Download - Experimental Robotics
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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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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
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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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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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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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Two decoupled Subsystems
*motionF *forceF
Unified Motion & Force Control
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ASIMO
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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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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
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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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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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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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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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( ) 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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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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