Oral Presentation in a Comm 2013

8/13/2019 Oral Presentation in a Comm 2013

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Vibration Suppression of a Cart-Flexible Pole

System using a Hybrid Controller

Presented by

Dr. Ashish SinglaAssistant Professor

Mechanical Engineering Department

Thapar University, Patiala

INDIA

Oral Presentation @ iNaCoMM-2013 Dec 19, 2013


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OUTLINE

Fig. Single-link flexible manipulator.

Dynamic Modeling

Command Shaping

Robustness of the Shaper

Controller Design

Results and Discussions

Shaper

RoboticManipulator

+ n( , ) + K

+

+

Kopt

K1C

-1

Ko2L

G

F

H

Compensator

Inverse Dynamics

xds

-

-

xd Inverse

Kinematics

TrajectoryGenerator

inCartesian

Space

xds

xd

yd


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Compensatorto PlantError Control Output

(Regulator + ROO) (Flexible Model)

Feedback Loop(L)

Input

-+

Inverse Dynamics

Feedforward Path (NL)

++

Ufb

Uff

to Input

Shaper

Input

Shaper

Reference

Trajectory

OVERALL SCHEME


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DYNAMIC MODELING

Small angle approximation.

Euler-Bernoulli beam theory

Lagrange approach

Each link - finitenumber of elements.

Fig. Moving cart with a flexible pole


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COMMAND SHAPING

Convolution


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Basis of Command Shaping


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Basis of Command Shaping . . .


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Basis of Command Shaping . . .


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System

s Response to Multiple Impulses (Superposition Principle)

Impulse Response of a Second Order Under-damped System

=

where

Development of Constraint Equations


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Residual Vibration Amplitude (in %)

Amplitude of Unit Impulse

Residual Vibration Amplitude


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(1)

(2)

(3)

(4)

Two Impulse Sequence

Constraints: 2 Eqs, 4 Unknowns


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Robustness of Input Shaper

Three Impulse Sequence (ZVD)

Four Impulse Sequence


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Sensitivity Curve


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CONTROLLER DESIGN

Linear observer based optimal feedback term

Nonlinear Feedforward Control

The feedforward control action is given as

Best utilized for fast and repetitive moves.Efficientsolution offlinecomputation (along the nominal trajectory).

Control action

Nonlinear feedforward Control

Input shaper

The controller consists of three parts


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Linear Feedback Control

estimation of the unmeasurable vector.

observer gain matrix solution of the followingstate equation

measured/unmeasured variables.

Compensator based on reduced-order observer

Plant state-vector

Plants state equation (partitionedform)

Estimated state-vector

ROO Dynamics


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Optimal feedback gain matrix Kopt is obtained by

Control Law

min

Error Dynamics

where e(t) = xd(t)x(t) = tracking erroreo2(t) = x2(t)xo2(t) = estimation error.


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Complete Tracking System with ROO and Shaper

Shaper

Robotic

Manipulator

+ n( , ) + K

+

+Kopt

K1C-1

Ko2L

G

F

H

Compensator

Inverse Dynamics

xds

-

-

xd Inverse

Kinematics

TrajectoryGenerator

inCartesian

Space

xds

xd

yd


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CASE STUDY : Cart with a flexible pole

Control objective is to move the cartby one meter, while not letting thependulum to fall i.e. xd= {1 0 0 0 0 0}

T

OL= [0, 0,5.89i,639i].

Double pole at origin- unstable plant -cannot be controlled using uff only.

ufb is essential and calculated usingoptimal control theory

The only input u(t) is the horizontalforce applied to the cart and threeoutputs are x, 1 and 2 .

Reduced-order observer is designed

by taking the carts position as theonly output variable. Fig. Moving cart with a flexible pole


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Mass of the cart in Kgs M 1

Mass of the first

(second) link in Kgs

m1 (m2) 0.05

Length of the first

(second) link in meters

l1 (l2) 1

Spring stiffness at first

joint in Nm/rad

k1 5

Spring stiffness at

second joint in Nm/rad

k2 500

State weighting

matrix

Q diag([5000 500 0 20 0 0])

Control cost

matrix

R [50]

Vector of desired

ROO poles

v [-20 -2260i -6069i]T

Optimal gain

matrix

Kopt [10 -1.29 -0.43 4.80 0.02 -0.01]

Impulse

First Mode Second Mode Third Mode

Mag. Time Mag. Time Mag. Time

1 0.9183 0.0000 0. 2792 0.0000 0.250 0.0000

2 0. 0799 1.4694 0. 4984 0. 5350 0.500 0.0049

3 0.0017 2.9387 0. 2224 1.0701 0.250 0.0098

System Parameters Controller Parameters

Wn= [ 3.02 5.87 638.92]T (Hz), = [0.706 0.0362 1e-5]TShaper Parameters

Parameters


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Plot 1(a) : One Mode Shaping Desired Position


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Plot 1(b) : One Mode Shaping Position Response


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Plot 1(c) : One Mode Shaping Velocity Response


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Plot 1(d) : One Mode Shaping Estimation Error


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Plot 2(a) : Two Mode Shaping Position Response


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Plot 2(b) : Two Mode Shaping Velocity Response


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Plot 3(a) : Three Mode Shaping Position Response


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Plot 3(b) : Three Mode Shaping Velocity Response


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ANIMATION

Unshaped Response 2-Mode Shaped Response


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Table : Level of Vibration/Force Reduction

Variable Bang-Bang

One Mode Shaping Two Mode Shaping Three Mode Shaping

Value % Reduction Value % Reduction Value % Reduction

|x(t)max| 1.0614 1.0112 4.73 % 1.0000 5.78 % 1.0000 5.78 %

|1(t)max| 13.0915 12.030 8.11 % 2.488 80.99 % 2. 488 81.00 %

|2(t)max| 0.0429 0.0393 8.32 % 0.0086 80.01 % 0.0078 81.83 %

|dx(t)max| 1.3069 1.2002 8.17 % 0.7903 39.53 % 0.7901 39.54 %

|d1(t)max| 1.1555 1.0613 8.16 % 0.2349 79.67 % 0.2272 80.33 %

|d2(t)max| 0.0616 0.0585 10.05 % 0.0148 75.97 % 0.0089 86.36 %

|u(t)max| 10.00 8.9206 10.79 % 4.3165 56.83 % 4.1920 58.08 %


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SUMMARY

The control scheme (LQR-ROO-CS) is sufficient to track large movements of flexiblerobotic manipulators.

The nonlinear feedforward control is derived using inverse dynamics, which provides

the major contribution in the control effort during tracking problems. The feedbackloop is designed with linear observer based optimal regulator which ensuresstabilization and performance objectives. Finally, the command shaping isincorporated to obtain the desired non-oscillatory response.

Large reductions in vibration levels as well as in input torque magnitudes are observed

when compared to controllers implemented without command shaping.


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FUTURE DIRECTIONS

The work has been implemented successfully on planar manipulators which can beextended to spatial manipulators.

The work can also be extended to redundant manipulators for Obstacle avoidance,

Increase in manipulability, Singularity avoidance,

Fault-tolerant design.

The feedback channel can be made more robust using Kalman filters.


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Thank You


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REFRENCES

M. O. Tokhi, Z. Mohamed, and M. H. Shaheed. Dynamic characterization of a flexiblemanipulator system. Robotica, 19(5):571580, 2001.

Z. Mohamed and M. O. Tokhi. Vibration control of a single-link flexiblemanipulator usingcommand shaping techniques. Proceedings of the Institution of Mechanical Engineers.

Part I: Journal of Systems and Control Engineering, 216(2):191

210, 2002.

NC Singer and WP Seering. Preshaping command inputs to reduce system vibration.Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME,112(1):7682, 1990.

WE Singhose, WP Seering, and NC Singer. Shaping inputs to reduce vibration: a vectordiagram approach. IEEE International Conference on Robotics and Automation, pages922927, 1990.W. Singhose, W. Seering, and N. Singer. Residual vibration reduction using vectordiagrams to generate shaped inputs. Journal of Mechanical Design, Transactions Of theASME, 116(2):654659, 1994.

J. Vaughan, A. Yano, and W. Singhose. Comparison of robust input shapers. Journal ofSound and Vibration 1 : 81 2008.


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Arun Banerjee and William Singhose. Command shaping in tracking control of a two-linkflexiblerobot. Journal of guidance, control, and dynamics, 21(6):10121015, 1998.

Z. Mohamed and M. O. Tokhi. Command shaping techniques for vibration control of aflexiblerobot manipulator. Mechatronics, 14(1):6990, 2004.

M. Z. M. Zain, M. O. Tokhi, and Z. Mohamed. Hybrid learning control schemes with inputshaping of a flexiblemanipulator system. Mechatronics, 16(3-4):209219, 2006.

WE Singhose and NC Singer. Effects of input shaping on two-dimensional tra jectoryfollowing. Robotics and Automation, IEEE Transactions on, 12(6):881887, 1996.

M. Romano, B.N. Agrawal, and F. Bernelli-Zazzera. Experiments on command shapingcontrol of a manipulator with flexiblelinks. Journal of Guidance, Control, and Dynamics,25(2):232239, 2002.

A. Tewari. Modern control design with MATLAB and SIMULINK. Chichester; Wiley.

J J Craig Introduction to Robotics: Mechanics and Control Addison Wesley LongmanP bli hi C I B MA USA 8

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