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CS225A

Experimental Robotics

Lecture 7 

Oussama Khatib

Project Groups

Project Students ‐ A Students ‐ B Students ‐ C Students ‐ D Students ‐E

Humanoid Valerie Sean Megan

Assembly Akram Ken Kathleen

Offshore Yutian Abdoul Chinmay

Service Andrew Kevin Carolyn

Sport ‐ I Sergio Courtney Bryce

Sport -II Rohan

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2

2

Kinematics

Dynamics

Jacobians

Inverses

Task

Representations

Equations of Motion

Operational Space Control

Dynamic

Models

Compliance

Force Control

Control

Modalities

Redundant

Robots

Posture

Null Space

Dynamic Behavior

Whole-Body Control

Menu

Redundancy

3

4

3

0 0 0( ) , ( ) contactox x p x F F

•Generalized Selection Matrix

•Dynamic Model (Homogeneity)

Unified Motion/Force Control

A Mass Spring System

System

smz k z f

s sf k z

1s s

s

m f f fk

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6

4

Control

fs f s d v sf f m k f f k f

s compf f m f

Control‐loop System

0fs s v s s f s df k k f k k f f

Static Equilibrium

s df f

1s s

s

m f f fk

System

End‐Effector/Sensor System

0 0 0 0( , ) ( ) contactFx p x F

0 motion forceF F F

Unified Control

*0 0 0ˆ ˆˆmotionmotion F PF

*0ˆ

forceforc sensore F FF

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8

5

0 0 0 0( , ) ( ) contactFx p x F

0 motion forceF F F

Unified Control

*0 0 0ˆ ˆˆmotionmotion F PF

*0ˆ

forf ce desorce iredF FF

End‐Effector/Sensor System

Unified Motion & Force Control

Two decoupled Subsystems

*motionF *forceF

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10

6

Two decoupled Subsystems

*motionF *forceF

Unified Motion & Force Control

11

12

7

ASIMO

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14

8

Equations of Motion

( ) ( , ) ( )A q q b q q g q Joint Space

( ) ( , ) ( )x x x x p x F Operational Space

) ( )( TJA b g x pq

RelationshipsT FJ

) ( )( TJA b xq Inertial forces

x p F

(joint dynamics)Aq b g TJ

TJ

(Task dynamics)

RedundancyNon

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16

9

x p F

(joint dynamics)Aq b g

(Task dynamics)

projectionTJ

Redundancy

1 1 1( )T TJ A J JA J where

dynamically consistent generalized inverse:J J J Iwhere

TJ

xF

q T q

Assuming a Virtual Displacement

# #0q J x I J J q

Redundancy

TF x

0J JJq x I q Dynamic Consistency:

Virtual Work

17

18

10

is the Dynamically ConsistentGeneralized Inverse

( )J q

1J J Non‐redundant

Dynamic Consistency

1 TJ A J J is unique and

Theorem (Consistency)

Null Space with Pseudo Inverse

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20

11

Dynamic Consistency

Velocity Force Duality

1q xJ T FJ

0J JJq x I q 0T T TJ J JF I

Velocity Force

Non Red.

Redundant

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22

12

Task dynamics

11 TJA J

( ) ( , ) ( )q x q q p q F

( , ) ( , ) ( ) ( )Tq q J b q q q J q q

( ) ( )Tp q J g q

Redundant Robot Control

dynamically decoupled

TN

0T TJ F N

Task Space:

Null Space:

TJ

1 2

Robot Control

N I JJ where

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24

13

Redundant Robot Control

TN

Task Space:

Null Space:

TJ

Robot Control

N I JJ where

0T TJ F N

Posture( )T TNF VJ

Asymptotic Stability

( ) TvD q k J J

disq0 ; 0T

dis q q for

T Tdis v vk J k J qx J

0) ; 0(Tq q qD q

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26

14

( ) 0Tq D q q

: is a n x n matrix of rank it is Positive Semi‐definite

0mTJ J

The System is Stable, but not asymptotically stable

Asymptotic Stability

Asymptotic Stability

0 ; 0Tdis q q

Positive definite

for

1[ ]T TvD k J J A J J A A

( ) vD q k A

( ) 0Tq D q q 0q for

N I JJ T Tdis v vk J J q k N Aq

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28

15

Asymptotic Stability

0 ; 0Tdis q q

Positive definite

for

T Tdis v vk J J q k N Aq

( ) vD q k A

( ) 0Tq D q q 0q for

N I JJ

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30

16

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