4/13/2021
1
CS225A
Experimental Robotics
Lecture 5
Oussama Khatib
Project ProposalsExperimental Robotics
1
2
4/13/2021
2
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
3
4
4/13/2021
3
Project ProposalsExperimental Robotics
.. motion in contact
Operational Space Framework
5
6
4/13/2021
4
Joint Space Control
Joint Space Control
7
8
4/13/2021
5
Joint Space Control
F
( )GoalV xF
( )GoalV x
T FJ
Task‐Oriented Control
9
10
4/13/2021
6
F
dynamics( )F F
x
x Fp
Task‐Oriented Control
Unified Motion & Force Control
motion contactF F F contactF
motionF
11
12
4/13/2021
7
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
13
14
4/13/2021
8
( ) ( , ) ( )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
1
2
T
Goal p Goal GoalV k x x x x
( )GoalV xF
15
16
4/13/2021
9
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
17
18
4/13/2021
10
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
19
20
4/13/2021
11
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
21
22
4/13/2021
12
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
2
2
1
1
d cx
d s
20
2
1 1
1 1
d s cJ
d c s
2 2q d
g
l1y
CI1
CI2
m1
m2
d2
x
23
24
4/13/2021
13
20
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
25
26
4/13/2021
14
20
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
27
28
4/13/2021
15
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
29
30
4/13/2021
16
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
31
32
4/13/2021
17
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
33
34
4/13/2021
18
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
35
36
4/13/2021
19
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
37
38
Top Related