feasibility of Hall sensor based magnetic force model is investigated. Moreover, with

direct measurement of the magnetic field, the secondary control loop can be established

to directly control the magnetic field of each pole.

(4). Dynamics force sensing and parameter estimation

Dynamics force sensing is desired to sense the force between the magnetic bead and

the biological samples in real-time. If the magnetic force can be accurately modeled, a

joint state-parameter estimator can be developed to simultaneously estimate the bead-

sample interaction force and the drag coefficient of the liquid environment. This

capability can transform the magnetic actuator into an active scanning probe. Combined

8

with the motion control capability, this magnetic actuator can achieve force control or

automatic scanning. For example, the bead-sample interaction force can map the

topography of an unknown part and the drag coefficient is an indication of local

environment change such as cytoplasm property and etc.

1.3. Dissertation Overview

This dissertation is organized as follow. Chapter 2 presents the design, force modeling

and inverse modeling of the hexapole magnetic actuator. The hardware design and

improved current allocation mechanism greatly increased the force generation capability.

Stabilization of the magnetic bead was achieved and the magnetic force model was

experimentally calibrated. Chapter 3 described the development of the optimal inverse

model in the sense of minimal current norm. This optimal inverse model solved

redundancy, nonlinearity, and position dependency simultaneously. Stabilization,

Brownian motion control and tracking control were achieved and the performance was

compared with the inverse model using constant constraints [1]. Chapter 4 presented the

so called Hall-sensor based magnetic force model. Hall sensors are introduced to directly

measure the magnetic field in each pole and the measurement result is lumped into a

magnetic force model. With Hall sensors, obvious magnetic hysteresis can be observed in

bead stabilization and motion control. By steering the magnetic bead in 3D space, the

Hall-sensor based model can be calibrated and compared with the current-based model in

term of force modeling accuracy. The result showed that the Hall-sensor based model

greatly outperformed the current-based model. With Hall sensor measurement, a

9

secondary control loop can be established to directly control the magnetic field in each

pole. When the magnetic field becomes larger, a more accurate Hall sensor based model

is proposed to consider the nonlinear magnetization effect of the magnetic bead. With the

establishment of Hall-sensor based magnetic force model, the dynamic force sensing and

parameter estimation capability are introduced in Chapter 5. This joint state-parameter

estimator enables the magnetic actuator to dynamically sense the interaction force

between the magnetic bead and biological samples. This capability will transform the

magnetic actuator from a simple force applier to a 3D active probing system. Conclusions

and future work are presented in Chapter 6.

10

Chapter 2: design, force modeling and inverse modeling of the

hexapole magnetic actuator

2.1 Introduction

Magnetic tweezers are widely used in biological engineering since the magnetic force

is a noncontact force with biocompatibility and susceptibility. However, most magnetic

tweezers are force appliers wherein the magnetic bead is anchored to biological samples

to avoid instability. One development of magnetic tweezers did use motion control to

achieve stabilization of a 4.5um magnetic particle in water is shown in [44]. However,

the feedback control is heuristic instead of model-based wherein large Brownian motion

is observed. Actively controlled multi-pole magnetic tweezers are developed by L. Chen

[63], but analytical force model is not proposed while the magnetic field comes from

interpolation and look-up table from FEM analysis. An improved 3-D hexapole magnetic

actuator was developed by Z. Zhang [2], wherein analytical magnetic force model is

proposed. Employing a visual particle tracking system [55], a visual servo control system

was developed to enable 3D motion control of the magnetically propelled particle in the

3-D workspace [1], wherein a simple inverse model was derived and used to implement a

nonlinear feedback control law to realize stable magnetic trapping.

Magnetic actuators in [1, 2] are made of thin permalloy film. However, there are

several limitations. First, specially made sample chambers are need in the experiment

11

since the poles are fixed. Therefore, standard culture dishes commonly used for live cell

experiments could not be employed due to space limitations. Second, the thin-film design

with small cross-sectional area will yield small magnetic flux and the saturation limit of

this material will be relatively low compared to other magnetic material.

For over-actuated multi-pole electromagnetic systems, inverse modeling is also

important, not only for feedback control, but also for force generation capability. The

issue raised by redundancy was solved through applying constant constraint in [1]. But

the use of constant constraints resulted in excessive actuation effort, severely limiting the

force generation capability.

This chapter presents the design, modeling, and calibration of an over-actuated

hexapole electromagnetic actuating system, which is used to stabilize and propel a

microscopic magnetic particle in the liquid environment. The design and synthesis of the

six electromagnetic poles, including geometric design and material selection, result in

high magnetic saturation limit, and thus larger force generation capability. An improved

current allocation scheme is derived and implemented to realize real-time current

allocation to enable the most effective manipulation of the 3-D magnetic force exerting

on the particle at the center of the workspace. Compared to the 3-D magnetic actuator in

[1], the hexapole electromagnetic actuating system achieves significantly greater force

generation capability and much improved controllability of 3-D force in the workspace.

12

2.2 Design of Hexapole Electromagnetic Actuator

2.2.1 Design, synthesis, and fabrication

Fig. 2.1. (a) CAD model of motion stage and lower poles. (b) Assemble yoke ring and three upper poles.

Fig. 2.2. (a) Fabricated prototype, integrated on an inverted microscope. (P2, P4, P5) forms the upper layer

and (P1, P3, P6) forms the lower layer. Each pole is associated with an actuation coil (for flux generation)

and a measurement coil (for flux measurement). P2→P1, P4→P3 and P6→P5 are +x, +y and +z directions

of the actuation coordinate system. (b) CAD model and meshing of the hexapole actuator

Since a single electromagnetic pole can only generate attractive force exerting on the

magnetic particle, three pairs of electromagnetic poles are employed and their pole tips

are placed symmetrically on three orthogonal axes to enable generation of 3-D forces. In

order to significantly increase the force generation capability, these sharp-tipped poles are

made of cone-shaped iron steel rods, and are assembled to concentrate the magnetic flux

13

into the workspace. Each electromagnetic pole is actuated through an individual coil. All

the coils and poles are then magnetically connected through a magnetic yoke, which

helps increase the magnetic flux density and the flux gradient in the workspace. The six

pole tips enclose the workspace, wherein the specimen and the magnetic particle are

placed. The hexapole electromagnetic actuator is assembled with two overlaid motion

stages to form a unique experimental apparatus. The apparatus is integrated with an

inverted microscope equipped with a visual particle tracking system [55]. In order to have

the optical path free of blockage, a rigid body rotation is applied to the three pairs of

electromagnetic poles along with their three orthogonal axes such that their tips are on

two parallel horizontal planes, i.e., one upper plane and the other lower plane. Fig.2.1 (a)

shows the design of a manual x-y stage and the lower three electromagnetic poles,

assembled on an x-y-z piezo motion stage. The coarse x-y stage achieves an 8mm×8mm

working range, and can be used to locate a specimen cell. The more detailed design of the

moving stage can be found in [64, 65]. The lower three poles are fixed under the culture

dish, which holds live sample cells so that the bottom cover glass of the culture dish,

whose thickness is approximately 100μm, can be placed in-between upper three and

lower three poles; the gap between them is as small as possible to generate strong

magnetic force. The upper three magnetic poles are assembled on the yoke ring and their

pole tips sunk into medium filled in the dish, as shown in Fig.2.1 (b). They can be easily

disassembled to replace culture dish to perform live cell experiments in practical use. The

arrangement also makes cleaning easy, necessary to avoid cell contamination. The

magnetic particle is to be stabilized and steered within the 3-D workspace, whereas the

14

position of the sample dish with respect to the 3-D workspace is controlled by the 3-axis

piezo stage.

The fabricated prototype is shown in Fig.2.2 (a). The electromagnetic poles and the

magnetic yoke are fabricated with 1018 steel, low-carbon (0.18% carbon) steel with high

saturation limit (over 2T). However, one drawback of using such material is the

hysteresis effect, which will be addressed in the future through hysteresis modeling or

real-time sensing and control. At the present time, superparamagnetism is assumed to

characterize force generation capability of the actuating system. The diameter of the

poles is about 6mm. Three upper poles are about 45mm long, whereas the lower poles are

42mm long and are milled to form a flat platform to support the culture dish. Whereas

sharper tips can be produced using advanced machining processes, the radius of the pole

tips of the prototype is 40μm, which is included in the finite element analysis (Fig.2.2

(b)). The distance from the workspace center to the tips of all six poles is adjustable due

to the flexibility achieved in the new design. Its nominal value of 500µm is used in all

analyses and experiments reported in this chapter. The actuation coils, realized for

experiments and winded around the protrusions of the yoke, have 70 turns. The whole

setup is integrated upon an inverted microscope (Olympus IX 81) with bright field

illumination and a dry 60x objective lens is used for visual measurement. A vision-based

measurement method with sub-nanometer resolution [55] is employed to provide position

feedback, wherein a speed CMOS camera (Mikrotron MC3010) and Image

grabbing/processing card (Matrix Odyssey XCL) are used. A 4.5um bead (Dynabead M-

450 Epoxy) is used in the experiment. The coils are driven by DA converters

15

(Measurement Computing, PCI-DAS6032), followed by six linear power amplifiers

(Micro Dynamics, BTA-28V-6A) working in current mode with 10K bandwidth. A piezo

positioner (PI P-721 PIFOC) is used to drive the lens for image calibration. The whole

setup is put on a Smart Table (Newport) and a vibration isolation table (Herzan TS-150)

to remove vibrations.

2.2.2 Finite element analysis of the magnetic field

Finite Element Method (FEM) is employed to analyze quantitatively the magnetic

field produced by the actuating system and to visualize its spatial distribution,

particularly within the workspace. A CAD model imitating the real setup is built and

meshed (Fig.2.2 (b)) using the ANSYS environment for FEM calculation. When applying

current to the coil, the magnetic field produced by the hexapole actuator can be computed

using FEM analysis.

Fig.2.3 shows the ANSYS analysis result by actuating coil 1 with 1A current. It can be

seen from Fig.2.3 (a) (top view of measurement coordinate) that the magnetic field forms

a closed loop under the guidance of magnetic poles and the yoke, which makes the

magnetic flux density and the flux gradient in the workspace significantly higher. Fig.2.3

(b) shows the magnetic flux density vectors, associated with the actuation coordinate

system, within a 100μm cube centered at the origin of the workspace. Due to the direction

of the input current, which is counter clockwise, these vectors all point to the tip of pole

1. The magnetic field vectors near the tip of the electromagnetic pole 1 are shown in

Fig.2.3 (c). It can be seen that they strongly converge to the tip of the electromagnetic

16

pole. Fig.2.3 (b) and Fig.2.3 (c) validate the assumption that sharp tip behaves like a point

charge. The magnetic flux density is measured in Tesla (104 Gauss) in Fig.2.3.

Fig. 2.3. (a) The top view (measurement coordinate) of the vector plot of the magnetic flux density

distribution (unit: Tesla). (b) The magnetic flux density vectors near the workspace center (actuation

coordinate). (c) The magnetic field vectors near the tip of pole 1.

Fig.2.4 shows the contour plot of the magnitude of the magnetic flux density

(measured in Gauss) when applying 1A current to coil 1. A similar analysis for the thin-

foil design was given in Fig. 6 of [2]. Specifically, the contour plots of two sections, i.e.

the horizontal plane and the vertical plane, are displayed and the gradient of the magnetic

field is clearly visualized. The magnetic flux density in the vicinity of the workspace

center of this hexapole actuator increases by twofold from that of the actuator with the

thin-foil design. Moreover, the saturation limit of the magnetic pole is over 2T, which is

much higher than that of permalloy pole with 0.9T saturation limit in the previous design.

This improvement allows each 70-turn coil to be actuated up to 3 Amperes, the force

17

generation capability is greatly increased. The detailed force generation calculation will

be addressed later in Chapter 2.6.

Fig. 2.4. Magnitude of the magnetic flux density, i.e., |B|, associated with the measurement coordinate

system. (a) |B| in the horizontal plane (top view). (b) |B| in the vertical plane (side view).

2.2.3 Hexapole magnetic field model

Fig. 2.5. (a) Hexapole magnetic actuator integrated on an inverted microscope. (P2, P4, P5) forms the upper

layer and (P1, P3, P6) forms the lower layer. Each pole is associated with an actuation coil (for magnetic

flux generation). P2→P1, P4→P3 and P6→P5 are +x, +y and +z directions of the actuation coordinate

system. (b) CAD model of six tips of the hexapole magnetic actuator and the associated Measurement

Coordinate {O;x ,y ,z }m m m and Actuation Coordinate {O;x ,y ,z }a a a

. (c) FEM analysis in ANSYS about

magnetic charge model of a sharp-tipped magnetic pole. (d) The sketch of the magnetic bead and six

magnetic charges.

18

Fig.2.5 shows the hexapole model associated with the measurement coordinate

system and the actuation coordinate system, wherein the three upper poles/charges are

associated with input currents I2, I4 and I5, and lower poles/charges I1, I3 and I6.

It can be seen from Fig.2.5(c) that the magnetic field produced by a sharp tipped pole

looks to the magnetic particle in the workspace of the actuator as though it is generated

by a point source [66],

0

iiq

(2.1)

Where qi is the magnetic charge of the ith pole, Φi is the magnetic flux going through the

corresponding pole, μ0 is the permeability of vacuum. The magnetic field at the location

of the magnetic bead results from six charges/poles (Fig.2.5 (d)), wherein the

contribution of each can be modeled by the following relationship,

2( , ) ( , )

( , )

ii i m i i

i i

qk

rB p b u p b

p b, i=1~6, (2.2)

where ( , )i iB p b is the magnetic flux density contributed by the ith magnetic charge qi.

( , )i iB p b is a function of the bead’s position p = [x,y,z]T and charge’s bias

=[ , , ]i i i ix y z b since the optimal location to model the magnetic charge iq is not

necessarily at the tip of the pole. 0mk =1.0×10-7 N/A2. ( , )i ir p b is the norm of the

displacement vector pointing from the ith magnetic charge iq to the location of the bead,

19

and ( , )i iu p b is the unit directional vector, i.e., ( , ) ( , ) ( , )i i i i i iru p b r p b p b (Fig.2.5 (d)).

The resulting magnetic field sensed by the magnetic bead can be obtained by superposing

the magnetic field generated by six magnetic charges/poles, i.e. 𝐁(𝐩, 𝐛) =

∑ 𝑘𝑚𝑞𝑖

𝑟𝑖2(𝐩,𝐛𝑖)

𝐮𝑖(𝐩, 𝐛𝑖)6𝑖=1 , where ( , )B p b is the total magnetic flux density at the location p

of the magnetic bead, 1 2 3 4 5 6=[ , , , , , ]b b b b b b b is defined as the 18-domensional Bias

Vector. The position p can be normalized with respect to , which is the radius of the

workspace. The total magnetic flux density can thus be expressed in the following form,

1

2

33 5 61 2 4

2 3 3 3 3 3 3

41 2 3 4 5 6

ˆ ˆ( , ) 5

6

ˆ ˆ ˆˆ ˆ ˆˆ( , )

ˆ ˆ ˆ ˆ ˆ ˆm

q

q

qk

qr r r r r r

q

q

R p b

Q

r r rr r rB p b (2.3)

Where ˆ ˆ( , ) ( , )i ir p b r p b , ˆ ˆ ˆˆ ( ) || ( )||i ir p r p . ˆ ˆ( , )R p b is a 3×6 Charge-Bead

Distribution Matrix depending on the special distribution of magnetic bead, six charges

and bias vector, Q is the Charge Vector.

For the current-actuated magnetic actuator, the magnetic flux in Eq. (2.1) is

determined by the magnetomotive force and the reluctance of the air according to

Hopkinson’s law: the magnetic Flux Vector 1 2 3 4 5 6[ , , , , , ]T Φ =

1 2 3 4 5 6[ , , , , , ]T

I aK F F F F F F , wherein the magnetomotive force is proportional to the input

current, i.e., i c iN IF , cN is the turns of the coil, a is the lumped magnetic reluctance

20

from the pole tip to the workspace center in the air, and IK is the 6×6 flux distribution

matrix describing the magnetic coupling among 6 poles since they are connected by a

magnetic yoke (Fig.2.2, Fig.2.3). The magnetic charges vector Q can thus be related to

the input currents, 1 2 3 4 5 6[ ]TI I I I I II , as described by the following equation,

0

0

cI

a

N

Q Φ K I (2.4)

The spatial distribution of the magnetic field within the workspace of the actuating

system and its dependence on the input currents can be determined using the above

equations once the reluctance and workspace radius, the distance from the magnetic

charge to the workspace center, of the actuating system are known. They are determined

by best fitting the magnetic flux density vector based on Eq. (2.3) with that calculated

using FEM analysis (Fig.2.3 (b)). The sum of the squared norm of error vectors is

selected as the objective function, which is minimized to determine the two optimal

values, i.e., workspace radius 900 m , which is the distance variable defining the

location of iq , and the air reluctance a =6.3×108A/Wb. The fitted result is shown in

Fig.2.6, wherein vector plots are compared in (a) and normalized norms of error vectors

in (b). It can be seen that norms of error vectors are mostly smaller than 1% of the norm

of the associated flux density. This validates the hexapole magnetic field model, i.e. Eq.

(2.3).

21

Fig. 2.6. Validation of the hexapole magnetic field model: (a) comparison of magnetic induction vectors,

and (b) normalized norms of error vectors. (c) the definition of the fitting error.

2.3 Force Model and Inverse Modeling

2.3.1 General magnetic force model

Without being magnetized to saturation, the magnetic force exerted on a

superparamagnetic microscopic particle placed in the field is (1 2) ( ) F m B , where F

is the gradient force, 0 0 0(3 ) [( ) ( 2 )]V m B is the effective magnetization of

the magnetic particle [67], μ is the permeability of the particle, and V is the volume of the

particle. An analysis beyond the linear magnetization of the particle can be found in [49].

Assume b 0 for the nominal magnetic force model. By substituting Eq. (2.3) into

the gradient force, the magnetic force can be expressed as

2

0 0 0(3 (2 )) [( ) ( 2 )] (|| || )V F B = 0 0 0(3 ) [( ) ( 2 )] || || (|| ||)V B B ,

where 2 1/2ˆ ˆˆ ˆ|| ||= ( ( ) ( ) )T T

mkB Q R p R p Q from Eq. (2.3). After manipulation, the magnetic

force can be expressed in the following form,

22

2

0

4

0 0

3 ( ) 1 ˆ ˆˆ ˆ( ) ( )2 ( 2 )

T Tm

f

V k

Q

F Q R p R p Q (2.5)

where Qf is called the Magnetic Charge Force Gain, the term 1 is due to that the

position is normalized with respect to . From Eq. (2.1), 0/ Q Φ , the force model can

therefore be lumped as a quadratic form of magnetic flux as follow,

2

0

1ˆ ˆ( , ) ( )T

i Q i

f

F f

p Φ Φ L p Φ , i=x,y,z (2.6)

where 2

0= Qf f is the Magnetic Flux Force Gain. Define ˆ( )xL p , ˆ( )yL p and ˆ( )zL p as

6×6 Charge-Bead Gradient Matrix and [ (m,n)ˆ( )xL p , (m,n)

ˆ( )yL p , (m,n)ˆ( )zL p ] =

(m,n)ˆ ˆˆ ˆ( ( ) ( ))T R p R p , which means the entries in the mth row and nth column of xL , yL

and zL consist the gradient of the entry in the mth row and nth column of ˆ ˆˆ ˆ( ( ) ( ))TR p R p .

Eq. (2.6) is called the General Magnetic Force Model.

2.3.2 Current-Based Magnetic Force Model

Substitute Eq. (2.4) into Eq. (2.6), the current-based magnetic force model can be

modeled as follow,

23

22

0

5

0 0 0

3 ( )ˆ ˆ( , ) ( ) , , ,

2 ( 2 )

I

T Tm ci I i I

a

g

V k NF i x y z

p I I K L p K I , (2.7)

whereIg is the force gain determined by the size and magnetic property of the magnetic

particle and that of the magnetic circuit, the 6×6 flux distribution matrix IK is adopted

from the same magnetic circuit analysis reported in [53],

5 6 1 6 1 6 1 6 1 6 1 6

1 6 5 6 1 6 1 6 1 6 1 6

1 6 1 6 5 6 1 6 1 6 1 6

1 6 1 6 1 6 5 6 1 6 1 6

1 6 1 6 1 6 1 6 5 6 1 6

1 6 1 6 1 6 1 6 1 6 5 6

I

K (2.8)

An normalized force gain NF can be normalized with the maximum input current, maxI ,

2

2 2 20max max5

0 0 0

32 ( 2 )

cN I m

a

NF g I Vk I

(2.9)

to characterize the force generation capability of a regular hexapole electromagnetic

actuating system. It is used to normalized the magnetic force and form a dimensionless

expression,

ˆ( , )ˆ ˆ ˆ ˆˆ ˆ( , ) ( )T Tii I i I

N

FF

F

p Ip I I K L p K I , i=x,y,z (2.10)

24

wheremax

ˆ II I is the normalized input current. It is evident that this dimensionless force

field characterizes the spatial distribution of the force field produced by the hexapole

electromagnetic actuating system.

2.3.3 Inverse Model Based on Constant Constraints

Whereas the force model described above establishes the relationship between the

input currents and the resulting magnetic force exerting on the magnetic particle, inverse

modeling is necessary for the practical use of the implemented electromagnetic actuating

system. Since the actuator is an over-actuated system, direct inverse of Eq. (2.10) is

impossible. It was shown in [1] that the redundancy could be removed by imposing three

constant constraints. This approach will be briefly summarized and the associated

limitations will be discussed.

Since three pairs of electromagnetic poles are placed symmetrically on the three

orthogonal axes of the actuation coordinate system, in the following, all analysis refer to

this coordinate system. Imposing three constant constraints, i.e., 1 2ˆ ˆ

xI I c , 3 4ˆ ˆ

yI I c ,

and 5 6ˆ ˆ

zI I c , and denoting three effective input currents, i.e., ˆ I = ˆ ˆ ˆ[ , , ]T

x y zI I I =

1 2 3 4 5 6ˆ ˆ ˆ ˆ ˆ ˆ[ , , ]TI I I I I I , an exact linear relationship between the effective input currents

and the dimensionless force at the center, i.e., ˆ ˆ ˆˆ( , )c F F p 0 I , is derived,

ˆ ˆ2c F A I (2.11)

25

where A is a constant actuation matrix,

[(2 ), ( 2 ), ( 2 )]x y z x y z x y zdiag c c c c c c c c c A (2.12)

The inverse model at the center can then be derived from Eq. (2.11) and Eq. (2.12),

1

1 3 5 0

1 1ˆ ˆ ˆ ˆ[ , , ]2 4

cI I I c A F , (2.13)

and

1

2 4 6 0

1 1ˆ ˆ ˆ ˆ[ , , ]2 4

cI I I c A F , (2.14)

where 0 [c ,c ,c ]T

x y zc is a constant offset introduced by the three constant constraints. It

is worth noting that whereas the use of three constant constraints leads to exact inverse

relationship at the center, the constant offset also results in unnecessarily large input

currents as evident in Eq. (2.13) and Eq. (2.14). It can also be seen from Eq. (2.11) that

due to the imposed constant constraints the magnetic force increases linearly, instead of

increasing quadratically as in Eq. (2.10), with the input currents, whereby the force

generation capability is severely degraded.

2.3.4 Optimal inverse model at the center of the workspace

It can be seen from Eq. (2.10) that the hexapole actuating system is capable of

generating 3-D force in arbitrary direction at any spatial position in the 3-D workspace,

26

and the resulting force is scalable, i.e., 2ˆ ˆ ˆˆ|| ( , )|| || ||F p I I . When the desired force is

expressed in magnitude and orientation in the spherical coordinate system (Fig.2.7), i.e.,

ˆ ˆ ˆ|| || ( , )d d F F r , where ˆ [cos cos ,cos sin ,sin ]T r is the unit force in the radial

direction, the optimal inverse solution can, therefore, be cast as,

1/2

ˆ ˆ ˆˆ ˆ ˆ( , ) ( , ),opt d d opt I F p F I r p (2.15)

Fig. 2.7. Illustration of the desired force in the spherical coordinate system.

At the center, the optimal current intˆ ( , )u

opt I associated with the unit force, i.e.,

ˆ ˆ( , )d F r , can be obtained by finding three optimal constraints, namely ( , )xc ,

( , )yc , and ( , )zc , and minimizing the squared norm of the input current vector, i.e.,

2ˆ|| ( , )|| I . This solution will improve current allocation of the over-actuated system. On

one hand, minimizing the norm of input current vector to generate the desired force will

reduce the heat generation from the coils. On the other hand, it will enhance the force

generation capability of the actuating system.

27

The squared norm of the input current vector is related to the effective input current

vector and three constraints as,

2 2

2 2 21ˆ ˆ( , ) ( , )2

x y zc c c I I (2.16)

where 1ˆ ˆ( , ) (1 2) ( , ) I A r according to Eq. (2.11). The objective function can then

be cast as

22 2 2 11 1

ˆ( , , ; , ) { } ( , )2 8

x x x x y zJ c c c c c c A r (2.17)

Minimizing this objective function yields the three optimal constraints, ( , )xc , ( , )yc

and ( , )zc , which are orientation dependent. They are compared with constant

constraints used in [1, 52] and displayed in Fig.2.8.

Fig. 2.8. Three orientation-dependent optimal constraints compared with their corresponding constant

constraints.

28

Let ˆ ˆ( , )c F r and substitute ( , )xc , ( , )yc and ( , )zc into 0c and A in Eq.

(2.13) and Eq. (2.14), the optimal current allocation,intˆ ( , )u

opt I , associated with the unit

force can be determined. The result is compared with that obtained using constant

constraints and displayed in Fig.2.9. It can be seen that the absolute value of each of the

six input currents resulted from optimal current allocation is significantly smaller than

that associated with constant constraints.

Fig. 2.9. Optimal current allocation compared with that obtained using constant constraints.

2.4. Experimental Verification of the Optimal Inverse Model

Fig. 2.10. Block diagram of the feedback motion control (

dP is the desired position, mP is the measurement

position, e is the error signal,dF is the desired force calculated by the controller,

' (t) (t )d d a F F is

caused by the actuator delay, MTF is the magnetic force, F is the modeling error,

TF is the thermal force,

a and m are the delay in actuator and measurement)

29

A feedback control system was implemented and used to stabilize the magnetic

particle placed in the workspace of the hexapole electromagnetic actuating system. A

block diagram of the control system is shown in Fig.2.10. The total delay D in the

feedback control system is contributed by actuator delaya and measurement delay

m ,

and Zero-Order-Hold delay ZOH due to digital control, i.e.

D a m ZOH . Denote

a ZOH as the effective actuator delayA . The position of the particle was measured

using a 3-D vision-based particle tracking system [55], wherein a CMOS camera was

employed to acquire the image of the particle at 200 frames per second (fps). The image

grabbing board is used to process the visual measurement algorithm and 200 fps is close

to the limit of calculation capability. Current allocation derived from inverse modeling

was implemented to overcome the issue raised by over-actuation and to achieve feedback

linearization. Together with the constant-gain feedback controller, it stabilized the

magnetic particle and suppressed the disturbances introduced by the random thermal

force. The bead dynamics can be described by the Langevin equation [68],

( ) ( )MT Tm t t P P F F , where m is the mass of the particle, is the drag coefficient of

the particle in aqueous solution. The inertia term mP is very small and can be neglected

compared to the damping forceP . Therefore, the bead dynamics can be described by a

1st order system.

( ), ( )MT d A D Tt t t P F I F p F (2.18)

30

Both methods for current allocation, i.e. the one based on optimal inverse modeling

and the other using constant constraints, were implemented. The feedback controller was

a constant-gain PI controller to stabilize the magnetic bead at the desired location.

Experiments were conducted to stabilize the particle in water and the results were shown

in Fig.2.11 and Fig.2.12 to compare the performance of the two methods.

It can be clearly seen from Fig.2.11 that when using optimal current allocation each

of the six input currents absolute value is significantly smaller than its counterpart. This

result validates the theoretical analysis developed for optimal inverse modeling.

Fig. 2.11. Six input currents of the hexapole actuator

The Brownian motion displayed in Fig.2.12 also shows improved performance of

stabilization when using optimal current allocation. As shown in the block diagram of the

feedback control system (Fig.2.10), the positioning fluctuation of the particle attributed to

at least two disturbances, i.e., the random thermal force FT and the force modeling error

ΔF. Whereas the two methods are based on the same force model and both yield

31

theoretically exact inverse solutions, the force error ΔF is likely different as the

realization of the two methods operate with different actuation currents (Fig.2.11). The

results shown in Fig.2.12 imply that the modeling error likely becomes greater when

increasing the actuation currents.

Fig. 2.12. Stabilization of the magnetic particle at the center of the workspace

2.5. Calibration and Validation of the Force Model

Whereas the hexapole magnetic force model has a sound theoretical basis, numerical

values of two model parameters, i.e., the force gain and the flux distribution matrix, are

not exactly known without experimental calibration. Any discrepancies introduce errors

in inverse modeling and degrade the effectiveness of current allocation for force

generation and control.

The nominal flux distribution matrix IK given in Eq. (2.8) was obtained through the

magnetic circuit analysis presented in [53]. Due to the magnetic leakage, which is not

32

negligible as can be seen from the results in Fig.2.3 (a), the exact values of IK differ

from the nominal values. An experimental setup along with an experimental procedure

was devised to calibrate the flux distribution matrix by utilizing the electromagnetic

induction, which is widely used to extract information from the magnetic field [69, 70].

In the experiment, each individual actuation coils were excited one by one sequentially

and the six induction voltages were measured simultaneously using the six measurement

coils (Fig.2.2 (a)). The current applied to the ith actuation coil together with the voltage

readings from the six measurement coils was used to determine the ith column of the flux

distribution matrix according to the Faraday’s Law, i.e., ( ) (t)mE t N d dt , where mN

is the number of turns of the measurement coil, (t) is the magnetic flux, and (t)E is the

induction voltage. A typical experimental result is shown in Fig.2.13, wherein a

sinusoidal actuation current applied to coil 1 and six measured induction voltages are

displayed.

Fig. 2.13. Actuation current applied to coil 1 and voltage readings from the six measurement coils.

33

The calibrated flux distribution matrix is denoted as ˆIK . It was determined after

completing the calibration procedure,

0.6022 0.0124 0.0285 0.1507 0.1668 0.0229

0.0103 0.9322 0.1740 0.0787 0.0680 0.1780

0.0294 0.1655 0.6291 0.0121 0.1458 0.0319ˆ

0.1540 0.0712 0.0112 0.9040 0.0746 0.1501

0.1805 0.0712 0.1521 0.0769 0.90

I

K

26 0.0095

0.0235 0.1726 0.0331 0.1506 0.0123 0.6122

(2.19)

Whereas the off-diagonal elements of the matrix quantify the degree of couplings

among the six electromagnetic poles, diagonal elements are actuation gains of the six

poles. It is seen that the actuation gains associated with the three lower poles are

significantly smaller than those with the three upper poles. This is due to the fact that

significant amount of the three lower poles’ material was removed to form a flat platform

to support the culture dish. It is worth noting that whereas the calibrated flux distribution

matrix in Eq. (2.19) precisely quantifies the effect of the geometric difference between

the upper and lower poles on the flux distribution in the six electromagnetic poles, it does

not fully quantify the effect in the workspace. The geometric difference also changes the

way that the magnetic field converges to the pole tip and thus the field distribution in the

workspace.

Another experiment was performed to calibrate the force gain of the actuating system

and to improve the result given by Eq. (2.19). In the experiment, the magnetic particle

was steered along a linear trajectory and propelled at a constant speed in viscous liquid.

34

Moreover, the setup is degaussed after each trajectory tracking. When traveling at

constant velocity, pv , the magnetic force exerting on the particle is balanced by the

viscous force, i.e. v p F v , where is the drag coefficient. The viscous fluid used in

the experiment was glycerol, which was much denser than water. Therefore, it required

greater magnetic force to balance the viscous force. The particle employed was 4.5μm

magnetic spherical bead and the drag coefficient was calibrated to be about 8.5×10-

6N.s/m.

Fig. 2.14. Twelve linear trajectories on the horizontal plane passing the center of the workspace (in

measurement coordinate)

With reference to the measurement coordinate, twelve linear trajectories, as shown in

Fig.2.14, having distinct directions on the horizontal plane passing the center of the

workspace were planned. In the experiment, the particle was controlled to travel along

each trajectory one by one at a speed of 6.5 m s . The viscous force was, therefore, 55pN

along the travel direction. The particle’s position in the workspace was measured using

the vision-based particle tracking system. With reference to the actuation coordinate, the

35

three components of the particle’s position, measured and target, are displayed as time

sequences in the left column of Fig.2.15. It can be seen that the particle followed the

target trajectories accurately. Three components of the calculated viscous force in the

actuation coordinate are displayed in the right column of Fig.2.15 as time sequences

synchronized with the particle’s motion. They were used to serve as measurements of the

magnetic force.

Fig. 2.15. Particle motion and viscous force displayed with reference to the actuation coordinate, wherein

black dashed lines are target motion and color solid lines are measured motion (plotted in time sequences).

Fig. 2.16. Six actuation currents associated with motion control

36

Associated with the real-time motion control in the experiment, six actuation currents

were known and displayed in Fig.2.16. Since the magnetic force is related to the

actuation currents through the hexapole magnetic force model, calibration can be

accomplished through best fitting by minimizing the following objective function,

2

1

( , ) (j) (j), (j); , (c)N

I v I I

j

J c

g F F p I g K , (2.20)

Where N denotes the number of total time instants, i.e., N=200Hz×72.6sec=14412, vF is

the viscous force. The magnetic force model is cast as ( (j), (j); , (c))I IF p I g K from the

point view of calibration. It differs from Eq. (2.7) in two ways. First, it adopts a force

gain vector with three components instead of a scalar gain. This allows the force model to

have different force gains in different directions. Second, the flux distribution matrix is

denoted as (c)IK to provide options for model calibration.

The results of three options are presented in this paper. First, the nominal flux

distribution matrix is used in Eq. (2.20), i.e., (c)I IK K , and the force gain vector Ig is

determined while minimizing the objective function. The calibrated force gain vector is

[5.37,6.82,6.93]T

I g , measured in pN , and the best-fitted results are shown in Fig.2.17.

Second, the measured flux distribution matrix is used, i.e., ˆ(c)I IK K as Eq. (2.19).

The force gain vector is [9.08,9.64,8.39]T

I g , and the fitted results are shown in

37

Fig.2.18. It can be seen that model calibration using the second option yields better

results.

Fig. 2.17. The best-fitted results when using the nominal flux distribution matrix to calibrate the force gain

vector.

Fig. 2.18. The best-fitted results when using the measured flux distribution matrix to calibrate the force

gain vector

Third, in order to further improve model calibration, a weighting coefficient, c , is

employed, i.e., row 1, 3, and 6 of ˆIK is scaled by c , to account for the difference

between the lower poles and the upper poles in the distribution of the magnetic flux

density in the workspace, and the modified flux distribution matrix is used in Eq. (2.20).

38

The improved results are shown in Fig.2.19. The value of c and the force gain vector Ig

are determined simultaneously, i.e., 1.2c and [7.56,8.55,7.62]T

I g .

Fig. 2.19. The best-fitted results when using the modified flux distribution matrix to calibrate the scaling

factor and the force gain vector simultaneously.

A quantitative measure, i.e., ( , ) (3 )Ie J c N g , is defined to evaluate and compare

the three calibration results. The average error reduces from 12.42pN to 10.04pN, and

further to 8.25pN. It is worth noting that a significant component of the average error is

attributed to the thermal force, which is about 4pN in our calibration experiments. The

actual modeling error is, therefore, significantly smaller than the calculated average error

and it is very small compared to the calibration force range, which is 110pN (±55pN).

2.6. Force Generation Capability

The force generation capability of a hexapole electromagnetic actuating system is

dictated by the design and synthesis of the electromagnetic poles and by the inverse

modeling derived for current allocation. According to Eq. (2.9), a design yielding a

higher force gain and allowing larger maximum input current will lead to greater

39

effective force possibly generated by the actuating system, i.e., 2

maxN IF g I . Compared

with the design of the 3-D actuator in [1], the new design of the hexapole electromagnetic

actuating system yields a fourfold increase in force gain. While the maximum input

current allowed by the 3-D actuator in [1] was 1.5 A, the new design was expected to

have larger maximum input current.

An experiment was conducted to determine the maximum input current allowed by the

newly developed actuating system. In the experiment, an actuation current was applied to

an individual electromagnetic pole and its strength was increased monotonously while a

Gaussmeter was used to measure the magnetic flux density near the pole tip. The test was

applied sequentially to two poles, namely P2 (an upper pole) and P3 (a lower pole). As

shown in Fig.2.20, the flux density increased linearly with respect to the input current and

neither test reached magnetic saturation while the actuation current was increased to 3A.

It is worth noting that the placement of the Gaussmeter sensor tip strongly affects the

absolute value of the readings of the hall-effect sensor. The objective of the test was to

examine the linear range of the electromagnetic actuation.

Fig. 2.20. Testing the linear range of electromagnetic actuation: P2 (left) and P3 (right)

40

The hexapole magnetic force model of the newly developed actuating system and that

of the 3-D actuator in [1] are used to calculate three force envelopes when a 4.5µm

magnetic bead is placed at the center of the workspace. Force generation capabilities are

compared and shown in Fig.2.21, wherein the three force envelopes are calculated using

nominal force models and they are spatially symmetric. It can be seen that the force

generation capability of the newly developed actuating system is significantly increased

through design and synthesis of the actuating system and the realization of optimal

current allocation. Fig.2.22 shows the comparison of three force envelopes, which are

calculated using calibrated force models. Whereas reduction of the force generation

capability of the newly developed system is noticeable due to the removing of significant

amount of the material from the three lower poles, significant improvement of force

generation capability using the optimal inverse model remains evident.

Fig. 2.21. Comparing three force envelopes calculated using nominal force models: 3-D actuator using

current allocation based on constraint constraints (blue), newly developed actuating system using current

allocation based on constraint constraints (red), and newly developed actuating system using optimal

current allocation (grey).

41

Fig. 2.22. Comparing three force envelopes calculated using calibrated force models: 3-D actuator using

current allocation based on constraint constraints (blue), newly developed actuating system using current

allocation based on constraint constraints (red), and newly developed actuating system using optimal

current allocation (grey).

2.7. Conclusion

An actively controlled hexapole electromagnetic actuating system was designed and

implemented for use with live cell experiments. It can be applied to stabilize and propel a

microscopic magnetic particle in aqueous solutions to serve as a measurement probe for

force sensing and mechanical property characterization. A hexapole magnetic force

model was employed to investigate the force generation capability of the actuating

system and to lay the foundation for inverse modeling. Optimal inverse modeling was

derived. It solved the redundancy problem of the over-actuated system and led to the

realization of optimal current allocation to enable the most effective manipulation of the

3-D magnetic force exerting on the particle at the center of the workspace.

Several aspects to improving the performance of the hexapole elesctromagnetic

actuating system are identified. First, optimal inverse modeling needs to be extended to

the entire workspace to enable superior motion control of the particle away from the

42

center and to enhance its applications that require larger workspace. Second, the delay in

the feedback control loop needs to be shortened to improve the control performance;

which can be realized through developing high-speed vision-based particle tracking

techniques. Third, it is necessary to investigate and reduce the effect of hysteresis on

force generation and control through modeling [58] or real-time estimation [24].

43

Chapter 3: The Optimal Inverse Model and Control of

Hexapole Electromagnetic Actuation

3.1 Introduction

Lumped-parameter analytical force models were presented in the previous chapter.

The performance of the hexapole magnetic actuator relies on accurate inverse modeling

of the electromagnetic force, which is essential for effective current allocation and

feedback control. The optimal inverse modeling at the workspace center is developed in

Chapter 2. However, the inverse modeling in the center cannot be directly used in the

workspace since the magnetic force model is position dependent.

A simplified inverse model was derived and employed to realize feedback

linearization in the implementation of active feedback control [1]. Whereas stable

magnetic trapping and motion control were successfully demonstrated, the simplification

and approximation adopted in inverse modeling severely limit the performance of the

developed actuating systems on two fronts. First, the use of constant constraints to derive

the simple inverse model, which is valid at the center of the workspace, results in

excessive current flow in the coil. It significantly degrades the force generation capability

of the actuating system. Second, extending the inverse model at the center to the entire

workspace through linear approximation in the spatial domain leads to significant error

44

when generating magnetic force exerting on the magnetic particle placed away from the

center of the workspace.

This chapter presents the derivation of an optimal inverse model of the over-actuated

hexapole electromagnetic actuating system to minimize a specially weighted norm of the

six input currents when applied to produce the desired 3-D magnetic force in the

workspace. The optimal inverse model is then employed to realize real-time current

allocation to enable feedback linearization in the implementation of active feedback

control. Stable magnetic trapping and precise motion control of the microscopic magnetic

particle in aqueous solutions are demonstrated in experiments.

3.2 Hardware Implementation of High Speed Control

Fig. 3.1. (a) Magnetic setup integrate with the inverted microscope (b) High speed embedded control

system using FPGA (c) real-time display of the reference bead and the control bead (the reference sticks to

the cover glass surface to provide the information such as drift, vibration and etc.)

45

It is widely known that the time delay in the system will greatly degrade the

performance of the feedback control. However, the feedback control implemented in PC

cannot achieve high sampling rate since the computation capability is limited and the

image grabbing card processing speed is under 300 fps. Moreover, the time consistency

of PC is not very good due to operation systems.

To achieve high sampling rate control, FPGA is used to calculate the control effort

and realize input/output functionalities. The FPGA based real-time image processing is

accomplished by P. Cheng [71], which can achieve 10,000 fps real-time image

processing with reduced image size. The complete FPGA system in shown in Fig.3.1. To

increase the image resolution, a superluminescent diode (SLD) illumination system

(QPhotonics) is used to improve the image sensitivity [72]. The visual sensing resolution

is about 1e-3 pixels in x and y direction, and sub nm in z direction. The CMOS camera

used in this system has 8um pixel size. Therefore, the visual sensing resolution is about

0.13 nm in x and y direction when 60x lens is used and 0.2 nm when 40x lens is used.

DA converters (DAC8814 EVM, Texas Instruments) are updated by the main FPGA to

drive linear power amplifiers (Micro Dynamics, BTA-28V-6A). USB modules (DLP-

USB1232H, FTDI) are connected and programmed to accomplish PC-FPGA

communication. In such a system, PC is used as a coordinator such as downloading

control parameters, data logging and etc. The real-time image is displayed on the monitor

by connecting it to a VGA port in the FPGA. A DE2-115 FPGA (Terasic Inc.) is used as

a main FPGA to calculate the control effort and output the actuation effort through DA

46

converters. The real-time image processing is performed in two TR4-230 FPGA (Terasic

Inc.) boards.

The calculation and input output functionalities are all realized in FPGA systems.

1606Hz sampling rate, which is the maximum achievable sampling rate for 512×512

image, is used in real-time control. Higher sampling rate can be easily achieved by

reducing the image size. A reference bead (Fig.3.1 (c)), which is attached to the coverslip

surface, is measured simultaneously to compensate the drift, such as structure thermal

drift.

3.3 The inverse model

3.3.1 The position dependent inverse model based on constant constraints

An inverse model based on constant constraints is proposed in [1] to remove

redundancies. It is briefly summarized here. It is already shown in Eq. (2.11) that the

force at the center can be expressed as ˆ ˆ ˆˆ( , )c F F p 0 I = ˆ2 A I , where 1 2ˆ ˆ

xI I c ,

3 4ˆ ˆ

yI I c , 5 6ˆ ˆ

zI I c , ˆ ˆ ˆ ˆ[ , , ]T

x y zI I I I =1 2 3 4 5 6ˆ ˆ ˆ ˆ ˆ ˆ[ , , ]TI I I I I I , A = diag[

(2 )x y zc c c , ( 2 )x y zc c c , ( 2 )x y zc c c ]. Denote ˆ 2δI

J A , then ˆˆ ˆ

c δIF J δI ,

whereas it is not straightforward to obtain the exact inverse solution for the entire

workspace. Nonetheless, by keeping the first-order terms in Taylor expansion of Eq.

(2.10), a linear approximation of the magnetic force around the center can be derived,

ˆ ˆˆ ˆ ˆˆ ˆ( , )

pδIF p δI J δI J p (3.1)

47

Where pJ = 2∙diag [ 2(2 )x y zc c c , 2( 2 )x y zc c c , 2( 2 )x y zc c c ]. The effective

actuation current required to produce the desired force is derived as follow,

1 1

ˆ ˆ ˆˆ ˆ ˆ ˆ( )d

pδI δI

δI J F p J J p (3.2)

The effective actuation current ˆ I along with the three linear constraints[ , , ]x y zc c c can be

employed to calculate the required actuation current, I ,

( , )dInverseI F p (3.3)

It is worth noting that the desired force,dF , and the spatial location, p , are associated

with the actuation coordinate frame. When employing Eq. (3.3) to realize inverse model

calculation, the rotational matrix a

m R between the two coordinate frames needs to be

applied to both vectors.

3.3.2 The Optimal Inverse Model in the Entire Workspace

The optimal inverse model in the center is presented in Chapter 2.3.4. However, the

magnetic force model strongly depends on the position of the magnetic bead as shown in

Eq. (2.10). The linear inverse model based on the 1st order Taylor expansion, i.e. Eq.

(3.2), will result in significant error for both constant constraints and optimal constraints.

For constant constraints, the error will grow when the desired force is large or the

48

position is further away from center (Fig.3.2). For optimal constraints, the error does not

depend on force magnitude due to scalable characteristics. However, the error does grow

significantly when the position is far away from center (Fig.3.3).

Fig. 3.2. Inverse model based on constant constraints. (a) The percent force error at (20,0,0)um, desired

force ˆdF =1; (b) The percent force error at (20,0,0)um, desired force ˆ

dF =10; (c) The percent force error at

(40,0,0)um, desired force ˆdF =1; (d) The percent force error at (40,0,0)um, desired force ˆ

dF =10

Fig. 3.3. Optimal inverse model. (a) The percent force error at (20, 0, 0) um; (b) The percent force error at

(40, 0, 0) um

Due to the insufficiency of the linear inverse model, the optimal inverse model in the

entire workspace is desired to solve the position dependency of the magnetic force model.

However, due to the asymmetrical magnetic charge configuration at a non-center point,

there is no analytical inverse model as Eq. (2.13) and Eq. (2.14). Therefore, the objective

49

function as Eq. (2.17) is not available. However, minimizing the norm of currents while

still satisfying the force model is essentially an optimization problem under nonlinear

equality constraints, and Lagrange multipliers ( , , )x y z can therefore be employed to

construct the following objective functions:

2ˆ ˆ ˆ ˆˆ( , , ) || || ( ) cos cos

ˆ ˆ ˆ ˆˆ ˆ( ) cos sin ( ) sin

T T

x I x I

T T T T

y I y I z I z I

J

P I I K L p K I

I K L p K I I K L p K I (3.4)

The force model Eq. (2.10) is used in Eq. (3.4). The position dependent optimal

inverse solution satisfying ˆmin( ( , , ))J P criterion in Eq. (3.4) is denoted as ˆ ˆ( , , )unit

opt I P

. Minimizing ˆ( , , )J P is an optimization problem with nine parameters, including six

currents and three Lagrange multipliers. The local minimum occurs at the values where

the gradient of ˆ( , , )J P is a zero vector, i.e. ˆ( ( , , ))J P 0 , which is still a nonlinear

functions. MATLAB optimization toolbox is used to minimize Eq. (3.4), wherein the

optimal inverse model at the center, i.e.intˆ ( , )u

opt I can serve as a good initial guess for the

numerical calculation of ˆ ˆ( , , )unit

opt I P at different locations P . In the first Octant, the

locations where ˆ ˆ( , , )unit

opt I P is numerically calculated are shown in Fig.3.4. The range is

a 45um×45um×45um cube in the first Octant, leading to a range of 90um×90um×90um

in the entire workspace.

50

Fig. 3.4. Locations where the optimal inverse model are obtained

After numerically obtaining the optimal inverse model at different locations, it is

desired that a usable function can be found to describe ˆ ˆ( , , )unit

opt I P so that real-time

control can be implemented. ˆ ˆ( , , )unit

opt I P can be fit with respect to position P at different

orientations ( , ) since this inverse model needs to capture the position dependent

information of the inverse model. A frequently used fitting scheme for capturing the

nonlinear trend is the Taylor expansion. Specifically, the 2nd order Taylor expansion

with respect to P is computed for each orientation ( , ) , which can be expressed in the

following quadratic form:

1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( , , ) ( , , ) ( ) ( ) = ( , )2

( , ) ( , ) ( , ) ( , )

( , ) ( , ) ( , ) ( , ) =

( , ) (

1

unit unit T T Taylor

opt i opt i i

T

xx i xy i xz i x i

yx i yy i yz i y i

zx i zy

c c c cx

c c c cy

c cz

I P I 0 G P P P H P P C+ P P

, ) ( , ) ( , )

( , ) ( , ) ( , ) ( , ) 1

i zz i z i

x i y i z i i

x

y

c c z

c c c c

(3.5)

51

where i=1, 2,… , 6 is the index of 6 currents, ˆ( )G P and ˆ( )H P are the gradient vector and

hessian matrix, [ , , ,1]Tx y zP is the augmented position vector, ( , )Taylor

i C is the Taylor

expansion matrix, ( , )ic , which is the last entry of matrix ( , )Taylor

i C , is just the

optimum inverse model at the center, ( , )k ic , k = x, y, z, is from the gradient vector,

( , )mn ic , where m,n = x,y,z, is from the Hessian matrix.

To test the performance of the inverse model, Eq. (3.5) is used to solve the actuation

current at different locations within the 45um×45um×45um cube in the 1st Octant,

wherein the desired forces are in different orientations. The real force generated by the

actuation current are calculated and compared with the desired force to evaluate the

performance of the inverse model in Eq. (3.5). Define the percent force modeling error

as | | | |*100dF F , which is shown in Fig.3.5 (a) ,the maximum force modeling error at

each location is selected and plotted in Fig.3.5 (b), where the x axis is ˆ|| ||P , denoted as R

and sorted in ascend order, y axis is the percent force error. It can be clearly seen that the

force modeling error grows with respect to R, which is expected for Taylor expansion.

Although the Eq. (3.5) captured the characteristics of Taylor expansion, the growing

error as in Fig.3.5 (b) is not desired. However, the format of the inverse model in Eq.

(3.5) can be kept due to its simplicity. Alternatively, the least square fitting can be

employed to fit a similar format but reduce the norm of error in the whole workspace:

ˆ ˆ( , , ) ( , )unit T LS

opt i i I P CP P (3.6)

52

The matrix ( , )LS

i C follows the same pattern as ( , )Taylor

i C in Eq. (3.5), but the

entries of ( , )LS

i C comes from Least-Square fitting. The force modeling error is

calculated again at each point and is plotted in Fig.3.5 (b), together with error from

Taylor expansion approximation. It can be seen that the force model error using Least-

Square fitting is almost always under 5%.

Fig. 3.5. Comparison of percentage force error between Taylor expansion and least-square fitting

The inverse model is thus constructed by following Eq. (3.6). Interpolation is used to

complete the inverse model since the inverse model is only fit at discrete orientations

where optimal solutions are numerically obtained.

The optimal inverse model can not only accurately solve the position dependency

issue in the force model, but also can greatly increase the force generation capability

since it minimized the norm of the actuation current. The comparison of the force

generation capability using the optimal inverse model (Eq. (3.6)) and the inverse model

based on constant constraints (Eq. (3.2)) are plot together in Fig.3.6.

53

Fig. 3.6. (a) Force generation envelops at (0,0,0)um. (b) Force generation envelops at (20,0,0)um. (c) Force

generation envelops (40,0,0)um

3.4 Active feedback control: stabilization, Brownian motion control and tracking

control

3.4.1 Stabilization and Brownian motion control

With the inverse model available, the nonlinear controller can be designed to achieve

motion control, wherein the nonlinearity between force and the actuation current can be

handled by the optimal inverse model. Therefore, the feedback controller design is easier.

Fig. 3.7. Block diagram of the feedback motion control using proportional controller (

dP is the desired

position, mP is the measurement position, e is the error signal, Kp is the proportional gain,

dF is the desired

force calculated by the controller, ' (t) (t )d d a F F is caused by the actuator delay,

a and m are the delay

in actuator and measurement, MTF is the magnetic force, F is the modeling error,

TF is the thermal force)

54

Looking at the feedback control block diagram in Fig.3.7, the real magnetic forceMTF

consists of desired force '

dF and modeling errorF . For the optimal inverse model, F is

the error that is not modeled by the nominal model, such as imperfect magnetic charge

approximation, hardware assembly and etc. For the inverse model based on constant

constraints, however, a great part ofF also comes from the linear approximation of the

nominal model of Eq. (3.1) and Eq. (3.2), the effect of which is shown in Fig.3.2.

The basic functionality to enable the magnetic actuator working as an active probing

system is the stabilization of the magnetic bead. A stabilization experiment using P

controller is performed. Similar to Eq. (2.18), the bead dynamics using proportional

control is described as follow,

p d A D Tt t t t t P K P P F F (3.7)

The magnetic bead is stabilized at different 3D positions as shown in Fig.3.8. There

are three layers in which each layer contains 25 points where the magnetic bead are

stabilized. At each layer, the magnetic bead is first stabilized at the lower left corner in

the xy plane. The magnetic bead is stabilized at each point for 10sec, and then is steered

at 5um/s to the next point, which is 10um away from previous point. The bead is steered

following a zigzag path until it reached the last point at the upper right corner of that

layer. After the transient response vanished at each position, the data corresponded to

7sec was selected and plotted. At each layer, the bead is steered back to the starting point,

i.e. the lower left point, after finishing the data collection of 25 points of that layer.

55

Fig. 3.8. 3D plot in bead stabilization experiment (the result in from The Optimal inverse model)

The optimal inverse model and the inverse model based on constant constraints are

compared to show the current allocation performance of the optimal inverse model. From

Fig.3.9 it can be seem that the current in the optimal inverse model is much smaller than

that based on constant constraints. It is worth mentioning that current I5 and I6 from

constant constraints are negative with much larger absolute value compared with the

optimal inverse model.

Experimentally, the optimal inverse model will not only lead to much smaller

actuation effort but also lead to better Brownian motion control with smaller standard

deviation. The Brownian motion standard deviations of 75 points in 3 layers (as Fig.3.8)

are plotted in Fig.3.10. It can be seen that the positioning standard deviation from the

optimal inverse model is obvious smaller compared to the inverse model based on

constant constraints, which will lead to better special resolution. It is likely that the larger

actuation current in the constant constraints will lead to larger modeling error, which can

lead to greater positioning fluctuation.

56

Fig. 3.9. Actuation effort using optimal inverse model and the inverse model based on the constant

constraints. (For the inverse model based on constant constraints, I5 and I6 are negative with much larger

absolute value than optimal invers model)

Fig. 3.10. The positioning standard deviation in x,y and z axis using the optimal inverse model and constant

constraints

Fig. 3.11. The positioning performance from Proportional controller, using optimal inverse model and

constant constraints.

The magnetic force modeling error in block diagram Fig.3.7 is inevitable in the real

system due to different factors such as hardware assembly, magnetic charge biases and

etc. From Eq. (2.10), with smaller actuation effort, the optimal inverse model will lead to

57

smaller modeling error. According to Eq.(3.7), the response from dP to e and F to e

are (s) ( )DS

d pe P s s K e

, (s) ( )m Ds s

pe F e s K e

. AssumingdP and

F are step input, the steady-state error caused by dP and F can be calculated by the

Final Value Theorem: | 0dPe , | 1F pe K . Where |

dPe and | Fe are steady-state errors

from dP and F respectively. The bead dyanmics is a type 1 system. Therefore, the

positioning error using P controller in the stabilization experiment is a good indication of

moeling error. The steady-state error | Fe is inverse proportional to the control gain pK .

Therefore, a larger gain can lead to smaller stedy-state error. The open loop transfer

function is (s) / ( )Ds

OL pG K e s , and the Nyquist stability criterion requires that

(2 )p DK . Therefore, the control gain has to be limited to avoid instability. From

Fig.3.11 it can be seen that the optimal invese model can lead to much smaller

positioning error, which indicate a much smaller modeling error. From the force model

Eq.(2.10), i.e. ˆ ˆˆF ( )T T

i I i I I K L p K I , the modeling error such as magnetic charge biases

and assembly error will be lump into ˆ( )iL p , which indicate that larger actuation current

will lead to larger modeling error.

3.4.2 Trajectory tracking

Tracking and steering capability is necessary to enable the magnetic actuator to

perform 3-D probe sensing tasks, such as cell scanning, binding a receptor with the

magnetic bead and etc. To demonstrate the capability of the inverse model performance

58

at different locations and directions, the magnetic particle are steered along different

trajectories as shown in Fig.3.12. The magnetic bead is steered at 5um/s along circular

trajectories at different locations in xy plane, yz plane and xz plane. In xy plane, the bead

is firstly steered to track the lowest 40um-diameter circle, then is moved to the plane in

the middle, which is 5um above the lowest one. After tracking the circle in the middle,

the bead is moved to the highest plane within which is another circle of the same shape.

After tracking 3 circles in xy planes, the bead is moved from the highest plane to the

middle plane, and is steered along 10um-diameter circles in yz plane and xz plane. First,

the bead is steered along circles in yz plane and xz planes in the center (labeled as 0 in

Fig.3.12 (b) and (c)). It is worth mentioning that the circles in yz plane and xz plane are

plotted separately in Fig.3.12 for clarity, while the yz circular tracking is followed by xz

circular tracking immediately. Then the magnetic bead is steered to position 1, 2, 3 and 4

sequentially, circular tracking in yz plane and xz plane is performance at each point.

Fig. 3.12. Trajectories in 3D tracking control.

59

Fig. 3.13. Actuation effort in 3D tracking control. (For the inverse model based on constant constraints, I5

and I6 are negative with much larger absolute value than optimal invers model)

It can be seen from Fig.3.13 that the optimal inverse model can achieve the 3D

trajectory tracking with much smaller actuation effort, which is as expected. Moreover,

the optimal inverse model can achieve much smaller tracking error, which is shown in

Fig.3.14. It can be seen that by using the optimal inverse model, the standard deviation of

the tracking error reduced to (57.82nm, 43.81nm, 42.01nm) compared with (123.10nnm,

63.08nm, 109.83nm) while using constant constraints.

Fig. 3.14. Tracking error using the optimal inverse model and inverse model based on constant constraints.

60

3.5 Comparison of High Speed Control in FPGA and Low Speed Control in PC

Due to the compact form of the inverse model Eq. (3.5) and Eq. (3.6). The inverse

model can be easily implemented in FPGA and the feedback control performance is

addressed in Chapter 3.4. The advantage of using high speed control is demonstrated in

this sections with a simple controller, i.e. the P controller. It is analyzed in Chapter 3.4.1

that (2 )p DK from Nyquist criterion, which means that the stability margin is

inversely proportional to the time delay in the feedback control loop. The time delay in

the control loop is directly determined by the sampling rate. Therefore, it is desired that

the sampling rate is high enough to reduce the time delay, thus better suppressing the

Brownian motion and increasing the stability margin.

Most researcher implemented the control algorithm in PC [1, 2, 52, 53, 56, 63].

However, the sampling rate in PC is limited by the computation capability and time

consistency of PC. The magnetic bead stabilization is performed in PC at 200 fps in

Chapter 2, before the FPGA system is developed. The data from the 200Hz control in PC

is compared with the high speed 1606Hz control in this Chapter.

Power Spectrum Density (PSD) curves [52] can be used to determine trapping

stiffness pK , damping ration and time delay D . From the block diagram Fig.3.7 it can

be seen that the transfer function from thermal force to bead position is

1 ( e )Ds

T ps K

P F . The PSD of the thermal force is known to be (F ) 4T BPSD k T

[73], where Bk is the Boltzmann constant, T is the absolute temperature. The PSD of the

magnetic bead can be formulated as follow,

61

2

4(P)

2 exp( 2 )

B

p D

k TPSD

i f K i

(3.8)

pK , andD can be known by comparing Eq. (3.8) with PSD of the measurement data.

The calibration result in x direction are shown in Fig.3.15. It is calibrated from the PSD

curves that the time delay D under 200Hz sampling rate is about 12ms, which is 2.4

steps of the sampling interval. Theoretically, 2 steps of the delay come from the

measurement and 0.5 step is caused by Zero-Order-Hold (ZOH) of the digital control.

Therefore, the calibrated 2.4 steps delay is very close to the theoretical prediction. The

calibrated drag coefficient is about 5.51e-8 N.s/m in x direction. There is a peak at PSD

curve at 71.37 Hz, which is due to structure vibration of the setup.

Fig. 3.15. The measurement PSD curves and calibrated PSD curves (The peak at 71.37Hz is due to

structure vibration)

62

For high sampling rate control in FPGA, the parameters can be calibrated with similar

method using PSD curves. From the PSD calibration, the time delay in the high speed

control system is dropped to 2.9ms, which is a great reduction compared with 12ms in PC

based control loop using 200Hz control. As expected, the standard deviation of the

Brownian motion is much smaller and the stability margin is greatly increased. Therefore,

a much larger control gain can be used (Fig.3.17). The minimal Brownian motion

standard deviations under 1606Hz control is 31.68nm, 30.65nm and 27.34nm in x, y and

z directions, while in 200Hz control the standard deviations are 51.97nm, 51.55nm and

36.32nm in three directions (Fig.3.16).

Fig. 3.16. The positioning result using the optimal trapping stiffness in 200Hz case and 1606Hz case

It is worth mentioning that for 1606Hz case the calibrated drag coefficient in x and y

direction is 3.71e-8 N.s/m and 6.43e-8 N.s/m in z direction, which is much smaller than

that in 200Hz case, i.e. 5.51e-8 N.s/m in x and y direction and 1.31e-7 N.s/m in z

direction. Therefore, the improvement of the high speed control is even better if the drag

coefficient is the same in these two cases.

63

The increase of sampling rate will reduce the measurement delay m and Zero-Order-

Hold delayZOH . The actuator delay

a , however, is limited by the actuator dynamics,

which cannot be changed by changing control sampling rate. If the actuator delay can be

further reduced, the increase of the control sampling rate is even more significant.

Fig. 3.17. Trapping stiffness vs. standard deviation of Brownian motion, 200Hz control and 1606Hz control

3.6. Conclusion

An accurate optimal current allocation scheme based on minimal 2-norm of currents of

an over-actuated hexapole electromagnetic actuator is accomplished, which solved the

redundancy, nonlinearity and position-dependency at the same time. Compared with

linearized inverse model by applying constant constraints, the optimized currents

allocation greatly increased the force accuracy and force generation capability, which is

demonstrated theoretically and experimentally. Stabilization and trajectory tracking are

experimentally studied. It can be seen that the optimal inverse model will reduce the

current magnitude, reduce the Brownian motion standard deviation and reduce the

64

tracking error. Due to the compact form of the optimal inverse model, the high speed

control can be implemented in the FPGA system and will greatly improve the feedback

control performance compared with the control implemented in PC.

There are still several aspects can be improved to fully enable the magnetic actuator to

perform 3-D scanning probe tasks. First, the magnetic force modeling accuracy need to

be improved. While the motion capability can be accomplished by feedback control

mechanism, the magnetic force modeling error, such as hysteresis, need to be reduced to

accurately know the interaction force between the bead and samples. Second, a dynamics

joint state-parameter estimator need to be developed to dynamically estimate the bead-

sample interaction force and the parameters such as drag coefficient, which can be time-

varying due to environment change.

65

Chapter 4: Hall Sensors Based 3-D Magnetic Force Modeling

and Experiment

4.1 Introduction

Since the soft magnetic material is used to fabricate the magnetic actuator, the

analytical magnetic force model based on actuation current in Chapter 2 and 3 usually

subjects to the hysteresis effect that will greatly degrade the magnetic force model

accuracy. For high speed actuated multi-pole magnetic actuator in which magnetic poles

are located close to each other, the magnetic field not only has rate-dependent hysteresis

effect [59, 60] that is difficult to model, the magnetic coupling among six pole is an even

bigger issue that is usually beyond the capability of modeling.

In this chapter, the difficulty of characterizing the relationship from input current to

output magnetic field is solved by introducing Hall Sensors. Six high bandwidth Hall

Sensors (Asahi Kasei EQ-730L), corresponding to six magnetic poles, are introduced to

directly measure the magnetic field of each pole and thus the total magnetic field. A so-

called Hall-Sensor-based magnetic force model, is proposed to describe the magnetic

force and completely solve the hysteresis issue. The Hall-sensor based model can be

experimentally calibrated by steering the magnetic bead in the 3-D workspace. By

comparing the viscous force and modeling force, the unknown parameters in the

magnetic force model can be calibrated. The Hall-Sensor-based force model is compared

66

with the current-based force model in Chapter 3. It is clearly seen that the Hall-Sensor-

based magnetic force model greatly outperforms the current-based magnetic force model

in term of force accuracy. Moreover, the calibrated Hall sensor based magnetic force

model can not only fit well with the measurement force, but can also be employed to

predict the magnetic force when different trajectories are traversed. Moreover, with Hall

sensor measurement capabilities, the magnetic field of each pole can be directly

controlled using a secondary control loop.

In most magnetic force models, the magnetic moment of the magnetic bead is

modeled linearly dependent on the external magnetic field. When the magnetic field

becomes large, however, the magnetic moment will be nonlinearly dependent on the

magnetic field. An accurate Hall sensor based model is proposed to characterize the

nonlinear effect of the magnetization and the force model is also calibrated by steering

the magnetic bead in Glycerol with large magnetic force.

4.2. The Magnetic force models

4.2.1 The Hall-sensor-based magnetic force model

From the analysis in Chapter 2.2 and Chapter 2.3, it can be seen that two factors

mainly determine the force modeling quality, 1) the accuracy of magnetic charge model,

i.e. Eq. (2.1), 2) the value of magnetic flux in Eq. (2.6). It is already verified in Chapter

2.2 that the point charge model can accurately model the magnetic field. However, the

uncertainties in the magnetic flux calculation, which is mainly caused by the hysteresis

effect, often result in significant modeling errors. The difficulty of modeling the

67

hysteresis in this hexapole magnetic actuator comes from three aspects: 1) the rate-

dependent hysteresis of the soft magnetic material [59, 60], 2) transient magnetization

effect such as accommodation effect [74-76], and 3) the magnetic coupling among six

poles.

Fig. 4.1. Magnetic field around the tip of the magnetic pole and a suppositional Hall Sensor.

To solve the hysteresis issue, suppose a miniature hall sensor is available at the tip of

the magnetic pole (Fig.4.1) to measure the magnetic field. Then the voltage measurement

of the Hall Sensor of the ith pole, denoted asiHv , is proportional to the corresponding

magnetic flux i ,

i ii H Hd v (4.1)

whereiHd is a constant gain from Hall sensor voltage to the magnetic flux. With the

measurement of each Hall Sensor, the Magnetic Flux Vector can be represented by the

following equation,

68

H HΦ D V (4.2)

where 1 2 3 4 5 6

= ( , , , , , )H H H H H H Hdiag d d d d d dD is defined as the Flux-Gain Matrix and

1 2 3 4 5 6=[ , , , , , ]T

H H H H H H Hv v v v v vV is the Hall Sensor Voltage Vector. Substituting Eq. (4.2)

into Eq. (2.6), the Hall sensor based force model can be written as,

( , , ) ( , )T T

H H H H HfF p b V V D L p b D V (4.3)

HD can be normalized with respect to the first entry1Hd , define the Normalized Flux-

Gain Matrix ˆHD =

1H HdD =2 3 4 5 6

ˆ ˆ ˆ ˆ ˆ(1, , , , , )H H H H Hdiag d d d d d , the force model can be

lumped as the following form,

1

2

ˆˆ ( , , )

ˆˆ ˆ ˆ( , , ) ( , ) ( , , )

H HH

T T

H H H H H H H H H

f

f d f

F p b V

F p b V V D L p b D V F p b V (4.4)

where ˆHf is the force gain of the quadratic form about voltage vector

HV .

4.2.2 The current-based magnetic force model

From Hopkinson’s Law, the theoretical value of the 6×6 flux distribution matrix IK

has 5/6 as the diagonal entries and -1/6 as the off-diagonal entries as in Eq. (2.8), but it is

shown in Chapter 2 that the real value does not follow the theoretical value due to the

69

magnetic leakage in the magnetic circuit and the geometry difference between upper

poles and lower poles, the following form is proposed to describeIK :

1

2

3

4

5

6

I

k s m m s

k m s s m

s m k m s

m s k s m

m s m s k

s m s m k

K (4.5)

Where k1~k6 means the portion of magnetic flux directly from the corresponding pole,

which takes the biggest weight among the 6 entries of each row/column vector. m , s and

describe the coupling between the neighboring poles, the poles that are separated by

one pole and the opposite poles. It is obvious that k1~k6 > m > s > , which can also be

inferred from the measurement IK in Eq. (2.19). Referring to Eq. (2.6) and Eq. (2.4), the

current-based magnetic force model can be written as:

2

( , , ) ( , )T TcI I

a

Nf

F p b I I K L p b K I (4.6)

Define1

ˆ =I I kK K , max

ˆ= II I the force model can be lumped in the following form,

2

2 2

max 1

ˆ

ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ( , , ) ( , ) ( , , )

I

T TcI I I I

a

f

Nf I k f

F p b I I K L p b K I F p b I (4.7)

70

Where ˆIf is the lumped force gain of the quadratic form about normalized current vector

I .

4.3 Hardware Integration and Validation of Hall Sensors based Hexapole Magnetic

Actuator

4.3.1 Hall sensors integrated with magnetic actuator

Fig. 4.2. (a). Hall sensors integrated with Hexaple magnetic actuator. (b) Zoom-in plot of Hall sensors

associated with upper poles. (c) Zoom-in plot of Hall sensors associated with lower poles.

The Hall Sensors utilized in this electromagnetic actuator, Asahi Kasei EQ-730L, are

high bandwidth Hall Sensors (100K Hz) with high sensitivity (130mv/mT typ.). It is

hoped that the magnetic field diverged from the tips of six poles can be directly measured

as in Fig.4.1. However, this cannot be realized since the size of the workspace enclosed

71

by the tips is 500um in radius, while the Hall Sensor is a 4.1mm×3mm×1.15mm cuboid.

Therefore, six Hall Sensors are placed at the surface of each magnetic pole, as in Fig. 4.2.

4.3.2 Validation of the Hall Sensor measurement

Fig. 4.3. (a). The setup for studying the relationship between the surface-mount Hall Sensor measurement

and the Hall Sensor measurement at the tip. P1 and P2 are actuated by current-driven coils. (b). Positions

and dimensions of hall elements.

The Hall Sensors are attached to the surface of six magnetic poles. Therefore, it is not

clear whether the measurement at the surface can represent the magnetic field at the tip.

This section is to investigate the possibility of modeling the magnetic field at the tip with

the measurement at the surface of the magnetic pole.

The setup for evaluation is in Fig.4.3. As shown in the figure, the Hall element within

the IC chip is a 0.3mm diameter plate, which is 0.41mm to the surface. The first Hall

Sensor H1 is attached to the surface of the first pole P1, the second Hall Sensor H2 is

placed near the tip of the P1. The second pole P2 is introduced to study the influence of

the magnetic coupling, which will happen in real situation since the six poles are placed

72

near each other. It is worth mentioning that the Hall Sensor element of H2 and the axis of

P2 are in the same plane so that H2 only measures the magnetic flux density diverged

from P1. The locations and dimensions of the Hall Sensor elements of H1 and H2 are

illustrated in Fig.4.3 (b). It is hoped that the measurement of H1 at the surface, denoting

as Vsurface, is proportional to the measurement of H2 at the tip, denoting as Vtip, so that

Vsurface can represent Vtip, which is used in the Hall-Sensor-based model Eq. (4.4).

The following two experiments are conducted:

1) Actuate P1 with sinusoidal input of different frequencies (1000Hz, 2000Hz, and

3000Hz).

2) Actuate P2 with sinusoidal input of different frequencies (1000Hz, 2000Hz and

3000Hz). In this case, P1 is magnetized by P2 instead of self-actuated. But H2 still only

measures the magnetic flux diverged from the tip of P1.

Fig. 4.4. Input-output plot between Vtip and Vsurface in two cases: 1). P1 is self-actuated, 2) P2 is actuated, P1

is magnetized by the magnetic coupling between P1 and P2.

The experiment result of above two cases is shown in Fig.4.4. First, when P1 is

actuated, it can be seen that the ratio between Vsurface and Vtip is almost the same when the

73

actuation frequency increased from 1000Hz to 3000Hz, which is obvious from the input-

output plot between Vtip and Vsurface. There is time delay between the measurement of Vtip

and Vsurface since they are non-collocated. However, the time delay is negligible since it is

μs scale (Fig.4.4) while the magnetic actuator system bandwidth is ms scale. Second,

when P2 is the actuation pole, P1 is magnetized through magnetic coupling. The

measurement result in Fig.4.4 shows that the ratio between Vsurface and Vtip almost has the

same value. Therefore, the surface-mount Hall Sensor can represent the magnetic field at

the tip of the magnetic pole.

Due to the proportionality between Vsurface and Vtip, the Hall-Sensor-based model as

Eq. (4.4) can be directly used, wherein the voltage vector VH stands for the voltage

measurement of six surface-mount Hall Sensors.

4.4 Hall Sensor Measurement in Magnetic Bead Control and Calibration Of

Magnetic Force Models

4.4.1 Hall Sensor measurement in bead trapping

With the hall-sensor measurement capability, the relationship between the input

current and the magnetic flux density of each pole can be studied since the hall sensor

voltage is proportional to the magnetic field in the corresponding pole. Therefore, the

current-voltage plot represents the relationship between the input current and resulting

magnetic field.

The bead trapping control is performed in water with 1606Hz sampling rate, using the

current-based optimal inverse model as described in Chapter 3.3.2. A proportional

controller is used in the bead trapping control and the positioning result will be shown

74

later in Chapter 4.5.2. Obvious positioning error is observed. Since the bead dynamics is

a type 1 first order system, the positioning error indicates that there is modeling error,

which can be clearly seen from Hall-Sensor measurement in Fig.4.5. It can be seen that

most poles have remanent magnetization since most input-output curves are biased. This

will result in modeling error since current-based actuation relies on the assumption that

the output magnetic flux of each pole is linearly dependent on the input current. The

steady-state error in bead trapping is most likely a result of the remanent magnetization.

Fig. 4.5. Input-output plot between current and Hall Sensor voltage.

4.4.2 Force model calibration

The positioning result in Chapter 4.4.1 is an obvious evidence that the hysteresis will

result in modeling error. Compared with the current-based model, the Hall-sensor based

model is a promising solution for modeling error caused by hysteresis since this model is

based on the direct measurement of the magnetic flux.

The Hall-Sensor-based model and current-based model are formulated in Eq. (4.4) and

Eq. (4.7). There are some parameters to be determined in both models. In Hall-Sensor-

75

based model Eq. (4.4), ˆHf , ˆ

HD andb are unknown parameters to be calibrated, while ˆIf ,

ˆIK and b are to be calibrated in current-based model in Eq. (4.7).

Fig. 4.6. Motion control result in trajectory tracking

To fully explore the magnetic force in the 3-D space, the magnetic bead is steered in

water along different trajectories in the 3-D space as shown in Fig.4.6. The inverse model

and control scheme is based on the nominal current-based magnetic force model, which is

demonstrated in Chapter 3.3.2. The magnetic bead is steered at 10μm/s along circular

trajectories at different locations in xy plane, yz plane and xz plane. In xy plane, the bead

is firstly steered to track the lowest 50μm-diameter circle, then is moved to the plane in

the middle, which is 5um above the lowest one. After tracking the circle in the middle,

the bead is moved to the highest plane within which is another circle of the same shape.

After tracking 3 circles in xy planes, the bead is moved from the highest plane to the

middle plane, and is steered along 10um-diameter circles in yz plane and xz plane. First,

the bead is steered along circles in yz plane and xz planes in the center (labeled as 0 in

76

Fig.4.6 (b) and (c)). It is worth mentioning that the circles in yz plane and xz plane are

plotted separately in Fig.4.6 for clarity, while the yz circular tracking is followed by xz

circular tracking immediately. Then the magnetic bead is steered to position 1, 2, 3 and 4

sequentially, circular tracking in yz plane and xz plane is performed at each point.

The motion control is performed twice, the second time is conducted by reversing the

sign of six actuation currents while the target trajectory is the same. According to Eq.

(4.7), reversing the sign of the current vector does not theoretically change the magnetic

force since the force is quadratic form about current vector. However, due to the

hysteresis effect, the hysteresis loop of each single pole is different after reversing the

signs of the current, which can be seen in Fig.4.7. Moreover, it can be seen that the

hysteresis loop is very complicated to model.

Fig. 4.7. The input-output of six poles in two tracking experiments, the second experiment is conducted by

reversing the sign of the actuation currents. (To make the input-output plot orientated with positive slope,

the signs of some plots are changed accordingly)

77

Fig. 4.8. Control loop for magnetic force analysis; (

dP is the desired position, mP is the measurement

position, pK is the proportional control gain, dF is the desired force calculated by the controller,

a and a

are the delay in actuator and measurement, MTF is the magnetic force,

TF is the thermal force, N is

measurement noise)

First, the influence of the thermal force and measurement noise are analyzed. The

signal processing scheme is studied for magnetic force analysis. The control loop for

magnetic force analysis is shown in Fig.4.8. To simplify the analysis, proportional control

is employed and modeling error is not included, i.e. (t- )= (t)d A MTF F . From the Langevin

equation [68], the bead dynamics can be modeled by the following 1st order equations,

1

m MT TF F Ns

P (4.8)

The bead dynamics can be rewritten as =m MT Ts F F s N P . It is hoped that the

magnetic force information can be obtained from position measurement. The drag

coefficient is calculated by comparing the measurement Power Spectrum Density (PSD)

and the theoretical PSD, as addressed in Chapter 3.5. To attenuate the effect of the

thermal force and the measurement noise, a low-pass-filter is applied on both sides of Eq.

(4.8),

78

1 1 1 1

1 1 1 1m MT Ts F F s N

s s s s

P (4.9)

According to Eq. (4.9), it is hoped that ( 1)MTF s ( 1)TF s + ( 1)s N s so

that the left side of Eq. (4.9) can represent the magnetic force.

Fig. 4.9. PSD analysis without low-pass-filter. var(.) operator means the variance of the corresponding

variable.

Fig. 4.10. PSD analysis with low-pass-filter. var(.) operator means the variance of the corresponding

variable.

A simple case is simulated to demonstrate the effect of low-pass-filter: particle tracking

speed is 5um/s along a circle with 50um diameter, control sampling rate is 1606Hz,

measurement noise is 1.5nm in standard deviation, low-pass-filter time constant =0.05,

79

the damping coefficient of water =3e-8 N.s/m, and proportional gain is pK =5e-6 N/m.

The PSD analysis and cumulative variance is plotted in Fig.4.9 and Fig.4.10, showing the

effect without and with low-pass-filter. It can be clearly seen that the effect of thermal

force and measurement noise are negligible after applying LPF and the magnetic force

can be measured accurately.

The calibration is accomplished by minimizing the following objective functions, i.e.

Eq. (4.10) and Eq. (4.11), for Hall-Sensor-based model and current-based model

respectively. The measurement position and force are filtered by the same low-pass-filter.

2

1

ˆ ˆˆ ˆ ˆ( , , ) (j) (j), , (j);N

H H H MT H H H H

j

J f f

D b F F p b V D (4.10)

2

1

ˆ ˆˆ ˆ ˆ ˆ( , , ) (j) (j), , (j);N

I I I MT I I I

j

J f f

K b F F p b I K (4.11)

Where HJ and IJ are objective functions of Hall-Sensor-based model and current-based

model, N is the number of sampling instants, MTF is the measurement magnetic force as

in Eq. (4.9). The Hall-Sensor-based magnetic force model at each sampling instant is

expressed as ˆ ˆ ˆ( (j), , (j); )H H H Hf F p b V D , where the force gain ˆHf , magnetic charge bias

vector b and the normalized Hall-Sensor-flux-gain matrix ˆHD are constants and position

(j)p and voltage (j)HV are from real-time measurement. The current-based model

ˆ ˆ ˆ ˆ( (j), , (j); )I I If F p b I K is defined similarly.

80

Both the Hall-Sensor-based magnetic force model and current-based model are

calibrated and their performances are compared. The calculation is in actuation

coordinate. The measurement force and modeling force are plotted in Fig.4.11 to

Fig.4.14.

Fig. 4.11. Hall-sensor-based magnetic force model along circles in xy plane (as Fig.4.6 (a))

Fig. 4.12. Hall-sensor-based magnetic force model along circles in yz and xz planes (as in Fig. 4.6 (b) and

Fig.4.6 (c))

From Fig.4.11, Fig.4.12 it can be seen that the Hall-Sensor-based magnetic force

model fits very well with the measured viscous force. The value of the objective function

Eq. (4.10) is that HJ = 2434.42, the error at each point is about √J/(3*N) = 0.087pN.

81

Fig. 4.13. Current-based magnetic force model along circles in xy plane (as Fig.4.6 (a))

Fig. 4.14. Current-based magnetic force model along circles in yz and xz planes (as in Fig.4.6 (b) and

Fig.4.6 (c))

From Fig.4.13, Fig.4.14 it can be seen that the current-based results in obvious

modeling error. The value of the objective function Eq. (4.11) is 10279.1, the error at

each point is about √J/(3*N) = 0.179pN, while the error in hall sensor based model is

about 0.087pN.

82

Fig. 4.15. Histogram of modeling error in force calibration (Hall-sensor-based model vs. Current-based

model; the black histogram are from Fig.4.11 and Fig.4.12, the green histogram are from Fig.4.13 and

Fig.4.14)

The Histogram of modeling errors in Fig.4.11 to Fig.4.14 are plotted in Fig.4.15. It

can be seen that the error in Hall-Sensor based model is not only smaller, but also follows

a reasonable Gaussian distribution, which means that it captured the physics of the

magnetic force model.

4.5 Application of Hall-Sensor-Based Model and application of Hall Sensors

4.5.1 Force Prediction

It is already shown in Chapter 4.4.2 that the Hall-Sensor-based force model can fit

well with the measured viscous force. But it is hoped the calibrated force model can not

only fit well with the measurement force but can also predict the magnetic force in the 3-

D space. Therefore, bead tracking control is conducted along circles and straight lines at

different directions as in Fig.4.16. For the circular trajectory of Fig.4.16, the bead is

steered at 10μm/s to follow circles at different height, similar to the trajectory in Fig.4.6

of force calibration. But the circle diameter is 25um instead of 50um. The bead is first

steered along the lowest circle, then the circle in the middle, followed by the highest

83

circular trajectory tracking. After finishing the circular tracking, the bead is steered from

the highest plane to the middle plane, wherein the bead is steered to follow several linear

motion as Fig.4.16 (b). Starting from the center of the middle plane (labeled as 0 in

Fig.4.16 (b)), the bead is steered from 0 to 1, where the bead stays for 2sec. Then the

bead is steered to 2, 3, 4 and 0 sequentially. The modeling force is calculated to compare

with the measured force, which is processed with the same LPF as in Chapter 4.4.2.

Fig. 4.16. Bead motion trajectory for force prediction

Fig. 4.17. Force prediction using Hall-Sensor-based model (motion along circles in xy plane)

The magnetic force prediction and the viscous force are plot together in Fig.4.17 to

Fig.4.20. From Fig.4.17 and Fig.4.18, it can be seen that the Hall-Sensor-based magnetic

force model can accurately predict the force experienced by the magnetic bead, while the

84

current-based force model has obvious error, which can be clearly seen in Fig.4.19 and

Fig.4.20.

Fig. 4.18. Force prediction using Hall-Sensor-based model (motion along straight line in xy plane)

Fig. 4.19. Force prediction using Current-based model (motion along circles in xy plane)

Fig. 4.20. Force prediction using Current-based model (motion along straight lines in xy plane)

85

Fig. 4.21. Histogram of modeling error in force prediction (Hall-sensor-based model vs. Current-based

model; the black histogram are from Fig.4.17, the green histogram are from Fig.4.19)

Fig. 4.22. Histogram of modeling error in force prediction (Hall-sensor-based model vs. Current-based

model; the black histogram are from Fig.4.18, the green histogram are from Fig.4.20)

The Histogram of force prediction errors, both in Hall sensor based model and

current-based model, are plotted in Fig.4.21 and Fig.4.22. It can be seen that the error in

Hall sensor based model is not only smaller but also resembles the normal distribution.

But the error in the current-based model is larger and there is no obvious statistics

pattern.

86

4.5.2 Magnetic Field Feedback Control Using Hall-Sensor inner loop control

The positioning performance Chapter 4.4.1 and the force calibration result in Chapter

4.4.2 indicate severe drawbacks of the hysteresis problem when the current-based model

is employed. However, with the Hall-sensor measurement capability, the Hall Sensor

voltage can be directly kept at desired value through so called Hall-sensor inner loop

control as in Fig.4.23.

Fig. 4.23. Magnetic bead position control integrated with Hall-sensor inner loop control.

dP andmP are

the desired and measurement position, e is the positioning error, dF is the desired force calculated from the

motion controller. dV and

mV are the desired and measured Hall-Sensor voltage.

First, the Hall-sensor inner-loop control performance is experimentally verified and

compared with current actuation without Hall-Sensor inner-loop control. Fig.4.24 is the

result of Hall-Sensor inner loop control for different step inputdV , i.e.

dV =0.1V, 0.2V

and 0.5V. The measurement voltage mV is normalized by dividing

mV withdV . The

normalized mV converge to 1 for different

dV , which means the Hall-Sensor inner loop

control can work as desired. Moreover, from the zoom-in plot in Fig.4.24, the raise time

is about 0.32ms. The transient response has slight difference since the magnetization of

the pole is a complicated nonlinear process due to hysteresis effect.

87

Fig. 4.24. Normalized Response of different desired voltage dV .

Fig. 4.25. Step response for different voltage inputs.

If Hall-Sensor inner loop control is not used, current/voltage input is used to drive the

coil for magnetic flux generation. Most researchers assume there is a linear relationship

between the input current/voltage and the output flux. A linear amplifier working in

current-mode is used to drive the coil, in which 1V input to the amplifier generates 0.3A

output. Two experiments are performed. First, decrease the voltage stepwisely,

specifically 0.1V step and 0.5V step and then increase voltages back to 0V again. Second,

88

input different step voltages and compared the output magnetic flux. The Hall-Sensor

measurement represents the magnetic flux in these two experiments.

From Fig.4.25 (a) and (b), it can be seen that the magnetic flux does not change in a

fixed increment when the input voltage changes in a fixed step. Moreover, the magnetic

flux does not return to the original status when the input voltage returned back to 0V. It is

worth mentioning that the starting point in Fig.25 (a) and (b) is not zero due to remnant

magnetization. From Fig.4.25 (c), normalized step responses are plotted for 0.1V, 0.5V

and 1V inputs. The responses are normalized by dividing the Hall-sensor measurement

mV by the input step size, i.e. 0.1V, 0.5V and 1V. From the result it can be seen that the

normalized output do not coincide with each other, which is an indication of nonlinear

effect of the magnetic pole.

The above experiment reveals that the Hall-Sensor inner loop control is a promising

method to solve the hysteresis in position control. A bead trapping experiment is

performed and the result will be compared with current-based stabilization in Chapter

4.4.1. The nominal optimal inverse model developed in Chapter 3.3.2 is used to calculate

the desired current and thus the desired Hall-sensor voltagedV . The whole setup is

controlled by an FPGA system as shown in Fig.3.1. The Hall-sensor voltage is measured

by an AD converter running at 200K Hz and the feedback control is running at 100K Hz.

Six coils are driven by DA converter followed by an amplifier. The DA converter is

running at 640K Hz while the bandwidth of the amplifier is 10K Hz.

89

Fig. 4.26. Positioning performance using Hall-Sensor inner loop control.

Fig. 4.27. dV is the desired voltage and mV is the measurement voltage)

Fig. 4.28. Step Response of Hall-sensor feedback control.

With the Hall-Sensor inner loop control introduced, it can be seen that the stabilization

steady state error in Fig.4.26 is very small compare with current-based feedback control,

since the remanent magnetization is removed. From the plot of dV and

mV in Fig.4.27, it

can be seen that Vd followed Vm accurately. To further demonstrate the Hall-sensor

90

feedback control performance, 3 step responses (denote at step 1, step 2 and step 3) at

different times are selected and the normalized response are plotted together in Fig.4.28.

It can be seen that the normalized step response for different step sizes agree with each

other. The feedback control output is a little different from the test in Fig.4.24 due to two

reasons: 1) different step size result in slight different transient response for nonlinear

systems with hysteresis, 2) the magnetic field in 6 poles are controlled simultaneously

while the magnetic coupling among 6 poles create some disturbances.

4.6 Accurate Hall sensor based Magnetic Force Modeling for large magnetic force

generation

The Hall-sensor-based magnetic force model presented in previous sections is capable

of solving complex hysteresis problem and achieve sub-pN accuracy. But the magnetic

field is relatively small and therefore the magnetization of the magnetic bead is modeled

as a linear function of the external magnetic field. When the external magnetic field is

large, however, the magnetization of the magnetic bead will be nonlinearly dependent on

the external magnetic field [63, 77-79]. An accurate magnetic force model is proposed in

this section wherein the Langevin-function [63] is used to model the nonlinear

relationship between the magnetization of the superparamagnetic magnetic particle and

the external magnetic field.

4.6.1 The modeling of the magnetic bead magnetization

The magnetic bead (Dynabead M450 Epoxy) is aggregated from nanoparticles of

2 3Fe O , which is experimentally verified to be superparamagnetic [77, 78]. This

91

property will greatly ease the modeling of the magnetic bead. But the magnetization

curve of this particle is only linear up to about 50 Gauss from different measurement

schemes [77-79]. Therefore, the bead magnetization cannot be modeled as a linear

function of the external magnetic field when the magnetic force is large. A Langevin

function is used to model the magnetization of this superparamagnetic particle since it is

a continuous function with relative simple form and therefore can greatly ease the

experimental calibration.

Fig. 4.29. Magnetization of the M450 superparamagnetic bead

The magnetization of the bead is illustrated in Fig.4.29, where B is the external

magnetic field, m is the magnetic moment of the magnetic bead, θ and ϕ are the direction

of the external magnetic field and the magnetic moment of the bead. For a general

ellipsoid bead, the directions of the external magnetic field and the magnetic moment

direction of the bead are not necessarily equal [49]. But since the M450 bead is a

superparamagnetic spherical bead with evenly distributed magnetic nanoparticle in the

bead, it can be assumed that the M450 bead is homogeneous in each directions.

Therefore, it is valid to assume that θ = ϕ.

92

The nonlinear magnetization of the superparamagentic object is usually modeled by

Langevin Function [63, 79]. More specifically, two parameters are employed to describe

the saturation limit and the shape of the Langevin function,

ˆ

1 ˆm , , coths s

m

a m aa

m B B BB

(4.12)

Where ms is the saturation limit of the magnetic bead, a controls the shape of the

Langevin Function, ˆ = / || ||B B B is the unit directional vector of the external magnetic

field, which is also the direction of the magnetic moment of the magnetic bead,

ˆ = coth 1m a aB B is defined as the Normalized Magnetization of the magnetic

moment. When a increases, the Langevin Function increases faster (Fig.4.30). The

external magnetic field B will be modeled in the following section as a function of Hall-

sensors measurement.

Fig. 4.30. The plot of the Langevin Function (assume the saturation limit sm =1)

93

Most researchers model the magnetization of the magnetic particle linearly dependent

on the external magnetic field [1, 2, 34, 52, 53]. And the hall-sensor-based model in

Chapter 4.4 has already shown that the linear model is accurate enough if the magnetic

field is relatively small, which means the magnetization is just a linear portion of the

Langevin function around ||B||=0,

m , , m ,3

s s

aa m B B B 0 (4.13)

Most researchers model the magnetization of the magnetic particle as

0 0 0(3 ) [( ) ( 2 )]V m B [67], which means m 3sa ≈

0 0 0(3 ) [( ) ( 2 )]V when || ||B 0 . From [77], the magnetic saturation of one

bead is about ms =1.50e-12 Am2. Therefore, the nominal value of a is about 227.8125.

4.6.2 The Modeling of The Accurate Hall Sensor Based Magnetic Force

From Eq. (2.1) and Eq. (4.1), the magnetic Charge Vector Q can be modeled as

01 H H Q D V . As presented in Chapter 4.2.1, 1

ˆH H HdD D . Therefore, the magnetic

flux density ( , )B p b can be modeled as follow, from Eq. (2.3)

1

2

0 ˆ

ˆ ˆˆ ˆ( , ) ( , )

H

H

m H

H H

b

k d

B

B p b R p b D V (4.14)

94

Where ˆ ˆ( , )R p b is the same as in Eq. (2.3), ˆHB is the Hall-Sensor-Based Normalized

Flux Density, Hb is the Hall-Sensor-Based Flux Density Coefficient. The force model

can be expressed by the gradient force, i.e. = 1 2 ( ) F m B =

ˆ ˆ2 || || || ||sm m m B B , where m = ˆ || || || ||m B B . Therefore,

ˆ

ˆ2

sm mm

F B BB

(4.15)

From Eq. (4.14), || ||B = 1/2

ˆ ˆT

H H Hb B B = 1/2

ˆ ˆ ˆ ˆˆ ˆ( , ) ( , )T T T

H H H H Hb V D R p b R p b D V . The

gradient B can be formulated as follow,

1/2

2

ˆ1 ˆ ˆ ˆ ˆ( , ) ( , )ˆ ˆ2

1 ˆ ˆ( , ), ( , ), ( , )2

T T T iH H H H H

Ti

H H

T THH H x y z H H

rb

r

b

B V D R p b R p b D V

B B

V D L p b L p b L p b D VB

(4.16)

The term i ir r exists because the position is normalized with respect to the

workspace radius . xL , yL and zL have the same meaning as that in Eq. (2.6). The

partial derivative of m with respect to || ||B is m B =22 ( ) 1 ( )a csch a a B B .

By utilizing Eq. (4.15) and Eq. (4.16), the force model can be formulated as follow,

95

2

2

ˆ ( )ˆ

ˆ

1 ˆ ˆˆcoth ( , )4

H

L

T Ts Hi H H i H H

if

f

m b aa csch a

a

F

F B B V D L p b D VB

,i = x,y,z (4.17)

Where ˆHF = [ ˆ ( )H xF , ˆ ( )H yF , ˆ ( )H zF ]is the Normalized Hall-Sensor-Based Force, f

is the Magnetic Force Gain and f = 1

2 2 5 2

04s m Hm k d a , ˆLf is defined as the Langevin

Force-Gain. The total force gain is the multiplication of f and ˆLf .

4.6.3 Calibration of the Magnetic Force Model

The Accurate Magnetic Force Model is described by Eq. (4.17). There are several

unknown parameters to be calibrated. First, f is a lumped force gain to be calibrated.

Second, The Langevin force-gain ˆLf is a function of || ||a B . From Eq. (4.14), || ||a B =

ˆ|| ||H Hab B , Hab can be lumped as another unknown parameter a , i.e. a = Hab . Third, ˆHD

and b need to be calibrated as well.

Fig. 4.31. Motion control result in trajectory tracking

96

The magnetic force model Eq. (4.17) is position dependent. The inverse model and

motion control capability is already achieved in Chapter 3.4. The magnetic bead can be

steered in Glycerol to obtain larger magnetic force. The bead motion trajectory is shown

in Fig.4.31, in xy plane, the bead is firstly steered to track the lowest 50um-diameter

circle at 2um/s, then is moved to the plane in the middle, which is 5um above the lowest

one. After tracking the circle in the middle, the bead is moved to the highest plane within

which is another circle of the same shape. After tracking 3 circles in xy planes at 2um/s,

the bead is moved from the highest plane to the middle plane, and is steered along 10um-

diameter circles in yz plane and xz plane at 1.5um/s instead of 2um/s, since the drag

coefficient in z direction is larger than that in xy plane. First, the bead is steered along

circles in yz plane and xz plane in the center (labeled as 0 in Fig.4.31 (b) and (c)). It is

worth mentioning that the circles in yz plane and xz plane are plotted separately in

Fig.4.31 for clarity, while the yz circular tracking is followed by xz circular tracking

immediately. Then the magnetic bead is steered to position 1, 2, 3 and 4 sequentially,

circular tracking in yz plane and xz plane is performance at each location.

The motion control is performed twice, the second time is conducted by reversing the

sign of six actuation currents while the target trajectory is the same. This is to explore

different part of the hysteresis loop. According to Eq. (4.17), reversing the sign of the

current vector does not theoretically change the magnetic force since the force is a

quadratic form about current vector. However, due to the hysteresis effect, the hysteresis

loop of each single pole is different for all six poles (Fig.4.32) after reversing the signs of

97

the current. Moreover, it can be seen that the hysteresis loop is very complicated to

model.

Fig. 4.32. The input-output of six poles in two tracking experiments, the second experiment is conducted by

reversing the sign of the actuation currents. (To make the input-output plots orientated with positive slope,

the signs of some plots are changed accordingly)

The calibration is accomplished by minimizing the following objective functions,

2

1

ˆ ˆ ˆˆ ˆ ˆˆ ˆ( , , , ) (j) ( ) (j), , (j);N

H H MT L H H H

j

J f a f f a

D b F F p b V D (4.18)

Where HJ is the objective functions of the Hall-Sensor-based model, N is the number of

sampling instants, MTF is the magnetic force measured with the same method as in

Chapter 4.4.2. The Hall-Sensor-based magnetic force model at each sampling instant is

expressed as ˆ ˆ ˆ ˆˆ (j), , (j);L H H Hf f a F p b V D , where f , a , the magnetic charge bias

98

vector b and the normalized Hall-Sensor-flux-gain matrix ˆHD are constants and position

(j)p and voltage (j)HV are from real-time measurement.

The Hall-Sensor-based magnetic force model using Langevin-function, i.e. Eq. (4.17),

and Hall-Sensor-based model in Chapter 4.2.1, i.e. Eq. (4.4), are calibrated and their

performances are compared. The measurement force and the modeling force are plot in

Fig.4.33 to Fig.4.36. From Fig.4.33 and Fig.4.35 it can be seen that the Hall-Sensor-based

magnetic force model using Langevin-function fits very well with the measured viscous

force, in all directions. The value of the objective function Eq. (4.18) is that HJ =

4.3063e+05, the error at each point is about (3*N)J =1.5774pN. From Fig.4.34 and

Fig.4.36 it can be seen that the Hall-Sensor-based magnetic force model without using

the Langevin-function followed the trend of the measured viscous force, but had error at

some orientations, especially when the force is large, which is an indication that the

magnetic moment of the magnetic bead cannot be model as an linear function. The value

of the objective function Eq. (4.10) is that HJ = 1.3859e+06, the error at each point is

about (3*N)J =2.8299pN. From the zoom-in plot in Fig.4.34 (b) and Fig.4.36 (b), the

error can be easily seen, while the Hall-sensor-based modeling using Langevin-function

can accurately model the measured force in Fig.4.33 (b) and Fig.4.35 (b).

99

Fig. 4.33. Calibration result of Hall-Sensor-based force model using Langevin function. (a) Force

calibration along circles in xy plane (as in Fig.4.31) (b) zoom-in plot of force calibration

Fig. 4.34. Calibration result of Hall-Sensor-based model without using Langevin-function. (a) Force

calibration along circles in xy plane (as in Fig.4.31) (b) zoom-in plot of force calibration

100

Fig. 4.35. Calibration result of Hall-Sensor-based force model using Langevin function. (a) Force

calibration along circles in yz and xy planes (as in Fig.4.31) (b) zoom-in plot of force calibration

Fig. 4.36. Calibration result of Hall-Sensor-based force model without using Langevin function. (a) Force

calibration along circles in yz and xy planes (as in Fig. 4.31) (b) zoom-in plot of force calibration

To better visualize the force model performance, the histogram of the force modeling

error is plot in Fig.4.37. It can be seen that the maximal force error in the generalized

101

Hall-sensor based model is smaller than the Hall-sensor based model. Moreover, the

modeling error in the Accurate Magnetic Force model is very close to the Gaussian

distribution.

Fig. 4.37. Histogram of the force calibration error for Hall-sensor-based model with Langevin-function and

without Langevin-function. (The red histograms are from the modeling errors of Fig.4.33 and Fig.4.35, the

blue histograms are from the modeling errors of Fig.4.34 and Fig.4.36)

4.6.4 Application of the Accurate Hall-Sensor-Based Model

4.6.4.1 Magnetic Force Prediction Using the Accurate Hall-Sensor-Base Force Model

Fig. 4.38. Bead motion trajectory for force prediction

It is already shown in Chapter 4.6.3 that the Hall-Sensor-based force model with

Langevin-function can fit well with the measured viscous force. But it is hoped the

calibrated force model can not only fit well with the target viscous force but can also

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predict the magnetic force in the 3-D space. Therefore, bead tracking control is conducted

along circles and straight lines at different directions as in Fig.4.38. The bead is steered at

2um/s to follow circles at different height, similar to the trajectory in Fig.4.31. But the

circle diameter is 25um instead of 50um. The bead is first steered along the lowest circle,

then the circle in the middle, followed by the highest circular trajectory tracking. After

finishing the circular tracking, the bead is steered from the highest plane to the middle

plane, wherein the bead is steered to follow several linear motion as Fig.4.38 (b). Starting

from the center of the middle plane (labeled as 0 in Fig.4.38 (b)), the bead is steered from

0 to 1, where the bead stays for 2sec. Then the bead is steered to 2, 3, 4 and 0

sequentially. The motion speed keeps at 2um/s. The modeling force is calculated to

compare with the viscous force. The modeling force is calculated to compare with the

viscous force.

Fig. 4.39. Force prediction of Hall-Sensor-based force model using Langevin function. (a) Force calibration

along circles in xy plane (as in Fig.4.38) (b) zoom-in plot of force calibration

103

Fig. 4.40. Force prediction of Hall-Sensor-based force model without using Langevin function. (a) Force

calibration along circles in xy plane (as in Fig.4.38) (b) zoom-in plot of force calibration

Fig. 4.41. Force prediction of Hall-Sensor-based force model using Langevin function. (a) Force calibration

along line trajectories in xy plane (as in Fig.4.38) (b) zoom-in plot of force calibration

The magnetic force prediction and the viscous force are plot in Fig.4.39 to Fig.4.42.

From Fig.4.39 and Fig.4.41 it can be seen that the Hall-Sensor-based magnetic force

model with Langevin-function can accurately predict the force experienced by the

magnetic bead, while the Hall-Sensor-based force model has obvious error, which can be

clearly seen in Fig.4.40 and Fig.4.42.

104

Fig. 4.42. Force prediction of Hall-Sensor-based force model without using Langevin function. (a) Force

calibration along line trajectories in xy plane (as in Fig.4.38) (b) zoom-in plot of force calibration

Fig. 4.43. Histogram of the force calibration error for Hall-sensor-based model with Langevin-function and

without Langevin-function. (The red histograms are from the modeling errors of Fig.4.39, the blue

histograms are from the modeling errors of Fig.4.40)

Fig. 4.44. Histogram of the force calibration error for Hall-sensor-based model with Langevin-function and

without Langevin-function. (The red histograms are from the modeling errors of Fig.4.41, the blue

histograms are from the modeling errors of Fig.4.42)

105

Again, the histogram of the force prediction error can be plot in Fig.4.43 and

Fig.4.44. It can be seen that the maximal force error of the Hall-Sensor-based model

without Langevin-function is larger than that of the Hall-sensor based model with

Langevin-function.

4.6.4.2 Parameter calibration and magnetization study

The calibrated parameters are as follow: f = 24.939pN (2.4939e-11N), a = 3.232.

Since ˆHa ab ,

Hb is therefore 1.412e-2. The Magnetic Force Gain2ˆ (4 )s Hf m b a , with

the values of sm , a and

Hb , the calculated Magnetic Force Gain f = 28.664pN, which is

close to the value from calibration, i.e. 24.939pN. The discrepancy may come from the

nominal value of magnetic bead saturationsm , the value of which is actually a statistical

average. From Chapter 4.6.1, m 3sa 0 0 0(3 ) [( ) ( 2 )]V , with

2ˆ (4 )s Hf m b a and ˆHa ab ,

sm is recalculated as 1.40e-12 instead of nominal value

1.50e-12. Based on this value, a is recalculated as 244.234 instead of 227.8125 as in

Chapter 4.6.1. Therefore, the recalculated Hb is 1.323e-2. Since

1

2

0( )H m Hb k d , the

Hall-sensor coefficient iHd can be calculated as 5.99e-8.

From the calibrated values and recalculated values, the magnitude of the external

magnetic field and magnetic bead moment can be calculated according to Eq. (4.14) and

Eq. (4.12). The magnetic bead moment with and without Langevin-function model are

both plotted in Fig.4.45, using experiment data. It can be seen that the external magnetic

field can reach about 260 Gauss (1 Gauss = 1e-4 T) in this experiment and the linear

106

model of the magnetization will lead to large modeling error. The linear range of

magnetization is up to 50 Gauss. This agrees with the information provided by the vendor

and other measurement result [77, 78].

Fig. 4.45. relationship between external magnetic field ||B|| (Gauss) and the magnetic moment ||m||.

4.6.4.3 Accurate Hall-sensor-based optimal inverse modeling

The magnetic force model Eq. (4.17) is a quadratic form of Hall-Sensor voltage

vector HV , with is similar to Eq. (4.4), the only difference is that the Langevin Force-Gain

ˆ || ||Lf a B is also a nonlinear function of the magnetic field B, and thus is a nonlinear

function of HV according to Eq. (4.14). The idea of the optimal inverse model in Chapter

3 can be applied with slight modification. In Eq. (4.17), ˆ || ||Lf a B is a gain that does not

change the direction of the force. Therefore, the optimal inverse model can be first

developed for the Normalized Hall-Sensor-Based Force ˆHF = [ ˆ ˆ( , )T T

H H x H HV D L p b D V ,

ˆ ˆ( , )T T

H H y H HV D L p b D V , ˆ ˆ( , )T T

H H z H HV D L p b D V ]. As shown in Fig.2.7, the desired forcedF

can be expressed in spherical coordinate as [cos cos ,cos sin ,sin ]T

d d F F . Using

similar method as describes in Chapter 3.3.2, the following objective function using

107

Lagrange multipliers can be used to find the optimal inverse model for unit desired force,

i.e. [cos cos ,cos sin ,sin ]T ,

2 ˆ ˆ( , , ) ( , ) cos cos

ˆ ˆ ˆ ˆ( , ) cos sin ( , ) sin

T T

H x H H x H H

T T T T

y H H y H H z H H z H H

J

p V V D L p b D V

V D L p b D V V D L p b D V (4.19)

For a particular desired unit force in a specific orientation, the unit inverse model

( , , )unit

H V p can be obtained by minimizing Eq. (4.19). The next step is to scale the

voltage vector unit

HV to make the force model magnitude equals the magnitude of desired

force, i.e. || ||dF .

For a particular position and dF orientation, the magnetic field norm || ||B and Hall-

Sensor voltage norm || ||V are linearly dependent, wherein V is a scaled vector of

( , , )unit

H V p . However, the inverse model is position dependent and orientation

dependent, therefore the ratio between || ||B and || ||V also depend on the position and

orientation. The plot of || ||B and || ||V in the experiment is shown in Fig.4.46.

Fig. 4.46. Plots of || ||B , || ||V and || || / || ||B V

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Assume that at a particular position p and orientation ( , ) , the ratio between || ||V

and || ||B (in Gauss) is 20, then the Langevin Force Gain can be written as ˆ 20 || ||Lf a V .

The force model can be expressed as,

2

2

|| ||ˆ ˆ ˆ ˆ20 || || ( ) ( , )|| ||

unit T T unit

L H H H Hunit

H

f f a V

F V V D L p b D VV

(4.20)

To satisfy the desired force magnitude, the lumped gain

2 2ˆ ˆ 20 || || || || / || ||unit

L Hf f a V V V must equal dF . Therefore,

2 2ˆ ˆ20 || || || || || || /unit

L d Hf a f V V F V and || ||V need to be solved to finalize the inverse

model. As shown in Fig.4.47, the Langevin Force Gain ˆ 20 || ||Lf a V is monotonically

decreasing due to the magnetic saturation of the magnetic particle, while

2ˆ 20 || || || ||Lf a V V is a monotonically increasing function. From Fig.4.47 (c),

2ˆ 20 || || || ||Lf a V V generally satisfies quadratic relationship when || ||V is small. When

|| ||V is growing, however, 2ˆ 20 || || || ||Lf a V V deviate from quadratic relationship and

resembles a linear relationship when || ||V is large, which is physically expected. Since

when magnetic bead magnetization is saturated, the magnetic force will only grow

linearly with respect to || ||B according to the force model 1 2 ( ) F m B . From

2ˆ 20 || || || ||Lf a V V =2 ˆ|| || /unit

d H fF V , || ||V can be easily solved from the inverse

function, which is shown in Fig.4.47 (d). With the magnitude || ||V known, the inverse

model can be expressed as || || / || ||unit unit

H HV V V .

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Fig. 4.47. (a). The plot of 2ˆ 20 || || || ||Lf a V V and ˆ 20 || ||Lf a V . (b). the plot of ˆ 20 || ||Lf a V . (c).

Zoom-in plot of 2ˆ 20 || || || ||Lf a V V when || ||V is small. (d). the plot of 2ˆ 20 || || || ||Lf a V V and the

inverse function 2ˆ( 20 || || || || )Linverse f a V V

4.7 Conclusion

Hall sensors are introduced to solve the complicated hysteresis issue, which is very

difficult to model due to the complicated hysteresis and magnetic coupling among six

poles. Six Hall Sensors are integrated with the hexapole electromagnetic actuator and the

validity of using surface-mount Hall Sensors is experimentally verified. The magnetic

hysteresis issue can be clearly observed during the bead motion control.

A Hall-Sensor-based magnetic force model is proposed to improve the accuracy of

the magnetic force model, wherein the Hall sensor measurement can accurately present

the magnetic field information. The magnetic force model is experimentally calibrated by

steering the magnetic bead in the 3-D space. The current-based magnetic force model,

which is conventionally used and subject to hysteresis issue is also calibrated and

compared with Hall-Sensor-based magnetic force model. The result clearly shows that

the Hall-sensor-based model greatly outperforms the current-based force model and can

110

achieve sub-pN resolution. With the Hall Sensor measurement capability, the magnetic

force can be accurately predicted, using the calibrated magnetic force model. The Hall

sensors can also be used in an inner control loop to directly control the magnetic field of

each pole, which can remove the modeling error caused by the hysteresis issue.

When the external magnetic field is large, the magnetization of the magnetic bead

cannot be modeled as a linear function of the magnetic field. The nonlinear effect in the

magnetic particle magnetization is modeled with the Langevin-function. The modeling

result is experimentally calibrated and compared with the model in which the particle

magnetization is modeled as the linear function of the magnetic field. The result showed

that the nonlinear magnetization effect exist and the improved model can model the

magnetic force more accurately.

With the introduction of hall sensors, the dynamic force sensing estimator can be

developed to dynamically estimate the interaction force between the magnetic bead and

other objects such as biological samples. This capability will separate this multi-pole

magnetic actuator from other magnetic force appliers. Moreover, parameter estimation

need to be accomplished since parameters such as drag coefficient may change greatly

due to local environment change.

111

Chapter 5: Dynamic force sensing and parameter estimation

5.1. Introduction

3D motion control and accurate magnetic force modeling are achieved through

optimal inverse modeling and Hall sensor measurement. However, these capabilities are

not enough to enable the magnetic actuator to perform active scanning tasks unless the

bead-sample interaction force can be sensed. Moreover, it is hoped that the force can be

sensed dynamically to map the transient force of many biological processes. This

capability will complete the magnetic actuator as an active scanning probe and can even

achieve automatic scanning by combining the motion control capability and dynamic

force sensing capability.

In conventional magnetic tweezers, the magnetic beads are often functionalized and

anchored to target bio samples to avoid instability. Under these circumstances, the

magnetic force sensing is mostly quasi-static, wherein the magnetic force is usually

calculated according to a pre-calibrated force function [29] or derived from Brownian

motion fluctuations through image measurement microscopy [61, 62]. Moreover,

automatic scanning is impossible since active control is not achieved.

There are actively controlled magnetic actuators that can stabilize the bead. In some

magnetic actuators [44, 63], the analytical magnetic force model is not proposed, which

makes the dynamic force sensing very difficult. For the magnetic actuator presented in

this project, Hall sensor based model can achieve sub-pN magnetic force accuracy

112

(Chapter 4). Moreover, the measurement resolution can reach sub-nm with high

bandwidth implementation in FPGA [71]. The Hall sensor measurement and the motion

measurement can be utilized by a Recursive Least Square Estimator to estimate the bead-

sample force in real-time. Moreover, the bead dynamics, which is describe by the

Langevin equation [68], can be a time-varying process since the drag coefficient can

change significantly as a result of local environment change, such as wall-effect,

temperature change, liquid property change and etc. Therefore, a joint state-parameter

estimation algorithm is proposed to simultaneously estimate the bead-sample interaction

force and the drag coefficient of the magnetic bead. In this estimation algorithm, a

disturbance observer is employed to estimate the external force dynamically and a 1st

order autoregressive (AR) model is utilized to estimate the drag coefficient. Specifically,

Kalman filter algorithm is implemented for the joint state-parameter estimator since the

system subject to random white noise such as thermal force and measurement noise and

the discrete dynamics model is time-varying.

Preliminary studies are performed theoretically and experimentally. From the force

sensing result it can be seen that the proposed joint state-parameter estimator using

Kalman filter algorithm is valid for dynamic force sensing and the Hall sensor based

magnetic force model will lead to improved force sensing result compared to current-

based model.

113

5.2 Joint State-Parameter Estimator Algorithm

5.2.1 Drag coefficient estimation based on the magnetic force model

Fig. 5.1. Joint state-parameter estimator (using the Hall sensor based force model) running simultaneously

with feedback controller. (dP is the desired position,

mP is the measurement position, e is the error signal,

a and m are the delay in actuator and measurement, I is the current from the nominal inverse model,

dF is

the desired force calculated by the controller, ' (t) (t )d d a F F is caused by the actuator delay, F is the

modeling error, MTF is the magnetic force,

TF is the thermal force, EF is the bead-sample interaction force,

B is the magnetic flex density, mV is the Hall sensor measurement voltage, ˆMTF is the modeling magnetic

force, is the estimated drag coefficient, ˆEF is the estimated external force)

According to the principle of separation of estimation and control, the observer

design is independent of the feedback controller design. Therefore, the feedback control

and the force sensing estimator can run simultaneously as shown in Fig.5.1. The bead’s

motion dynamics is needed for the estimator design. Since the motion in x, y and z

directions are all governed by the Langevin equation, only the model in x direction is

presented here while the model in y and z direction can be derived similarly. As

mentioned in Chapter 2.4, the inertia term in the Langevin equation can be neglected.

114

Therefore, the Langevin function from the block diagram of Fig.5.1 can be described by

the following function:

( ) (t) ( ) ( )MT E Tx t F F t F t (5.1)

The visual measurement has delay due to image transfer and calculation (1 sampling

for exposure, 1 sampling for image transfer, and 1 sampling for calculation). As a result,

the measurement position ( ) ( )m mx t x t . The actuator also has delay due to

magnetization dynamics, i.e. ˆ(t) (t )MT MT aF F . The dynamics based on measurement

position is as follow after considering time delays,

ˆ( ) (t ) ( ) ( )m MT D E m T mx t F F t F t (5.2)

According to the sources of the force, ( )mx t can be decomposed into three parts, i.e.

( )MTx t , ( )Ex t and ( )Tx t due to the magnetic force, the external force and the thermal force

respectively. The decomposition is described as follow,

'

'

ˆ( ) (t ), 5.3.1

( ) ( ), 5.3.2

( ) ( ), 5.3.3

MT MT d

E E

T T

x t F

x t F t

x t F t

Where '( ) ( )E E mF t F t and '( ) ( )E T mF t F t . In discrete domain, the increment from

sampling instance 1kt to kt is consist of three different parts:

115

ˆ[ ] [ 1] [ ] [ ] [ ] (t 1 ) [ ] [ ]sm m MT E T MT E Tx k x k x k x k x k F x k x k

(5.4)

Where [ ] [ 1]m mx k x k is the increment within one sampling interval, s is the digital

control sampling interval, /D s is the number of delay in digital control. [ ]MTx k ,

[ ]Ex k and [ ]Tx k are the motion of the magnetic bead within the time interval1[t , t ]k k

,

caused by the magnetic force, the external force and the thermal force. One assumption is

that the force dynamics is much lower compared to the sampling rate and can be

considered constant within 1[t , t ]k k . That is the reason [ ]MTx k can be described by

ˆ( ) (t 1 )s MTF in Eq. (5.4). Moreover, if [ ]Ex k is known, the external force can be

similarly describes as ' [ ] [ ]E E sF k x k .

Since the exact dynamics of [ ]Ex k is very difficult to know, an autoregressive (AR)

model is used to describe the disturbance dynamics. Specifically, the 2nd order AR model

is used to describe [ ]Ex k ,

[ ] (1 ) [ 1] [ 2] [k]E E E Ex k x k x k w (5.5)

where is a weighting factor close to 1 but smaller than 1, [k]Ew is the process noise

representing the discrepancy between the actual dynamics of [k]Ex and the 2nd order AR

model. The z-transform from [k]Ew to [k]Ex is therefore 2[z] (( 1)( ))E Ex w z z z ,

which is an integrator combined with a 1st order low-pass-filter. Another unknown in Eq.

116

(5.4) is the drag coefficient , the value of which can not only indicate the environment

change such as wall-effect or temperature change but also is needed for external force

estimation since ' [ ] [ ]E E sF k x k . The change of is often slower, therefore, 1st order

AR is used to model . Specifically, 1 is chosen as the state variable,

1 1

[k] [k 1] [k]w

(5.6)

where [k]w is the process noise of 1 . Therefore, the state-space presentation of [k]Ex

and 1 [k] is a 3rd order dynamics model as follow,

[ ] [ 1] [k]

[ ] 1 0 [ 1] [ ]

[ 1] 1 0 0 [ 2] 0

(1 )[ ] 0 0 1 (1 )[ 1] [ ]

E E E

E E

k k

x k x k w k

x k x k

k k w k

X Φ X W

(5.7)

The observation model is derived from Eq. (5.4) and is formulated as follow,

[k][ ]

[ ]

[ ]

ˆ[ ] [ 1] 1 0 (t 1 ) [ 1] [ ]

1 [ ]m

E

m m s MT E T

Ok

k

x k

x k x k F x k x k

k

H

X

(5.8)

where [ ]kX is the state variable, Φ is the state-space transition matrix, [ ]kW is the process

noise vector, [k]mO , which is the measurement increment between 1[t , t ]k k , is the

117

observation variable, [ ]kH is the observation matrix with respect to [ ]kX . Kalman filter

algorithm is selected to estimate the state variable since 1) the system subject to random

process noise [k]W and thermal noise [ ]Tx k , 2) the matrix [ ]kH is time varying therefore

linear time invariant (LTI) analysis method such as pole placement is not suitable in this

situation. However, the convergence of the Kalman Filter does require that [ ]kH satisfies

the persistent condition [80], which is naturally satisfied in this system since the magnetic

bead is constantly perturbed by the random thermal force. Therefore, the modeling force

ˆMTF is always changing randomly, which will naturally satisfy the persistent excitation

condition.

Similar to the Current Estimator in LTI system [81], the Kalman filter algorithm is a

two-step process, i.e. time update and measurement update. In the time update step, the

Kalman filter predicts the value of the state variables as well as their uncertainty

covariance matrix. When the new measurement information is available, the Kalman

filter gain will be updated and the state variables are corrected based on the new

measurement data. The time update and measurement update algorithm between time

interval 1[t , t ]k k are as follow, wherein Eq. (5.7) and Eq. (5.8) are used.

Time Update:

| 1 1

| 1 1

ˆ ˆ (5.9.1)

[k] (5.9.2)

k k k

T

k k k

X ΦX

P ΦP Φ Q

118

Measurement Update:

| 1

| 1

1

| 1

| 1

| 1

ˆ[k] [k] (5.10.1)

[k] [k] (5.10.2)

[k] (5.10.3)

ˆ ˆ (5.10.4)

[k] (5.10.5)

k m k k

T

k k k k

T

k k k k

k k k k k

k k k k

y O

S R

S

y

H X

H P H

K P H

X X K

P I K H P

In Eq. (5.9),1

ˆkX , which is available at the beginning of

1[t , t ]k k, is the state variable

estimation from previous step, | 1ˆ

k kX is the state variable prediction using Eq. (5.7), [k]Q

is the process noise covariance, i.e. [k] ( [k])covQ W , | 1k kP characterizes the uncertainty

of | 1ˆ

k kX using the uncertainty information of 1

ˆkX ,i.e.

1kP , and covariance [k]Q . In Eq.

(5.10), ky is called the innovation or measurement residual,

kS is the innovation

covariance, kK is the Kalman filter gain, ˆkX is the measurement update of the state

variable, kP characterize the uncertainty of ˆkX . This Kalman filter algorithm can run

recursively when the new measurement information comes in.

5.2.2 Drag coefficient estimation based on the thermal variance measurement

In Eq. (5.8), the state relates to drag coefficient gamma, i.e.1 , is multiplied by

ˆs MTF , in which s is the sampling interval and ˆ

MTF is the magnetic force. Therefore, the

accuracy of the magnetic force is crucial for the drag coefficient estimation. Alternatively,

119

the drag coefficient can also be determined according to the innovation information (Eq.

(5.10.1)) since the innovation information is the motion caused by the thermal force (Eq.

(5.8)). A recursive form for estimating the variance of [ ]Tx k in Eq. (5.8) in as follow [21],

2

2 2

| 1ˆ[k] [k 1] 1 [k] [k]T T m k kO

H X (5.11)

Where [k]T is the thermal variance. The thermal variance is inverse proportional to

the drag coefficient of the environment,

22 /B s Tk T (5.12)

The thermal force is determined by the drag coefficient, which will determine the

Brownian motion fluctuation. This is the physical reason that the drag coefficient can be

determined from the thermal variance. As long as the drag coefficient is known, the net

force and thus the external force can be known from the bead dynamics (Eq. (5.1)). The

external force can be modeled similarly as that in Chapter. 5.2.1.

However, it is worth mentioning that the measurement information in Eq. (5.8) also

contains the measurement noise. Therefore, the estimation of 2

T in Eq. (5.11) is likely

larger than the actual value of the thermal variance due to measurement noise, which will

lead to smaller estimation value of the drag coefficient according to Eq. (5.12). In dense

liquid where the thermal variance is very small, the additional measurement noise will

result in great estimation bias of the drag coefficient.

120

5.3 Simulation result the joint state-parameter estimator

A simulation is performed to test the estimation algorithms in Chapter 5.2. It is

assumed in the simulation program that the magnetic force in Eq. (5.8) is accurately

known and a three-step delay exists in the control loop. The standard deviation of the

measurement noises in x,y and z directions are assumed to be 0.6nm, 0.6nm and 1.2nm.

The thermal force used in the simulation program is a white Gaussian noise, the power of

which is described by the theoretical PSD of the random thermal force [73].

5.3.1 The simulation of the drag coefficient estimation in water

To test the performance of the estimators in Chapter 5.2.1 and 5.2.2, a magnetic bead

stabilization simulation is performed wherein the bead-sample interaction force is set to

zero. The drag coefficient in the simulation changed step-wisely every 10 seconds.

Physically, the change of the drag coefficient will change the random thermal force and

thus the Brownian motion. As described in Chapter 5.2.1 or Chapter 5.2.2, the drag

coefficient can be estimated by using the magnetic force model or the thermal variance.

The performance of these two methods are shown in Fig.5.2 and Fig.5.3. It can be seen

that the drag coefficient estimation can converge to the nominal value using both

methods. The drag coefficient estimation from the thermal variance can converge to the

nominal value since the measurement noise is negligible compared with the thermal

variance. The estimated external force fluctuates around zero in both methods, which is

as expected since the bead-sample interaction force in the simulation is zero. The

parameter estimation using the thermal variance fluctuate more rapidly since this

121

estimation is directly from the measurement innovation while the parameter estimation

using the magnetic force model varies slower since the parameter is lumped as a state

variable.

Fig. 5. 2. Estimation result in water using the magnetic force model to estimate the drag coefficient

Fig. 5. 3. Estimation result in water using the thermal variance to estimate the drag coefficient

5.3.2 The simulation of the drag coefficient estimation in Glycerol

It can be seen that in Chapter 5.3.1 that the drag coefficient estimation in water can

converge to the nominal value using both methods. However, it is mentioned in Chapter

5.2.2 that the thermal variance estimation will be greatly biased if the measurement noise

is comparable to the thermal motion. In dense liquid, such as glycerol, the motion caused

122

by the thermal force is comparable to the measurement noise. It is therefore expected that

the estimated drag coefficient is smaller than the nominal value since the estimated

thermal variance is larger than the nominal value due to the measurement noise. Similar

simulation as that in Chapter 5.3.1 is performed. It is assumed that the bead is stabilized

in glycerol, the drag coefficient of which is two orders larger than that in water. The drag

coefficient estimation using the magnetic force can converge to the nominal value as

shown in Fig.5.4. When using the thermal variance, however, the estimated drag

coefficient value is much smaller than the nominal value due to the measurement noise,

which can be seen in Fig.5.5. To further confirm the effect of the measurement noise,

another simulation is performed wherein the measurement noise is set to zero and it can

be seen that the drag coefficient can converge to the nominal value in Fig.5.6.

In real situations, the measurement noise can already achieve sub-nm resolution,

which is still not enough for the thermal variance estimation if the magnetic bead is

stabilized in a dense liquid such as glycerol.

Fig. 5. 4. Estimation result in glycerol using the magnetic force model to estimate the drag coefficient

123

Fig. 5. 5. Estimation result in glycerol using the thermal variance to estimate the drag coefficient

Fig. 5. 6. Estimation result in glycerol using the thermal variance to estimate the drag coefficient (no

measurement noise)

5.3.3 The simulation of the simultaneous estimation of the drag coefficient and the

external force

From the simulation result in Chapter 5.3.1 and 5.3.2 it can be seen that the estimator

in Chapter 5.2.1 is more robust to measurement noise in different liquids. This estimator

is selected in the following simulation. The purpose of this simulation is to test if the joint

state-parameter estimation algorithm can converge to the nominal value when the drag

coefficient and the bead-sample interaction force change simultaneously. In this

124

simulation, the drag coefficient changes in a similar way as that in Chapter 5.3.1 and

5.3.2 and the external force applied on the bead has a step change of 1pN at 10 second. It

can be seen from Fig.5.7 that both the drag coefficient and the external force can

converge to the nominal value, which is due to that the persistent excitation condition is

satisfied. Since this observation matrix Eq. (5.8) is a time-varying matrix, theoretical

estimation bandwidth is difficult to obtain. However, using the method of evaluating 1st

order linear system, the estimation bandwidth of the external force is about 4.42Hz and

the drag coefficient estimation bandwidth is about 1Hz. The estimation bandwidth of the

drag coefficient is enough since the drag coefficient is usually slow varying. The

estimation bandwidth of the external force can be made even higher by implementing

adaptive Kalman filter [82, 83] wherein the process noise [ ]kW in Eq. (5.7) can change

adaptively. However, the process noise [ ]kW is set as a constant here.

Fig. 5. 7. Estimation result using the magnetic force model when the drag coefficient and external force are

changing simultaneously.

125

5.4 Experiment results

The estimator mentioned in Chapter 5.2.1 is used here since it is more robust to

measurement noise in different liquids. The motion data of the magnetic bead in water

environment, which is shown in Fig.4.16 of Chapter 4.5.1, is used to test the joint state-

parameter estimator algorithm. The algorithm is not implemented in FPGA yet, but the

position data, Hall sensor measurement are all from experiment and the algorithm update

follow the same way in real-time implementation.

Fig. 5.8. Joint state-parameter estimation using Hall sensor based model (a) estimation result in x direction

(b) estimation result in y direction (c) estimation result in z direction (d) trajectories used to test joint state-

parameter estimation algorithm.

126

The Hall sensor based magnetic force model is employed to calculate ˆMTF in Eq. (5.8).

The Kalman filter estimation result is shown in Fig.5.2 (a), (b) and (c), which are the

result in x, y and z direction respectively, and the bead’s motion trajectory as in Fig.4.16

is plot again in Fig.5.2 (d). The plot in Fig.5.2 (a), (b) and (c) contains the result from

different trajectories as in Fig.5.2 (d) and are concatenated into one curve. The drag

coefficient [k] , disturbance motion due to external force and [k]EF are plotted. The

nominal drag coefficient which is from PSD calibration (using the method in Chapter 3.5)

can serve as a reference to compare the accuracy of the estimation result. Physically,

bead-sample interaction force is zero when the bead is moving freely. Therefore, the

estimation result of [k]EF represents the modeling error. From Fig.5.2 (a) it can be seen

that the drag coefficient [k] in x direction has a little discrepancy from the nominal

value, which is due to the slight modeling error in x direction that can be seen from Fig.

4.17. From Fig.5.2 (b) and (c) it can be seen that the drag coefficient [k] in y and z

directions fluctuate around the nominal value, which is a natural result of the accurate

magnetic force modeling using the Hall sensor based model. The estimated external

forces [k]EF in all directions are almost always under 0.2pN, which means that the joint

state-parameter estimator is a promising method to achieve sub-pN dynamic force

sensing.

From the estimation result in Fig.5.2 it can be seen that the Hall-sensor based model

can be employed in the joint state-parameter estimator (Fig.5.1) to achieve high accuracy

dynamic force sensing. Moreover, it can also predict the drag coefficient very well, which

127

will make the magnetic bead a sensor to detect the local environment change. For

comparison, the current-based model is used in the joint state-parameter estimator, and

the estimator structure is shown in Fig.5.3. The estimator is updated in the same way as

Eq. (5.9) and Eq. (5.10), and the only difference is that the modeling force ˆMTF in the

[k]H matrix is form the current-based magnetic force model.

Fig. 5.9. Joint state-parameter estimator (using current-based magnetic force model) running

simultaneously with feedback controller. (The meaning of all parameters is the same as that in Fig.5.1, the

only difference is that the modeling force ˆMTF is from current-based model instead of Hall-sensor based

model).

128

Fig. 5.10. Joint state-parameter estimation, Hall sensor based force model vs. Current-based force model. (a)

Joint state-parameter estimation result in x direction. (b) Joint state-parameter estimation result in y

direction. (c) Joint state-parameter estimation result in z direction. (d) Trajectories used to test joint state-

parameter estimation algorithm.

The estimation result using current-based magnetic force model is shown in Fig.5.4.

For comparison, the estimation result using Hall sensor based model (Fig.5.2) is plotted

together. From the estimation result in Fig.5.4, it can be seen that the joint state-

parameter estimator using current-based magnetic force model has much degraded

performance. There is a large discrepancy between the estimated drag coefficient [k] and

the nominal value, especially in y and z direction. In the x direction, even though the

value of the estimation result [k] fluctuate around the nominal value due to the fact that

the current-based magnetic force fitting result in x direction is not as worse as y and z

129

directions (Fig.4.19 and Fig.4.20), but the estimation error in x direction at certain

location has a large jump and the estimated external force [k]EF can be very large.

Moreover, the estimation result of [k]EF in y and z directions is not as large as x direction

simply because the estimated drag coefficient [k] is very small compared to the nominal

value since the external force is linearly dependent on [k] , i.e. ' [ ] [ ]E E sF k x k .

It is also worth mentioning that the estimation result using the current-based model

will not be as consistent as estimation using Hall-sensor based model since the hysteresis

history can be very different in different experiments.

5.5 Conclusion

A joint state-parameter estimator is developed to dynamically estimate the bead-

sample interaction force and the drag coefficient, wherein the measurement position and

modeling force are employed to recursively estimate the desired values. The preliminary

experimental study validated the real-time estimation algorithm. Since the bead-sample

interaction force estimation is greatly determined by the magnetic force model accuracy,

the estimator using Hall sensor based model greatly outperforms that using current-based

force model. It can be clearly seen that the estimator using the Hall-sensor based model

can accurately estimate the bead-sample interaction force as well as the drag coefficient,

which can indicate the local environment change such as wall effect. The estimator based

on the current-based model, however, result in large discrepancy for drag coefficient

estimation.

130

In the future, the dynamic force sensing algorithm can be implemented in FPGA to

monitor the bead-sample interaction force and drag coefficient in real-time. Moreover, by

combing the motion control capability, the force control can be developed in the future,

which will make the automatic scanning possible.

131

Chapter 6: Conclusion and future works

6.1 Conclusion

In this research project, an over-actuated hexapole electromagnetic actuator is

developed, wherein a micro magnetic bead can be stably controlled by actively changing

six input currents. Stabilization, Brownian motion control, trajectory tracking, accurate

magnetic force modeling and dynamic force sensing are accomplished, which will

transform the magnetic actuator into an active scanning probe.

The magnetic actuator design and force modeling are analyzed first. Six sharp-tipped

poles, i.e. three pairs, are aligned in three orthogonal directions to generate 3D magnetic

forces. Six magnetic charge are employed to describe six magnetic poles and the

magnetic charge model are verified in FEM analysis. This magnetic charge model is then

employed to derive the magnetic force model, which can be lumped into a quadratic form

about the current. This force model describes the redundancy, position dependency and

nonlinearity. Moreover, the magnetic actuator can generate larger magnetic force

compared to most existing magnetic actuators since the poles are made of soft magnetic

material, which will lead to larger saturation limit.

The inverse model is very important for facilitating feedback linearization and real-

time feedback control. The inverse model needs to solve redundancy, nonlinearity and

position-dependency. Due to over-actuation, the inverse modeling greatly determines the

force generation capability since the inverse solution is not unique. Even though the

132

redundancy/nonlinearity/position-dependency can be solved by linearizing the quadratic

form after imposing constant constraints, the constant constraints and linearization will

greatly limit the force generation capability and result in force generation error. A so-

called optimal inverse model is proposed such that the norm of the actuation current is

minimized while satisfying the desired force at different orientations at different locations.

For each orientation of the desired force, a compact form is proposed to describe six

optimal actuation currents and capture the position dependency. Experiment is conducted

to compare the performance of linear inverse model using constant constraints and the

optimal inverse model. It can be seen that the optimal inverse model will lead to smaller

current consumption and better position control in term of smaller positioning error,

smaller Brownian motion and smaller tracking error. It is also widely known that the time

delay in the feedback control loop can be harmful and reduce the stability margin.

However, the feedback control implemented in PC has limited sampling rate due to

limited computation capability and time consistency. Since the inverse model has a very

compact form for implementation, FPGA board can be used to compute the control

algorithm and coordinate input/output signals. 1606Hz sampling rate can be achieved

with 512×512 image size using FPGA while 200Hz sampling rate is used in PC. The

experimental result showed that high speed control in FPGA can greatly reduce the time

delay in the feedback control, which will greatly increase the gain margin and reduce the

Brownian motion fluctuation.

The soft magnetic material can greatly improve the force generation capability since

the saturation limit is very high. However, the drawback is that the hysteresis problem is

133

obvious in such material. The hysteresis is very difficult to model since the hysteresis is

rate-dependent. Moreover, the magnetic coupling among six poles will make the

hysteresis very complicated. By introducing surface mount Hall sensors, the magnetic

field can be directly measured and it is verified that the measurement from this surface

mount Hall sensor can represent the magnetic field diverge from the tip of the magnetic

pole. A so-called Hall sensor based magnetic force model, which is based on the Hall

sensor voltage, is proposed to solve the complicated hysteresis issue. It can be clearly

seen that the hysteresis issue exist both in stabilization and tracking. Moreover, the

hysteresis is very difficult to model. Owning to the motion control capability, the Hall

sensor based magnetic force model is experimentally calibrated by steering the magnetic

bead in the 3D space. From the calibration result if can be seen that the Hall sensor based

model can accurately model/predict the magnetic force at sub-pN scale, while the

current-based model results in significant errors due to the hysteresis problem. In most

magnetic force models, the magnetic moment of the superparamagnetic bead is modeled

as a linear function of the external magnetic field. When the magnetic field is large,

however, the nonlinear magnetic saturation effect begins to emerge. An accurate Hall

sensor based model is proposed to model the bead magnetization using the Langevin

function. This model is also experimentally verified by steering the magnetic bead in 3D

space using large magnetic force.

Force sensing capability is very important to enable the magnetic actuator performing

probe scanning tasks. Specifically, dynamic force sensing capability is desired to sense

the transient bead-sample interaction force. Since the accuracy of the bead-sample

134

interaction force totally depends on the magnetic force model accuracy, the accurate

force modeling is a very important achievement for force sensing capability. A joint

state-parameter estimation algorithm, which employed the measurement position and

magnetic force model, was developed to simultaneously estimate the bead-sample

interaction force and drag coefficient. The value of the drag coefficient can indicate the

local environment change, such as wall effect, cytoplasm properties and etc. A

disturbance observer is employed to estimate the motion caused by the external force. A

1st order autoregressive (AR) model is used to estimate the drag coefficient. A Recursive

Least Square estimator is used to update the state variables. Specifically, the Kalman

filter algorithm is used since this system subjects to random noises such as thermal force

and measurement noise, and the discrete model is a time-varying process. A preliminary

experimental study is conducted to show the validity of the estimation algorithm. It can

be clearly seen that the estimator using the Hall sensor based model can accurately

estimate the state variables with very small error while the estimator using the current-

based model results in significant errors.

6.2 Future Works

There are several tasks can be worked on to complete this magnetic actuator as a

scanning probe microscopy and improve the performance of this magnetic actuator.

First, the joint-state parameter estimator algorithm need to be implemented in FPGA

to estimate and monitor the state variables in real-time. The algorithm is validated in

Chapter 5 and the state variables can converge to the nominal values. By implementing

135

the estimator in FPGA, not only the real-time information of the state variables can be

known but also the force control can be accomplished by combining the motion control

capability.

Second, biological experiment can be done to study the cell properties. With motion

control and dynamic force sensing capability, the automatic scanning can be performed to

study the unknown environment, for example, intracellular topography scanning. Since

the joint state-parameter estimator can sense the drag coefficient in real time, the

magnetic bead can also be a sensor to sense the local environment change such as

cytoplasm properties.

Third, the optimal inverse model based on calibrated force model can be developed. It

can be seen from the calibrated force model that there are biases between the equivalent

magnetic charge location and the tips of the magnetic pole. The inverse model in the

project is based on the nominal force model, which can already serve for the feedback

linearization purpose. However, if high bandwidth motion control is desired, such as fast

steering, the modeling error will be the drawback of the high speed tracking.

136

BIBLIOGRAPHY

[1] Z. P. Zhang, F. Long, and C. H. Menq, "Three-Dimensional Visual Servo Control

of a Magnetically Propelled Microscopic Bead," Ieee Transactions on Robotics,

vol. 29, pp. 373-382, Apr 2013.

[2] Z. P. Zhang and C. H. Menq, "Design and Modeling of a 3-D Magnetic Actuator

for Magnetic Microbead Manipulation," Ieee-Asme Transactions on

Mechatronics, vol. 16, pp. 421-430, Jun 2011.

[3] S. Suresh, "Biomechanics and biophysics of cancer cells," Acta Biomaterialia,

vol. 3, pp. 413-438, Jul 2007.

[4] E. L. Florin, V. T. Moy, and H. E. Gaub, "Adhesion Forces between Individual

Ligand-Receptor Pairs," Science, vol. 264, pp. 415-417, Apr 15 1994.

[5] P. Cluzel, A. Lebrun, C. Heller, R. Lavery, J. L. Viovy, D. Chatenay, et al.,

"DNA: An extensible molecule," Science, vol. 271, pp. 792-794, Feb 9 1996.

[6] J. T. Finer, R. M. Simmons, and J. A. Spudich, "Single Myosin Molecule

Mechanics - Piconewton Forces and Nanometer Steps," Nature, vol. 368, pp. 113-

119, Mar 10 1994.

[7] K. C. Neuman and A. Nagy, "Single-molecule force spectroscopy: optical

tweezers, magnetic tweezers and atomic force microscopy," Nature Methods, vol.

5, pp. 491-505, Jun 2008.

[8] G. Binnig, C. F. Quate, and C. Gerber, "Atomic Force Microscope," Physical

Review Letters, vol. 56, pp. 930-933, Mar 3 1986.

[9] Z. Liu, Y. Jeong, and C. H. Menq, "Calibration of measurement sensitivities of

multiple micro-cantilever dynamic modes in atomic force microscopy using a

contact detection method," Review of Scientific Instruments, vol. 84, Feb 2013.

[10] Y. Jeong, G. R. Jayanth, and C. H. Menq, "Control of tip-to-sample distance in

atomic force microscopy: A dual-actuator tip-motion control scheme," Review of

Scientific Instruments, vol. 78, Sep 2007.

[11] Y. Jeong, G. R. Jayanth, S. M. Jhiang, and C. H. Menq, "Design and Fabrication

of an Active Multiaxis Probing System for High Speed Atomic Force

Microscopy," Ieee Transactions on Nanotechnology, vol. 9, pp. 392-399, May

2010.

[12] Y. Jeong, G. R. Jayanth, S. M. Jhiang, and C. H. Menq, "Direct tip-sample

interaction force control for the dynamic mode atomic force microscopy," Applied

Physics Letters, vol. 88, May 15 2006.

[13] D. J. Muller, J. B. Heymann, F. Oesterhelt, C. Moller, H. Gaub, G. Buldt, et al.,

"Atomic force microscopy of native purple membrane," Biochimica Et Biophysica

Acta-Bioenergetics, vol. 1460, pp. 27-38, Aug 30 2000.

137

[14] M. Carrion-Vazquez, A. F. Oberhauser, S. B. Fowler, P. E. Marszalek, S. E.

Broedel, J. Clarke, et al., "Mechanical and chemical unfolding of a single protein:

A comparison," Proceedings of the National Academy of Sciences of the United

States of America, vol. 96, pp. 3694-3699, Mar 30 1999.

[15] S. Sen, S. Subramanian, and D. E. Discher, "Indentation and adhesive probing of

a cell membrane with AFM: Theoretical model and experiments," Biophysical

Journal, vol. 89, pp. 3203-3213, Nov 2005.

[16] A. L. Stout, "Detection and characterization of individual intermolecular bonds

using optical tweezers," Biophysical Journal, vol. 80, pp. 2976-2986, Jun 2001.

[17] A. Caspi, R. Granek, and M. Elbaum, "Diffusion and directed motion in cellular

transport," Physical Review E, vol. 66, Jul 2002.

[18] Z. W. Li, B. Anvari, M. Takashima, P. Brecht, J. H. Torres, and W. E. Brownell,

"Membrane tether formation from outer hair cells with optical tweezers,"

Biophysical Journal, vol. 82, pp. 1386-1395, Mar 2002.

[19] K. C. Neuman and S. M. Block, "Optical trapping," Review of Scientific

Instruments, vol. 75, pp. 2787-2809, Sep 2004.

[20] A. Ashkin, J. M. Dziedzic, and T. Yamane, "Optical Trapping and Manipulation

of Single Cells Using Infrared-Laser Beams," Nature, vol. 330, pp. 769-771, Dec

24 1987.

[21] J. F. Wan, Y. A. Huang, S. Jhiang, and C. H. Menq, "Real-time in situ calibration

of an optically trapped probing system," Applied Optics, vol. 48, pp. 4832-4841,

Sep 1 2009.

[22] M. L. Bennink, S. H. Leuba, G. H. Leno, J. Zlatanova, B. G. de Grooth, and J.

Greve, "Unfolding individual nucleosomes by stretching single chromatin fibers

with optical tweezers," Nature Structural Biology, vol. 8, pp. 606-610, Jul 2001.

[23] C. C. Huang, C. F. Wang, D. S. Mehta, and A. Chiou, "Optical tweezers as sub-

pico-newton force transducers," Optics Communications, vol. 195, pp. 41-48, Aug

1 2001.

[24] Y. A. Huang, P. Cheng, and C. H. Menq, "Dynamic Force Sensing Using an

Optically Trapped Probing System," Ieee-Asme Transactions on Mechatronics,

vol. 16, pp. 1145-1154, Dec 2011.

[25] E. J. G. Peterman, F. Gittes, and C. F. Schmidt, "Laser-induced heating in optical

traps," Biophysical Journal, vol. 84, pp. 1308-1316, Feb 2003.

[26] Y. Huang, J. Wan, M.-C. Cheng, Z. Zhang, S. M. Jhiang, and C.-H. Menq,

"Three-axis rapid steering of optically propelled micro/nanoparticles," Review of

Scientific Instruments, vol. 80, pp. 063107-1-063107-8, Jun 2009.

[27] Y. N. Huang, Z. P. Zhang, and C. H. Menq, "Minimum-variance Brownian

motion control of an optically trapped probe," Applied Optics, vol. 48, pp. 5871-

5880, Oct 20 2009.

[28] S. Earnshaw, "On the nature of the molecular forces which regulate the

constitution of the luminiferous ether," Trans. Camb. Phil. Soc, vol. 7, pp. 97-112,

1842.

138

[29] P. Kollmannsberger and B. Fabry, "High-force magnetic tweezers with force

feedback for biological applications," Review of Scientific Instruments, vol. 78,

pp. 114301-1-114301-6, Nov 2007.

[30] C. Haber and D. Wirtz, "Magnetic tweezers for DNA micromanipulation," Review

of Scientific Instruments, vol. 71, pp. 4561-4570, Dec 2000.

[31] J. Zlatanova and S. H. Leuba, "Magnetic tweezers: a sensitive tool to study DNA

and chromatin at the single-molecule level," Biochemistry and Cell Biology-

Biochimie Et Biologie Cellulaire, vol. 81, pp. 151-159, Jun 2003.

[32] C. H. Chiou and G. B. Lee, "A micromachined DNA manipulation platform for

the stretching and rotation of a single DNA molecule," Journal of

Micromechanics and Microengineering, vol. 15, pp. 109-117, Jan 2005.

[33] C. H. Chiou, Y. Y. Huang, M. H. Chiang, H. H. Lee, and G. B. Lee, "New

magnetic tweezers for investigation of the mechanical properties of single DNA

molecules," Nanotechnology, vol. 17, pp. 1217-1224, Mar 14 2006.

[34] J. K. Fisher, J. Cribb, K. V. Desai, L. Vicci, B. Wilde, K. Keller, et al., "Thin-foil

magnetic force system for high-numerical-aperture microscopy," Review of

Scientific Instruments, vol. 77, pp. 023702-1-023702-9, Feb 2006.

[35] F. J. Alenghat, B. Fabry, K. Y. Tsai, W. H. Goldmann, and D. E. Ingber,

"Analysis of cell mechanics in single vinculin-deficient cells using a magnetic

tweezer," Biochemical and Biophysical Research Communications, vol. 277, pp.

93-99, Oct 14 2000.

[36] A. R. Bausch, F. Ziemann, A. A. Boulbitch, K. Jacobson, and E. Sackmann,

"Local measurements of viscoelastic parameters of adherent cell surfaces by

magnetic bead microrheometry," Biophysical Journal, vol. 75, pp. 2038-2049,

Oct 1998.

[37] H. Huang, C. Y. Dong, H. S. Kwon, J. D. Sutin, R. D. Kamm, and P. T. C. So,

"Three-dimensional cellular deformation analysis with a two-photon magnetic

manipulator workstation," Biophysical Journal, vol. 82, pp. 2211-2223, Apr 2002.

[38] A. R. Bausch, W. Moller, and E. Sackmann, "Measurement of local

viscoelasticity and forces in living cells by magnetic tweezers," Biophysical

Journal, vol. 76, pp. 573-579, Jan 1999.

[39] B. G. Hosu, K. Jakab, P. Banki, F. I. Toth, and G. Forgacs, "Magnetic tweezers

for intracellular applications," Review of Scientific Instruments, vol. 74, pp. 4158-

4163, Sep 2003.

[40] A. H. B. de Vries, B. E. Krenn, R. van Driel, and J. S. Kanger, "Micro magnetic

tweezers for nanomanipulation inside live cells," Biophysical Journal, vol. 88, pp.

2137-2144, Mar 2005.

[41] J. S. Kanger, V. Subramaniam, and R. van Driel, "Intracellular manipulation of

chromatin using magnetic nanoparticles," Chromosome Research, vol. 16, pp.

511-522, May 2008.

[42] M. Barbic, J. J. Mock, A. P. Gray, and S. Schultz, "Scanning probe

electromagnetic tweezers," Applied Physics Letters, vol. 79, pp. 1897-1899, Sep

17 2001.

139

[43] F. Amblard, B. Yurke, A. Pargellis, and S. Leibler, "A magnetic manipulator for

studying local rheology and micromechanical properties of biological systems,"

Review of Scientific Instruments, vol. 67, pp. 818-827, Mar 1996.

[44] C. Gosse and V. Croquette, "Magnetic tweezers: Micromanipulation and force

measurement at the molecular level," Biophysical Journal, vol. 82, pp. 3314-

3329, Jun 2002.

[45] T. Honda, K. I. Arai, and K. Ishiyama, "Micro swimming mechanisms propelled

by external magnetic fields," Ieee Transactions on Magnetics, vol. 32, pp. 5085-

5087, Sep 1996.

[46] K. Ishiyama, K. I. Arai, M. Sendoh, and A. Yamazaki, "Spiral-type micro-

machine for medical applications," Mhs 2000: Proceedings of the 2000

International Symposium on Micromechatronics and Human Science, pp. 65-69,

2000.

[47] K. B. Yesin, K. Vollmers, and B. J. Nelson, "Modeling and control of untethered

biomicrorobots in a fluidic environment using electromagnetic fields,"

International Journal of Robotics Research, vol. 25, pp. 527-536, May-Jun 2006.

[48] L. Zhang, J. J. Abbott, L. X. Dong, B. E. Kratochvil, D. Bell, and B. J. Nelson,

"Artificial bacterial flagella: Fabrication and magnetic control," Applied Physics

Letters, vol. 94, Feb 9 2009.

[49] J. J. Abbott, O. Ergeneman, M. P. Kummer, A. M. Hirt, and B. J. Nelson,

"Modeling magnetic torque and force for controlled manipulation of soft-

magnetic bodies," Ieee Transactions on Robotics, vol. 23, pp. 1247-1252, Dec

2007.

[50] M. P. Kummer, J. J. Abbott, B. E. Kratochvil, R. Borer, A. Sengul, and B. J.

Nelson, "OctoMag: An Electromagnetic System for 5-DOF Wireless

Micromanipulation," Ieee Transactions on Robotics, vol. 26, pp. 1006-1017, Dec

2010.

[51] C. Bergeles, B. E. Kratochvil, and B. J. Nelson, "Visually Servoing Magnetic

Intraocular Microdevices," Ieee Transactions on Robotics, vol. 28, pp. 798-809,

Aug 2012.

[52] Z. P. Zhang, Y. N. Huang, and C. H. Menq, "Actively Controlled Manipulation of

a Magnetic Microbead Using Quadrupole Magnetic Tweezers," Ieee Transactions

on Robotics, vol. 26, pp. 531-541, Jun 2010.

[53] Z. P. Zhang, K. Huang, and C. H. Menq, "Design, Implementation, and Force

Modeling of Quadrupole Magnetic Tweezers," Ieee-Asme Transactions on

Mechatronics, vol. 15, pp. 704-713, Oct 2010.

[54] Z. P. Zhang, Y. H. Shi, S. M. Jhiang, and C. H. Menq, "Mechanical anisotropy

and adaptation of metastatic cells probed by magnetic microbeads," Imaging,

Manipulation, and Analysis of Biomolecules, Cells, and Tissues Viii, vol. 7568,

2010.

[55] Z. P. Zhang and C. H. Menq, "Three-dimensional particle tracking with

subnanometer resolution using off-focus images," Applied Optics, vol. 47, pp.

2361-2370, May 1 2008.

140

[56] R. Probst, J. Lin, A. Komaee, A. Nacev, Z. Cummins, and B. Shapiro, "Planar

steering of a single ferrofluid drop by optimal minimum power dynamic feedback

control of four electromagnets at a distance," Journal of Magnetism and Magnetic

Materials, vol. 323, pp. 885-896, Apr 2011.

[57] A. Komaee and B. Shapiro, "Steering a Ferromagnetic Particle by Optimal

Magnetic Feedback Control," Ieee Transactions on Control Systems Technology,

vol. 20, pp. 1011-1024, Jul 2012.

[58] S. Mittal and C. H. Menq, "Hysteresis compensation in electromagnetic actuators

through Preisach model inversion," Ieee-Asme Transactions on Mechatronics,

vol. 5, pp. 394-409, Dec 2000.

[59] J. H. Oh and D. S. Bernstein, "Semilinear Duhem model for rate-independent and

rate-dependent hysteresis," Ieee Transactions on Automatic Control, vol. 50, pp.

631-645, May 2005.

[60] M. Al Janaideh, S. Rakheja, and C. Y. Su, "A generalized rate dependent play

operator for characterizing asymmetric and symmetric hysteresis nonlinearities,"

2008 American Control Conference, Vols 1-12, pp. 1911-1916, 2008.

[61] A. J. W. T. Velthuis, J. W. J. Kerssemakers, J. Lipfert, and N. H. Dekkert,

"Quantitative Guidelines for Force Calibration through Spectral Analysis of

Magnetic Tweezers Data," Biophysical Journal, vol. 99, pp. 1292-1302, Aug 18

2010.

[62] M. Kruithof, F. Chien, M. de Jager, and J. van Noort, "Subpiconewton dynamic

force spectroscopy using magnetic tweezers," Biophysical Journal, vol. 94, pp.

2343-2348, Mar 15 2008.

[63] L. Chen, A. Offenhäusser, and H.-J. Krause, "Magnetic tweezers with high

permeability electromagnets for fast actuation of magnetic beads," Review of

Scientific Instruments, vol. 86, p. 044701, 2015.

[64] D. Matsuura and C.-H. Menq, Development of 3-Axis Flexure Stage for Bio-

Cellular Experimental Apparatus: Springer, 2013.

[65] D. Matsuura, Z. Zhang, and C.-H. Menq, "Development of Noncontact Micro

Particle Manipulation System Using Three-Dimensional Magnetic Tweezer and

Three Axis Piezo Stage," in Proceedings of the 15th International Conference on

Mechatronics Technology, 2011, pp. 514-519.

[66] W. M. Saslow, Electricity, Magnetism, and Light. New York: Academic Press,

2002.

[67] S. Chikazumi, Physics of magnetism. New York: Wiley, 1964.

[68] W. Coffey, Y. P. Kalmykov, and J. Waldron, The Langevin Equation: With

Applications to Stochastic Problems in Physics, Chemistry and Electrical

Engineering, 2nd ed. Singapore: World Scientific Publishing Company, 2004.

[69] H. Himberg, Y. Motai, and A. Bradley, "Interpolation Volume Calibration: A

Multisensor Calibration Technique for Electromagnetic Trackers," Ieee

Transactions on Robotics, vol. 28, pp. 1120-1130, Oct 2012.

[70] S. A. Hanna, Y. Motai, W. J. Varhue, and S. Titcomb, "Very-Low-Frequency

Electromagnetic Field Detector With Data Acquisition," Ieee Transactions on

Instrumentation and Measurement, vol. 58, pp. 129-140, Jan 2009.

141

[71] P. Cheng, S. M. Jhiang, and C. H. Menq, "Real-time visual sensing system

achieving high-speed 3D particle tracking with nanometer resolution," Applied

Optics, vol. 52, pp. 7530-7539, Nov 1 2013.

[72] B. M. Lansdorp, S. J. Tabrizi, A. Dittmore, and O. A. Saleh, "A high-speed

magnetic tweezer beyond 10,000 frames per second," Review of Scientific

Instruments, vol. 84, Apr 2013.

[73] F. Gittes and C. F. Schmidt, "Thermal noise limitations on micromechanical

experiments," European Biophysics Journal with Biophysics Letters, vol. 27, pp.

75-81, 1998.

[74] B. Tellini, R. Giannetti, S. Lizon-Martinez, and M. Marracci, "Characterization of

the Accommodation Effect in Soft Hysteretic Materials Via Sensorless

Measurement Technique," Ieee Transactions on Instrumentation and

Measurement, vol. 58, pp. 2807-2814, Aug 2009.

[75] S. E. Zirka, Y. I. Moroz, and E. Della Torre, "Combination hysteresis model for

accommodation magnetization," Ieee Transactions on Magnetics, vol. 41, pp.

2426-2431, Sep 2005.

[76] L. Yanik, A. E. Yarimbiyik, and E. Della Torre, "Comparison of the differential

equation accommodation model with experiment," Journal of Applied Physics,

vol. 99, Apr 15 2006.

[77] G. Fonnum, C. Johansson, A. Molteberg, S. Mørup, and E. Aksnes,

"Characterisation of Dynabeads® by magnetization measurements and Mössbauer

spectroscopy," Journal of Magnetism and Magnetic Materials, vol. 293, pp. 41-

47, 2005.

[78] J. Xu, K. Mahajan, W. Xue, J. O. Winter, M. Zborowski, and J. J. Chalmers,

"Simultaneous, single particle, magnetization and size measurements of micron

sized, magnetic particles," Journal of Magnetism and Magnetic Materials, vol.

324, pp. 4189-4199, Dec 2012.

[79] G. Mihajlovic, K. Aledealat, P. Xiong, S. Von Molnar, M. Field, and G. J.

Sullivan, "Magnetic characterization of a single superparamagnetic bead by

phase-sensitive micro-Hall magnetometry," Applied Physics Letters, vol. 91, Oct

22 2007.

[80] P. A. Ioannou and J. Sun, Robust adaptive control: Courier Corporation, 2012.

[81] G. F. Franklin, J. D. Powell, and M. L. Workman, Digital control of dynamic

systems vol. 3: Addison-wesley Menlo Park, 1998.

[82] J. Han, Q. Song, and Y. He, Adaptive unscented Kalman filter and its applications

in nonlinear control: INTECH Open Access Publisher, 2009.

[83] J. Qi, D. Song, C. Wu, J. Han, and T. Wang, "KF-based adaptive UKF algorithm

and its application for rotorcraft UAV actuator failure estimation," Int J Adv

Robotic Sy, vol. 9, 2012.