By - Automation, Robotics, & Mechatronics Laboratory

i

ANALYSIS OF PARALLEL MANIPULATOR ARCHITECTURES

FOR USE IN

MASTICATORY STUDIES

By

MADUSUDANAN SATHIA NARAYANAN

September 2008

A thesis submitted to the Faculty of the Graduate School of the State University of New York at Buffalo in partial fulfillment of the requirements

for the degree of MASTER OF SCIENCE

Department of Mechanical and Aerospace Engineering State University of New York at Buffalo

Buffalo, New York 14260

ii

To my family and friends,

Without whom, I am nothing

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Acknowledgement

I want to express my sincerest gratitude to my advisor, Dr. Krovi for constantly

motivating and supporting me, and for being a good friend. This work would be highly

impossible without him. I grateful to have worked under his guidance for the last two

years and I thank him for providing me the wonderful opportunity of being a part of

ARMLAB, which actually became my second home in the due course. I also thank Dr.

Mendel, my co-advisor throughout this work for his motivation and support. I enjoyed

working with him.

I also thank Dr. Tarunraj Singh for being a part of my thesis committee as well as

a former instructor. His classes and the projects were very useful.

I also thank all the ARMLAB members for having our shared of enjoyment and

entertaining experiences. I also thank CP and Leng-Feng, the current senior lab members

for helping and supporting me since the day I joined the lab. Also special thanks to Pat,

Hao, Qiushi, Yao, Anand, Kun as well as Rajan for lending helping hands whenever I

needed them. I also thank Srikanth for being my partner not only in this project but

throughout my graduate course work and lab activities, and his good company.

I also thank all my friends in Buffalo- Parthi, Amol, Govind and others, who made my

stay in the U.S. more comfortable and happy. Special thanks to all my friends in India as

well especially, Karthik, Arun, Nisha, Deepika and many more, with whom I have

cherished memories that will never fade. Lastly, but most importantly, I would like to

dedicate this work to my parents and my sister without whom I would not be where I am

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now. They constantly motivated and supported me from the beginning, and cherished my

growth and achievements, especially while I was away from home.

Thank you— Mom, Dad and Sis for everything.

Thank you everyone.

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Table of Contents

List of Figures............................................................................................... xi

List of Tables ............................................................................................ xxiv

Abstract..................................................................................................... xxvi

1. Introduction ........................................................................................ 1

1.1. Background .......................................................................................................... 2

1.2. Problem Statement ............................................................................................... 4

1.2.1. Research Goals........................................................................................ 4

1.3. Research Issues .................................................................................................... 6

1.4. Thesis Organization.............................................................................................. 7

2. Literature Survey ............................................................................... 8

2.1. Related Work........................................................................................................ 8

2.1.1. Biomechanical Model of the Jaw............................................................ 8

2.1.2. Masticatory Robotic Manipulators ....................................................... 10

2.1.3. Parallel Actuated Robotic Manipulators............................................... 15

2.1.4. Jaw Motion Analysis, Imaging and Experiments ................................. 16

2.2. Computational Tools .......................................................................................... 18

2.2.1. Musculoskeletal Analysis Tools: .......................................................... 18

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2.2.2. Dynamics Simulation/ Analysis Tools: ................................................ 22

2.2.3. CT Scanning and Imaging .................................................................... 25

2.3. Motion Capture Technology .............................................................................. 26

2.3.1. Motion Capture and analysis Systems: ................................................. 26

2.3.2. Optical Markers: ................................................................................... 27

2.3.3. Non-optical markers: ............................................................................ 29

2.3.4. Markerless tracking devices:................................................................. 30

2.3.5. Motion Capture beyond Markers - 3D Scanners .................................. 31

2.3.6. Issues with MoCap Systems ................................................................. 33

2.3.7. SimiMotion System .............................................................................. 35

2.3.8. Modules................................................................................................. 36

2.3.9. MoCap Transformation and Synchronization....................................... 38

3. Biomechanics of Masticatory Motion............................................. 41

3.1. Human Masticatory System ............................................................................... 41

3.1.1. Temporo-Mandibular Joint (TMJ)........................................................ 42

3.1.2. Musculoskeletal Modeling.................................................................... 42

3.1.3. Dynamic modeling of Mastication........................................................ 44

3.1.4. Inverse Dynamic Analysis and Muscle Actuation................................ 45

3.1.5. Muscle Modeling (Active and Passive): ............................................... 47

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Passive Elements: ...................................................................................................... 48

3.1.6. Posselt Envelope: .................................................................................. 49

4. Mathematical Background .............................................................. 51

4.1. Kinematics of R-U-S Configuration .................................................................. 51

4.2. Position Kinematics: .......................................................................................... 53

4.3. Velocity Kinematics to find Link Jacobian Matrix:........................................... 54

4. 3.1. Link and Manipulator Jacobian Matrix................................................. 56

4.4. Screw Theoretic Jacobian Matrix and Singularity Analysis .............................. 57

4.5. Singularity Analysis of the Jacobian of R-U-S Platform ................................... 58

4.6. Kinematic of P-U-S Platform............................................................................. 59

4.7. Velocity Kinematics of P-U-S Platform............................................................. 61

4.8. Denavit Hartenberg Parameterization ................................................................ 62

4.9. P-U-S Manipulator Parametric Study ................................................................ 65

4.9.1. Maximum Reachable Workspace, Velocity and Force......................... 65

4.10. Jacobian-Based Performance Measures (JBPM) .......................................... 69

4.10.1. SVD and Manipulability ellipsoid ........................................................ 70

4.10.2. Yoshikawa Measure.............................................................................. 71

4.10.3. Condition Number ................................................................................ 72

4.10.4. Isotropy Index ....................................................................................... 72

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4.11.1. Transformation Matrix between MoCap coordinates and Platform

coordinates ............................................................................................................ 73

5. Technological Tools .......................................................................... 76

5.1. CT-Scan to CAD Conversion............................................................................. 76

5.2. SimiMotion MoCap Station ............................................................................... 78

5.2.1. Workflow for Motion Capture using SimiMotion system:................... 78

5.2.2. Experimental Setup............................................................................... 82

5.3. Inverse Dynamics Analysis of Musculoskeletal Models ................................... 83

5.3.1. Human Jaw Model (Mark de Zee Model)............................................. 83

5.3.2. Human Jaw Model (14 Muscle Actuator Model) ................................. 85

5.3.3. Labrador Jaw Model ............................................................................. 87

6. Musculoskeletal Modeling Simulation and Results ...................... 91

6.1. Work envelope study using Posselt diagram...................................................... 91

6.1.1. Human Jaw Model ................................................................................ 91

6.1.2. Labrador Jaw Model ............................................................................. 92

6.2. Musculoskeletal Model Case Studies- Human Jaw ........................................... 93

Case A.I.1: Simple Muscle Model, No bite force (free chewing motion) ............ 93

Case A.I.2: Simple Muscle Model, Constant Bite Force (200 N in –z) ............... 95

Case A.I.3: Simple Muscle Model, Realistic Bite Force ...................................... 96

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Case A.II.1: Muscle Model 3E, No Bite Force (free chewing motion) ................ 97

Case A.II.2: Muscle Model 3E, Constant Bite Force (100 N in –z) ..................... 98

Case A.II.3: Muscle Model 3E, Realistic Bite Force............................................ 99

6.3. Musculoskeletal Model Case Studies- Labrador Dog...................................... 100

Case B. I. 1: Simple Muscle Model, no bite force (free chewing motion) ......... 100

Case B.I.2: Temporalis Muscle Tendon Unit, Constant Bite Force ................... 101

Case B.I.3: Temporalis Muscle Tendon Unit, Realistic Bite Force.................... 102

7. Masticatory Simulator- Analysis and Results ............................. 103

7.1. Inverse Kinematics and Jacobian Based Workspace Analysis of 6-DOF R-U-S

Manipulator.............................................................................................................. 103

7.1.1. Inverse Kinematics Simulation ........................................................... 103

7.2. Validation using Visual Nastran for R-U-S Case............................................. 104

Case 1: Point ....................................................................................................... 105

Case 2: Straight line ............................................................................................ 105

Case 3: Circle...................................................................................................... 105

7.3. Constant Orientation Workspace Analysis ...................................................... 106

7.4. Inverse Kinematics and Jacobian Based Workspace Analysis of 6-DOF P-U-S

Manipulator.............................................................................................................. 111

7.4.1. Inverse Kinematics Simulation ........................................................... 111

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7.4.2. Constant Orientation Workspace Analysis ......................................... 113

7.5. Visual Nastran Implementation P-U-S Manipulator ........................................ 116

Case 1: Human Jaw Motion Trajectory .............................................................. 116

Case 2: Labrador Jaw Motion Trajectory ........................................................... 117

Case 2: Bulldog Jaw Motion Trajectory ............................................................. 118

7.6. Parametric Study of P-U-S Manipulator .......................................................... 119

7.6.1. Simplified Representation of Parametric Analysis ............................. 119

7.6.2. Parametric Study I: Effects of R/r and r on Fz, Pz and Vz ................. 120

7.6.3. Parametric Study II: Effects of /l r and r on Fz, Pz and Vz.............. 122

7.6.4. Parametric Study III: Effects of q0/ Tq0 and r on Fz, Pz and Vz ....... 123

7.6.5. Parametric Sweep of Workspace Variables against r and r/R ratio: ... 125

7.6.6. Parametric Sweep of Workspace Variables against r and r/R ratio: ... 125

8. Conclusion:...................................................................................... 128

8.1. Summary .......................................................................................................... 128

8.2. Future Work: .................................................................................................... 129

Bibliography .............................................................................................. 131

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List of Figures

Figure 1-1: Engineering Paradigm..................................................................................... 3

Figure 1-2: Virtual to Physical Prototyping Cycle ............................................................. 3

Figure 1-3: Project Flowchart ............................................................................................ 5

Figure 2-1: Hinge Joint Axis Of TMJ For Left Side Movement ........................................ 9

Figure 2-2: Modeling Of Muscles As Threads ................................................................... 9

Figure 2-3: Anteriolateral View Of The Basic Model ........................................................ 9

Figure 2-4: Jaw Model With TMJ And Incisor Point Envelope......................................... 9

Figure 2-5: Oblique And Side View Of Jaw And Larynx Model In “Artisynth”............. 10

Figure 2-6: WY-5RII Mastication Robot......................................................................... 11

Figure 2-7: Commercial Mastication Robot For Use (WOJ1).......................................... 11

Figure 2-8: Food Texture Measurement Robot WWT-1 .................................................. 11

Figure 2-9: Jaw Robot 3D Simulation WOJ-1RII ............................................................ 11

Figure 2-10: Masticatory Robot Covered By The Skull In SolidWorks .......................... 12

Figure 2-11: 3-D Model Of The Mandible- The Actuators’ Attaching Points (Mi), And

The Reference Points ........................................................................................................ 12

Figure 2-12: Robotic Model In The Form Of Platform Mechanism Nomenclature And

Coordinate Systems .......................................................................................................... 13

Figure 2-13: 3D Kinematic Jaw Model ............................................................................ 13

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Figure 2-14: Robotic Model Of Crank Actuation............................................................. 14

Figure 2-15: Physical Prototype Of The Dental Simulator............................................... 14

Figure 2-16: Four Bar Mechanism Simulator (MS Thesis, Darren Lewis, University Of

Massey) ............................................................................................................................. 14

Figure 2-17: Schematic Diagram Of Stewart Platform..................................................... 15

Figure 2-18: Force Analysis On Leg ................................................................................ 15

Figure 2-19: 2D CT scans and 3D STL Model Of The Human Jaw ............................... 18

Figure 2-20: AnyBody Software Interface ....................................................................... 19

Figure 2-21: Inverse Dynamics of a Redundant Musculoskeletal System (Konakanchi

[24])................................................................................................................................... 20

Figure 2-22: SIMM Virtual Model ................................................................................... 21

Figure 2-23: Double Pendulum Model implementation in DFP....................................... 22

Figure 2-24: SolidWorks- SimMechanics- VRML Framework ....................................... 23

Figure 2-25: SolidWorks Model of Parallel Manipulator................................................. 24

Figure 2-26: Open Loop Kinematics and Dynamic Analysis in Visual Nastran.............. 24

Figure 2-27: CT Scan Slices to STL Conversion.............................................................. 25

Figure 2-28: Working with STL in Rhinoceros................................................................ 25

Figure 2-29: OptiTrack Motion Capture........................................................................... 28

Figure 2-30: Peak Motus MoCap Equipment .................................................................. 28

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Figure 2-31:Time Modulated Active Marker ................................................................... 29

Figure 2-32: Semi Perceptible Marker.............................................................................. 29

Figure 2-33: Electromagnetic Motion Capture System .................................................... 30

Figure 2-34: Exoskeleton Mocap System......................................................................... 30

Figure 2-35: Inertial Motion Capture Systems ................................................................. 30

Figure 2-36: Inertial Motion Capture Systems at NASA (Miller, Jenkins et al. [42]) ..... 30

Figure 2-37: Markerless Motion Capture Systems by Organic Motion ........................... 31

Figure 2-38: Markerless Motion Capture Systems by Noraxon ....................................... 31

Figure 2-39: Facial Motion Capture at CMU Robotics Institute ...................................... 32

Figure 2-40: Facial Motion capture at IBM...................................................................... 32

Figure 2-41: PONTOS Scanners for Crash testing ........................................................... 32

Figure 2-42: PONTOS for Car Body testing .................................................................... 32

Figure 2-43: Errors in Motion Capture Process Gonzalez-Morcillo, Jimenez-Linares et al.

[49].................................................................................................................................... 34

Figure 2-44: Frames Captured at Different Instant during Motion Capture of a Mobile

Robotic Platform with Traces of the Markers at all times ................................................ 35

Figure 2-45: 2D/ 3D Kinematics Module ......................................................................... 36

Figure 2-46: Human Avatar within SimiMotion .............................................................. 36

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Figure 2-47: 3D Visualization of MoCap data of Jaw Motion using Stick Diagrams and

Plots of 3D coordinates of the incisor point tip ................................................................ 37

Figure 2-48: Simi Motion System..................................................................................... 38

Figure 2-49: Configure Trigger Signal in SimiMotion..................................................... 39

Figure 2-50: Triggering Unit in SimiMotion .................................................................... 39

Figure 3-1: Human Mandible............................................................................................ 42

Figure 3-2: Temporomandibular Joint .............................................................................. 42

Figure 3-3: Masticatory Muscular Architecture (Xu, Lewis et al. [11]) .......................... 43

Figure 3-4: Anatomical Planes- used to describe the positions of the muscles or organs in

musculoskeletal systems (Shin [53]) ................................................................................ 44

Figure 3-5: Six degrees of freedom for jaw movement. ................................................... 45

Figure 3-6: Forces acting on the lower jaw in the Sagittal plane...................................... 46

Figure 3-7: Schematic of force vector diagrams for each muscle groups......................... 47

Figure 3-8: Sagittal Plane- Posselt Envelope.................................................................... 49

Figure 3-9: Frontal Plane : Posselt Envelope.................................................................... 50

Figure 4-1: Schematic diagram of R-U-S configuration.................................................. 51

Figure 4-2: Kinematics of leg I of the R-U-S manipulator (front and side view) ........... 52

Figure 4-3: Base- Revolute Joint frame A ................................................................... 52

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Figure 4-4: Front and Side Orthogonal View of ith Leg Configuration of the R-U-S

Manipulator....................................................................................................................... 55

Figure 4-5: Schematic diagram showing the DH parameterization for each joint

coordinate to determine the end effector reference frame ................................................ 62

Figure 4-6: Top view of base platform ............................................................................ 66

Figure 4-7: Front view of the manipulator at the two extreme positions to find Pz ......... 66

Figure 4-8: Top view of the base platform with the linear actuators – FBD to calculate

maximum workspace forces and torques.......................................................................... 67

Figure 4-9: Front and side view of the manipulator with the linear actuators – FBD to

calculate maximum workspace forces and torques........................................................... 68

Figure 4-10: Manipulability ellipsoid: mapping joint space velocities (hyper sphere) to

task space velocities (hyper ellipsoid) (Manipulability Index [61]) ................................. 70

Figure 4-11: Plane of Jaw Motion Measured using MoCap System with marker positions

identified ........................................................................................................................... 73

Figure 4-12: Transformation between MoCap and Platform Reference Frames............. 74

Figure 5-1: Different Views of Labrador Specimen ........................................................ 76

Figure 5-2:CT Scanned Images Import and Pixels Information Dialog ........................... 76

Figure 5-3: Segmentation Options .................................................................................... 77

Figure 5-4: Segmentation and 3D Calculation.................................................................. 77

Figure 5-5: STL file of CT scans ...................................................................................... 78

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Figure 5-6: Digitizing the Human Jaw Movement ........................................................... 79

Figure 5-7: Digitizing the Dog Jaw Movement ................................................................ 79

Figure 5-8: Typical Camera Setup for 2D Motion Capture Analysis............................... 79

Figure 5-9: Calibration of MoCap Region- Camera1 ....................................................... 80

Figure 5-10: Calibration of MoCap Region- Camera2 ..................................................... 80

Figure 5-11: Marker Occlusion Problems in Animals...................................................... 81

Figure 5-12: Digitization of the sequence of frames and automatic tracking of 2D

coordinates- Camera1 and 2.............................................................................................. 81

Figure 5-13: 3D Data Calculation and Stick Diagram Representation of Jaw for Cameras

1 and 2............................................................................................................................... 82

Figure 5-14: Experimental Setup for Animal Motion Capture with Cameras.................. 83

Figure 5-15: MoCap Workstation and Camera Setup for Animal Subjects ..................... 83

Figure 5-16: Muscle Actuators of a Jaw Model................................................................ 84

Figure 5-17: Lateral View of the Mark de Zee Human Jaw Model.................................. 84

Figure 5-18: Lower Mandible with the muscles attached used in Mark de Zee Model and

Structural Properties.......................................................................................................... 84

Figure 5-19: Temporalis Muscle Model (Green Colored Muscle Tendon Units) ............ 86

Figure 5-20: Aerial View of Mandible with Different Muscle Groups ............................ 86

Figure 5-21: Side and Top view of Musculoskeletal Jaw Model of Labrador Dog ........ 87

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Figure 5-22: Measurement of fiber length for Masseter muscle....................................... 88

Figure 5-23: Digastric Muscle (Jaw opener)- dissected out of the jaw ............................ 88

Figure 6-1: X and Z coordinates forming a Posselt Envelope in Sagittal Plane- Human

Jaw .................................................................................................................................... 91

Figure 6-2: X and Y coordinates forming a Posselt Envelope in Transverse Plane- Human

Jaw .................................................................................................................................... 91

Figure 6-3: Velocity trajectory of incisor tooth tip point for Human Jaw........................ 92

Figure 6-4: Acceleration trajectory of incisor tooth tip point for Human Jaw ................. 92

Figure 6-5: X and Z coordinates forming a Posselt Envelope in Sagittal Plane- Labrador

Jaw .................................................................................................................................... 92

Figure 6-6: X and Z coordinates forming a Posselt Envelope in transverse plane-

Labrador Jaw..................................................................................................................... 92

Figure 6-7: Velocity trajectory of incisor tooth tip point - Labrador Jaw ........................ 93

Figure 6-8: Acceleration trajectory of incisor tooth tip point - Labrador Jaw.................. 93

Figure 6-9: (a), (b) Elevator Muscle forces and activities (c) Muscle Lengths of RHS

Muscles ............................................................................................................................. 94

Figure 6-10: (a), (b) Depressor Muscle forces and activities (c) Muscle Lengths of LHS94

Figure 6-11: Parametric Studies for Mastication Musculoskeletal Analysis.................... 94

Figure 6-12: (a), (b) Elevator Muscle forces and activities for Case A.I.2...................... 95

Figure 6-13: (a), (b) Depressor Muscle forces and activities for Case A.I.2.................... 95

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Figure 6-14: TMJ Reaction forces for Case A.I.2 ............................................................ 95

Figure 6-15: (a), (b) Elevator Muscle forces and activities for Case A.I.3...................... 96

Figure 6-16: (a), (b) Depressor Muscle forces and activities Case A.I.3.......................... 96

Figure 6-19: (a) TMJ Reaction forces (b) Simulated bite force for Case A.I.3................ 96


Muscles for Case A.I.2...................................................................................................... 97

Figure 6-19: (a), (b) Depressor Muscle forces and activities (c) Muscle Lengths of LHS

for Case A.I.2.................................................................................................................... 97

Figure 6-20: (a), (b) Elevator Muscle forces and activities of RHS Muscles for Case A.I.2

........................................................................................................................................... 98


Figure 6-22: TMJ Reaction forces Case A.II.3................................................................. 98


........................................................................................................................................... 99


Figure 6-25: (a) TMJ Reaction forces (b) Simulated bite force for Case A.II.3............... 99


Muscles for Case B.I.1.................................................................................................... 100


for Case B.I.1 .................................................................................................................. 100

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Figure 6-28: (a), (b) Elevator Muscle forces and activities of RHS Muscles for Case B.I.2

......................................................................................................................................... 101

Figure 6-29: (a), (b) Depressor Muscle forces and activities for Case B.I.2 .................. 101

Figure 6-30: TMJ Reaction forces for Case B.I.2........................................................... 101


......................................................................................................................................... 102

Figure 6-32: (a), (b) Depressor Muscle forces and activities for Case B.I.3 .................. 102

Figure 6-33: (a) TMJ Reaction forces (b) Simulated bite force for Case B.I.3 .............. 102

Figure 7-1: R-U-S Manipulator Configuration for Validation Test: (a) Line along Z (b)

3D Sine curve.................................................................................................................. 103

Figure 7-2: R-U-S Joint trajectory for Validation Test: (a) Line along Z axis (b) 3D Sine

curve................................................................................................................................ 103

Figure 7-3: R-U-S Manipulator Configuration for Validation Test: (a) Circle in YZ plane

(b) Circle in XY (c) Ellipse in XZ plane......................................................................... 104

Figure 7-4: R-U-S Joint trajectory for Validation Test: (a) Circle in YZ plane (b) Circle in

XY (c) Ellipse in XZ plane ............................................................................................. 104

Figure 7-5:Point Tracking Simulation of R-U-S in Visual Nastran ............................... 105

Figure 7-6:Point Tracking Joint Angle Trajectories from MATLAB Code ................... 105

Figure 7-7: Line Tracking Simulation of R-U-S in Visual Nastran................................ 105

Figure 7-8: Line Tracking Joint Angle Trajectories from MATLAB Code ................... 105

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Figure 7-9: Circle Tracking Simulation of R-U-S in Visual Nastran ............................. 105

Figure 7-10: Circle Tracking Joint Angle Trajectories from MATLAB Code............... 105

Figure 7-11: (i), (ii), (iii): Condition number based measure of manipulability plotted

along the vertical z-axis for every point in the 2D circular region: 0<r<0.06 at z=0.20

plane................................................................................................................................ 107

Figure 7-12: (i), (ii), (iii): Yoshikawa measure of manipulability plotted along the vertical

z-axis for every point in the 2D circular region: 0<r<0.06 at z=0.20 plane ................... 107

Figure 7-13: (a), (b), (c): Condition number based measure of manipulability plotted


plane................................................................................................................................ 108

Figure 7-14: (a), (b), (c): Yoshikawa measure of manipulability plotted along the vertical




plane................................................................................................................................ 109





plane................................................................................................................................ 110

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Figure 7-18: (i), (ii), (iii): Yoshikawa measure of manipulability plotted along the vertical


Figure 7-19: P-U-S Manipulator Configuration for Validation Test: (a) Line along Z axis

(b) Straight line (c) 3D Sine curve.................................................................................. 111

Figure 7-20: P-U-S Joint trajectory for Validation Test: (a) Line along Z axis (b) Straight

line (c) 3D Sine curve ..................................................................................................... 111

Figure 7-21: Maximum Reachable Characteristic Values of the Manipulator 6-71: P-U-S

Manipulator Configuration for Validation Test: (a) Circle in XY Plane (b) Circle in YZ

Plane (c) Ellipse in YZ plane .......................................................................................... 112

Figure 7-23: P-U-S Joint trajectory for Validation Test: (a) Circle in XY Plane (b) Circle

in YZ Plane (c) Ellipse in YZ plane................................................................................ 112



plane................................................................................................................................ 113





plane................................................................................................................................ 114



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plane................................................................................................................................ 115



Figure 7-30: Visual Nastran Implementation of P-U-S with Human Jaw motion

Trajectory Input .............................................................................................................. 116

Figure 7-31: Visual Nastran Implementation of P-U-S with Labrador Jaw motion


Figure 7-32: Visual Nastran Implementation of P-U-S with Bulldog Jaw motion


Figure 7-33: Workspace in Z, m..................................................................................... 120

Figure 7-34: Speed in Z, m/s........................................................................................... 120

Figure 7-35: Maximum Force in Z, Newton................................................................... 120

Figure 7-36: Pitch Torque, Nm...................................................................................... 121

Figure 7-37: Roll Torque, Nm ........................................................................................ 121

Figure 7-38: Yaw Torque, Nm........................................................................................ 121


Figure 7-40: Speed in Z, m/s........................................................................................... 122

Figure 7-41: Force in Z, Newton .................................................................................... 122

xxiii


Figure 7-43: Speed in Z, m ............................................................................................. 123

Figure 7-44: Workspace in X, m/s.................................................................................. 124

Figure 7-45: Force in Z, N .............................................................................................. 124

Figure 7-46: Pitch/ Roll angles, rad and velocities, rad/s ............................................... 125

Figure 7-47: Yaw angles, rad and velocities, rad/s......................................................... 125

Figure 7-48: Pitch angles, rad and velocities, rad/s ........................................................ 125

Figure 7-49: Yaw angles, rad and velocities, rad/s......................................................... 126

Figure 7-50: Torque about X, Nm .................................................................................. 126

Figure 7-51: Torque about Y, Nm .................................................................................. 126

Figure 7-52: Torque about Z, Nm................................................................................... 126

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List of Tables

Table 2-1: Brief Survey on Commercially Available 6 DOF Parallel Architecture

Simulator........................................................................................................................... 17

Table 4-1 DH Parameterization of R-U-S Manipulator.................................................... 64

Table 4-2 DH Parameterization of P-U-S Manipulator .................................................... 64

Table 4-3 List of Parameters............................................................................................. 65

Table 5-1 CT Scan Images Information............................................................................ 77

Table 5-2: Muscle Model Parameters [Koolstra, 2002].................................................... 85

Table 5-3: Muscle Fiber Length, Insertion and Origin Points of Human Jaw Model ...... 86

Table 5-4: Muscle Mass based on Turnbull [66] .............................................................. 87

Table 5-5: Muscle Model Parameters [Koolstra, 2002].................................................... 89

Table 5-6: Muscle Fiber Length, Insertion and Origin Points of Labrador Jaw Model ... 89

Table 7-1: Manipulator Parameters for Set 1: (all linear dimensions in meters and angular

dimensions in radians) .................................................................................................... 106

Table 7-2: Maximum Reachable Characteristic Values of the Manipulator .................. 113

Table 7-3: Parametric Studies Conducted to Identify Optimal Values for Jaw Motion

Simulation ....................................................................................................................... 119

Table 7-4: Values of Platform Parameters...................................................................... 120

Table 7-5: Preliminary Specification Values of the Manipulator................................... 124

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Table 7-6: Final Specification of the Manipulator.......................................................... 127

Table 7-7: Maximum Reachable Characteristic Values of the Manipulator .................. 127

xxvi

Abstract

The goal of this work is to (i) quantitatively analyze the jaw motions using a

variety of quantitative engineering tools and (ii) create/ design a virtual jaw motion

simulator based on parallel architecture manipulator that can reproduce these recorded

jaw motions of vertebrates/ animals accurately. Such an implementation could provide a

test bed to quantitatively characterize the jaw motion based on “chewability index” factor

for wide range of applications. To this end we will examine two sets of case studies—

human jaw motion and canine jaw motion.

In the first phase, we begin with initially developing the biomechanical models of

a human and canine jaw in AnyBody. The human model is developed using elements

from the AnyBody repository while the canine jaw was developed from scratch. . For this

purpose, the muscle model parameters as well as kinematic modeling of temporo-

mandibular joint are ascertained by a real dog cadaver dissection. Detailed case studies

were conducted to validate the inverse dynamics analyses and work envelope of both the

models— human and labrador dog. We also artificially simulated the bite force in these

biomechanical models as a part of this case study for further validation.

In the second phase, we examine the different parallel architecture manipulator

systems that would suit our desired application resulting in final selection of two

manipulator configurations for our application- 6 DOF R-U-S (revolute- universal-

spherical with active revolute joints) and P-U-S (prismatic- universal-spherical with

active prismatic joints) manipulators. Accurate kinematic models of these manipulators

were developed in MATLAB and these were used to evaluate the manipulators based on

xxvii

Jacobian based measures. Kinematics of Parallel manipulators is, in general, modeled

using simple loop closure technique and Jacobian matrices were derived using screw-

theoretic modeling. We then completed a comprehensive parametric studies based on

maximum workspace and force limits for P-U-S manipulator case. Motion capture

trajectories for the jaw motion are obtained using the high speed SimiMotion motion

capture system with the real subjects (humans and dogs). For ascertaining the model of

simulator, we finally use the motion capture trajectories to drive the AnyBody as well as

the parallel manipulator model and verify the workspace envelope to match with the jaw

trajectories. Finally, we briefly discuss the dynamic modeling of the system for real-time

with hardware-in-loop simulation and physical prototype implementation using rapid

prototypes as well as cast dentitions.

1

1. Introduction Use of mathematics, mechanics and numerical simulation to build sophisticated

dynamic models has played a major role in the bridging different fields- physics based

modeling, rigid body dynamics, nanotechnology and many more. Engineering related

fields benefited most from these advances but application to other areas are fast

approaching. Developing computational tools for anatomists and biologists for rapid

hypotheses testing and “what-if” type analyzes is already an active area of research. In

fact, it is this need, which provide enormous amount of scope for collaboration between

engineering and biological sciences.

Similar collaboration between biological researchers and engineers can be found

in the field of bioengineering and biomedical applications. Recently, there is increase in

the interest in bioengineering/ biomimetic systems and human system-based simulators.

Presently, bioengineered models are used with increasing frequency to study structural

and functional interactions in the human musculoskeletal system. They provide a way to

create virtual anatomical models accurate properties and functional attributes, and to

express these relationships quantitatively. As a result, mathematical models that are

developed help to demonstrate, and explain the causes-and-effects within a given

physiological or pathological environment. When used in what-if scenarios their ability

to demonstrate relationships among system components during all phases of a simulation

can be valuable both from research and educational perspective.

In this thesis, we will concentrate on musculoskeletal modeling of jaw motions

based on muscle grouping and modeling, rigid body dynamics, their mathematical

modeling and simulation of the whole simulator. Most of currently available

mathematical and biomechanical models of the masticatory system have been useful in

simulating the human chewing process. However, it is hard for them to apply to study the

jaw motions in the context of assessing the “chewability or performance index”,

especially with numerous variations in food properties, dentitions, and physiological

structure of human jaws. Thus, we will seek to arrive at such a “performance index”-

based assessment of mastication process. Further, we will seek to develop a test-bed for

continuous and detailed monitoring of bolus sizes and mandibular motion during the

2

entire mastication cycle. But this is difficult and time consuming in the real world. Not

only these models need to be developed, refined and validated, but additionally must be

designed to automatically adjust the model parameters for any given species. Building

such a generic engineering model that can be used to study the masticatory motion across

different species is a challenging goal. Moreover, these well-engineered models also aid

in creating an ideal platform for designing and prototyping prosthetic additions to the

masticatory apparatus. Hence, the pursuance of this objective suggests the need for the

development of a robotic manipulator by means of which the mastication process can be

reproduced in a mechanically controllable way while the “chewability index” and/or food

dynamics determined quantitatively.

1.1. Background

To aid us in this process we will engage current technological tools and

paradigms in an engineering paradigm of measure-estimate-test cycle (Figure 1-2). The

main idea is to test an accurate virtual prototype before building a physical prototype. It is

based on this phenomenon there is greater emphasis and interest on the need to build an

accurate virtual prototype compared to real physical prototype testing. (See Figure 1-1:

Engineering Paradigm ).

Currently, a number of definitions for VP exist in the literature and in industry. To

have a clear insight about the ideas that will be discussed in the subsequent sections, it is

better to understand the meaning of “Virtual Prototyping (VP)” Wang [1] in the present

context. Here, we refer to VP as a process of simulating the product and their physical

interaction (whether or not involving humans) in virtual environment by means of

software across different stages of product design and conducting virtual quantitative

performance analysis of the product before physical prototyping. Other synonyms, for VP

are— “digital mock-up”, “software prototypes”, ‘software mock-up” and simulation

based design (SBD). Usually VP techniques aim at building products with the notions of

design and product optimization. It can however be used just for concept verification,

presentation, and training. Secondly, design optimization based on virtual prototypes

entails many new and challenging issues. Thus, VP-based design optimization deserves

3

to be an independent topic and such distinction will help to address research issues of

different nature which are not discussed here.

Courtesy: Web-Based Self-Paced Virtual Prototyping Tutorials,

ARMLAB

Figure 1-1: Engineering Paradigm Figure 1-2: Virtual to Physical Prototyping Cycle

The preliminary measurement phases of engineering paradigm as outlined in

Figure 1-1 involve non-invasive measurements to study—geometric parameters of

musculoskeletal systems (segments mass properties, muscle properties etc.), calculation

of 3D motion trajectories (position, velocity and accelerations of the joints or any

superficial points) using high end motion capture (MoCap) systems and physical

parameters of the manipulators and actuators. In particular, to analytically characterize

masticatory efficiency, the measurements must include: frequency, length of chewing,

tracking of jaw movement, force distribution, application of compression and shear

forces on the food, and particle size and structure of the bolus just prior to swallowing.

We seek to develop dynamic musculoskeletal models during the estimation phase

to accurately analyze the joint and structural behaviors and estimate the muscle forces for

different types of food, cycles etc. This also involves implementing kinematic and

dynamic models of different parallel manipulators for the masticatory motion analysis

and simulation.

Measure Estimate Test

4

For the final testing phase, models of the parallel manipulators are used to draw macro

level comparisons between the both— musculoskeletal systems as well as robotic

manipulators. These models are validated using forward and inverse dynamics tools

before implementing these models for our application. We analyze and study the

behavior of the systems based on standard “what-if’ type tests in a virtual environment

prior to the physical prototyping phase.

1.2. Problem Statement

The goal of this thesis is to analyze the masticatory jaw motions of animals

(including humans) and establish the quantitative relationship between relevant

mechanical parameters, for example joint motions, forces, pressures, work envelope and

muscle or actuator parameters, for example maximum muscle forces, optimal activity

regions for muscles during the mastication process. Such an understanding would be

useful from different perspectives. From a biological science perspective, it would be

useful to know how various animals process food by chewing and biting. From an

economic perspective, such a study on masticatory performance would:

I. enable designing animal foods based upon the “chewability index”

II. enable design of dental orthosis based on the knowledge of mechanical signals-

motions, forces, pressures etc.

1.2.1. Research Goals

We propose as a part of this thesis to develop a mathematical model of a jaw

simulator for dogs that has the capability to mimic the trajectory of the actual jaw motion.

For this purpose, we intend to perform a biomechanical analysis on a musculoskeletal jaw

model to determine muscle behavior, dynamic parameters, joint motions, and the work

envelope that the jaw can cover by its incisor tooth tip point (ITP). We intend to

implement human and dog jaw case studies to study the behavior in detail, determine

muscle parameters (actuators in musculoskeletal systems), evaluate motion of the ITP

(end-effector point) and biting forces that could be achieved for different types of jaw

motions.

5

To develop a jaw motion simulator, we examine various design variants of

parallel manipulators, keeping in mind the high force requirements for simulating jaw

biting actions. We plan to develop a trajectory tracking inverse kinematics model for the

application. To ensure the manipulator possess the desired workspace a detailed

workspace analysis based on the Jacobian measures is imperative. We will then drive the

kinematic model with the jaw motion trajectories obtained from the Motion Capture

(MoCap) Analysis System. A detailed kinematic study between two parallel manipulators

(6 DOF R-U-S: revolute-universal-spherical, and P-U-S: prismatic-universal-spherical

joint with active revolute and prismatic joints respectively) will be presented later in this

thesis. Finally, we develop parametric analysis for building the prototype and provide

insight on developing dynamic models for 6 DOF P-U-S parallel manipulators.

Figure 1-3: Project Flowchart To tie all these disparate aspects together, we will consider specific case studies of

dog and human jaw motion in this thesis (refer Figure 1-3). We will primarily focus on

the following major research challenges in this regard:

6

a. Virtual prototyping using CT scan to CAD Model to Dentition

b. Motion capture of jaw motions of humans and animals

c. Musculoskeletal and muscle modeling

d. Mathematical modeling of parallel manipulators- forward and inverse

kinematics

1.3. Research Issues

To run the biomechanical analyzes, we develop a simplified virtual model with

minimum number of muscle groups (favorably six muscle groups- three on each side) to

achieve the necessary jaw motion and work envelope. Next, we analyze the masticatory

biomechanics of different animals and create a dynamic musculoskeletal model to

estimate the muscle activity levels and biting forces for the case of labrador breed of dog.

To accurately calculate the 3D jaw motion trajectories which can drive these virtual

models, and later the mechanical jaw simulator, we record the jaw motions of different

subjects using high-speed video cameras— a SimiMotion- a semi portable 3D motion

capture system.

Based on the conclusions ascertained by this study, we propose a jaw simulator

based on the spatial parallel-actuated mechanism and implemented the inverse kinematics

for accurate trajectory tracking. For subsequent study and analyzes, we narrow our

choices into two parallel-actuated mechanisms and conduct a detailed kinematics analysis

of these two manipulators— 6-DOF R-U-S manipulator and P-U-S manipulators for our

application. We validate the generated mathematical models against standard case studies

and past literature for desired performance.

To this end, we discuss the screw-theoretic modeling and other methodologies that

provide a convenient way of modeling these platforms at kinematics and dynamics level.

The relevant details encompassed with the modeling framework will be addressed in the

latter portions of this work. For the above mentioned application, we develop a detailed

kinematics and workspace analysis on the specific class of manipulators.

7

1.4. Thesis Organization

The rest of the thesis is organized as follows: Section 2 discusses about the past

and present research efforts relevant to developing a jaw simulator, and virtual

prototyping and testing, and biomechanical jaw models. A brief overview of different

scientific and computational tools as well as other equipment that were considered in the

process of designing the simulator is also discussed in Section 2. Section 3 gives a

detailed background on different musculoskeletal modeling aspects and masticatory

biomechanics. The mathematical modeling and kinematic derivations and basic concepts

on measure of manipulability and other mathematical derivations are presented in Section

4. Section 5 covers the equipment setup, dynamic model creation, and experiments

conducted, and Section 6 provides simulation results of our biomechanical and

manipulator models and discusses the results of kinematic and dynamic analyzes.

8

2. Literature Survey

2.1. Related Work

2.1.1. Biomechanical Model of the Jaw

Since the early 1980s many researchers in the field of biomechanics have worked

in the areas of redundant musculoskeletal systems, muscle modeling, and mathematical

modeling for dynamic simulation. Realizing the significance of developing an accurate

kinematic model, numerous researchers have attempted to model the jaw kinematics (i.e.)

describing the geometrical and analytical connections between rotation of the joints and

the actual motion of all parts of the jaw. Some of these results formed the basis for the

dynamic simulation of the masticatory simulation model, which is an active research

topic today.

The biomechanical model of the jaw developed by Koolstra and Eijden [2] can be

regarded as the first reliable and realistic model built. The model is unique in the way that

each muscle is able to influence all six degrees of freedom, which makes the system

kinematically, and mechanically indeterminate. This work further provides insight into

procedure to determine muscle forces affecting each degree of freedom.

Another notable effort from University of Karlsruhe, Germany by Weingartner,

Hassfeldet al. [3] where they developed an accurate kinematic model of the complex

temporomandibular joints (TMJs). Such realistic models of TMJ are necessary to

simulate the jaw motion perfectly (Figure 2-1, Figure 2-2, Figure 2-3 and Figure 2-4). For

dynamic simulation of muscles, they modeled each muscle as threads, signifying the

major force vector of the muscles and implemented an open loop robotic model for

different mandibular motions— opening/ closing, protrusions and lateral movements.

9

Figure 2-1: Hinge Joint Axis Of TMJ For Left Side Movement

Figure 2-2: Modeling Of Muscles As Threads

Muscle groups, 1-anterior digastric; 2-superficial

masseter; 3-medial pterygoid; 4-deep masseter; 5-

lateral pterygoid; 6-posterior temporalis; 7-middle

temporalis; 8-anterior temporalis. 9-gravity

Figure 2-3: Anteriolateral View Of The Basic Model

Figure 2-4: Jaw Model With TMJ And Incisor Point Envelope

Peck, Langenbachet al. [4] developed jaw models to determine the external force

required to reach maximum gape in five relaxed participants, and used this information,

with other musculoskeletal data, to construct a dynamic, muscle-driven, three-

dimensional mathematical model of the craniomandibular system in ADAMS. It allowed

six degrees-of-freedom shaped by forces from 16 craniomandibular muscle groups, two

TMJs, and gravity.

10

Figure 2-5: Oblique And Side View Of Jaw And Larynx Model In “Artisynth”

This model also had a more realistic way of modeling TMJs and the contact of

condyles with functional fossae. A follow up of this work was published by Enciso,

Memonet al. [5] wherein they detail about the forward dynamic simulation of the jaw

model in the context of swallowing using an open source platform “ArtiSynth” for

running the biomechanical simulation (Figure 2-5).

2.1.2. Masticatory Robotic Manipulators

The advent of the latest technologies and advanced computational tools marked a

new era in human masticatory robotic simulation. Many research groups have

concentrated on accurately simulating jaw motion by building a physical robotic

manipulator and conducting real time studies on biting and food texture properties.

However, the first ever device built for tracking jaw motion was in the mid 1950s by

Posselt [6]. The apparatus (known as the Gnatho-thesiometer) permitted measurements

(at three points) in the three main planes on a freely movable cast of a lower jaw. He also

presented a simple comparative study on error obtained between mounting various casts

manufactured by different techniques.

The research group at Waseda-University [7], Japan led by Atsuo Takanishi

actually marked the beginning of this new era. The group has developed a “Waseda Jaw’

series of masticatory robots named as WJ-0, WJ-1, WJ-2 and WJ-3 as in Figure 2-7, Figure

2-8 and Figure 2-9, whose mechanical structures resemble those of the human masticatory

11

system, especially muscle positions. They placed sensors in muscles and under the teeth and

in the mandible to measure forces and movements. The same group also succeeded in

extending their application of WJ series of robots to treating TMJ disorders with their new

WY series of robots (Figure 2-6).

Even though their robots are predominantly 3-DOF, it is mentioned in Takanobu,

Kuchikiet al. [8] one of their works that the human masticatory system is similar in

configuration to a parallel mechanical manipulator driven by linear or rotary actuators, and so

resembles closely a 6-DOF Stewart Platform. We would like to make use of this

interpretation in our thesis to model a jaw simulator for a generic case.

The research group named BioMouth led by Dr. Xu from Massey University,

New Zealand is actively involved in jaw modeling and dynamic analysis at present. They

Figure 2-6: WY-5RII Mastication Robot Figure 2-7: Commercial Mastication Robot For Use (WOJ1)

Figure 2-8: Food Texture Measurement Robot WWT-1

Figure 2-9: Jaw Robot 3D Simulation WOJ-1RII

12

implemented a jaw model prototype Figure 2-10 based on parallel manipulator analogy.

Xu, Bronlundet al. [9]

They studied muscle parameters and their functioning and reduced the complex

jaw model shown in Figure 2-11 to a simple model with 6-muscle groups. This simplified

model was then mapped into a 6-DOF dental simulator similar to a Stewart Platform. In

this work, they tried to represent the major muscle groups as double acting linear

actuators (Figure 2-12 and Figure 2-13) and hence just included the jaw closing muscles

in their model which can pull and push the jaw. An interesting aspect of this work is that

this group has used currently available mechanical simulation package, namely

SolidWorks/ Cosmos Motion, for dynamic analyzes of their simulators and compared the

performance of the manipulator with that of the musculoskeletal model (Figure 2-10 and

Figure 2-11). The same group implemented a jaw simulator using the existing Stewart

Platform model built in SimMechanics, and developed a dynamic simulation setup for the

dental simulator, and conducted studies on jaw kinematics, actuator forces and food

textures.

Figure 2-10: Masticatory Robot Covered By The Skull In SolidWorks

Figure 2-11: 3-D Model Of The Mandible- The Actuators’ Attaching Points (Mi), And The

Reference Points

13

Figure 2-12: Robotic Model In The Form Of Platform Mechanism Nomenclature And Coordinate Systems

Figure 2-13: 3D Kinematic Jaw Model

They have also come up with a life-sized masticatory robot (Figure 2-15), which

is intended to “chew” foods in a human way while the food properties are evaluated using

a 6RSS parallel mechanism. It is based on a robotic mechanism whose kinematic

parameters (Figure 2-14) are defined according to the biomechanical findings and

measurements of the human masticatory system by Xu, Torrenceet al. [10]. For a given

mandibular trajectory to be tracked, the closed-form solution to inverse kinematics of the

robot is found for joint actuations, whereas differential kinematics is derived in Jacobian

matrices. Experimental results for free chewing, soft-food chewing, and hard-food

chewing are given where the foods are simulated by foam and hard objects. They also

studied crank actuations and driving torques (an indication of muscular activities) and

compared them for chewing different foods.

Relevant to this discussion is another robotic device that was built by Lewis— a

Master’s student Xu, Lewiset al. [11] from Massey University based on a four bar

linkage mechanism. It has adjustable link lengths to simulate chewing trajectories only in

the frontal plane of different subjects. The work also involved building a physical

prototype of the simulator and the enclosure and a comparison study with the virtual

prototype.

14

Figure 2-14: Robotic Model Of Crank Actuation Figure 2-15: Physical Prototype Of The Dental Simulator

Most of the research on jaw modeling and simulation is concerned with human jaws and

biting force analysis based on the food texture. Only a few of those focus on other

mammals. Hence, in this thesis we would like to address this topic and arrive at a way to

model and build a generic dental simulator.

Figure 2-16: Four Bar Mechanism Simulator (MS Thesis, Darren Lewis, University Of Massey)

15

2.1.3. Parallel Actuated Robotic Manipulators

Based on the biomechanical studies just describe, we could restrict our choices for

building a dental simulator to a set of parallel-actuated platform manipulators. Parallel

manipulators offer many advantages as they can withstand high-end effector forces as

well as satisfy trajectory space requirements. They also possess high system stiffness and

high load/weight ratio, which makes it possible to describe the given trajectories

precisely, even under heavily alternating loads and large accelerations. However, the

major limitation of these types of manipulators is the limited workspace compared with

those of serial manipulators.

Figure 2-17: Schematic Diagram Of Stewart Platform

Figure 2-18: Force Analysis On Leg

There have been considerable developments in this field especially related to

developing inverse dynamic models of Stewart platform since its birth in the mid 1950s.

Notable among these are inverse dynamics equations obtained using virtual work

principle by Tsai [12] (Figure 2-17 and Figure 2-18) as well as Lagrangian modeling by

Guo and Li [13]. The virtual work as well as Lagrangian modeling procedures can be

extended to most variants of Stewart platforms. Tsai’s details about the standard

methodology for deriving dynamical equations of motion for parallel architectural

manipulators at the joint space Figure 2-17 and Figure 2-18. In the other work, a closed-

form explicit dynamic equation in task space using Lagrangian method is derived. The

work also provides benchmark results that can be used for validation of similar models.

Both this work considered the leg mass and inertial effects in the dynamic formulation.

16

For symbolic computation based on Tsai’s work, interested readers are referred to Wang

[14].

Parallel manipulators also offer advantages from a manufacturing point of view as

similar parts are used to fabricate the overall system resulting in cost reduction. However,

most of the commercially available motion simulators are customized devices, which

ultimately make these costlier. These simulators are commonly used in motion generator

for flight simulators, surgical devices, high-speed machine heads and for augmented VR

applications. A brief survey of commercial simulators and their features are displayed in

the Table 2-1.

2.1.4. Jaw Motion Analysis, Imaging and Experiments

Several implementations of the jaw motion capture and design of fixtures for

accurate motion capture has already been implemented in the context of medical

evaluation of injuries, fractures, tumors etc. In Enciso, Memonet al. [5], the author

describes the process of CT scan and segmentation of the human jaw for 3D model

reconstruction. The process of motion capture using an ultrasonic capture device to

construct patient specific models is also discussed to evaluating the medical condition for

specific patients based on the model reconstruction techniques. Weingartner a dental

scientist from Switzerland explained in Gallo [15] the wrench axis representation for the

jaw motion about which the rotation as well as translation takes plane. He concluded that

the joint axis seldom passes through the centers of the condyles even in the case of free

opening and closing motion. The work also details about the dynamic stereometry of the

TMJ for 3D reconstruction and animation of the joint using real time tracking of jaw

motion.

Dentitions for dental simulators can be built using the virtual prototypes of human

jaws. These virtual prototypes are obtained using high-end 3D digital laser scanners that

convert the point cloud data to a 3D CAD model (easily accessible in most CAD

platforms). This 3D CAD model is used in rapid prototyping machines to obtain the

dentition in physical form. To create dentitions of different materials, casting of the wax

model can be undertaken.

17

Company Name and Features Applications

Moog [16] Motion Base Non customizable

Payload: 1000-14500 kgs MATLAB RT control High fidelity and scalability

Servos [17] Simulation Electric actuators only, portable system Computer control

Weight: 500 lbs. Miniature type available Cost: 25k- 75k USD Workspace: 0.38” (X-Y-Z)

Sarnicola [18] Simulation Systems Electric and hydraulic Computer control(no MATLAB)

Weight: 500 lbs Workspace: 0.3m (X-Y-Z) Velocity: 0.45 m/sec (X-Y-Z) Payload: 2000 lbs

Mitsubishi Motion Base Electric and hydraulic actuators

Driver training as VR applications Workspace: 0.3m (X-Y-Z)

In Motion Simulation:http://inmotionsimulation.com/ [19] Electro pneumatic actuator

Workspace: 0.38m (X-Y-Z) Payload: 3000 lbs Cost: 75k USD

Aeronumerics, Inc.: http://www.aeronumerics.com/ [20] 6 DOF Customizable system Hydraulic and Electromechanical actuators

Car and flight simulators Cost: 19k USD

Alio Industries Hexapods: http://www.alioindustries.com/stages_hexapods.html [21] C++/ Labview Compatible Open architecture in Windows and intuitive GUI

Medical and micromachining Workspace: 0.025-0.096 m Velocity: 0.2 m/s

Table 2-1: Brief Survey on Commercially Available 6 DOF Parallel Architecture Simulator

18

Figure 2-19: 2D CT scans and 3D STL Model Of The Human Jaw

The research team in University of Illinois working on Mandible-Reconstruction-Project

[22] has done CT scan to CAD model conversion for different mandibles (Figure 2-19) as

a part of a bone implant project, and in fact, developed a virtual database of mandibles.

2.2. Computational Tools

2.2.1. Musculoskeletal Analysis Tools:

AnyBody

This musculoskeletal modeling software (AnyBody Modeling Software Manual

[23]) was developed as part of the “The AnyBody Project” at Aalborg University in

Denmark. The software was developed to parametrically analyze detailed

musculoskeletal systems of both humans and animals. Figure 2-20 shows a picture of a

musculoskeletal simulation performed within AnyBody. AnyBody is a script based

analysis program so that writing code is necessary to develop the musculoskeletal system

models. It uses its own scripting language, AnyScript, which is an object-oriented

language similar to C++ or Java Script. This software also has the capability to perform

analyses on complex musculoskeletal models. This complexity encompasses model

geometry, number of muscles, dynamic changes in muscle position, and degrees of

freedom of the system. Within the program musculoskeletal and physiological properties

such as muscle forces, joint reactions, metabolism, mechanical work, and efficiency can

19

be examined for a given system. Several musculoskeletal system simulations have been

performed using this software and example simulations can be seen on the AnyBody

website (anybody.auc.dk). More specifically, in conjunction with this research effort, the

capabilities and application of this software toward modeling and simulating the

skull/mandible musculoskeletal system of an extinct saber-toothed cat (Smilodon-Fatalis)

was explored in depth by Konakanchi [24].

Inverse dynamics analysis (IDA) can be thought as the main process of

calculating muscles forces within AnyBody Modeling Software system. Given the

motion of the musculoskeletal model, muscle and reaction forces are calculated by setting

up the equations of motion. Problems pertaining to static indeterminacy, limitations on

the maximum muscle forces need to be resolved before the analysis. Because there are

more muscles than degrees of freedom (see Figure 2-21), redundancy posed by the

system indicates that there is no unique solution to the inverse dynamics problem.

Figure 2-20: AnyBody Software Interface

While performing a body motion the muscles collaborate according to some

rational criteria. These criteria when combined with the fact that muscles can only pull

and not push results in a unique recruitment pattern. This suggests that the central

nervous system applies some sort of “optimality” criteria to determine muscle activation

-0.1-0.08

-0.06-0.04

-0.020

0.12

0.14

0.16

0.18-28.9

-28.8

-28.7

-28.6

-28.5

-28.4

-28.3

-28.2

Pterygoid 'X'

Surface plot of Bite force

Pterygoid 'Y'

Bite

forc

e

20

order. If such an “optimal criteria” is combined with the equilibrium equations, we can

have unique solutions for a problem. The basic optimality assumption is that “the body

attempts to use its muscles in such a way that fatigue is postponed as far as possible”.

Hence, in our optimization problem we would minimize the maximum muscle activity

subject to equilibrium constraints and positive muscle force constraint. Hence, the

optimization problem for calculating the muscle forces Rasmussen, Damsgaardet al. [25]

can be mathematically written as:

,

1

,

:

:

0 : 1,...,

pnM i

i i

M i

FMinimize V

N

Subject toCf dF i n

=

⎛ ⎞= ⎜ ⎟

⎝ ⎠

=≥ =

∑ (2.2.1)

Various forms of this optimization problem may be created raising the power of the

individual muscle activity to a polynomial power, ‘p’. With increasing value of ‘p’, the

criteria tend to distribute the relative load evenly between the muscles.

The min/max-objective function is non-differentiable and therefore appears to complicate

the practical solution of the optimization problem. However, by using bound formulation,

which is widely used and well tested, in the field of optimum engineering design we can

easily solve the min/max problems. By introducing a new artificial variable β and an

artificial criterion function ( )B β and the new criterion can be a monotonic function ofβ .

By choosing the ( )B β = β we can reformulate the above problem as:

Figure 2-21: Inverse Dynamics of a Redundant Musculoskeletal System (Konakanchi [24])

21

,

,

: ,

:

0

; 1,...

M i

M i

i

Minimize

Subject Cf d

FF

i nN

β

β

=

≥

≤ ∈

(2.2.2)

In the min/max optimization problem we are looking at the muscle recruitment that

balances the exterior loads and minimizes the largest relative load on any muscle in the

system, thereby postponing fatigue of the muscle as far as possible. This is the approach

used within the AnyBody software.

SIMM

SIMM or Software for Interactive Musculoskeletal Modeling is a biomechanics

software toolkit developed by MusculoGraphics Inc. Within this software, the user has

the ability to construct, model, animate, and analyze a musculoskeletal system in a three-

dimensional environment. This software differs from the traditional articulated

mechanical system analysis packages in that is was designed specifically to simulate

systems consisting of bones, muscles, ligaments, and tendons. Figure 2-22 below depicts

an example simulation performed using SIMM (Musculographics-Inc. [26]).

Figure 2-22: SIMM Virtual Model

Within a SIMM model, each musculoskeletal system consists of representations

of bones, muscles, ligaments etc. SIMM enables the analysis of a musculoskeletal system

by calculating the joint moments that each muscle can produce at any given body

22

position. The resultant motion and muscle force properties can then be analyzed and

visualized with the SIMM environment. The exploration of the capabilities of the SIMM

musculoskeletal analysis software has been examined by many authors both in the

context of musculoskeletal system analysis(Yamaguchi, Moranet al. [27]), and its

adaptation and expansion into the development other musculoskeletal system analysis

tools (Konakanchi [24, Davoodi, Brownet al. [28]).

2.2.2. Dynamics Simulation/ Analysis Tools: MAPLE/ DynaFlexPro with MATLAB/ Simulink

DynaFlexPro (DFP) is a Maple package (Bhatt and Krovi [29]) for modeling and

simulating the dynamics of mechanical multibody systems. DFP uses a graph theoretic-

modeling approach to create kinematic and dynamic EOMs within a systematic and

automated symbolic implementation. While a symbolic or numeric study of the system is

possible using Maple’s built-in ODE solver dsolve, DFP also offers the capability to

export the EOMs to other platforms (C, Fortran, and Matlab) using code-generation tools.

Figure 2-23 illustrates the basic functionality of DFP for the double pendulum example.

SolidWorks-SimMechanics- VRML

SimMechanics software (Mathworks [30]) is a block diagram-modeling

environment for the engineering design and simulation of rigid body machines and their

motions, using the standard Newtonian dynamics of forces and torques. With

SimMechanics software, one can model and simulate mechanical systems with a suite of

tools to specify bodies and their mass properties, their possible motions, kinematic

Figure 2-23: Double Pendulum Model implementation in DFP

23

constraints, and coordinate systems, and to initiate and measure body motions. In

SimMechanics, a connected block diagram represents a mechanical system as with any

Simulink models, and complex mechanical systems can be modeled as hierarchical

subsystems.

The visualization tools of SimMechanics software display simplified renderings

of 3-D machines, before and during simulation, using the MATLAB Graphics system and

with some additional effort the system can be even visualized using VRML engine- in-

built in MATLAB.

Figure 2-24: SolidWorks- SimMechanics- VRML Framework

Additionally, any 3D CAD models created in SolidWorks can directly be

imported into VRML model files by saving the model assemblies as *.xml format. This

file is subsequently imported into VRML environment for visualization and animation.

To generate SimMechanics models from SolidWorks, “import_physmod” model builder

in SimMechanics, SolidWorks assemblies can be used to seamlessly convert into

Simulink block diagrams with the exact geometric relationship between different

components (Figure 2-24). In SimMechanics, one can implement task space or joint

space control by adding Sensor and Actuator blocks to joints or to the end effector

directly and drive the joints using the appropriate control algorithms. Following is a

figure that shows the different steps involved from building an assembly in SolidWorks

24

to running simulation in SimMechanics environment using VRML engine for animation/

visualization.

SolidWorks- Visual Nastran

MSC.Software Visual Nastran vN4D MSC.visualNastran 4D Manual:

www.mae.virginia.edu/meclab/images/visualNastran4D.pdf [31] simulates 3D motion

with dynamic finite element analysis FEA on Windows and is intended mainly for

engineering education and professionals. MSC.visualNastran4D (vN4d) merges the

technologies from motion, animation, and FEA simulation into a single functional

modeling system. It allows one to simulate his/her mechanical designs dynamically, to

determine if the products will function as expected. vN4d also has the capability to

simulate a mechanical system running in an open loop and monitor the output positions,

velocities, actuator forces/ torques etc. A block representing the vN4d mechanical model

can be inserted into Simulink to expand beyond mechanical simulation to system-level

simulation. Using this block in Simulink one can represent the mechanical system in

vN4D, that allows one to simulate an entire system, including hydraulics, electronics, and

controls. Mechanical parameters in the vN4d mechanical model, such as velocity,

position, or torque, can be linked between vN4d and MATLAB or Simulink for control

system design or processing to allow closed loop simulation functionality (Figure 2-25

and Figure 2-26).

Figure 2-25: SolidWorks Model of Parallel Manipulator

Figure 2-26: Open Loop Kinematics and Dynamic Analysis in Visual Nastran

25

2.2.3. CT Scanning and Imaging

Mimics

Mimics is the standard imaging software that is used for image processing and

converting scanner data (CT scans/ MRI data) into 3D CAD models. Based on the

segmentation mask, Mimics automatically generates the contours (polylines) of the mask.

The MedCADMaterialize [32] Module allows the user to grow polylines into a subset of

polylines based on a specified tolerance. The subset of polylines can then be used to fit

different CAD objects to the anatomical geometry. One can either build CAD objects

interactively or create the objects parametrically by specifying the different parameters

for the object (e.g. the location of the center point and the radius of a circle). The objects

can also be adjusted interactively with the mouse after they are created. All of these

entities can be exported as IGES files. The files are directly usable for the design of

custom-made prostheses in any CAD system. Another typical application is the use of the

MedCAD Module for statistical analyses (e.g. it is possible to do measurements on a

number of different femurs and use these measurements for the design of a set of

standardized implants). STL module (Figure 2-27) also allows one to create STL files

from the scanner data which can then be transferred to any rapid prototyping machine for

creating the parts slice by slice.

Figure 2-27: CT Scan Slices to STL Conversion Figure 2-28: Working with STL in Rhinoceros

It is possible to export from a mask, 3D object or 3dd file. The available export formats

are ASCII STL, Binary STL, DXF, VRML 2.0 and Point Cloud. Several calculation

26

parameters can be specified for building an STL file that gives the possibility to reduce

the triangles of the exported files, to interpolate the images and to do smoothing on the

3D files.

Rhinoceros

Rhinoceros [33], Rhino for short is mainly a NURB surfaces modeling tool used

to create, edit, analyze, document, render, animate, and translate NURBS curves,

surfaces, and solids with no limits on complexity, degree, or size. It also supports

polygon meshes and point clouds. It allows uninhibited free-form 3D modeling as well as

repairs any sort of complex surfaces from IGES or mesh files. The main advantage of

using Rhino is it is compatible with most of the 3D scanners currently available in the

market today and can be obtained at a low cost. Hence, their applications are mainly

found in reverse engineering of complex designs. (See Figure 2-28)

2.3. Motion Capture Technology

2.3.1. Motion Capture and analysis Systems:

In the 1970s, motion tracking or motion capture (WikiPedia [34]) started as a

photogrametric analysis tool in biomechanics research and since then the research

community has seen tremendous advances in this field. These tools are not only used by

biomechanical community for research, product design or ergonomics study purposes but

also by graphical programmers prevailing in animation-movie, sports and video gaming

industries. Present automated motion capture systems record the positions, angles,

velocities, accelerations and impulses that can provide an accurate digital representation

of the motion in real time. Biomechanists can use this real time data to diagnose problems

or suggest ways to improve performance, requiring motion capture technology to record

motions up to 140 miles per hour for a golf swing. These motion capture systems are

broadly classified into optical and non-optical (video- markerless motion capture, electro-

magnetic and mechanical) types and are discussed in detail below.

27

2.3.2. Optical Markers:

Optical systems triangulate the 3D position of a marker between one or more

cameras calibrated to provide overlapping projections. Tracking a large number of

markers or multiple performers or expanding the capture area is accomplished by the

addition of more cameras. These systems produce data with 3 degrees of freedom for

each marker, and rotational information must be inferred from the relative orientation of

three or more markers; for instance shoulder, elbow and wrist markers providing the

angle of the elbow. The performer is free to move unencumbered by cables or harnesses

typically associated with magnetic trackers.

a) Passive markers:

Passive optical system use markers coated with a Retro-reflective material to reflect light

back that is generated near the cameras lens. Cameras sensitivity can be adjusted taking

advantage of most cameras’ narrow range of sensitivity to light, so only the bright

markers will be sampled ignoring skin and fabric. This type of system can capture large

numbers of markers at frame rates as high as 2000fps. SimiMotion system is an example

for this type and will be discussed in detail later in this section. Other systems of this type

are Vicon-Peak-Motus [35, OptiTrack [36].

OptiTrack (see Figure 2-29) is a complete motion capture and analysis studio with

limited features and low cost. The refreshing rate of the camera is about 100 fps and is a

gray scale camera. With up to four cameras per USB hub, one can cover a large range

area by using multiple cameras / hubs. Vicon Motus/ Vicon MX (Figure 2-30) is one of

the fully equipped motion capture systems with broader features for different applications

in the industry and research. It has a high-speed video cameras operating up to 10,000

frames per second with a resolution of 2-4 million pixels, which can track intricate

movements in 2-D and 3-D motion analysis.

28

Figure 2-29: OptiTrack Motion Capture Figure 2-30: Peak Motus MoCap Equipment

b) Active markers:

Active optical systems triangulate positions by illuminating one LED at a time very

quickly or multiple LEDs, but sophisticated software is required to identify them by their

relative positions, somewhat akin to celestial navigation. The markers themselves are

powered to emit their own light. Active markers can further be refined as Semi

perceptible and time modulated. Phoenix technologies and PhaseSpace Inc. are the major

players in this category and normally provide active markers of both these types.

PhaseSpace [37]

Time Modulated Active Marker (Figure 2-31): The motion capture is handled by strobing

one marker on at a time, or tracking multiple markers over time and modulating the

amplitude or pulse width to provide marker ID. The unique ID of the markers reduces the

turnaround, by eliminating marker swapping and providing much cleaner data than other

technologies. These motion capture systems are typically under $50,000 for an eight

camera, 12-mega pixel spatial resolution 480-hertz system with one actor.

Semi Perceptible Active Marker (Figure 2-32): These systems use inexpensive multi-

LED high-speed projectors that optically encode the space. The system uses

photosensitive marker tags to decode the optical signals. By attaching tags with photo

sensors to scene points, the tags can compute not only their own locations of each point,

but also their own orientation, incident illumination, and reflectance.

29

Figure 2-31:Time Modulated Active Marker Figure 2-32: Semi Perceptible Marker

These tracking tags that work in natural lighting conditions can be imperceptibly

embedded in attire or other objects. The system supports an unlimited number of tags in a

scene, with each tag uniquely identified to eliminate marker reacquisition issues. The tags

also provide incident illumination data, which can be used to match scene lighting when

inserting synthetic elements. The technique is therefore ideal for on-set motion capture or

real-time broadcasting of virtual sets

2.3.3. Non-optical markers:

a) Mechanical

These trackers directly track body joint angles and are often referred to exo-skeleton

motion capture systems (in Figure 2-34) due to the way the sensors are attached to the

body. Most of them operate in real-time, are relatively low-cost, free-of-occlusion, and

use wireless (untethered) systems that have unlimited capture volume. Typically, they are

rigid structures of jointed, straight metal or plastic rods linked together with

potentiometers that articulate at the joints of the body. Some examples are GolfMotion by

METAMotion [38], MiniRobot XBot by Nuzoo [39].

b) Magnetic

Magnetic systems calculate position and orientation by the relative magnetic flux of

three orthogonal coils on both the transmitter and each receiver. The relative intensity of

the voltage or current of the three coils allows these systems to calculate both range and

orientation by meticulously mapping the tracking volume as an e-Motek [40] MoCap

system Figure 2-33.

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Figure 2-33: Electromagnetic Motion Capture System

Figure 2-34: Exoskeleton Mocap System

c) Inertial

Based on miniature inertial sensors, biomechanical models and sensor fusion algorithms

the data from inertial sensors can be transmitted to PC across a wireless network for

recording of motion/ position. MoCap systems Moven-Inc [41] are of inertial types.

The inertial motion trackers give absolute orientation estimates, which can also be used to

calculate the 3D linear accelerations in world coordinates, which in turn give translation

estimates of the body segments (Figure 2-35 and Figure 2-36).

Figure 2-35: Inertial Motion Capture Systems Figure 2-36: Inertial Motion Capture Systems at NASA (Miller, Jenkinset al. [42])

2.3.4. Markerless tracking devices:

The necessity for capturing animal motions without use of trackers led to a wide

range of software and camera systems that allows one to capture and analyze motion

using markerless tracking softwares. These systems are suitable for biomechanical

experiments to be performed on animals. However, the accuracy of the markerless

tracking devices depend on the accuracy of selecting the points on each frame of the

video, efficiency of the automatic tracking algorithm implemented on the system and

31

various other factors. As a general case, based on the current trends, it is better to opt for

motion capture systems using makers rather than markerless devices for accuracy and

usefulness. Among the very few companies, Noraxon-Inc,

http://www.noraxon.com/index.php3 [43] (commercially called- Functional Assessment

of Biomechanics- FAB, see Figure 2-37) and Organic Motion [44] (see Figure 2-38:

Markerless Motion Capture Systems by Noraxon) are the commercially successful

systems.

Figure 2-37: Markerless Motion Capture Systems

by Organic Motion Figure 2-38: Markerless Motion Capture Systems

by Noraxon

2.3.5. Motion Capture beyond Markers - 3D Scanners

Usually 3D scanners of this kind create point cloud data of geometric samples on

the surface of the subject. These points can then be used to extrapolate the shape of the

subject (a process called reconstruction). If color information is collected at each point,

then the colors on the surface of the subject can also be determined. In short, 3D

scanners digitize real objects, which can then be used for any type of virtual analysis/

visual applications.

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Figure 2-39: Facial Motion Capture at CMU Robotics Institute

Figure 2-40: Facial Motion capture at IBM

Facial motion capture is the process of electronically converting the movements

of a person's face into a digital database, and cameras or laser 3D scanners are used for

this purpose. Active LED Marker technology is currently being used to drive facial

animation in real-time to provide user feedback. Markerless technologies use the features

of the face such as nostrils, the corners of the lips and eyes, and wrinkles and then track

them. This technology is discussed and demonstrated at The Robotics Institute, CMU

[45], IBM [46], from University of Manchester (where facial capture actually started)

using active appearance models, principle component analysis, eigen-tracking and other

techniques to track the desired facial features from frame to frame. This technology is

much less cumbersome, and allows greater expression for the actor.

Vision based approaches also have the ability to track pupil movement, eyelids,

tooth occlusion on the lips and tongue, which are obvious problems in most computer

animated features. Typical limitations of vision based approaches are resolution and

frame rate, both of which are decreasing as issues as high speed, high resolution CMOS

cameras become available from multiple sources. Structured-light scanners like PONTOS

from G-O-M [47] (Gesellschaft für Optische Messtechnik) based in Germany are used to

capture precise position, motion and deformation calculation of structures and

components. These are normally used for in a system for optical, dynamic 3D analysis

like car crash tests, vibrations in the structures, component tests, etc. For more detailed

survey of digital 3D scanners readers are referred to Kannan [48].

Figure 2-41: PONTOS Scanners for Crash testing Figure 2-42: PONTOS for Car Body testing

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2.3.6. Issues with MoCap Systems

In spite of their advantages, motion capture systems require considerable effort in

post processing, because the capture process is commonly associated with four basic

error-types (Figure 2-43)— discrete channel error, interchange of channels, erroneous

trajectory and precision mistakes. The correction of these capture errors is automatically

completed by the use of cubic splines or another type of interpolation curve, like cardinal

splines. Nevertheless, the detection of this variety of errors is frequently made with

human intervention, so that it develops into the essentially speed problem of this type of

systems as described by Gonzalez-Morcillo, Jimenez-Linareset al. [49].

The main disadvantage of magnetic motion capture systems is the “fails” and

“distortions” when the operational environment has external noise or disturbance of other

magnetic fields. On the other hand, this kind of systems operates better than other

systems in real time and interactive environments due to their good behavior at

identifying the marks. Optical motion capture systems do not present any limitation about

the number of markers, and they are usually applied in a complex movement capture. As

we can see in Figure 1, there are four kinds of usual errors in optical systems that we

must repair before making any kind of biomechanical analysis with the data.

Data Transmission Losses

One of the most important issues in real-time video capturing is that the MoCap

workstation should be capable of storing the data onto the harddrive at very high rates. In

full size and full speed (NTSC mode) the amount of data even at the slowest frame rate of

30 frames per second (fps) that has to be streamed onto the harddrive is approximately

27Mb (24bit color * 640 * 480 * 30 frames/sec). Hence, the drive should be capable of

handling large amounts of data even at 30 fps. It is common to record videos at 100 fps

even for the simplest motion capture analysis. Hence, harddrive capacities of about 1 TB

and speed of about 10000 rpm should be used. To avoid data transmission or frame

capture losses at such high rates, the firewire ports are commonly used, which have the

capability to synchronize the data received at 100 fps from all the cameras at every

instant. The workstation must have data buses to support one firewire port for each

camera for such applications.

34

Figure 2-43: Errors in Motion Capture Process Gonzalez-Morcillo, Jimenez-Linareset al. [49]

Marker swapping and Occlusion

The motion capture data obtained with this class of systems presents a group of problems

represented in Figure 2-43. Each graph in Figure 2-43 represents the time in X-axis and

the position of the marker in the space along Y-axis (we use, of course, one graph per

trajectory and axis). One type of problem is the occlusion of the mark through the

subject's body. If the subject hides some markers in a zone where at least two cameras are

not grabbing the marker (we need two simultaneous cameras to achieve stereoscopic

vision), the system will produce discrete errors called Lost Marks.

By way of a set of 2D captured images, the system is not able to obtain the correct 3D

position for the intermediate frames. This incident (called Erroneous Trajectory) is

represented in Figure 2-43, left bottom. When two markers are close to each other, the

capture system may swap each of their values at a time interval and there on may

continue with the swapped values. This is called marker swapping.

The Channels Interchange error (Figure 2-43, center) is difficult to detect only with

the trajectories of the marker and hence, some extra information is needed to accomplish

it. Eventually, there are Precision Errors in the calculation of the position in the space.

This event produces a shaky trajectory effect. In Figure 2-43, and on the right part, the

top graph represents a trajectory with precision errors and the bottom graph is the same

35

trajectory after smooth handling. This is caused by the lens curvature and the imperfect

markers stuck to the subject’s joint. Usually, the detection of these errors is made

manually. An expert observes the captured data and analyses the input channels

identifying the problems in the capture phase. The cleaning process is usually made

through curve interpolation (generally cubic splines), or curve smoothing. Sometimes,

repositioning of the markers may be involved in each frame.

2.3.7. SimiMotion System

Based on the detailed survey and careful analysis of our application which

pertains to jaw motion studies, the choice of MoCap system for our application was

narrowed down to SimiMotion capture system SimiMotion [50]. This is an optical retro-

reflective-passive marker based semi portable MoCap system. Since most of the

experiments will be conducted in a controlled environment (either in the case of humans

or animals), the problem of safety in operational environment is ensured. The video

capture speed that is required for a mastication motion analysis is in the range 60- 80 Hz

and hence, the system required the video cameras to record at maximum rate of 100 Hz.

For efficient data transmission between cameras and workstation, high-speed firewire

ports are plugged in each data bus.

Figure 2-44: Frames Captured at Different Instant during Motion Capture of a Mobile Robotic Platform with Traces of the Markers at all times

Simi Motion has been designed for professional 2D or 3D motion analysis in the

fields of sports, biomechanics, veterinary medicine, rehabilitation, industry, biology and

entertainment. Due to long cooperation with movement scientists and biomechanical

institutes, the software has become an ideal tool for motion capture and analysis. High

36

flexibility and accuracy combined with an easy to use interface are the strengths of this

product. Capturing of the movement is not time-limited and after digitization, all data can

be edited and visualized in many ways. 3D coordinates are calculated by synchronizing

data from other devices and can be exported in various formats. The following section

explains different modules of the motion capture system.

2.3.8. Modules

a) 2D/ 3D Kinematics:

SimiMotion 2D has been designed to analyze planar movements. First the

movement is digitized by placing markers on the measured object. In this case, automatic

tracking uses image-processing algorithms to detect the markers. Pattern matching

algorithms are also available and allow tracking of any object without markers

(markerless tracking). Raw data can be filtered or smoothed and interpolated to add

missing information. The 3D kinematics module includes all the features of 2D

kinematics, in particular the automatic tracking of moving objects (with or without

markers). A spatial analysis requires at least two cameras from different perspectives to

reconstruct 3D coordinates. One or more video clips are captured from different angles

and processed using the software for this purpose.

Figure 2-45: 2D/ 3D Kinematics Module Figure 2-46: Human Avatar within SimiMotion

b) Inverse Kinematics/ Dynamics:

Using a pre-defined marker set which is similar to 'Helen Hayes', this module

calculates joint centers, joint and segment rotation as well as axes. The resulting data

37

complies with the worldwide typical data collections for clinical gait analysis. This

module provides data like inverse kinematics and additionally computes the joint forces

and muscle moments. It requires data from one or two force plates and the results make

up a complete set of data for professional and detailed motion analysis (e.g. clinical gait

analysis).

All results can be displayed in graphs and stick diagrams for visualization.

Additionally, virtual reality representations can be created using a skeleton model or

another 3D model and forces and moments can be simultaneously visualized.

Figure 2-47: 3D Visualization of MoCap data of Jaw Motion using Stick Diagrams and Plots of 3D coordinates of the incisor point tip

c) Cameras:

Basler 602f cameras can be connected to standard FireWire (IEEE-1394)

interfaces, which are inexpensive and available for all computers, including notebooks.

With 100 frames per second (non interlaced) and a resolution of 656x492 pixels, these

cameras play in higher league than standard DV cameras. For special purposes, the time

resolution can be increased by reducing the image size: More than 200 Hz for 400x300 or

more than 300 Hz for 400x200 pixels. Simi integrated these cameras seamlessly into the

Simi Motion software.

38

A single computer can be used to capture 4 or even 6 cameras and all video

streams are automatically synchronized with analog data (force platforms, EMG etc.). All

video clips are saved in AVI format and can be processed with standard video editing

software and of course with all built-in features of Simi Motion (cropping, rotation,

mirror etc.).

These modules provide support for simultaneous capturing with Windows 2000

and Windows XP of multiple digital video cameras (DV). They are often combined with

the 2D and 3D motion analysis (kinematics) modules, but you may also extend your

EMG or force plate module with synchronized video analysis from several perspectives

without performing kinematic measurements.

2.3.9. MoCap Transformation and Synchronization

The computation phase of analysis is performed after all camera views have been

digitized. The purpose of this phase is to compute the three-dimensional image space

coordinates of the subject's body joints from the relative two-dimensional digitized

coordinates of each camera's view. Transformation is the process of converting two or

more, two-dimensional digitized views into a three-dimensional image sequence. The

transformation option is also available to convert a single, two-dimensional digitized

view into a two-dimensional image sequence. In either case, the process involves

automatic transforming of the relative digitized coordinates of each point in each frame to

absolute image space coordinates but for some initial timing information.

(a) Workstation and Camera Connection

(b) Basler Camera (c) Synchronization of 6 DV Cameras using DV Capture

Figure 2-48: Simi Motion System

39

If a three-dimensional transformation is performed, an additional operation must be

performed on the individual camera views to synchronize them. This process is called

time matching. Since each digitized camera view may start at a different point in time,

frame one of the first view may not correspond to frame one of the second view. The

transformation will only yield accurate results if digitized coordinates from simultaneous

frames are used. For synchronization, image sequences do not have to have the same

frame rate as the individual views, as the software module will automatically interpolate

linearly between digitized frames to create any resulting frame rate desired. In case, if the

cameras are not synchronous then following options may be considered:

Triggering the cameras

By connecting all the cameras through special cables and triggering them by one signal

can help avoid such problems. This is often the case with digital cameras that are

activated by a signal ("trigger") and can begin simultaneously with the recording. If all

the cameras are started in this way at the same time, on the one hand it is guaranteed that

the moment of exposure will be simultaneous and on the other hand it is not necessary to

identify the reference frame because all the camera sequences begin with the same

picture.

Figure 2-49: Configure Trigger Signal in SimiMotion

Figure 2-50: Triggering Unit in SimiMotion

Measuring time discrepancy between the cameras

Using the appropriate electronic equipment, it is possible to measure the discrepancy in

time between cameras, which are not synchronized (with an accuracy of 0.0001s). This

40

time lag must then be taken into account in the mathematical computation of 3-D data.

The error can thus be greatly reduced.

Finding synchronous frames

If there is no exact information available and the equipment used does not guarantee

synchronous recording, then the frames with the smallest possible time lag must be

looked for in each camera. A signal is generated which is visible from every camera

angle in order to indicate that the measurement is beginning. The recording of the

movement by the individual cameras begins with this optically marked frame or is

displaced by a constant time factor. This signal can also often be used to control other

measuring instruments or can be generated by these when measurement begins.

In spite of all these options, the automatic triggering control of the cameras is taken care

by SimiMotion software and hence automatic synchronization of the cameras is ensured

via firewire ports. In SimiMotion, the triggering unit sends a signal continuously within

short time periods to all the cameras that are connected to it and aids in transmitting the

data to the computer with synchronization.

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3. Biomechanics of Masticatory Motion The biomechanics of masticatory motion is an important part of the thesis which

needs a detailed description here. Actually, the biomechanical literature on human

masticatory system occupies a relatively small place owing to its relative complexity,

which makes it more difficult to analyze than, for instance, the system of the shoulder,

arm, hip, knee, or leg. Reasons that render the dynamic simulation of jaw motion difficult

are:

a. Large number of muscles of different size and shapes, making the system

kinematically indeterminate

b. Complex architecture of the jaw muscles making it impossible to determine the jaw

configuration only from orientation

c. Complexity of TMJ architecture that helps in articulating lower jaws with respect to

upper jaws- separation of the articular surfaces by cartilaginous disc that moves freely

between these surfaces that introduce complexity in motion and muscle actuation

d. Limited capability of experimental data collection pertaining to jaw motion

In the next few sections, we will discuss most of these issues including the

anatomical relationship, muscular system architecture and muscle models of jaw

mastication (Koolstra [51]).

3.1. Human Masticatory System The structural elements of the human mastication system consists of a skull and

mandible articulated at right and left TMJs. Dynamic balance of the system is provided

by complex architecture of the muscle elements and their actuations. The complexity of

the TMJs was discussed in the literature survey Section 2.1.1, but a detailed description

of the TMJ is provided here.

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Figure 3-1: Human Mandible Figure 3-2: Temporomandibular Joint

3.1.1. Temporo-Mandibular Joint (TMJ) Mandibular movements are guided by the articular surfaces fossae of TMJs that

reside on the temporal bones of the skull. Cartilaginous articular discs of variable

thickness separate the articular surfaces. These discs can move against fossae along the

fossa while interior surfaces of the discs simultaneously rotating on the condyles. The

discs attach to the ligaments binding mandible to skull. Together, these structures make

up the articular capsule. This articular capsule is slack.

Some consequences of this architecture are as follows. First, the articulating surfaces of

the TMJs are incongruent to each other, which allow a large a range of motion, but at the

cost of smaller joint contact and less joint stability. Second, the joint axis about which the

jaw rotates is not fixed in space or relative to the skull. Consequently, motion of an

incisor tooth cannot be related to TMJ position or condylar motion. Put it simply, the

paths that an incisor path can track is indefinite (analogous to redundant parallel

manipulators), so in reality, there are infinite solutions to this inverse kinematics/

dynamics problem.

3.1.2. Musculoskeletal Modeling The importance of complex muscle architecture in maintaining the jaw system in

dynamic balance was already mentioned earlier. Muscles that move the jaw are classified

as elevators (jaw closers) and depressors (jaw openers). Elevators include temporales,

masseters, and pterygoids (medial) and run from the skull to mandible. Depressor groups

technically include geniohyoid, mylohyoid, digastric muscles and lateral pterygoids, but

only the digastrics by far the most powerful depressors, were included in analyzes.

Elevator muscles differ from the depressor muscles. The former have larger pinnate

43

angles, higher physiological cross sectional areas (PCSA), with shorter fibers compared

to depressors.

Figure 3-3: Masticatory Muscular Architecture (Xu, Lewiset al. [11])

From Figure 3-3 (right-top corner), one can see that the temporalis is a flat, fan-

shaped muscle that originates in the temporal fossae and inserts on the coroniod process and

the anterior edge of the ramus of the mandible. The masseter (left top of Figure 3-3) is an

elevator muscle providing much of the power required for crushing food. It also assists in

protrusion. In addition to functioning as an elevator of the mandible, the medial pterygoid

aids in lateral positioning and is active during protrusion as explained by Lehman-Grimes

[52]. Similarly, other muscles can be easily identified from the Figure 3-3.

It is clear that the masticatory system is redundant system (more actuators than

degrees of freedom). This means that the controlling central system could potentially

control the actuation of the muscles to optimally reduce the energy consumption. As a

result, the muscles can move the jaw in various ways, in fact in infinite ways. Although

the system is able to generate cyclic movements controlled by a central pattern generator,

its muscles cannot be lumped into a limited number of alternating muscle groups. One of

the reasons for this is that they have to adapt constantly to the texture of the food between

the teeth. Because the muscle attachments are spread over wider regions (i.e., not at

specific points) and given the adaptability of the muscles to exert forces based on the

motion activity, it is certain that muscle lines of action will vary for different jaw

44

motions. Other points to be noted before considering the problem of the determining the

muscle forces for a given jaw motion are— inherent capability of fine tuning of muscles

by selective activation of motor units, relative large attachment area of muscles, presence

of spatially distant fibers to shorten by varying degrees for the mandibular movements.

All these may cause a shift in muscle’s action of force, which does not depend on the

central pattern generator (or central nervous system).

Figure 3-4: Anatomical Planes- used to describe the positions of the muscles or organs in musculoskeletal systems (Shin [53])

3.1.3. Dynamic modeling of Mastication

To model the mastication, we need to determine degrees of freedom for jaw

movements and then try to form a geometrical relationship to express joint positions in

defining the incisor tooth tip (end effector) position, the occlusal edged of lower central

incisor. Considering the mandible to be a body in space, it is considered a 6-DOF body,

but it is constrained and balanced by joints, discs and muscular architecture. In

mathematical terms, the position and orientation of the jaw can be given in terms of 3-

translational and 3-rotational coordinates— anteroposterior or X, mediolateral or Y, and

supero-inferior or Z giving a orthogonal set of planes to define the motion in anatomy,

see Figure 3-4 for the explanation of the anatomical planes.

There are different ways to represent jaw motions but a convenient way is to use

the screw axes. That is movement as a translation along and rotation about the same axis.

To account for 6-DOF, a set of 3-screw axes must be determined corresponding to

45

movement of the jaw. However, in this case the screw axes vary for each movement and

hence must be determined at every instant to completely define the position and

orientation of the jaw. By this token, we can express the jaw motion using 6-coordinates

and the location of the joint axes depends on the type of motion described along the axes.

We can now proceed to describe the position, velocity and acceleration of the

mandible, which will be parameterized using six independent variables as previously

explained. As in Koolstra [51], we use Newton’s law to determine the equations of

motion including active and passive muscle forces generated by ligaments, tendons and

dental elements. These forces and the resulting torques about the points of interest can be

used in Newton’s equation to determine accelerations. The calculated acceleration terms

induce changes in velocities and positions.

Dashed lines: principal axes, a: (linear) accelerations, F: (linear) forces, m: mass, α: angular accelerations.

M: torques, I: moments of inertia

Figure 3-5: Six degrees of freedom for jaw movement.

3.1.4. Inverse Dynamic Analysis and Muscle Actuation

Any musculoskeletal system in general is considered a redundant mechanical

system because there are more actuators to cause movement of the segments than there

are the actual degrees of freedom for the system considered. As mentioned before, this

statement is true for the masticatory system jaw as it has as many as 24 actuators (muscle

portions), which can be actuated independently to cause mandibular movements in 6-

46

DOF space. Because these muscles (passive and active) generate forces with higher

degrees of interdependency, it is impossible to formulate a clear-cut inverse dynamic

problem and determine the influence of individual forces. In addition, the fact that these

muscles attach over wide areas makes it difficult to accurately determine lines of action

of forces. For most jaw movements, active elements of muscles play a dominant role and

in general, used as jaw motion determinants however, along the boundaries, passive

muscles become dominant.

Generally, the contribution of a muscle to jaw movements can be established by

the direction of its line of action, and the position of this line with respect to the center of

gravity of the lower jaw. It accelerates the jaw in the direction of the line of action

according to F = ma, where a is the linear acceleration vector, F is the muscle force

vector, and m is the mass of the jaw. Also, an angular acceleration about the center of

gravity occurs according to M = Iα, where α is the angular acceleration vector, M the

muscle torque vector about the center of gravity, and I the moment of inertia vector. The

actual movement, then, is determined by the resultant instantaneous linear and angular

accelerations initiated by the forces of all active and passive structures. These combined

factors determine the effect of muscle contraction and, consequently, the contribution of

each single muscle to jaw movement.

Crosshairs: center of gravity, Fclosers: mean force of the jaw-closing muscles, Fopeners: mean force of jaw-opening muscles, Fjoint: joint force, Fbite: bite force, a: moment arm of the different forces.

Figure 3-6: Forces acting on the lower jaw in the Sagittal plane

47

In a sagittal plane analysis, the lines of action of most jaw-closers are directed

upward, and those of the jaw-openers, downward and backward as in Figure 3-6.

1. MA- Masseter

2. TE- Temporalis

3. LP- Lateral Pterygoid

4. Medial Pterygoid

JF- Joint Forces

BF- Bite forces

Courtesy: Koolstra 2002

Figure 3-7: Schematic of force vector diagrams for each muscle groups (force on both sides will be included for mathematical modeling)

However, in both cases, each line of action has a similarly directed moment with

respect to the sagittal axis through the center of gravity of the lower jaw. Jaw closers and

openers are able to produce a similarly directed torque about this axis, which leads to an

angular acceleration in the “negative elevation” (opening) direction. Consequently,

almost every muscle pair that is activated symmetrically attempts, aside from its specific

action, to perform an opening rotation about the center of gravity. It is through this

mechanism that both jaw closers and openers, despite their difference in orientation, are

able to maintain articular contact while performing unloaded (symmetrical) jaw

movements.

3.1.5. Muscle Modeling (Active and Passive):

The factors affecting optimal forces produced by muscle are given below:

a. Physiological cross sectional area (PCSA) - It is the area of a transverse section of

muscle and can be calculated as

( )cosmuscle

fibre density

mPCSA

lθ

ρ= , (3.1.1)

48

where muscle mass is the wet weight of the muscle, theta is the angle of fiber pinnation,

fiber length is the mean fiber length within the muscle and muscle density is assumed to

be a constant (1.067 g cm−3).

b. Muscle activity level (0 to 1) -. If the activities of a set of muscles are close to 1, then

it is said to be in maximum activation and can cause serious damages i.e. failure of

muscle or other passive elements.

c. Force- length relationship- This relationship maintains the ability of the muscles to

generate forces when their sarcomeres (sliding elements) are within a specific range

of lengths (lmin and lmax), and are not extended beyond or contracted below this range.

It has been found that is an important limiting factor in jaw mastication movements.

d. Muscle contraction shortening velocity- It is not a major limiting factor for normal

jaw movements. However, during deceleration of the jaw after a sudden

disappearance of resistance during forceful biting the relationship between force-

velocity appears.

Passive Elements:

These elements have the ability to resist jaw motion in one or more degrees of

freedom by generating reaction forces/ torques. Their influence is evident when the jaw

deviates from the midline and serve to prevent joint dislocation in these off-centered

movements. Ligaments involved in temporomandibular joints have three main functions:

stabilization, guidance of movement, and limitation of movement. The temporomandibular,

stylomandibular, and sphenomandibular ligaments are the three primary ligaments of the

TMJ. In most cases, however, the ratio between linear and angular accelerations effected

by a muscle is subtly dependent on the mass and moments of inertia of the jaw, and all

structures that are more or less rigidly attached to it. This attachment may include the part

of the masticatory muscles attached to the mandible, the tongue, skin, and other soft

tissues. However, the influence of these inertial properties on the final movement is small

and can be neglected in the final dynamic model of mastication.

Mathematical models applied to the study of the passive forces of the masticatory

muscles have been unable to open the jaw more than about 3 cm, whereas an opening of

49

6 cm is frequently observed in vivo. Therefore, the quantitative nature of these

predictions is disputable. Due to the proposed exponential relationship between the

passive muscle forces and their sarcomere lengths, small errors in the constants that

determine this relationship may lead to relatively large errors in the projected passive

forces. As long as there are no accurate quantitative data on the relationship between

sarcomere length and passive force of the human masticatory muscles, this issue remains

uncertain.

3.1.6. Posselt Envelope:

The main motions of the jaw, namely opening and closing, are a combination of

both translation and rotation. The translational and rotational components are not

combined equally through the whole of the motion. In the initial phase of the opening

cycle, the movement is primarily rotational, but after approximately the first 20 mm of

jaw opening, translation becomes pronounced. Maximum opening is reached when the

distance between occlusal edges of the upper and lower incisors is about 50 mm, which is

actually limited by the ligament and joint capsules (Zantos, Tavareset al. [54]). Closing

movement begins with a phase in which posterior translation predominates. The jaw

closes with translation as its major component until about two-thirds of maximal opening.

At this time the condyles and discs have returned to the posterior slope of the articular

eminence. Once this happens, closing occurs as a smooth combination of translatory and

rotary motion. Occlusal position is then attained primarily, though not entirely, by

rotational motion.

Jaw Motion Cycles

a. 1-2: MRO

b. 2-3- FRO

c. 3- MO

d. 3-4: MFP

e. 4-1- RP

Figure 3-8: Sagittal Plane- Posselt Envelope

50

A point located between the incisal edges of the lower central incisors, termed the

incisal point, is usually used to describe the movement of the mandible. The Posselt diagram

(Figure 3-9, Figure 3-8) outlines the border movements of the incisal point. The initial

position of the incisal point is called centric occlusion, and at this position, the occlusal

surfaces are in maximum contact. The sagittal Posselt figure can be divided into 4 segments.

In the first segment, called the maximal rear opening (MRO, 1-2) period, the jaw rotates

approximately 10 degrees about an axis that intersects the center of the condyles. If the

mandible opens farther, protrusion starts, and the final rear opening (FRO, 2-3) period can be

considered as the combined movement of the rotation about the axis and protrusion. After

this period the maximal opening (MO- 3) is reached. The maximal frontal path (MFP- 3, 4) is

described as a rotation around the axis accompanied by maximal protrusion. At the upper

border, the mandible is only in maximum protrusion and can return to its initial position of

centric occlusion by a retrusive path (RP, 4-1).

Figure 3-9: Frontal Plane : Posselt Envelope

51

4. Mathematical Background

In this section, we will explain the mathematical derivations and results obtained

by numerical simplifications that will be used in the section 6 and Error! Reference

source not found. for simulating the musculoskeletal mastication model as well as jaw

masticators.

4.1. Kinematics of R-U-S Configuration

The kinematics of a 6 DOF R-U-S platform is explained below. The reference

frames at the base and platform are as chosen at points B and P respectively— shown in

figure (1).

Figure 4-1: Schematic diagram of R-U-S configuration

The first frame (frame O) is defined by the center of the base-platform- shown by a red

circle in the figure (1). Following description pertains to ith leg of the manipulator:

OBi- base platform radius: l0- link length between the frame O and frame B

AB- leg1: l1- link length between the origin of the frame A and frame B

BiCi- leg 2: l2- link length between the frame B and frame C

52

CiPi- moving platform radius: l3- link length between the frame C and frame P

Figure 4-2: Kinematics of leg I of the R-U-S manipulator (front and side view)

Types of joints located at different points are given below:

Ai- revolute joint (θ1i) - connects (link 1)i to the base platform- acts like crank to the ith leg

Bi- universal joint - connects (leg 1) i and (leg 2) i

Ci- spherical joint - connects (leg 2) i with the upper platform frame P.

l3- link length between the frame C and frame P

x –axis is aligned along iOA→

z-revolute joint axis

y- axis selected to form a RH triad (parallel to global y axis)

Figure 4-3: Base- Revolute Joint frame A

53

The third frame (Frame P) is defined on the center of the top platform. Angles θ2i and

θ3i are the orientation of the link2 as shown in the figure (3) above. These angles were

included to develop the inverse kinematics code in MATLAB and can be used to

represent the universal joint angles. The orientations of the frame B and C is aligned

based on their respective joint configuration.

4.2. Position Kinematics:

The loop closure equation given by Tsai [55]of an ith leg of the R-U-S manipulator yields:

OA AB BC CP OP→ → → → →

+ + + = (4.2.1)

; ;a OA c CP p OP→ → → → → →

= = = - where all the vectors are defined in base reference frame.

1 1 2 3 1 2

1 1 2 3 1 2

2 3

cos( ) sin( ) cos( )sin( ) sin( )sin( ) ( )

0 cos( )

i i i iB

i i i i

i

AB BC p a cl ll l p a CP

l

θ θ θ θθ θ θ θ

θ

→ → → → →

→ → →

+ = − −

+⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥+ + = − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(4.2.2)

1 1 2 3 1 2

1 1 2 3 1 2

2 3

cos( ) sin( )cos( )sin( ) sin( )sin( )

0 cos( )

i i i i i

i i i i i

i i

l l cxl l cy

l cz


θ

+⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ + =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(4.2.3)

where [ , , ]P

Ti i ic cx cy cz

→⎛ ⎞ =⎜ ⎟⎝ ⎠

Equating 3rd row of the equation(4.2.3), we get 3 2cos( / )ia cz lθ =

Substituting that in the following equation, we can find 2θ as follows:

2 2 2 2 21 2 1 2 3 22 sin( ) cos( )i i icx cy cz l l l l θ θ+ + = + − (4.2.4)

2

2 2 2 2 21 2

1 2

cos( ),( )

2i i i

a Kcx cy cz l lwhere K

l l

θ =

+ + − −=

(4.2.5)

54

From 2θ and 3θ we can determine 1θ from linear equations in 1cos( )θ and 1sin( )θ obtained

from 1st and 2nd row of loop closure equation(4.2.3).

4.3. Velocity Kinematics to find Link Jacobian Matrix:

From the loop closure equation (4.2.1),

OA AB BC CP OP→ → → → →

+ + + = (4.3.1)

; ;a OA c CP p OP→ → → → → →

= = = - where all the vectors are defined in base reference frame.

To find the link Jacobian matrix, we consider ith leg of the R-U-S manipulator and

express top platform points, Ci in the base frame O.

1 1 2 3 1 2

1 1 2 3 1 2

2 3

,

cos( ) sin( ) cos( )sin( ) sin( )sin( ) .

0 cos( )

PAi

i i i iO P

i i i i P

i

AB BC c c p R c

l ll l p R CP

l


θ

→ → → → → →

→→

⎛ ⎞+ = = + ⎜ ⎟⎝ ⎠+⎡ ⎤ ⎡ ⎤

⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥+ + = + ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(4.3.2)

where AiOR is the rotation transformation matrix from base O with reference to Ai and R is

obtained from the end effector position and orientation as:

cos( )cos( ) cos( )sin( )sin( ) sin( )cos( ) cos( )sin( )cos( ) sin( )sin( )sin( )cos( ) sin( )sin( )sin( ) cos( )cos( ) sin( )sin( ) cos( ) cos( )sin( )

sin( ) cos( )sin( ) cos( )cos( )

OP

y p y p r y r y p r y rR y p y p r y r y p r y r

p p r p r

− +⎡ ⎤⎢ ⎥= + −⎢ ⎥⎢ ⎥−⎣ ⎦

(4.3.3)

Differentiating (4.3.2) w.r.t time t, we get,

2 3 1 2 1 2 2 3 1 2 31 1

1 1 1 2 3 1 2 1 2 2 3 1 2 3

2 3 3

sin( )sin( )( ) cos( ) cos( )( )sin( )cos( ) sin( ) cos( )( ) cos( )sin( )( )

0 sin( )( )

i i i i i i i i ii

i i i i i i i i i i i

i i

l lll l l

l

p

θ θ θ θ θ θ θ θ θθθ θ θ θ θ θ θ θ θ θ θ

θ θ

• • •

• • • •

•

→•

⎡ ⎤− + + + +⎢ ⎥−⎡ ⎤⎢ ⎥⎢ ⎥ + + + + + +⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ −⎢ ⎥⎣ ⎦

⎡=⎣

.( )OPR c

→⎤⎢ ⎥ +⎢ ⎥⎦

i

(4.3.4)

55

2 3 1 2 1 2 2 3 1 2 31 1

1 1 1 2 3 1 2 1 2 2 3 1 2 3

2 3 3

sin( )sin( )( ) cos( ) cos( )( )sin( )cos( ) sin( ) cos( )( ) cos( )sin( )( )

0 sin( )( )

i i i i i i i i ii

i i i i i i i i i i i

i i

l lll l l

l

p

θ θ θ θ θ θ θ θ θθθ θ θ θ θ θ θ θ θ θ θ

θ θ

• • •

• • • •

•

→•

⎡ ⎤− + + + +⎢ ⎥−⎡ ⎤⎢ ⎥⎢ ⎥ + + + + + +⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ −⎢ ⎥⎣ ⎦

⎡=⎣

cω→⎤

⎢ ⎥ + ×⎢ ⎥⎦

(4.3.5)

1

0

, .( ) .( ) .( ).( ) . , 0

0

, .( )

O O O O O OP P P P P P

O OP P

as R c R R R c c where

extracting from R c c

γ β

γ α

β α

α

ω β ω

γ

−

→ →

⎡ ⎤−⎢ ⎥⎢ ⎥

= = Ω Ω = −⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥⎢ ⎥

= Ω = ×⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

i i

i i i i

i i

i

ii

i

Figure 4-4: Front and Side Orthogonal View of ith Leg Configuration of the R-U-S Manipulator

where on assuming ω→

is obtained by considering the Euler angles α, β, γ are represented

w. r. t. to global x, y, z axes respectively and so rearranging the terms in (4.3.5) becomes,

56

1 1 2 3 1 2 2 3 1 2

1 1 2 3 1 2 1 2 3 1 2 2

2 3 1 2

2 3 1 2

2 3

sin( ) sin( )sin( ) sin( )sin( )cos( ) sin( ) cos( ) sin( )cos( )

0 0

cos( ) cos( )cos( )sin( )

sin( )

i i i i i i i

i i i i i i i i i

i i i

i i i

i

l l ll l l

ll

l

θ θ θ θ θ θ θθ θ θ θ θ θ θ θ θ

θ θ θθ θ θ

θ

• •− − + − +⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥+ + + +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

+⎡+ +

−[ ]3 3 3

00

0

z y

i z x

y x

x c cI y c c

c cz

α

θ β

χ

• •

• • •

×

• •

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎡ ⎤−⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ = + −⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ −⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(4.3.6)

1 1 2 3 1 2 2 3 1 2

1 1 2 3 1 2 2 3 1 2

12 3 1 2

2 3 1 2 2

2 33

sin( ) sin( )sin( ) sin( )sin( )cos( ) sin( ) cos( ) sin( ) cos( )

0 0

cos( )cos( )cos( )sin( )

sin( )

i i i i i i i

i i i i i i i

ii i i

i i i i

ii

l l ll l l

ll

l

θ θ θ θ θ θ θθ θ θ θ θ θ θ

θθ θ θθ θ θ θ

θ θ

•

•

•

− − + − +⎡⎢ + + +⎢⎢⎣

⎡ ⎤⎢+ ⎤⎢⎥+ ⎢⎥⎢⎥− ⎦ ⎢⎣ ⎦

~

~

1 0 0 00 1 0 00 0 1 0

z y

z x

y x

qi xi

c cc c

c c

J q J

X

x

•

•

⎥ ⎡ ⎤−⎥ ⎢ ⎥ ⎡ ⎤= −⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎥ ⎢ ⎥−⎣ ⎦⎥

=i

(4.3.7)

where Jqi is joint space jacobian matrix and Jxi is the task space jacobian matrix that can

be used to determine the iVci— velocity of point ci w.r.t the ith reference frame located at

the corresponding base revolute joint of the leg in the base plate (Ai).

4. 3.1. Link and Manipulator Jacobian Matrix

To obtain the manipulator Jacobian matrix, we need to first form link Jacobian matrix

from equation (4.3.7) as follows in the local frame at ith revolute joint. For this we need to

multiply the RHS by 1( )O AA OR R− = to express the link Jacobian matrix in the local leg

frame at Ai. So, let Jq1 be defined as

(4.3.8)

1 1 2 3 1 2 2 3 1 2

1 1 1 2 3 1 2 2 3 1 2

2 3 1 2

2 3 1 2

2 3

sin( ) sin( ) sin( ) sin( ) sin( )cos( ) sin( ) cos( ) sin( ) cos( )

0 0

cos( ) cos( )cos( ) sin( )

sin( )

i i i i i i i

q i i i i i i i

i i i

i i i

i

l l lJ l l l

ll

l

θ θ θ θ θ θ θθ θ θ θ θ θ θ

θ θ θθ θ θ

θ

− − + − +⎡⎢= + + +⎢⎢⎣

+ ⎤⎥+ ⎥⎥− ⎦

57

From the link jacobian matrix (Jq) i, we can obtain the manipulator jacobian matrix JP by

extracting the first row of (Jq) i for each leg and equating to 1iθ•

(manipulator active joints)

for each leg. Finally we arrive at the relation:

1 ,1~

( )i q iJ xθ••

= , (4.3.9)

where (Jq, 1) i is the first row of (Jq)i obtained for ith leg. Therefore, we get the manipulator

jacobian matrix JP as

,1 1

,1 2

,1 3

~ ~,1 4

,1 5

,1 6

( )( )( )( )( )( )

q

q

qpa

q

q

q

JJJ

q JJJJ

x x• ••

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= =⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

, where 1 2 3 4 5 6 ,T

aq θ θ θ θ θ θ• • • • • • •⎡ ⎤= ⎢ ⎥⎣ ⎦

active joint rates for

the manipulator system considered

4.4. Screw Theoretic Jacobian Matrix and Singularity Analysis From the loop closure equation (4.3.1), we have for this case as derived by Filho and

Cabral [56]

1 1

11 2 2 1

~ ~ ~

3 3

, ( ) ( ) [( ) ] ( )

i i

C Cq i A ci i q A ci q i

i i

Then J R J J R J Jx x xθ θ

θ θ

θ θ

• •

• • •• •−

• •

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

= ⇒ = =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

i i

58

OA AB BC CP OP

OA AB BC OP PC

→ → → → →

→ → → → →

+ + + =

+ + + = + (4.4.1)

Differentiating this equation w.r.t time,

( ), 1,.....,6i i

i iP P bi i i B C i iV P l l B C for iω θ θ ω+ × = × + × + × =i i

(4.4.2)

Since the velocity of the passive link, BC is irrelevant to our present analysis we can

eliminate by cross- multiplying the above equation by BiCi to obtain:

1

11

1

. .( ) 2 .( ), 1,.....,6

[ ] 2 .( ) , 1,.....,6

ii i P P i i i i i i i

iPii i i i i i i i i

P i

B C V b B C A B B C for i

VB C b B C A B B C for i

ω θ

θ θω θ

+ × = × =

⎧ ⎫× = × =⎨ ⎬

⎩ ⎭

i

ii

i

(4.4.3)

which on combining for all the six legs becomes,

11

1

[ ] 2 .( ) , 1.....6x

q

iPii i i i i i i i i

PJ i q

XJ

VB C b B C diagonal A B B C for iθ θ

ω θ

⎡ ⎤⎢ ⎥⎧ ⎫

× = × =⎢ ⎥⎨ ⎬⎢ ⎥⎩ ⎭⎢ ⎥⎣ ⎦

i

i

ii

i (4.4.4)

4.5. Singularity Analysis of the Jacobian of R-U-S Platform Once we obtain the Jacobian matrix by the above formulation, we can now find when the

manipulator reaches singular positions by analyzing the forward and backward dynamic

jacobian matrices and finding the determinant of these matrices. Thus we obtain by

expanding along the 3rd column,

1 2 3 1 1 2 3 1 2 2 3 1 2

1 1 2 3 1 2 2 3 1 22 2

1 2 3 2

det( ) *sin( )*[( sin( ) sin( )sin( ))( sin( ) cos( ))

( cos( ) sin( ) cos( ))( sin( )sin( ))]

sin( ) sin( ) 0

q i i i i i i i i

i i i i i i i

i i

J l l l l

l l l

l l

θ θ θ θ θ θ θ θ

θ θ θ θ θ θ θ

θ θ

= − − − + + −

− + + − +

= − = (4.5.1)

One of the solutions for the above equation is θ2i = nπ, n=-α ...-3,-2,-1, 0, 1, 2, 3… α,

which occurs when the passive link comes in line with the crank (active link).

59

4.6. Kinematic of P-U-S Platform Similar to the loop closure equation (4.3.1) of R-U-S platform, it can obtained for P-U-S

platform as in Kim [57],

, ,i o i o iiOA A A A B BP OP

→ → → → →

+ + + = (4.6.1)

The start and end points of the rails of the prismatic joint is given by Ai,o and Ai,1 the

center of the universal joint i, which lies on the line segment Ai,o Ai,1 (rail axis i), will be

denoted by the base joint Ai. A right handed base reference frame with center O is

attached to the base. The center of the spherical joint i will be denoted by a platform joint

Bi. The mobile reference frame is attached to the point Ci on the platform (CG of the

platform). The distance between point Ai,0 and Ai,1 will be denoted by articular coordinate

λi, changing the articular coordinates will change the pose of the platform, P. The position

of the platform is denoted by point P measured from the origin O and the orientation

angles (Euler angles) are expressed as [α, β, γ]. For inverse kinematic problem, the

position and orientation of the platform will be given. Euler angles are defined w.r.t.

global X, Y, Z-axes respectively and hence the same rotation matrix (4.3.3) as in the

previous case with R-U-S manipulator can be used.

Suppose the position of P is given at any instant, then we can determine the as AiBi. Let ai

be the unit vector along the rail axis slider Ai,0 Ai,1. To find λi the following procedure is

followed. From the position vector of P, the joint center position OBi can be determined

as PBi is known and remains constant in the platform reference frame. Then AiBi is

determined in the global reference frame.

,0 ,i i i i i o i i iiA B A B A A d aλ→ → → → →

= − = − (4.6.2)

But, ,i iA B l→

= of the constant magnitude (4.6.3)

2( ) ( )Ti i i i i id a d a lλ λ

→ → → →

− − =i (4.6.4)

Solving the equation(4.6.4) for the articular variable λi, we get,

60

2( )T T Ti i i i i i ia d a d d d lλ = − − + (4.6.5)

Therefore, λi is obtained by knowing the vector di vector (i.e. Bi) from the equation

(4.6.1)

,i i o i iA A aλ→

= + (4.6.6)

The following method is actually used to implement the inverse kinematics of the 6-DOF

P-U-S system. We simplify (4.6.1) for this purpose to obtain the following:

, ,

, ,

,,

00

i o x i i xO O O

i o y A B i i y P

i i zi o z

A A B xA R R A B y R PC

qi A B zA

→⎧ ⎫ ⎧ ⎫⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪+ + = +⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

⎩ ⎭ ⎩ ⎭⎩ ⎭⎩ ⎭

(4.6.7)

Since Ai,o as well as OAR , we proceed with the equation to solve for 4 unknown

variables- , , ,, , intO O Oi x i y i y iB B B and q prismatic jo position−

From first 2 rows of equation(4.6.7), we solve for , ,O

i i x i i yA B and A B as,

, , ,

, , ,

Oi i x i x i x

Oi i y i x i y

A B C A

A B C A

= −

= − (4.6.8)

Since we know that length of link AB is constant= l2. We obtain ,i i zA B as

( )

( )

2 2 22 , , ,

2 2 2, 2 , ,

i i i i x i i y i i z

i i z i i x i i y

A B l A B A B A B

A B l A B A B

→

= = + +

= − + (4.6.9)

which again on substituting in the 3rd row of the equation(4.6.7), we can solve for qi.

Thus we, see that there are two values of AiBi, z (+/ -) which actually results in 2 solutions

for each leg that produces 26 solutions for the Stewart platform manipulator. For practical

application purposes, positive values are commonly used.

61

4.7. Velocity Kinematics of P-U-S Platform In their work by Zhao and Gao [58], the Jacobian matrix using the screw theoretic

formulation technique was derived which can be used conveniently to derive Jacobian

matrices for any parallel manipulators. We start from differentiating the loop closure

equation of P-U-S as in equation(4.6.1) w.r.t. time t as follows:

^

i ii i i ia A B V cλ ω ω→ → → →

+ × = + ×i

(4.7.1)

Dot multiplying by AiBi, we get

^

. . .i ii i i i i i ia A B V A B c A Bλ ω→ → → → → →⎛ ⎞ = + ×⎜ ⎟

⎝ ⎠

i (4.7.2)

^

1

.i ii i i

i i i

VA B c A B

a A Bλ

ω

→→ → →

→ →

⎛ ⎞⎡ ⎤⎜ ⎟ ⎡ ⎤ ⎢ ⎥⎜ ⎟= ×⎢ ⎥ ⎢ ⎥⎛ ⎞ ⎣ ⎦⎜ ⎟⎜ ⎟ ⎣ ⎦⎜ ⎟⎝ ⎠⎝ ⎠

i (4.7.3)

^

1. , ,.

i ii i i

i i i

VJ where J A B c A B

a A Bλ

ω

→→ → →

→→

⎛ ⎞⎡ ⎤ ⎜ ⎟ ⎡ ⎤⎢ ⎥ ⎜ ⎟= = ×⎢ ⎥⎢ ⎥ ⎛ ⎞ ⎣ ⎦⎜ ⎟⎜ ⎟⎣ ⎦ ⎜ ⎟⎝ ⎠⎝ ⎠

i

(4.7.4)

is the manipulator jacobian matrix for P-U-S degree of manipulator, with i=1…6.

62

4.8. Denavit Hartenberg Parameterization

Figure 4-5: Schematic diagram showing the DH parameterization for each joint coordinate to

determine the end effector reference frame

The DH parameterization CMU [59] requires four quantities to be determined at each

joint by attaching the reference frames along each joint DOF whose names are generally

given as link length(ai), link twist(αi), link offset(di) and joint angle (θi). These are called

as DH parameters and can be obtained for each joint DOF frame as follows.

Initially, a base reference (zeroth ) frame is attached to first joint, in this case at the point

Ai for each legs. Then the DH parameters can be determined as follows:

o Assign Zi axis of i-th reference frame along the axis of joint i.

For a revolute joint, the joint axis is along the axis of rotation.

For a prismatic joint, the joint axis is along the axis of translation.

o Choose Xi axis to point along the common perpendicular of Zi and Zi+1

pointing towards the next joint.

63

if Zi and Zi+1 intersect, then choose Xi as normal to the plane of intersection.

o Choose Yi to round out a right hand coordinate system.

The Y-axis is not used for DH transformation matrices so it is not usually displayed in

DH parameterized systems. So based on these guidelines, we can determine the

transformation matrix of i-th reference frame w.r.t the (i-1)th reference frame as follows:

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

1, , , ,. . .

cos sin cos sin sin cossin cos cos cos sin sin

0 sin cos0 0 0 1

i i i i i

ii z z d x a x

i i i i i i i

i i i i i i i

i i i

A R Trans Trans R

aa

d

θ α

θ θ α θ α θθ θ α θ α θ

α α

− =

⎡ − ⎤⎢ ⎥− −⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

(4.8.1)

Subsequently, by deriving mathematically or numerically the homogeneous

transformation by DH parameterization, we can finally obtain the global transformation

matrix at the end effector 0EEA as:

0 0 1 2 1 11 2 3 2........... EE EE

EE EE EEA A A A A A− −−= (4.8.2)

where 1iiA − is the transformation matrix as explained before and 1

EEEEA − − is the

transformation matrix of end effector (EE) reference frame w.r.t. its previous joint axis

frame (EE-1).

Once the transformation matrix 0EEA is obtained, we can find its differentiation w.r.t time

symbolically and use that to compute the twist matrix 0EET :

( )

. 1

0 0 0.EE EE EET A A−⎛ ⎞

= ⎜ ⎟⎝ ⎠ (4.8.3)

This twist matrix actually comprises of angular velocity matrix 0

0EE⎡ ⎤Ω⎣ ⎦ as well as the

translational velocity components 0

0EEV⎡ ⎤⎣ ⎦ in the base reference frame and can be

extracted as below:

64

( )0 00 0 0

00 0

EE EEEE VT

⎡ ⎤⎡ ⎤ ⎡ ⎤Ω⎣ ⎦ ⎣ ⎦⎢ ⎥=⎢ ⎥⎣ ⎦

(4.8.4)

Where,

03 2

0

0 3 1

2 1

00

0

EE

ω ωω ωω ω

−⎡ ⎤⎢ ⎥⎡ ⎤Ω = −⎣ ⎦ ⎢ ⎥⎢ ⎥−⎣ ⎦

and 0 0

0 ,0 ,0 ,0[ , , ]EE EE EE EE Tx y zV V V V⎡ ⎤ =⎣ ⎦

DH Parametric table for the R-U-S and P-U-S configuration is given below:

Table 4-1 DH Parameterization of R-U-S Manipulator

link length(ai) link twist(αi) link offset(di) joint angle (θi) 1 l1 0 0 1θ 2 0 90 0 2θ 3 l2 0 0 3θ 4 0 90 0 4θ 5 0 90 0 5θ 6 l3 0 d6 6θ

Table 4-2 DH Parameterization of P-U-S Manipulator

link length(ai) link twist(αi) link offset(di) joint angle (θi)

1 0 0 di 0

2 0 90 0 2θ

3 l2 0 0 3θ

4 0 90 0 4θ

5 0 90 0 5θ

6 l3 0 d6 6θ

65

4.9. P-U-S Manipulator Parametric Study

We will see from the workspace analysis of P-U-S manipulator that it is capable of

tracking the jaw motion trajectories. Before building the physical prototype, we need to

estimate the maximum limits on workspace, velocities as well as forces in these

manipulators, especially to obtain the geometrical and actuator specifications of the

system. For this purpose, we proceed with a top down approach by which we derive the

equations for the maximum reachable workspace, speed as well as end effector force

given the leg lengths, force and travel and sizes of the base plate and top plate.

We use the following nomenclature in the subsequent derivations:

Table 4-3 List of Parameters

Geometric Parameters

r Base platform radius, m

R Top platform radius, m

l Length of each leg, m

Actuator Parameters

q0 Nominal length of the linear actuator, m

Tq0 Travel of linear actuator on either side (+/ -), m

Vq0 Maximum actuator speed, m/s

Fq Actuator maximum force, Newton

4.9.1. Maximum Reachable Workspace, Velocity and Force This can be obtained by assuming the initial configuration of the manipulator when the

actuators are at q0-Tq0 position and are free to travel to other extreme position q0+Tq0 as

in the following figure.

66

Figure 4-6: Top view of base platform Figure 4-7: Front view of the manipulator at the two extreme positions to find Pz

From the figure, it is clear that as the actuator 1 and 2 travel between two extreme

positions, the workspace covered by end effector frame in X and Y axes can be given as,

02x yP P Tq= = (4.9.1)

Similarly, due to the travel of the actuator between two extreme positions, we get the

maximum reachable workspace in the Z direction as,

2 2 2 20 1 0 2 0 0 0 0( ) ( )zP A A A A l q Tq l q Tq→ →

= − = − − − − + (4.9.2)

2 2 2 20 1 0 2 0 0 0 0( ) ( )zP A A A A l q Tq l q Tq→ →

= − = − − − − + (4.9.3)

The maximum attainable orientation- roll, pitch, yaw of the platform can be

geometrically obtained as

1tan zroll pitch yaw

Pr

θ θ θ − ⎛ ⎞= = = ⎜ ⎟⎝ ⎠

(4.9.4)

We can also determine the maximum reachable speed in translational X, Y and Z

directions simply as follows:

67

0x yV V Vq= = (4.9.5)

00 2 2

0

zqV Vq

l q

⎛ ⎞⎜ ⎟=⎜ ⎟−⎝ ⎠

(4.9.6)

Since we know that linear velocities are related to angular velocity as V rω= , we obtain

roll, pitch and yaw velocities as,

0roll pitch yaw

Vqr

ω ω ω= = = (4.9.7)

Figure 4-8: Top view of the base platform with the linear actuators – FBD to calculate maximum

workspace forces and torques

To calculate the maximum forces, we draw a FBD of the base platform along with the

actuators. From the figure, we can identify the lines of action of actuator forces along the

linear actuators’ axes. Resolving the forces in the X and Y direction, we find the

maximum possible forces in the X and Y directions as,

( )

( )4 cos / 6

2 4 sin / 6x q

y q q

F F pi

F F F pi

=

= + (4.9.8)

To determine the maximum exertable force in the Z- direction, a three-dimensional FBD

is used to determine the orientation of the force vector along the leg, Fi.

( ), cosz i iF F φ= (4.9.9)

68

where φ is defined as the angle between the leg vector and the position vector of Ai,0

which is given as,

Figure 4-9: Front and side view of the manipulator with the linear actuators – FBD to calculate maximum workspace forces and torques

( ) 1

1 1

sin sini i

R r R r

P A P Aφ φ −

→ →

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟− −⎜ ⎟ ⎜ ⎟= ⇒ =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(4.9.10)

Substituting (4.9.9) in (4.9.10), we get

( ) 1,

1

cos cos sinz i i i

i

R rF F FP A

φ −→

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟= =⎜ ⎟⎜ ⎟

⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

(4.9.11)

We can now find the relationship between Fi and Fq based on the simple trigonometric

relationship as,

( ) ( )tan tanii

F F FqFq

θ θ= ⇒ = (4.9.12),

where θ can be obtained as,

69

2 2

01

0

tanl q

qθ −

⎛ ⎞−⎜ ⎟=⎜ ⎟⎝ ⎠

(4.9.13)

Substituting equation (4.9.13) in(4.9.11), we get

( )1,

1

cos sin tanz i q

i

R rF FP A

θ−→

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟

⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

(4.9.14)

Hence, the maximum exertable force in the Z direction by all the legs will be summation

of the equation (4.9.14) for i=1 to 6,

( ) ( )6

, ,1

6 6 cos tanz z i z i qi

F F F F φ θ=

= = =∑ (4.9.15)

( )2 cos / 63roll zF r piτ = (4.9.16)

( )1 2 sin / 63 3pitch z zF r F r piτ = + (4.9.17)

6yaw qF rτ = (4.9.18)

Thus, we have derived all the equations to obtain maximum reachable workspace, speeds

as well as forces w.r.t. the base reference frame axes and expressed them in terms of

geometrical and actuator parameters. The setup now is ideal to conduct a series of

parametric studies to determine the optimum values for these undecided parameters that

would satisfy our application requirements which will be dealt in detail in Section 5.

4.10. Jacobian-Based Performance Measures (JBPM)

As noted from above equations, the Jacobian matrix offers a configuration dependent

linear relationship between the joint and task space velocities. This matrix has played an

important role in developing metrics for evaluating and characterizing the performance of

robotic systems. Such Jacobian-Based Performance Measures (JBPM), including

70

manipulability, isotropy index, condition number, dexterity and singularity, have been

employed in many robotic applications. By quantitatively evaluating the qualitative

characteristics, such measures play a critical role in design, evaluation and optimization

of a robotic system.

In this thesis, we mainly utilize the manipulability measures to evaluate the performances

of our systems. Manipulability is defined as the measure of the flexibility of the

manipulator to transmit the end-effector motion in response to a unit norm motion of the

rates of the active joints in the system. We will briefly review some of the literature

pertaining to the research on manipulability.

4.10.1. SVD and Manipulability ellipsoid

The Singular Value Decomposition (SVD) of the Jacobian matrix and its geometric

relationship offer further insights in characterizing the “manipulability” of a manipulator.

We here briefly summarized the major mathematical and geometrical aspects that are

relevant to our work, and interested readers are referred to Nakamura [60] for more

details.

Consider an m×n Jacobian matrix PJ , it can always be transformed into the form of

m mI

TP

T T

J U V

with U U UU ×

= ∑

= = (4.10.1)

and T Tn nV V VV I ×= = (4.10.2)

Figure 4-10: Manipulability ellipsoid: mapping joint space velocities (hyper sphere) to task space

velocities (hyper ellipsoid) (Manipulability Index [61])

71

and Σ is an m×n matrix with singular values σi’s on the diagonal, where

1 1

1 2 3

... 0 0 ... 0........

m n k

k

diag σ σ σσ σ σ σ

×∑ =

≥ ≥ ≥ (4.10.3)

The rank of PJ is k, and k ≤ min (m, n)

The columns of U are the ortho-normal eigenvectors of TP PJ J , while the columns of V

are the ortho-normal eigenvectors of TP PJ J . In this thesis, we employ the svd command

in MATLAB for our computation.

4.10.2. Yoshikawa Measure

The manipulability is defined in Yoshikawa [62]

( ) det( )Ty P P PJ J JΓ = (4.10.4)

In the context of SVD, if

T TPJ U V= ∑ (4.10.5)

Since U and V are ortho-normal, det (U) =det (V) =+1, we can simplify equation

(4.10.4) to

det( )TyΓ = ∑∑ (4.10.6)

1 2. ...........Y Kσ σ σΓ = (4.10.7)

Hence, the measures is nothing more than the product of all the singular values of PJ

1 2. ...........Y Kσ σ σΓ = (4.10.8)

Geometrically, such product is directly proportional to the volume VE of the

manipulability ellipsoid

72

/ 2

(1 / 2)

k

E yVk

π= ΓΓ +

(4.10.9)

where [ ]yΓ i is the gamma function. The proportional coefficient is constant and depends

only on the rank of PJ , k.

4.10.3. Condition Number

Salisbury and Craig [63] define the manipulability to evaluate the workspace quality by

utilizing the condition number of PJ , which is given by

1( )CN Pk

J σσ

Γ = (4.10.10)

where σ1 and σk are the minimum and maximum singular values of JP, and 1 1 ,σ λ=

k kσ λ= and λ1 and λk are the minimum and maximum of the eigenvalues of TP PJ J ,

respectively.

Geometrically, it is the ratio of the length of the semi-major axis to the length of the

semi-minor axis of the manipulability ellipsoid. Such measure has a lower bound of 1,

but it grows out of bound and tends to infinity when the manipulator is near the singular

configuration. The reader is referred to Section 4.9 in [37] for more mathematical

treatment on this topic.

4.10.4. Isotropy Index

Zanganeh and Angeles [64] define the manipulability measures by the reciprocal of the

condition number of JP, which is given by

1( )Ty P

k

J σσ

Γ = (4.10.11)

Geometrically, it is the ratio of the length of the semi- minor axis to the length of the

major axis of the manipulability ellipsoid. Such a measure is better behaved compared

with the condition number, since the values remain bounded between 0 and 1.

73

4.11.1. Transformation Matrix between MoCap coordinates and Platform coordinates

The 3D coordinates of the three points of the mandible (left TMJ, right TMJ and incisor

tooth tip) obtained from motion capture system defined with respect to its own global

reference frame (reference frame of the calibration grid used for motion capture

experiments) should be transformed to the base reference frame of the platform. The

transformed coordinates will be used as kinematic motion trajectories to drive the end-

effector platform in the final simulation.

Figure 4-11: Plane of Jaw Motion Measured using MoCap System with marker positions identified

Let 1v→

and 2v→

be any two vectors along the plane containing the points— left TMJ, right

TMJ and incisor tooth tip defined in the reference frame of the calibration grid (A0) as

shown in the figure. We define the unit vectors along the triad attached with the incisor

tooth tip as follows

1 23

1 2

v vev v

→ →→

→ →

×=

× (4.11.1)

74

2 3 1

1 1 2 3 1 2 3

| | | | | |

, ,| | | | | |

e

e

e

e e e

xe e e e y where R e e e

z

→ → →

→ → → → → → →

= ×

⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥= =⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(4.11.2)

1 2 3 1 2 3

| | | | | |

, ,| | | | | |

e

O e

e

xX e e e y where R e e e

z

→ → → → → → →

⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥= =⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(4.11.3)

where T- transformation matrix of the jaw reference frame with respect to the motion

capture reference frame,

1

2

3

| | || || | |0 0 0 1

dR d

Td

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦ (4.11.4)

Let A(0) be the transformation matrix of the initial jaw reference frame to the origin

reference frame A0 and A(t) be the transformation matrix of the moving jaw reference

frame with time to the origin reference frame, A0. Using these relationship, we can

obtain the jaw motion trajectories w.r.t the A(0) reference frame as:

1( ) (0) ( ) ( )Pr eA t A A t X t−= = (4.11.5)

where, Ar(t) is the relative jaw motion defined w.r.t the initial frame A(0).

MoCap transformation Platform Transformation

Figure 4-12: Transformation between MoCap and Platform Reference Frames

75

This Ar(t) is equivalent to the relative jaw motion trajectory defined in the platform frame

P which can be converted to the global reference frame B.

For the platform, we can find the global transformation matrix B0 that relates the base

frame B to P based on the end effector position and orientation using the equation

(4.11.4) discussed before in Sec. 7.3.

0

0, 00

0

| | || || | |0 0 0 1

PE

P PB E

PE

xR y

Bz

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

(4.11.6)

So using this matrix, we can define the global motion trajectories of the platform using

(4.11.6) and (4.11.5) as,

0( ) ( )BP rX t B A t= (4.11.7)

which can be used to drive the platform in global frame B,

0, 0 0 0

PTP P P

E E E EX x y z→

⎡ ⎤= ⎣ ⎦

76

5. Technological Tools

5.1. CT-Scan to CAD Conversion

CT scans of the mandible and skull of labrador dog were obtained from the vendor which

is used to build 3D CAD models in STL format using Mimics software. The STL

conversion process includes the following steps, which can now be used in the CAD

model as well as musculoskeletal models. The original specimens of skull and mandible

are shown in Figure 5-1.

Dorsal view Postero-Lateral View Ventral View

Figure 5-1: Different Views of Labrador Specimen

The CT scans of the Labrador’s skull and mandible are first imported into a project in

Mimics to start the conversion process. Each of the CT scan slices must occupy same

memory in the disk to proceed with the 3D data calculation. Once the slices are imported,

we need to enter the slice thickness and pixels information for accurate conversion.

Figure 5-2:CT Scanned Images Import and Pixels Information Dialog

77

The pixels and image information is normally provided by the vendor. The following

table shows the CT scanned image information. These values must be entered in the

pixels and images information dialog.

Table 5-1 CT Scan Images Information

File Output Type 16bit: 1024x1024 16-bit TIFF images

Slice Thickness 0.25 mm

Slice Spacing 0.2 mm

Total number of slices 746

The slices will be seen in the three orthographic projection views and proper

nomenclature must be assigned to each of these (anterior-posterior, left- right and top-

bottom) before proceeding further. Then, we need to properly segment the sliced images

with an appropriate threshold value. The segmented image at one specific slice is shown

in the Figure 5-4.

Segmented CT slice 3D Calculate Dialog

Figure 5-4: Segmentation and 3D Calculation

Figure 5-3: Segmentation Options

78

Figure 5-5: STL file of CT scans

Next, the 3D mesh surfaces are calculated using the 3D Calculate option. Finally, we

obtain a 3D cad model of the Labrador skull, which will be used in musculoskeletal and

virtual analyses.

5.2. SimiMotion MoCap Station

5.2.1. Workflow for Motion Capture using SimiMotion system: Specification: The markers are selected based on the analysis from pre-defined skeleton

landmarks or user-defined points or imported marker sets from previous projects or from

a template file. The number of markers is decided based on the degrees of freedom of the

subject under study. In our case, we are interested in determining 3D coordinates of jaw

motion. As explained before, the jaw model has six motion-DOFs and hence in our

model, we use three markers (reflective type) to accurately determine the positions and

orientations of the jaw in the global x-, y- and z- axes respectively. The three markers are

attached to the left TMJ, right TMJ and frontal point of the chin.

The main idea of using three markers for this study is that determining the coordinates of

three points in jaw allows us to not only define the exact position of the incisor tooth tip

(which in our case is the frontal point of chin) but also calculate the orientation of the

plane. Another reason is that we would like to use minimum number of markers making

it convenient to conduct the motion analysis especially on animals.

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Figure 5-6: Digitizing the Human Jaw Movement Figure 5-7: Digitizing the Dog Jaw Movement

Camera-setup: The number and the position of the cameras are determined based on the

testing requirements. For 2D analyzes, at least one camera is required and for 3D at least

two cameras are required. However, most 3D movements require three or more cameras

to avoid occlusion.

Figure 5-8: Typical Camera Setup for 2D Motion Capture Analysis

Calibration: It is the process of capturing a known calibrated object (e.g. a cube for 3D).

This is generally used to define the value of one unit distance that which the software can

use to quantify the positions in the camera views. This way, the system can reconstruct

the position and settings of your cameras. In addition, the calibration can be checked for

accuracy and validated for further applications.

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Figure 5-9: Calibration of MoCap Region- Camera1

Figure 5-10: Calibration of MoCap Region- Camera2

Video capturing and analog data acquisition: With cameras positioned to cover the

whole motion of the test subject, it is synchronized using firewire ports at maximum rate

of 100 Hz. The capture process is initiated by an analog signal (e.g. touching a force plate

or triggering a light barrier) using a pulsed trigger. Once recorded, SimiMotion provides

tools to crop and save the video clips of all cameras in a single step. Other methods of

achieving synchronization are already explained in Literature Survey section of this

thesis.

Tracking / digitization: The markers are then digitized in each frame of the video by

manually selecting in first few frames. Afterwards, using the automatic tracking

algorithm will be used to identify white and colored markers automatically and the

system will be able to track the markers for remaining frames. To overcome problems of

marker occlusion or falling down of makers from skin, the pattern-matching algorithm

for markerless tracking or manual tracking option in SimiMotion is used. A typical

scenario of such an occurrence is shown in the figure below.

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Figure 5-11: Marker Occlusion Problems in Animals

Reconstruction of 2D or 3D coordinates: Once completing the 2D tracking for all

frames in all the cameras, 3D coordinates of the markers can be calculated. This can be

followed by a post-processing step to smoothen out the disturbances using filter and

interpolation settings from your raw data.

Figure 5-12: Digitization of the sequence of frames and automatic tracking of 2D coordinates-

Camera1 and 2

Analysis/ Visualization: From the calculated 3D coordinates (post-processed data), other

geometrical parameters like angles, distances and centers of mass can be determined. We

can also use arithmetic, trigonometric and a rich set of other functions (integration,

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frequency analysis, transformations etc.) to get further information. For visualization

purposes, SimiMotion provides diagrams, stick figures and virtual reality representations.

Figure 5-13: 3D Data Calculation and Stick Diagram Representation of Jaw for Cameras 1 and 2

5.2.2. Experimental Setup

To conduct motion capture on animals an experimental setup is required that not only

restricts the motion of animals to track the markers accurately but also to ensure the

safety of the equipments involved. For this purpose, we used the standard sized cage with

nearly double the maximum volume estimate of a dog. All the subject studies presented

here are conducted only after explanation of all the procedures involved in the motion

capture to the owners concerned and providing the training to the subject for feeding

inside the cage for a certain period.

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The experimental setup used for carrying out the motion studies is shown below. As the

analysis is three dimensional, we used three cameras with appropriate lighting struts for

illumination.

Figure 5-14: Experimental Setup for Animal Motion Capture with Cameras

Figure 5-15: MoCap Workstation and Camera Setup for Animal Subjects

5.3. Inverse Dynamics Analysis of Musculoskeletal Models

5.3.1. Human Jaw Model (Mark de Zee Model) The jaw model developed by Zee, Dalstraet al. [65] contained two rigid bodies, the skull

and the mandible. Mass and mass moments of inertia of the mandible are based on the

work of Koolstra and Van Eijden (2005) and are listed in Table 1.

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Figure 5-16: Muscle Actuators of a Jaw Model Figure 5-17: Lateral View of the Mark de Zee Human Jaw Model

In this model, the mandibular fossa was modeled as a planar constraint, which was angled

30° downwards relative to the occlusal plane and canted 5° medially (Langenbach and

Hannam, 1999). Each TMJ had, therefore, the possibility to rotate in all three dimensions

and to translate in the specified plane (i.e. five degrees-of freedom). This resulted in a

mandible with four degrees of freedom. In reality, the condyles move between the

articular discs and TMJ is not constrained to move in a plane. However, this assumption

proved to be satisfactory for normal clenching and chewing jaw motions in humans and

in effect can be extended to animals.

Mass: 0.44 Kg

Ixx: 0.00086 kg m2

Iyy: 0.00029 kg m2

Izz: 0.00061 kg m2

Figure 5-18: Lower Mandible with the muscles attached used in Mark de Zee Model and Structural Properties

This model included 24 muscle-tendon units as discussed before that include— masseter

(superficial, deep anterior and posterior), temporalis (anterior and posterior), medial

pterygoid, lateral pterygoid (superior and inferior), anterior digastric, geniohyoid,

mylohyoid (anterior and posterior) on each side. The muscle model parameters for the

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model was obtained as reported in Zee, Dalstraet al. [65]. The muscle cross sectional area

and hence, their maximal forces can be obtained from the slices of CT scans available

online (http://vhp.med.umich.edu/HeadNeck.html).

Table 5-2: Muscle Model Parameters [Koolstra, 2002]

Muscles Max Force (N)

Muscle-length (mm)

CE length (mm)

SE length (mm)

Masseter 272.8 48 22.6 25.8

Anterior Temporalis 308 57.4 30.7 24.2

Temporalis 222 62.9 31.3 28.8

Posterior Temporalis 250 60.0 30.5 27

Medial Pterygoid 240 43.3 14.1 27.6

Lateral Pterygoid 112 27.2 22.3 9

Digastrics 46.4 51.9 42.6 3

There are also certain limitations that persist in this model. In reality, the TMJ condyles

are not constrained to move in a plane but in curvilinear paths. In addition, the model

does not take into account of ligaments, which play a major role only near the joint limits.

As long as, the jaw motion that we try to simulate do not necessitate the actuation near

the joint limits, ligaments and their contribution to joint forces can be ignored safely. We

ultimately would like to simulate this model for even abnormal chewing trajectories.

However, presently we use the MoCap data obtained by motion capture of a human

subject to drive the musculoskeletal jaw model as illustrated in the previous chapter.

5.3.2. Human Jaw Model (14 Muscle Actuator Model) We recreated the human jaw model with only those musculotendon actuators that connect

the head and skull and ignored the geniohyoid and mylohyoid muscles that connect the

mandible from the neck region to reduce the complexity as well as limit our study to

these muscles alone. Hence the simplified version of our model comprises of 14

musculotendon actuators— temporalis (anterior, medial and posterior), masseter,

digastric, medial and lateral pterygoid on each side. We implemented a TMJ model

similar to the one explained in Mark de Zee model. The insertion and origin points of the

model were obtained by approximate measurements made on a human skull and

mandible specimen and locations of the muscle attachment sites.

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The muscle model parameters and insertion and origin points that we used are given in

the table below:

Table 5-3: Muscle Fiber Length, Insertion and Origin Points of Human Jaw Model

Muscles Insertion Origin

Masseter 35.0,-49.0,-49.0 37.9,-52.3,20.0

Anterior Temporalis 36.0,-42.0,-6.0 -20.0,-50.0,67.0

Temporalis 36.0,-42.0,-6.0 -38.0,-60.0,45.0

Posterior Temporalis 36.0,-42.0,-6.0 -43.0,-70.0,0.0

Medial Pterygoid 25,-50,-3 30.0,-22.0,-19.0

Lateral Pterygoid 20.0,-50,-5.0 37.0,-18.0,-25.0

Digastric 85.0,-7.0,-84.0 -20.0,-50.0,-20.0

In our case, the tendon lengths were assumed to be around 4-5% of the overall

fiber length and calibrated for accurate results within AnyBody at one of the extreme

positions obtained from jaw trajectory data. We also tried to incorporate the wing shaped

temporalis muscle by approximating it to a three muscle-tendon units with varying lines

of action as shown in figure.

Figure 5-19: Temporalis Muscle Model (Green

Colored Muscle Tendon Units) Green: Temporalis (3 muscle-tendon units)

Blue: Medial Pterygoid, Grey: Lateral Pterygoid Red: Masseter, Pink: Digastric

Figure 5-20: Aerial View of Mandible with Different Muscle Groups

Green: Temporalis (3 muscle-tendon units) Blue: Medial Pterygoid, Grey: Lateral Pterygoid

Red: Masseter, Pink: Digastric

The jaw model was simulated for two different muscle models- simple muscle model and

Hill type muscle model (passive and active elements). A brief comparative study of these

two models for both the cases will be presented below mainly based on the muscle

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activities and forces are given below. To simulate the bite forces without any

measurements, we introduce a constant bite force and time varying bite force at the

incisor tooth tip and studied the increase in muscle forces.

5.3.3. Labrador Jaw Model A labrador dog jaw model was developed to realize our ultimate goal of

masticatory study in animals. The TMJ model was adopted from the human jaw studies

and applied to the labrador model. The STL file for visualization was obtained as

explained in the first section of this chapter.

Muscle Nomenclature: Green: Temporalis, Red: Masseter, Blue- Lateral Pterygoid White: Medial Pterygoid, Pink: Anterior Digastric

Figure 5-21: Side and Top view of Musculoskeletal Jaw Model of Labrador Dog

As before, the muscle insertion and origin points were measured on a real

specimen of labrador skull and mandible. The muscle model parameters implemented for

this case study is shown below in the table as well as other geometric parameters. Some

of these muscle model parameters are obtained by conducting the dissection experiment

on the real animal cadaver. The following section discusses the dissection experiments in

detail.

Table 5-4: Muscle Mass based on Turnbull [66]

Subject Masseter Anterior Temporalis0

Temporalis Posterior Temporalis

Medial Pterygoid

Lateral Pterygoid

Digastrics

Canine 20% 20% 20% 20% 4% 6% 10 %

Muscle Mass (gm)

40.185

40.185

40.185

40.185

8.307

12.056

21.7

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Real Animal Cadaver Dissection Experiment

The dissection experiment was conducted on the cadaver of the dog subject that has been

kept under the care of Laboratory Animal Facilities department, UB South Campus.

Proper approval forms were filed prior to performing dissection on the cadaver (dog)

under the supervision of Dr. Frank Mendel, Associate Professor in Pathology and

Anatomical Sciences. At the end of this cadaver study, we were able to visually study the

muscle architecture in the real musculoskeletal system, layer by layer attachments of

muscles, region of muscle attachments sites as well as we were able to determine the

muscle model parameters required to build our model, especially length of muscle fibers

and physiological cross section area. Also, to some extent, the approximate lines of action

of muscle forces could also be evaluated.

Figure 5-22: Measurement of fiber length for Masseter muscle

Figure 5-23: Digastric Muscle (Jaw opener)- dissected out of the jaw

This study also helped understand the level of approximations of our musculoskeletal

model. In reality, the jaw comprises of nearly 5-6 layers of muscles of each type

separated by cleavage planes that act independently at certain range of motion of the

joint. For example in Figure 5-22 , the masseter muscle comprised of 6 layers which in

effect act as each independent actuator to the TMJ joint and consequently the overall jaw

motion. But only one or utmost a few of these single muscle layer units are active during

a certain part of the cycle and the active layer switches from one to another (above or

below) based on the current joint position. The sequence of their actuation is in effect

controlled by the central nervous system- a complicated control system by itself. Hence

the real musculoskeletal jaw comprises of the large number of muscle layers which act

independently but not simultaneously, thereby increasing the complexity of the modeling

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techniques involved. However, in our model we use only one muscle fiber for the

masseter. Though using such a model enabled us to achieve comparable results with the

previous studies by Mark de Zee.

To better model the muscle parameters, we used the equation in (3.1.1)

Table 5-5: Muscle Model Parameters [Koolstra, 2002]

Muscles Max Force (N)

Muscle-length (mm)

CE length (mm)

SE length (mm)

Masseter 272.8 48 0.066 0.0033

Anterior Temporalis 308 57.4 0.035 0.00175

Temporalis 222 62.9 0.06 0.0030

Posterior Temporalis 250 60.0 0.065 0.00325

Medial Pterygoid 240 43.3 0.1 0.005

Lateral Pterygoid 112 27.2 .072 0.00360

Digastrics 46.4 51.9 0.185 0.00925

We also estimated the physiological cross sectional area property for all the muscles

based on the relative weights estimation in Horsman, Koopmanet al. [67]. To estimate the

mass of all the muscles, the anterior digastrics muscle was separated from the cadaver

and its wet mass was measured. Using the relative weights scaling given in Turnbull [66]

the mass of all other masticatory muscles were determined and provided the base to build

a more realistic model of the canine jaw.

Table 5-6: Muscle Fiber Length, Insertion and Origin Points of Labrador Jaw Model

Muscles Insertion Origin

Masseter 35.0,-50.0,-20.0 40,-65,30.0

Anterior Temporalis 25.0,-42.0,45.0 15.0,-25.0,65.0

Temporalis 25.0,-42.0,45.0 -20.0,-25.0,50.0

Posterior Temporalis 25.0,-42.0,45.0 -40.0,-25.0,35.0

Medial Pterygoid 0.0,-40.0,-35.0 37.0,-15.0,26.0

Lateral Pterygoid 0.0,-25,0.0 24.0,-10.0,0.0

Anterior Digastrics 155.0,-5.0,-5.0 -40.0,-20.0,-0.0

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In our model, we also simulate the biting force of the incisor by artificially introducing a

force vector at the tip- a constant force vector and a more realistic bite force model.

These bite force simulations can be considered similar to the real subjects chewing foods

of different texture and hardness. Hence, the importance of realistic modeling of bite

force is realized but it is left as a part of the future work. One such example can be found

as a part of Mark de Zee model. Other possibilities in this direction can be conducting

real subjects testing with implant force transducers in the incisor tooth tip to allow

accurate measurement of the biting forces, which can now be transferred to AnyBody as

a force and trajectory tracking optimization problem.

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6. Musculoskeletal Modeling Simulation and Results

6.1. Work envelope study using Posselt diagram 6.1.1. Human Jaw Model

In order to develop a dental simulator, it is significant to study the motion ranges

of the jaw in all the above cases. Since we are driving these models using motion capture

data, we conduct some preliminary analyzes on those trajectories first.

As discussed before about the Posselt diagram, we plot the path of incisor tooth tip in

sagittal, transverse as well as coronal plane. Thus, we will obtain 2D graphs of the work

envelope for the jaw motion case studies, which should be comparable with the Posselt

diagram. By this way, we can not only validate the accuracy of the jaw model but also

use that result while charting out the specifications for the parallel manipulator.

Figure 6-1: X and Z coordinates forming a Posselt Envelope in Sagittal Plane- Human Jaw

Figure 6-2: X and Y coordinates forming a Posselt Envelope in Transverse Plane- Human

Jaw

Other than the actuator forces and workspace envelope, it is also necessary to

have a quantitative estimate of jaw motion velocity and acceleration in each direction—

X, Y and Z. For this purpose, we will again monitor the incisor tooth tip point and

graphically obtain the motion rates using AnyBody plot tools as below.

From the graph (Figure 6-3), we see that velocity of the human jaw case study is about 4-

5 mm/s and acceleration is about 5 mm/s2. This value however, is obtained for normal

chewing action in humans. In order to build a generic simulator, we need to have similar

estimates for jaw models of different vertebrates performing under varying conditions.

However, we do not concentrate on such detailed studies in this thesis.

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To provide motivation for such a study in future, we will next analyze the jaw

model of the labrador and study its work envelope. It should be noted that Posselt

diagram is generally studied in case of human models only.

Figure 6-3: Velocity trajectory of incisor tooth tip point for Human Jaw

Figure 6-4: Acceleration trajectory of incisor tooth tip point for Human Jaw

6.1.2. Labrador Jaw Model We see that Posselt diagram for Labrador jaw model in sagittal plane as well as in

transverse plane as in Figure 6-6 and Figure 6-5. it can seen that the velocities and

accelerations of the jaw motion in both the cases- human jaw as well labrador jaw model

is less and hence these are not considered to be an important design variable for our study

in building the simulator.

Figure 6-5: X and Z coordinates forming a Posselt Envelope in Sagittal Plane- Labrador Jaw

Figure 6-6: X and Z coordinates forming a Posselt Envelope in transverse plane- Labrador

Jaw

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Figure 6-7: Velocity trajectory of incisor tooth tip point - Labrador Jaw

Figure 6-8: Acceleration trajectory of incisor tooth tip point - Labrador Jaw

6.2. Musculoskeletal Model Case Studies- Human Jaw We will discuss the parametric analysis of the musculoskeletal models and conduct a

series of studies based on the following figure. So to start with, we will first create a 3

temporalis muscle-tendon unit using Simple Muscle Model. We then proceed to do

systematic parametric studies as highlighted in the figure.

Case A.I.1: Simple Muscle Model, No bite force (free chewing motion)

Figure 6-9 (a)

Figure 6-10 (a)

Figure 6-9 (b)

Figure 6-10 (b)

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Figure 6-9 (c)

Figure 6-10 (c)

Figure 6-9: (a), (b) Elevator Muscle forces and activities (c) Muscle Lengths of RHS Muscles


After carefully examining the muscle forces and actuators, we found the muscle

activities of 14 muscle model in case A.I is close to the Mark de Zee Model as forces

obtained in both these models are in the similar range. Therefore, we can conclude that

the jaw model built using 14-muscle actuator is a good approximation and gives a better

base to study the effects of only those muscles attached to the skull and mandible which

actually are major players in masticatory motion. Hence, we progress to implement a

series of studies based on type of subjects, bite force simulation and muscle models as

indicated in Figure 6-18.

Figure 6-11: Parametric Studies for Mastication Musculoskeletal Analysis

The bite forces will be exerted on the pre molar node on the right hand side of the

mandible. We will implement a constant bite force as well as a realistic bite force in a

negative z direction. By realistic bite force, we mean it is maximum at minimal gape and

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minimum at maximal gape. These bite forces in effect actually simulate the different food

textures and by accurate modeling of this for different food and chewing cycles, we will

be able to obtain the muscle force estimates in a real sense. Obtaining accurate models of

the bite forces will be a part of our future work.

Case A.I.2: Simple Muscle Model, Constant Bite Force (200 N in –z)

Figure 6-12 (a)

Figure 6-13 (a)

Figure 6-12 (b)

Figure 6-13 (b) Figure 6-12: (a), (b) Elevator Muscle forces and

activities for Case A.I.2 Figure 6-13: (a), (b) Depressor Muscle forces and

activities for Case A.I.2

Figure 6-14: TMJ Reaction forces for Case A.I.2

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Case A.I.3: Simple Muscle Model, Realistic Bite Force

Figure 6-15 (a)

Figure 6-16 (a)

Figure 6-15 (a)

Figure 6-16 (b)

Figure 6-15: (a), (b) Elevator Muscle forces and activities for Case A.I.3

Figure 6-16: (a), (b) Depressor Muscle forces and activities Case A.I.3

Figure 6-17 (a)

Figure 6-17 (b)

Figure 6-17: (a) TMJ Reaction forces (b) Simulated bite force for Case A.I.3

We see from the plots that most of the jaw opening action is performed by the pair

of digastric muscles and muscles effort for the jaw closing action is distributed between

the three temporalis units, medial pterygoid as well as the masseter. Also the model

response for the simulated chewing activities provided realistic muscle force and activity

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plots. From the cases Case A.I.1, Case A.I.2 and Case A.I.3 we notice that muscle forces

and activity levels have increased due to the biting force at the incisor tooth tip and more

realistic muscle forces can be realized in the later case using the Simple Muscle model.

The model response proved to be better for hill type muscle models compared to simple

muscle model.

Case A.II.1: Muscle Model 3E, No Bite Force (free chewing motion)

Figure 6-18 (a)

Figure 6-19 (a)

Figure 6-18 (a)

Figure 6-19 (b)

Figure 6-18: (a), (b) Elevator Muscle forces and activities (c) Muscle Lengths of RHS Muscles for

Case A.I.2

Figure 6-19: (a), (b) Depressor Muscle forces and activities (c) Muscle Lengths of LHS for

Case A.I.2

Using the Hill type muscle models for the human mastication models, provide a more

accurate way to model the passive elements especially tendons, ligaments etc. and

monitor the passive muscle forces to better understand the model response for jaw

trajectories.

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Case A.II.2: Muscle Model 3E, Constant Bite Force (100 N in –z)

Figure 6-20 (a)

Figure 6-21(a)

Figure 6-20 (b)

Figure 6-21(b)


Figure 6-21: (a), (b) Depressor Muscle forces and activities for Case A.I.2

Figure 6-22: TMJ Reaction forces Case A.II.3

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Case A.II.3: Muscle Model 3E, Realistic Bite Force

Figure 6-23 (a)

Figure 6-24(a)

Figure 6-23(b)

Figure 6-24(b)


Figure 6-24: (a), (b) Depressor Muscle forces and activities for Case A.I.3

Figure 6-25 (a)

Figure 6-25 (b)

Figure 6-25: (a) TMJ Reaction forces (b) Simulated bite force for Case A.II.3

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6.3. Musculoskeletal Model Case Studies- Labrador Dog

Case B. I. 1: Simple Muscle Model, no bite force (free chewing motion)

Figure 6-26 (a)

Figure 6-27 (a)

Figure 6-26 (b)

Figure 6-27 (b)

Figure 6-26 (c)

Figure 6-27 (c)

Figure 6-26: (a), (b) Elevator Muscle forces and activities (c) Muscle Lengths of RHS Muscles for

Case B.I.1

Figure 6-27: (a), (b) Depressor Muscle forces and activities (c) Muscle Lengths of LHS for Case B.I.1

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Case B.I.2: Temporalis Muscle Tendon Unit, Constant Bite Force The muscle behavior for labrador case did not vary considerably for Hill type

muscle model (Muscle model 3E in AnyBody) in our preliminary analyses and hence the

results for the first case only are shown below.

Figure 6-28 (a)

Figure 6-29 (a)

Figure 6-28 (b)

Figure 6-29 (b)


Figure 6-29: (a), (b) Depressor Muscle forces and activities for Case B.I.2

Figure 6-30: TMJ Reaction forces for Case B.I.2

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Case B.I.3: Temporalis Muscle Tendon Unit, Realistic Bite Force

Figure 6-31 (a)

Figure 6-32(a)

Figure 6-31 (b)

Figure 6-32 (b)


Figure 6-32: (a), (b) Depressor Muscle forces and activities for Case B.I.3

Figure 6-33 (a)

Figure 6-32 (b)

Figure 6-33: (a) TMJ Reaction forces (b) Simulated bite force for Case B.I.3

Similar to human jaw case, the muscle activities seem to increase proportional to

the bite forces exerted at the incisor tooth tip of the labrador jaw. Three cases are

implemented for labrador mastication model— no bite force, constant bite force and a

realistic bite force.

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7. Masticatory Simulator- Analysis and Results

7.1. Inverse Kinematics and Jacobian Based Workspace Analysis of 6-DOF R-U-S Manipulator

7.1.1. Inverse Kinematics Simulation The 6 DOF crank type parallel manipulator (otherwise known as 6 DOF R-U-S

manipulator) is not so widely used among the variants of the Stewart platform

manipulator. In this work however, we will concentrate on the implementation of inverse

kinematics for such manipulator. The inverse kinematics routine developed in MATLAB

is first tested against standard trajectories— point, line, circle and ellipse. After carrying

our the validation, the manipulator is driven using the kinematic jaw trajectories obtained

from MoCap system after carefully transforming the points to the platforms as explained

in the Sec. 4.11.1.

Case 1: Line and Curve Tracking:

Figure 7-1 (a)

Figure 7-2 (a)

Figure 7-1 (b)

Figure 7-2 (b)

Figure 7-1: R-U-S Manipulator Configuration for Validation Test: (a) Line along Z (b) 3D Sine

curve

Figure 7-2: R-U-S Joint trajectory for Validation Test: (a) Line along Z axis (b) 3D Sine curve

104

Case 2: Circle and Ellipse Tracking

Figure 7-3(a)

Figure 7-4(a)

Figure 7-3(b)

Figure 7-4(b)

Figure 7-3 (c)

Figure 7-4(c)

Figure 7-3: R-U-S Manipulator Configuration for Validation Test: (a) Circle in YZ plane (b) Circle in

XY (c) Ellipse in XZ plane

Figure 7-4: R-U-S Joint trajectory for Validation Test: (a) Circle in YZ plane (b) Circle in XY (c)

Ellipse in XZ plane

7.2. Validation using Visual Nastran for R-U-S Case A similar system is also deployed in SolidWorks/ Visual Nastran framework for open

loop simulation of tracking desired jaw motion trajectories. We start with the sample case

studies before implementing the jaw motion trajectory.

105

Case 1: Point

Figure 7-5:Point Tracking Simulation of R-U-S in

Visual Nastran Figure 7-6:Point Tracking Joint Angle

Trajectories from MATLAB Code

Case 2: Straight line

Figure 7-7: Line Tracking Simulation of R-U-S in

Visual Nastran Figure 7-8: Line Tracking Joint Angle


Case 3: Circle

Figure 7-9: Circle Tracking Simulation of R-U-S

in Visual Nastran Figure 7-10: Circle Tracking Joint Angle


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We were able to validate our mathematical model based on the joint trajectories’

plots obtained from visual Nastran with some angle offsets that was necessary to be given

for creating the CAD model.

7.3. Constant Orientation Workspace Analysis By constant orientation workspace, it is implied as a set of permissible positions

for the centre of the mobile platform while the platform is kept at a constant orientation,

normally at zero roll-pitch-yaw orientation. We will look at this type of workspace

analysis of the 6-DOF R-U-S manipulator. The workspace considered is the 3D

cylindrical space swept by the circle of maximum radius 0.06 m with center at origin (0,

0, 0) and height ranging from 0.20 to 0.45 m.

The jacobian analysis is performed for the following set of geometric parameters

and the jacobian-based measures of the manipulability of the 6-DOF manipulator are

determined namely the Yoshikawa Measure and Condition number. From the plots we

were able to verify that within the region considered the manipulator does not attain

singular configuration (i.e. jacobian matrix does not loose rank). It means that each

revolute joint is able to actuate independently and affect the each dof of the top platform.

Table 7-1: Manipulator Parameters for Set 1: (all linear dimensions in meters and angular dimensions in radians)

Base plate radius, l0 0.1 AB Link length, l1 0.1 BC Link length, l2 0.1693 Top plate radius, l3 0.1

107

Figure 7-11 (a)

Figure 7-12(a)

Figure 7-11 (b)

Figure 7-12(b)

Figure 7-11 (c)

Figure 7-12(c)

Figure 7-11: (i), (ii), (iii): Condition number based measure of manipulability plotted along the

vertical z-axis for every point in the 2D circular region: 0<r<0.06 at z=0.20 plane

Figure 7-12: (i), (ii), (iii): Yoshikawa measure of manipulability plotted along the vertical z-axis

for every point in the 2D circular region: 0<r<0.06 at z=0.20 plane

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Figure 7-13 (a) Figure 7-14 (a)

Figure 7-13 (b)

Figure 7-14 (b)

Figure 7-13 (c)

Figure 7-14 (a)

Figure 7-13: (a), (b), (c): Condition number based measure of manipulability plotted along the vertical z-axis for every point in the 2D circular region: 0<r<0.06

at z=0.30 plane

Figure 7-14: (a), (b), (c): Yoshikawa measure of manipulability plotted along the vertical z-axis for every point in the 2D circular region: 0<r<0.06 at

z=0.30 plane

109

Figure 7-15 (a) Figure 7-16 (a)

Figure 7-15 (b) Figure 7-16 (b)

Figure 7-15 (c) Figure 7-16 (c) Figure 7-15: (a), (b), (c): Condition number based

measure of manipulability plotted along the vertical z-axis for every point in the 2D circular

region: 0<r<0.06 at z=0.40 plane

Figure 7-16: (a), (b), (c): Yoshikawa measure of manipulability plotted along the vertical z-axis


110

Figure 7-17(a) Figure 7-18 (a)

Figure 7-17(b)

Figure 7-18 (b

Figure 7-17(b)

Figure 7-18 (c)

Figure 7-17: (a), (b), (c): Condition number based measure of manipulability plotted along the

vertical z-axis for every point in the 2D circular region: 0<r<0.06 at z=0.45 plane

Figure 7-18: (i), (ii), (iii): Yoshikawa measure of manipulability plotted along the vertical z-axis


111

7.4. Inverse Kinematics and Jacobian Based Workspace Analysis of 6-DOF P-U-S Manipulator

7.4.1. Inverse Kinematics Simulation For P-U-S manipulators we implement the kinematics as shown in Figure 7-19, Figure

7-20, Figure 7-21 and Figure 7-23. Based on the inverse kinematics equations derived in

Sec. 4.6, the kinematic models were implemented and validated as before for standard

cases. We then proceed to do a workspace analysis based on jacobian based measures to

assess the manipulability of the P-U-S manipulator.

Line and Curve Tracking:

Figure 7-19 (a)

Figure 7-20 (a)

Figure 7-19 (b)

Figure 7-20 (b)

Figure 7-19 (c)

Figure 7-20 (c)

Figure 7-19: P-U-S Manipulator Configuration for Validation Test: (a) Line along Z axis (b)

Straight line (c) 3D Sine curve

Figure 7-20: P-U-S Joint trajectory for Validation Test: (a) Line along Z axis (b) Straight line (c) 3D

Sine curve

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Circle and Ellipse Tracking

Figure 7-21 (a)

Figure 7-23 (a)

Figure 7-21 (b)

Figure 7-23 (a)

Figure 7-21 (c)

Figure 7-23 (c) Figure 7-21: Maximum Reachable Characteristic

Values of the Manipulator 7-22: P-U-S Manipulator Configuration for Validation Test: (a) Circle in XY Plane (b) Circle in YZ Plane (c)

Ellipse in YZ plane

Figure 7-23: P-U-S Joint trajectory for Validation Test: (a) Circle in XY Plane (b) Circle in YZ Plane

(c) Ellipse in YZ plane

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7.4.2. Constant Orientation Workspace Analysis Table 7-2: Maximum Reachable Characteristic Values of the Manipulator

Base plate radius, l0 0.1 m BC Link length, l2 0.35 m Top plate radius, l3 0.1 m Offset angle for top and base plate gimbal point, theta0 pi/4 rad

The jacobian analysis is performed for the following set of geometric parameters and the

jacobian-based measures of the manipulability of the 6-DOF manipulator are determined

namely the Yoshikawa Measure and Condition number.

Figure 7-24 (a)

Figure 7-25 (a)

Figure 7-24 (b)

Figure 7-25 (b)

Figure 7-24 (c)

Figure 7-25 (c) Figure 7-24: (a), (b), (c): Condition number based

measure of manipulability plotted along the vertical z-axis for every point in the 2D circular region: 0<r<0.06

at z=0.45 plane


z=0.45 plane

114

Figure 7-26(a)

Figure 7-27(a)

Figure 7-26(b)

Figure 7-27(b)

Figure 7-26(c)

Figure 7-27(c)


at z=0.60 plane

Figure 7-27: (a), (b), (c): Yoshikawa measure of manipulability plotted along the vertical z-axis for every point in the 2D circular region: 0<r<0.06 at z=0.60 plane

115

Figure 7-28 (a)

Figure 7-29 (a)

Figure 7-28 (b)

Figure 7-29 (b)

Figure 7-28 (c)

Figure 7-29 (c)


at z=0.75 plane


z=0.75 plane

116

7.5. Visual Nastran Implementation P-U-S Manipulator We created a CAD model of a P-U-S manipulator in SolidWorks and used visual Nastran

to run an open simulation project for our analysis. We initially validated the model

against standard trajectories before driving the platform using our jaw motion trajectories.

We show the implementation of the series of test for the jaw motion case studies in

Figure 7-30, Figure 7-31 and Figure 7-32.

Case 1: Human Jaw Motion Trajectory

Figure 7-30 (a)

Figure 7-30 (b)

Figure 7-30: Visual Nastran Implementation of P-U-S with Human Jaw motion Trajectory Input (a) Visual Nastran model imported from SolidWorks (b) Actuator force trajectory Vs time

117

Case 2: Labrador Jaw Motion Trajectory

Figure 7-31 (a)

Figure 7-31 (b)

Figure 7-31: Visual Nastran Implementation of P-U-S with Labrador Jaw motion Trajectory Input (a) Visual Nastran model imported from SolidWorks (b) Actuator force trajectory Vs time

118

Case 2: Bulldog Jaw Motion Trajectory

Figure 7-32 (a)

Figure 7-32 (b)

Figure 7-32: Visual Nastran Implementation of P-U-S with Bulldog Jaw motion Trajectory Input (a) Visual Nastran model imported from SolidWorks (b) Actuator force trajectory Vs time

119

7.6. Parametric Study of P-U-S Manipulator Implementing dynamic model of these types of manipulator which have a passive

link in each of their legs is complicated. Hence in our study, we explore an alternative

method to specify the parallel manipulator based on finding the limits on workspace,

velocity and forces by considering the simple geometry as well as the free body diagram

at extreme positions. However, for implementing inverse dynamic models for P-U-S

manipulators interested readers are referred to Zhao and Gao [58] based on virtual work

principle to determine the actuator forces.

In section 4.9 we have derived all the equations to obtain maximum reachable workspace,

speeds as well as forces about the base reference frame axes and expressed them in terms

of geometrical and actuator parameters. The setup now is ideal to conduct a series of

parametric studies to determine the optimum values for these undecided parameters that

would satisfy our application requirements.

7.6.1. Simplified Representation of Parametric Analysis We choose two sets of parameters— geometric and actuator type as discussed

before in the table to study their effects on the output variables namely Fz, Pz and Vz. To

keep our parametric study generalized for manipulator of different sizes, we seek to use

dimensionless parameters in our analysis. Before arriving at the final specification for the

manipulator system, we conduct the parametric studies based on these dimensionless

quantities— (a) R/r ratio, (b) l/r ration, (c) q0 and Tq0. Following figure illustrates

the types of parametric studies we have carried out for this purpose.

As a part of validation of the values of geometric and actuator parameters, we will

conduct a final parametric studies to study the effects of the dimensionless quantities on

workspace, speed and forces in other two directions (X and Y) for validation.

Table 7-3: Parametric Studies Conducted to Identify Optimal Values for Jaw Motion Simulation Fz (N) Pz (m) Vz (m/s)

R/r ratio I.a I.b I.c l/r ration II.a II.b II.c

qo and Tqo III.a III.b III.c

120

To start with, we assume the values of the actuator and geometric parameters as below:

Table 7-4: Values of Platform Parameters Parameters Values

Fq (N) 20 Vqo (m/s) 0.2 m/s qo (m) 0.08

Actuator

Tqo (m) 0.05 Geometric l (m) 0.3 m

7.6.2. Parametric Study I: Effects of R/r and r on Fz, Pz and Vz For this parametric study, the ranges of values used were— r/R ratio ∈[1, 3] and r

∈[0.05, 0.35]. We can see from the equations (4.9.3) and (4.9.6) that there is no effect of

R/r ratio and r on Pz and Vz, hence they remain constant at 0.04575 m and 0.08729 m/s

respectively as shown in the surface plot.

Figure 7-33: Workspace in Z, m Figure 7-34: Speed in Z, m/s

Figure 7-35: Maximum Force in Z, Newton

121

However, Fz varies according to the equation (4.9.15) and has a non-linear

relationship with R/r as well as r. From the plot, we see that Fz continuously decreases

and does not exist when the inverse of sine or tangent of the resulting angle is not real or

invalid as indicated by the flat region. With the knowledge of our workspace as well as

the approximate estimation of the ground area, we fix the value of r at about 0.2- 0.3 m.

We also choose r/R ratio of 1 as the maximum attainable force in Z direction is highest

along the line r/R=1. The value of Fz also falls in the desired range of above 200 N.

Similar trends are also observed with torque about X and Y axes as they depend on Fz

and r.

Figure 7-36: Pitch Torque, Nm Figure 7-37: Roll Torque, Nm

Hence, these torques also are at maximum when r/R ratio is equal to one. For yaw

torque (torque about Z axis), we see that it stays constant with r/R ration but increases as

r increases.

Figure 7-38: Yaw Torque, Nm

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7.6.3. Parametric Study II: Effects of /l r and r on Fz, Pz and Vz Since we fixed the value of r/R ratio at 1, we proceed further to the second parametric

study where we look for l/r and r in the ranges of [1, 2] and [0.15, 0.45] respectively.

Giving l/r value greater than 1, maximum force in Z direction is higher. Now, as evident

from the equations (4.9.3) and(4.9.6), Pz and Vz depend on l alone. However, for Pz to be

positive and Vz to exist, l needs to satisfy the condition—

00 0

0 0

,l ql q Tq

l q Tq⎫> ⎪ > +⎬> + ⎪⎭

Therefore, as long as the second condition is satisfied, Pz and Vz can be calculated from

the equation to complete a 3D plot. Using this, we will be able to determine the optimal

value of l/r.

Figure 7-39: Workspace in Z, m Figure 7-40: Speed in Z, m/s

Figure 7-41: Force in Z, Newton

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However, Fz increases continuously and non-linearly with both l/r as well as r. Based on

our bite force estimate, we can now find the value of l/r as 1.5 or higher corresponding to

r=0.25 (according to parametric study I) and Fz = 489.5 N app. and maximum value of

Pz= 0.1996 m.

7.6.4. Parametric Study III: Effects of q0/ Tq0 and r on Fz, Pz and Vz To determine the actuator parameters— nominal length as well as the travel of the

actuator, we use the geometric parameter values obtained from the previous studies. From

Fz equation derived before, for lower values of q0, the force in Z direction increases but

smaller q0 ( / 2l< ) would result in lower Vz. Hence, an optimal value of q0 should be

reached based on this parametric study.

So, we chose to vary Tq0 from 0.25l to 0.707l and studied the effects of Fz, Vz and Pz.

From the figure we see that, when the top platform aligns itself with the base platform,

0 0l q tq= + (4.11.8)

However, in our case, we would give a minimum height between the top and base

platform by limiting the value of tq0 as,

( )0 01.25l q tq= + (4.11.9)

By this way, we can make sure that the values of Tq0 and q0 calculated are real and valid.

Plotting the values of Px and Pz against q0 we can find the optimal value for the nominal

length of the actuator (maximum value is 0.2192 m).

Figure 7-42: Workspace in Z, m Figure 7-43: Speed in Z, m

124

Figure 7-44: Workspace in X, m/s Figure 7-45: Force in Z, N

From the plots, we can see the optimal value of q0 is approximately 0.66 (2l )= 0.175 m

and Tq0 is approximately 0.15 where Fz = 227 N. Combining all the above, we now have

formed a initial set of specifications for the P-U-S manipulator case which needs to be

validated for workspace, speed and forces in other directions (X and Y).

Table 7-5: Preliminary Specification Values of the Manipulator Symbol (units) Description Values

r (m) Platform radius 0.25

R(m) Base radius 0.25

L (m) Arm length 0.375

h (m) Actuator nominal height 0.33

q0 (m) Actuator nominal length 0.175

Tq0 (m) Actuator travel +/- 0.15

Fq (N) Actuator maximum force 20

Similar to above, we created the parametric studies for each of the variables- roll, pitch,

yaw and the corresponding velocities against:

a. r/R ratio and r

b. l/r ratio and r

125

7.6.5. Parametric Sweep of Workspace Variables against r and r/R ratio:

Figure 7-46: Pitch/ Roll angles, rad and velocities, rad/s

Figure 7-47: Yaw angles, rad and velocities, rad/s

7.6.6. Parametric Sweep of Workspace Variables against r and r/R ratio:

Figure 7-48: Pitch angles, rad and velocities, rad/s

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Figure 7-49: Yaw angles, rad and velocities, rad/s

In all these analysis, the forces in X and Y direction remain constant at 80 and 69.282 N

respectively as they depend only on Fq as well as the geometry of the base platform. For

the torques about X, Y and Z-axes, we get the following plots.

Figure 7-50: Torque about X, Nm Figure 7-51: Torque about Y, Nm

Figure 7-52: Torque about Z, Nm

127

Since for our application, high velocities are not required, all the workspace velocities as

well as forces are compatible with the jaw motions and forces. Thus we, arrive at the final

specification for the P-U-S manipulator as:

Table 7-6: Final Specification of the Manipulator Symbol (units) Description Values

r (m) Platform radius 0.25

R(m) Base radius 0.25

L (m) Arm length 0.375

h (m) Actuator nominal height 0.311

q0 (m) Actuator nominal length 0.208

Tq0 (m) Actuator travel +/- 0.125

Fq (N) Actuator maximum force 20

The output values table corresponding to these geometric and actuator parameters

obtained from the equations derived in Section 4.9.1 are given below:

Table 7-7: Maximum Reachable Characteristic Values of the Manipulator

X Y Z Roll Pitch Yaw

Maximum Force, Torque (N, N.m) 69.282 80.000 227.411 32.824 37.902 30.000

Workspace (m, deg) 0.300 0.300 0.187 36.809 36.809 50.193

Maximum Speed (m/s, deg/s) 0.200 0.200 0.106 45.837 45.837 24.187

Thus we obtained the set of geometrical and actuator parameters’ values based on our

requirements of tracking the jaw motions of different vertebrates precisely.

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8. Conclusion:

8.1. Summary We developed a biomechanical model with accurate model of TMJ for

performing inverse dynamics analysis on animal as well as human jaws. This helped us to

perform quantitative analysis on these models and also study the muscle behavior under

different circumstances. The necessity of dissection study was realized to estimate muscle

parameters for the animals which was conducted in Animals Facility, UB South campus.

With a more realistic model, we were able to study the jaw motion envelope of human

and canine jaws as a part of our model validation process.

This model provided a basis on which we were able to implement parallel

manipulator architectures for the jaw motion simulators. To improve the performance of

the manipulator, we setup a motion capture system and conducted real subjects testing to

extract the jaw motion trajectories. We conducted on field experiments to capture the

motions of animals to obtain the kinematic trajectories. These trajectories were then used

to drive the musculoskeletal model case studies with different biting forces models to

understand the jaw behavior. We also created 3D CAD models of dentitions using laser

3D Scanner as well as from CT scan slices which were used for biomechanical analyzes.

This dentition can also be used while building the physical prototype of the jaw

simulator. For analysis of parallel manipulator architectures, we implemented the

kinematics for R-U-S and P-U-S manipulator and validated the model to accurately track

the jaw trajectory. We also developed the model in SolidWorks – VisualNastran that

helped us to validate our kinematics equations which in turn helped us to track the jaw

trajectories accurately. Finally to conclude the work, a parametric analysis on workspace,

129

velocities as well as forces parameters were conducted to determine the physical

specification of the jaw simulator.

8.2. Future Work: As a part of future work, we would like to have the inverse dynamic model of P-

U-S manipulator as well as R-U-S manipulator implemented in real time to allow us to

work out a much more comprehensive analysis of the system for jaw motion tracking.

With proper development of the dynamics code, simultaneous force and trajectory

tracking can also be accomplished which will be very useful for the concerned

application (i.e.) the implementing the dynamic level control of manipulator so that they

can track the desired jaw trajectory as well as apply the necessary force to bite food

particles of different texture. Such setting would enable us to conduct mastication

experiments of food chewing as well food texture for better understanding of the chewing

behavior.

However, for this we need to know the accurate estimate of the bite force which

will also be a part of our proposed future work. This can be implemented by using the

implant force transducer devices for the subjects testing so that an accurate model of

force estimate can be obtained. As mentioned in the section 5.3.3, the biomechanical

model developed in this work does not exactly resemble the real jaw model where the

muscles are attached as layers to themselves before attaching to the corresponding

segments. in order to achieve such an accurate biomechanical model, a comprehensive

dissection study is really important by which we can determine each individual muscle

parameters separately. In short, improving the model realism as well as accuracy of

muscle parameters for the model will be included in our future work.

130

Also to realize the goal of building a generic jaw simulator, the motion capture

experiments should be performed across different subjects as well as in more numbers so

as to study the cumulative muscle models and behaviors for each kind of subjects which

would help the work envelope study for the manipulator and hence, improve the design

and control of the simulator.

131

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