operational space dynamics-exp-2021
TRANSCRIPT
2
Joint to Operational Space Relationships
𝐽 𝑞 𝐸 𝑥 𝐽 𝑞
𝐽# 𝑞 𝐽# 𝑞 𝐸 𝑥𝛿𝑞 𝐽# 𝑞
𝛿𝑥𝛿𝜙
𝛿𝑥𝛿𝜙 𝐽 𝑞 𝛿𝑞𝛿𝑥 𝐸 𝑥
𝛿𝑥𝛿𝜙
Task Representation
Forward Relationships
Inverse Relationships 𝛿𝑥𝛿𝜙 𝐸 𝑥 𝛿𝑥
𝑥 Operational point 𝐽 𝑞 Representation specific task Jacobian
𝑥 Op. point orientation in task space 𝐽 𝑞 Basic (rep. ind.) task Jacobian
𝛿𝑥 Change in op. point config. 𝐽# 𝑞 Generalized Inverse of 𝐽
𝛿𝑥 Change in cartesian point position 𝐽# 𝑞 Generalized Inverse of 𝐽
𝛿𝜙 Instantaneous angular error 𝐸 𝑥 Transforms Basic Jacobian to Task Specific Jacobian
𝛿𝑞 Change in joint value 𝐸 𝑥 Inverse of the 𝐸 matrix
( )v
x E x
0
vJ q
P
R
xx
x
Position Representations
Representation 𝑬𝑷 Matrix 𝑬𝑷𝟏 Matrix
Cartesian 𝒙,𝒚, 𝒛1 0 00 1 00 0 1
1 0 00 1 00 0 1
Cylindrical 𝝆,𝜽, 𝒛
𝑐𝑜𝑠 𝜃 𝑠𝑖𝑛 𝜃 0𝑠𝑖𝑛 𝜃
𝜌𝑐𝑜𝑠 𝜃
𝜌 00 0 1
𝑐𝑜𝑠 𝜃 𝜌 𝑠𝑖𝑛 𝜃 0𝑠𝑖𝑛 𝜃 𝜌 𝑐𝑜𝑠 𝜃 0
0 0 1
Spherical 𝝆,𝜽,𝝓
𝑐𝑜𝑠 𝜃 𝑠𝑖𝑛 𝜙 𝑠𝑖𝑛 𝜃 𝑠𝑖𝑛 𝜙 cos 𝜙sin 𝜃
𝜌sin 𝜙𝑐𝑜𝑠 𝜃
𝜌sin 𝜙 0
𝑐𝑜𝑠 𝜃 𝑐𝑜𝑠 𝜙𝜌
𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜙𝜌
𝑠𝑖𝑛 𝜙𝜌
𝑐𝑜𝑠 𝜃 𝑠𝑖𝑛 𝜙 𝜌 𝑠𝑖𝑛 𝜃 𝑠𝑖𝑛 𝜙 𝜌 𝑐𝑜𝑠 𝜃 𝑐𝑜𝑠 𝜙𝑠𝑖𝑛 𝜃 𝑠𝑖𝑛 𝜙 𝜌 𝑐𝑜𝑠 𝜃 𝑠𝑖𝑛 𝜙 𝜌 𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜙𝑐𝑜𝑠 𝜙 0 1 𝜌 𝑠𝑖𝑛 𝜙
3
4
3
Orientation Representations
Rep. 𝑬𝑹 Matrix 𝑬𝑹𝟏 Matrix 𝜹𝝓 𝑬𝑹 𝒙𝑹 𝜹𝒙𝑹
Direction Cosines
𝒙𝒓 𝒓𝟏𝑻, 𝒓𝟐
𝑻, 𝒓𝟑𝑻 𝑻
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�̂��̂��̂�
12�̂� 𝑟 �̂� 𝑟 �̂� 𝑟
Euler Angles𝒙𝒓 𝝍,𝜽,𝝓
𝑠𝑖𝑛 𝜓 𝑐𝑜𝑠 𝜃𝑠𝑖𝑛 𝜃
𝑐𝑜𝑠 𝜓 𝑐𝑜𝑠 𝜃𝑠𝑖𝑛 𝜃 1
𝑐𝑜𝑠 𝜓 𝑠𝑖𝑛 𝜓 0𝑠𝑖𝑛 𝜓
𝑠𝑖𝑛 𝜃𝑠𝑖𝑛 𝜓
𝑠𝑖𝑛 𝜃 0
0 𝑐𝑜𝑠 𝜓 𝑠𝑖𝑛 𝜓 𝑠𝑖𝑛 𝜃0 𝑠𝑖𝑛 𝜓 𝑐𝑜𝑠 𝜓 𝑠𝑖𝑛 𝜃1 0 𝑐𝑜𝑠 𝜃
0 𝑐𝑜𝑠 𝜓 𝑠𝑖𝑛 𝜓 𝑠𝑖𝑛 𝜃0 𝑠𝑖𝑛 𝜓 𝑐𝑜𝑠 𝜓 𝑠𝑖𝑛 𝜃1 0 𝑐𝑜𝑠 𝜃
𝜓𝜃𝜙
EulerParameters
𝝀 𝝀𝟎,𝝀𝟏,𝝀𝟐,𝝀𝟑
12
𝜆 𝜆 𝜆𝜆 𝜆 𝜆𝜆 𝜆 𝜆𝜆 𝜆 𝜆
2 𝜆 𝜆 𝜆 𝜆𝜆 𝜆 𝜆 𝜆𝜆 𝜆 𝜆 𝜆
2 𝜆 𝜆 𝜆 𝜆𝜆 𝜆 𝜆 𝜆𝜆 𝜆 𝜆 𝜆
𝜆𝜆𝜆 𝜆
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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6
11
TennisBot Gordon GroceryBot
William KoskiDimitri PetrakisChenkai
Ewurama KarikariZhengqiu LouJoseph WangJiaqiao Zhang
Gabriela Bravo IllanesMax FarrChun Ming Zhang
RehaBot DocBot DrawBot
Eleonore JacquemetRuta JoshiJuhi Madan
Yuxiao ChenDan FanShivani Guptasarma
Kaojun HeAce HuRan Le
CS225A – Spring2021 Projects
Sports EnvironmentCookingGroceries
Human Robot InteractionMedical
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22
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F
( )GoalV xF
( )GoalV x
T FJ
Task‐Oriented Control
F
dynamics( )F F
x
x Fp
Task‐Oriented Control
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28
15
Unified Motion & Force Control
motion contactF F F contactF
motionF
Equations of Motion
d L LF
dt x x
with ( , ) ( , ) ( )gravityL x x T x x V x
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30
16
End‐Effector Control
( )TJ q F F
1
2
T
goal p g gV k x x x x
goalVF
1
2
T
goal p g gV k x x x x
System gravityT Vd T
Fdt x x
ˆ
goal gravityF V VX
Passive Systems
0
goalT Vd T
dt x x
StableConservative Forces
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Asymptotic Stability
is asymptotically stable if
0 ; 0TsF x for x
0s v vF k x k
ˆp g vF k x x k x p Control
s
goalT Vd T
dt x xF
a system
sFx
Artificial Potential Field
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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
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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36
19
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
( ) ( , ) ( )A q q b q q g q
Joint Space Dynamics
Centrifugal and Coriolis forces( , ) :b q q( ) :g q Gravity forces
: Generalized forces
( ) :A q Kinetic Energy Matrix
:q Joint Coordinates
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38
20
Lagrange Equations
( )d L L
dt q q
( ) ( , ) ( )A q q b q q g q
(( )) ,A b q qq 1
2( ): TA q qAqT
Equations of Motionvcii
Pci
Link i
TTotal Kinetic Energy:
1
2Lin i
T
kAqT T q
39
40
21
1( )2 i i
T T Ci i C C i i iT mv v I
1
n
ii
T T
vcii
Pci
Link i
Explicit Form
Total Kinetic Energy
Equations of Motion
Generalized Coordinates q
Kinetic EnergyQuadratic Form of
Generalized Velocities
1
1(
1
2)
2 i i
nT T C
i C C i iT
ii
m v vq A q I
q
1
2Tq qT A
vcii
Pci
Link i
Explicit Form
Generalized Velocities
Equations of Motion
41
42
22
i ivC Jv q
1
1( )
2 i i i i
T Tv v
nT T C
i ii
m q q qJ J J JI q
Explicit Formvcii
Pci
Link i
iiCJ q
1
1(
1
2)
2 i i
nT T C
i C C i iT
ii
m v vq A q I
Equations of Motion
vcii
Pci
Link i
Explicit Form
1
2Tq qA
1
( )1
2 i i i i
nT T
i vi
T Cv imq J J I J qJ
1
( )i i i i
nT T C
i v v ii
A m J J J I J
Equations of Motion
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44
23
( )
11 12 1
21 22 2
1 2
( )n n
n
n
n n nn
a a a
a a aA q
a a a
Christoffel Symbols 1( )
2ijk ijk ikj jkib a a a
ij
k
a
q
2( , ) ( ) ( )b q q C q q B q qq
B
b b b
b b b
b b b
q q
q q
q qn
n n n n
n n
n n
n n n n n n n
( ) [ ]
(
( )) (( )
)
, , ,(
, , ,(
, , ,( (
q qq
L
N
MMMM
O
Q
PPPP
L
N
MMMM
O
Q
PPPP
1
2
1
21
2 2 2
2 2 2
2 2 2
1 12 1 13 1 1)
2 12 2 13 2 1)
12 13 1)
1 2
1 3
1)
C
b b b
b b b
b b b
q
q
qn n n
nn
nn
n n n nn n
( )[ ]
( ) ( )
, , ,
, , ,
, , ,
q q
L
N
MMMM
O
Q
PPPP
L
N
MMMM
O
Q
PPPP2
1 11 1 22 1
2 11 2 22 2
11 22
12
22
2
1
45
46