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CS225A
Experimental Robotics
Lecture 5
Oussama Khatib
Project ProposalsExperimental Robotics
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Spring2021 Projects
Sports EnvironmentCookingGroceries
Human Robot InteractionMedicalService
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
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Project ProposalsExperimental Robotics
.. motion in contact
Operational Space Framework
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Joint Space Control
Joint Space Control
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Joint Space Control
F
( )GoalV xF
( )GoalV x
T FJ
Task‐Oriented Control
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F
dynamics( )F F
x
x Fp
Task‐Oriented Control
Unified Motion & Force Control
motion contactF F F contactF
motionF
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Effector Equations of Motion
Non‐Redundant Manipulator ; 0n m
01 2 T
mx x x x
1 2 T
nq q q q x G q
0R1nR
10n
Lagrange Equations in Operational Space
( ) Fd L L
dt x x
L VT
( ) GravityVF
d T T
dt x x x
Inertial forces Gravity vector
( )GravityV V q
Kinetic Energy
Potential Energy (Gravity)
Lagrangian
Since
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( ) ( , ) ( )x x x x p x F
Operational Space Dynamics
End‐Effector Centrifugal and Coriolis forces
( , ) :x x
( ) :p x End‐Effector Gravity forces
:F End‐Effector Generalized forces
( ) : x End‐Effector Kinetic Energy Matrix
:x End‐Effector Position andOrientation
Operational Space Control
( )TJ q F
F
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2
T
Goal p Goal GoalV k x x x x
( )GoalV xF
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1
2
T
Goal p Goal GoalV k x x x x
System GravityT V
Fd T
dt x x
ˆGoal GravityF V V
X
Passive Systems
0GoalTd T
dt x x
V Stable
Conservative Forces
Asymptotic Stability
is asymptotically stable if
0 ; 0TsF x for x
0s v vF k x k
ˆp Goal vF k x x k x p Control
s
GoalTd T
dt x xF
V a system
sFx
p̂ Estimate of p
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Artificial Potential Field
Example: 2‐d.o.f arm
ˆ ( )p g vF k x x k x p x
( ) ( , ) ( )x x x x p x F
1q1l
2q2l
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Closed loop behavior
111*
1( ) v p gm q x k x k mx yx
122*
2( ) v p gm q y k y k my xy
* 2 *1 2 1 112 p g vm c m x m y k x x k x
* 2 *1 2 1 212 p g vm c m y m x k y y k y
Joint Space/Operational SpaceRelationships
( , ) ( , ) x qT x x T q q
1 1( ) ( )
2 2T Tx X x q A q q
1 1
2 2T TTq qJ J Aq q
Using ( )x J q q
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where ( , ) ( )h q q J q q
1( ) ( ) ( ) ( )Tx J q A q J q
( , ) ( ) ( , ) ( ) ( , )Tx x J q b q q q h q q
( ) ( ) ( )Tp x J q g q
Joint Space/Operational SpaceRelationships
Example
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2
1
1
d cx
d s
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2
1 1
1 1
d s cJ
d c s
2 2q d
g
l1y
CI1
CI2
m1
m2
d2
x
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2
1 1
1 1
d s cJ
d c s
21 1 0 1;
1 0
dJ
11 21
2 22
0 1 0 0 1
1 0 0 1 0
m d
d m
0
2
0 11 1
01 1
c sJ
ds c
1 J
21
2 2
0
0
m
m m
2221 222 1 1
2 22
I I m l
md
2 2m m
1x1y
1m
2m
2m
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2
01 1 1 1
1 1 1 10
mc s c s
s c s cm
2 2 2m m m
20 2 2 2
22 2 2
1 1
1 1
m m s m sc
m sc m m c
2m2 2m m
20 2 2 2
22 2 2
1 1
1 1
m m s m sc
m sc m m c
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Nonlinear Dynamic Decoupling
with TJ F
( ) ( , ) ( )x x x x p x F Model
*( ) ( ,ˆ ˆ) ( )ˆF px x x xF Control Structure
*I x FDecoupled System
Dynamic Decoupling
( ) ( , ) ( )x x x x p x F
*ˆ ˆ ˆ( , ) ( )F F x x p x
0
1 * 1 1ˆ ˆˆmI X F P P
( )G x ( , )x x ( )P x
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0
*) ( , ) ( )(mI x x dx tGx F
1 ˆ( )G x I
1( , )x x P
( ) :d t unmodeled disturbances
Dynamic Decoupling: Closed Loop
Perfect Estimates
0
*mI x F
input of decoupled end‐effector*F
Goal Position Control
*v p gF k x k x x
Closed Loop
0m v p p gI x k x k x k x
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Closed Loop0m v p p gI x k x k x k x
t
x
max
2gx
1gx
PD Control
*v p gF k x k x x
Velocity‐Like Control
* pv g
v
kF k x x x
k
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dx
pd g
v
kx x x
k
dx
* pv g
v
kF k x x x
k
* v dF k x x
withmax
d
Vsat
x
1
0 1max
dx
1
( ) 1
x if xsat x
sign x if x
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Trajectory Tracking
Trajectory: , , d d dx x x
0
* ( ) ( ) m d v d p dF I x k x x k x x
( ) ( ) ( ) d v d p dx x k x x k x x
with
or
X dx x
0 X v X p Xk k
In joint space
0 q v q p qk k
with q dq q
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