3ppr_mechanism[1].pptx

7/26/2019 3ppr_mechanism[1].pptx

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Dynamic model and disturbance

observer based nonlinear positiontracking control of a new XYz motion

platform

M.Santhakumar , Ramavatar Meena, Jayant Kr. Mohanta and Sandip patidarCenter for robotics and control ,Indian Institute of Technology ,Indore


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Introduction

What are XYz motion platforms?

Difference between serial and parallel motion

platforms

Need(s) of a proposed platform

Advantages and limitations of the proposed

platform

2Santhakumar


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Conceptual and frame diagram

3

P(x,y)

O(0,0)

z

r2

r1

r3

Passive joint

Prismatic

joint 2

Prismaticjoint 2

Work

table

Fixed

base

Rotary

joints

Guide ways for

prismatic joints

Guide ways for

prismatic joints

Prismatic joint

1

s

Inertial

frame

endeffector

frame

Number of links = 6

Number of joints = 6

Degrees of freedom = 3 (6-1)2 (6) = 3

Santhakumar


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Kinematic model

Forward kinematic model

Inverse kinematic model

4

s

rr

s

rrrry

rx

z231

231

2

1

tan

z

z

xsyr

xyr

xr

tan

tan

3

2

1

where,

r1, r2 and r3are prismatic joint

displacements (joint parameters)

sis the fixed distance (horizontal span

of the platform)

xand yare the endeffector (work table

/ tool) positions

zis the yaw angle of the endeffectorfrom the inertial frame

x, yand zare the task space

parameters

Santhakumar


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Dynamic model

Euler-Lagrangian formulation method

It is an energy based approach.

Since, it is planar platform potential energy of the

robotic platform considered to be zero. Total kinetic energy (KE) of the robotic platform is

sum of individual prismatic (slider) joint kinematicenergies and work table kinetic energy.

5

zzzw

w

wT

z

sss y

x

I

m

m

y

x

rmrmrmKE

00

00

00

2

1

2

1

2

1

2

1 211

2

11

2

11

Santhakumar


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Dynamic model continued...

Jacobian (velocity mapping) matrix Joint space velocities to task space velocities

Use this relation for deriving kinetic energy of the roboticplatform. Therefore, total kinetic energy is in terms of jointspace variables

6

3

2

1

22

1123

coscos0

1001

r

rr

ss

s

r

s

r

s

rryx

zzz

Santhakumar


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Dynamic model continued...

Euler-Lagrangian formulation method continued... Joint space input variables (joint forces) can be derived from the

following relation as given below:

By combining all input variables and formulating the equationsof motion of the platform in state space form as follows:

where,

7

i

iifr

KE

r

KE

dt

d

disCM ,

edisidisdis

T

TTT

fff

rrrrrrrrr

vectoreDisturbanc

,,

321

321321321

matrixlcentripetaandCoriolistheis,Cmatrixinertiatheis

M

Santhakumar


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Disturbance observer based nonlinear

position tracking controller

Proposed control law based on disturbance

observer as given below:

Disturbance observer scheme as follows:

8

disPDd CKKM ,~~

0,0i.e.,constants,positiveareand

,,,simplicityfor

matrix,gainobserveranis,ector,arbitary vanis,,,

0,~~~,~

~~

,,,,

,

1

1

1

M

zMdt

d

MMCzz

z

disdisdisdisdis

dis

Santhakumar


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Block diagram of the proposed

controller

9

Robotic

platform

Disturbance

observer

M

DK

PK

Trajectory

planner

Inverse

kinematic

model

,Cx(t),

y(t),

z(t)d

d

d

~

~ dis

Disturbances

dis

,

s

zsss yx ,,

+

+-

-

-+

+

Sensor

systems

User input block Proposed controller block

Robotic System

Santhakumar

EKF


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Stability proof of the proposed

controller Applying the proposed nonlinear disturbanceobserver-based control law to the robot platformdescribed by its equations of motion results in thefollowing closed-loop equation:

We can show that all the tracking errors will be

converged to zero asymptotically by the Lyapunovsdirect method. i.e., The proposed controller stability

(closed loop) can be proved using the Lyapunovsdirect method.

10

disPD MKK ~~~~ 1

Santhakumar


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Stability proof continued...

Consider the following positive Lyapunovs candidate function as given

below:

Time derivative of the above function along with its state variable

trajectories, and using closed-loop equation and time derivative of

disturbance error dynamics, we get

11

disTdisDPTT

dis IKKV ~~

2

1~~

2

1~~~~

2

1~,~,~2

0i.e.,dynamics,errorestimationn with thecomparisioin

negligibleisplatformroboticon theactingedisturbanctheofchangeofrateThe

thatsatisfyingisand,0,0i.e.,

matrices,definitepositiveandsymmeticconstantareand

dis

DDDP

DP

IKKKK

KK

disT

disP

T

D

T MKIKV ~~~~~~ 1

0i.e.,property,bymatrixdefinitepositiveais 11 MM Santhakumar


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Stability proof continued...

Lyapunovs function is positive definite and its time derivative isnegative definite in the entire state space.

It is assumed that, the estimated state vectors based on extend

Kalman filter (EKF) are converged to their true state vector values.In this research, the EKF convergence is not discussed.

Therefore, based on LaSalles invariance theorem and Lyapunovsdirect method the closed-loop equation is globally asymptotically

stable. i.e., the velocity, position and disturbance tracking errorsconverge to zero.

12

0~and0~,0~ then,if dist

Santhakumar


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Numerical Simulations

Robotic platform details

Joint limits

Horizontal span (s) = 20 cm

Controller details

Controller gains

Observer details

Observer constants

13

cm20cm0

cm20cm0

cm20cm0

3

2

1

r

r

r

5,

5,

33

33

ppP

ddD

kIkK

kIkK

2,2

EKF details Slider displacements, work table

positions and turning angle can bemeasure using linear encoders (orpotentiometers) and vision system

No velocity and accelerationfeedback

Slowly varying Gaussian noises andinternal disturbances are considered

Simulation cases Case 1: without any disturbances and no

sliding friction forces

Case 2: with disturbances, parameteruncertainties and sliding friction forces

Task 1: Rectangular trajectory

following Task 2: Circular trajectory

following

Comparisons With disturbance compensation and

without disturbance compensation

Santhakumar


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Simulation Case 1: without any disturbances and no sliding frictionforces

14

Proposed controller without

disturbance compensation

Proposed controller with

disturbance compensation

XY task space motion of the robotic

platform

Simple rectangle trajectory has

considered for the analysis with

four straight line segments.

Results are in satisfactory level and

the system behaviors are almost

equal for both controllers.

Santhakumar

Trajectories of task space tracking

errors

Trajectories of task space tracking

errors

Trajectories of joint space tracking errorsTrajectories of joint space tracking errors


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Simulation case 2: with disturbances, parameter uncertainties and sliding friction

forces

Simple rectangular trajectory with constant joint disturbances and sliding

friction forces on their prismatic joints.

Desired task space trajectory details:

Santhakumar 15

seconds10050

102.3024.010

cm1028.10096.05

cm13

seconds500

106.1012.0

cm5

cm1028.10096.05

342

342

342

342

t

tt(t)

tty(t)

x(t)

t

tt(t)

y(t)

ttx(t)

z

z

seconds200150

106.1012.010

cm1028.10096.013

cm5

seconds150100

102.3024.010

cm13

cm1028.10096.013

342

342

342

342

t

tt(t)

tty(t)

x(t)

t

tt(t)

y(t)

ttx(t)

z

z

XY task space motion trajectories of the robotic platform for

a given rectangular trajectory in the presence of disturbances


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Simulation Case 2: with disturbances, parameter uncertainties and sliding friction

forces continued

Santhakumar 16

Time trajectories of the joint space position tracking errors

(without disturbance compensation)

Time trajectories of the joint space position tracking errors (with

disturbance compensation)


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forces continued

Santhakumar 17

Time trajectories of the task space position tracking errors


Time trajectories of the task space position tracking errors (with



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forces continued

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Time trajectories of the estimated joint disturbances for the given rectangular

trajectory


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Simulation Case 2: with disturbances, parameter uncertainties and sliding

friction forces

Simple rectangular trajectory with constant

joint disturbances and sliding friction forces

on their prismatic joints.

Desired task space trajectory details:

Santhakumar 19

4cos10

cm2sin510

cm2cos510

t(t)

ty(t)

tx(t)

z

XY task space motion trajectories of the robotic platform for a given

circular trajectory in the presence of disturbances


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forces continued

Santhakumar 20

Time trajectories of the joint space position tracking errors


Time trajectories of the joint space position tracking errors (with



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forces continued

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Time trajectories of the task space position tracking errors


Time trajectories of the task space position tracking errors (with



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forces continued

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Time trajectories of the estimated joint disturbances for the given circular

trajectory


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Conclusions and future work

The effectiveness of the proposed scheme wasdemonstrated using extensive numerical simulations withsuitable manipulation tasks.

The results confirmed the effectiveness and robustness ofthe proposed scheme in terms of tracking errors in thepresence of unknown external disturbances and parameteruncertainties.

The proposed controller is simple structure, ease ofcomputation and positional feedback inputs are sufficient.Therefore, it can be implemented easily and extended tospatial platforms as well.

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3ppr_mechanism[1].pptx

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