![Page 1: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/1.jpg)
CS225A
Experimental Robotics
Oussama Khatib
![Page 2: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/2.jpg)
ASIMO
![Page 3: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/3.jpg)
Kinematics
![Page 4: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/4.jpg)
Operational Coordinates
A set of independent configuration parameters0m
01 2, , , mx x x
number of degrees of freedomof the end‐effector.
0 :m
Operational point1 :nO
10n{0}
![Page 5: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/5.jpg)
0n m
Redundancy
Degrees of redundancy: 0n m
A robot is said tobe redundant if
![Page 6: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/6.jpg)
degrees of redundancy/task:taskn mtaskn m
Task Redundancy
![Page 7: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/7.jpg)
• Forward Kinematics
• Inverse Kinematics
![Page 8: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/8.jpg)
Denavit‐Hartenberg (DH) Parameters
1iR
1iz
1iy
ix
iR
1ix
iz
ii ia
1i
1iLink
1ia
ijoint
iLink
1ijoint
![Page 9: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/9.jpg)
Homogeneous Transformation
( 1) ( 1) 1 1, , ,i i i i i i i iT T a
( 1)
( 1) ( 1) ( 1) ( 1)( 1)
( 1) ( 1) ( 1) ( 1)
cos sin 0sin cos cos cos sin sinsin sin cos sin cos cos
0 0 0 1
i i i
i i i i i i ii i
i i i i i i i
a
T
![Page 10: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/10.jpg)
0( 1) 01 1 12 2 ( 1) ( 1)( ) ( ) ( ) ( )n n n n n nT q T q T q T q T
Geometric ModelForward Kinematics
1nR
10n0R
![Page 11: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/11.jpg)
( 1) ( 1)0( 1)
( ) ( )( )
0o n o n
n
R q qT q
I
m equations
( )x G q
( )( )
p
r
x qx
x q
Geometric ModelForward Kinematics
1nR
10n0R
![Page 12: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/12.jpg)
Representations
p
r
xx
x
•Cartesian•Spherical•Cylindrical•….
•Euler Angles•Direction Cosines•Euler Parameters
1nR
10n0R
![Page 13: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/13.jpg)
Position Representations
is obtained from( )px q 0( 1) ( )n q
, ,r •Spherical
•Cylindrical , , z
z
r10n
x
y
•Cartesian , ,x y z
![Page 14: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/14.jpg)
Orientation Representation
is obtained from( )rx q 0( 1) ( )nR q
0( 1)nR
1nR
10n0R
![Page 15: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/15.jpg)
Direction Cosines
11 12 13
0( 1) 21 22 23
31 32 33
n
r r rR r r r
r r r
0( 1) 1 2 3( ) ( ) ( ) ;nR r q r q r q
1
2
3
( )( )( )
r
r qx r q
r q
0( 1)nR
1nR
10n0R
Orientation Representation
![Page 16: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/16.jpg)
Euler Angles
0z x
0( 1)nR R R R
( 1)nx z
( 1)nz
0x
0z
( 1)ny
x
0y
( 1)nx
( 1)nz
![Page 17: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/17.jpg)
0( 1) ( )n
c c s c s c s s c c s sR q s c c c s s s c c c c s
s s s c c
( )( )( )
r
qx q
q
11 12 13
0( 1) 21 22 23
31 32 33
n
r r rR r r r
r r r
0x
0z
( 1)ny
x
0y
( 1)nx
( 1)nz
Euler Angles
![Page 18: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/18.jpg)
Singularity of the representation
213 23 33( ) sgn arccos 1 ;q r r r
33 1r
33( ) arccos ;q r
231 32 33( ) sgn arccos 1 ;q r r r
k:(integer)k 33 1r
or are defined‐
Euler Angles
0x
0z
( 1)ny
x
0y
( 1)nx
( 1)nz
YB
XA
YA
ZB
XB
( )
( )
ZA
![Page 19: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/19.jpg)
Quaternion
Rotations:Product of two plane symmetries
Euler Parameters
![Page 20: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/20.jpg)
wv
u2
cos 2u v sin 2u v w
Euler Parameters
![Page 21: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/21.jpg)
Normality condition
0 cos 2 ;
2 2 2 20 1 2 3 1
1 1 sin 2 ;w 2 2 sin 2 ;w 3 3 sin 2 ;w
Euler Parameters
![Page 22: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/22.jpg)
Rotation Matrix
2 20 1 1 2 0 3 1 3 0 2
2 20( 1) 1 2 0 3 0 2 2 3 0 1
2 21 3 0 2 2 3 0 1 0 3
2 1 2 2
( ) 2 2 1 2
2 2 2 1
nR q
11 12 13
0( 1) 21 22 23
31 32 33
n
r r rR r r r
r r r
![Page 23: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/23.jpg)
Sign Determination
0 1 0 2 0 3 1 2 1 3 2 3, , , , and
0 32 23 11 22 33sgn 1 ;2
r r r r r
1 13 31 11 22 33sgn 1 ;2
r r r r r
2 21 12 11 22 33sgn 1 ;2
r r r r r
3 11 22 33 1 ;2
r r r
where 1
![Page 24: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/24.jpg)
Lemma: For all rotations, at least one of theEuler Parameters has a magnitude larger than or equal to 1/2.
Algorithm
with
![Page 25: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/25.jpg)
Basic Jacobian
F Ix
H K( 6 1)
vJ q qxn nxG J
( ) (( ) 0 6 1)
{0}
angular velocity
linear velocity
v
![Page 26: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/26.jpg)
Jacobian for a Task X
( )xx J q q
0( ) ( ) ( )xJ q E x J q
p
r
xx
x
For a representation
0 ( )v
J q q
Basic Jacobian
The corresponding Task Jacobian
Decomposed
![Page 27: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/27.jpg)
00
p p vx
rr
J E JJEJ J
0( ) ( ) ( )xJ q E x J q
0 ( )v
J q q
Jacobian and Basic Jacobian
![Page 28: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/28.jpg)
Position Representations
3( )P pE x ICartesian Coordinates ( , , )x y z
cos sin 0
( ) sin cos 0
0 0 1
P pE x
Cylindrical Coordinates ( , , )z Using ( ) ( cos sin )T Tx y z z
![Page 29: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/29.jpg)
cos sin sin sin cos
( ) sin cos 0sin sincos cos sin cos sin
p pE x
Spherical Coordinates ( , , )
( ) ( cos sin sin sin cos )T Tx y z Using
![Page 30: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/30.jpg)
Euler Angles
Singularity of the representationfor k
1
; ( ) 0
0
r r r
s c c cs s
x E x c ss cs s
![Page 31: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/31.jpg)
Revolute Joint i i iZ q
i i iV Z q Prismatic Joint
jVi
The Jacobian Explicit Form
![Page 32: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/32.jpg)
Revolute
Linear Vel: jVAngular Vel:
jV
i inP
inP
i inP
Prismatic
none
Effector Linear Velocity
Effector Angular Velocity
1[ ( )]
n
i i i i ini
v V P
1i
n
ii
Effector
v
i
i
i
i i iV Z q
i i iZ q
i
jV
The Jacobian Explicit Form
![Page 33: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/33.jpg)
1( ) i
n
i ii
Z q
1[ ( )]
n
i i i i ini
iv Z P qZ
Revolute
Linear Vel: jVAngular Vel:
jV
i inP
inP
i inP
Prismatic
none
Effector
i i iV Z q
i i iZ q
v
i
i
i
Effector Linear Velocity
Effector Angular Velocity
i
jV
The Jacobian Explicit Form
![Page 34: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/34.jpg)
1( ) i
n
i ii
Z q
1
ˆ( )n
i i i i iini
v P qZ Z
Revolute
Linear Vel: jVAngular Vel:
jV
i inP
inP
i inP
Prismatic
none
Effector
i i iV Z q
i i iZ q
v
i
i
i
Effector Linear Velocity
Effector Angular Velocity
i
jV
The Jacobian Explicit Form
![Page 35: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/35.jpg)
1
21 1 1 1 1 2 2 2 2 2( ) ( )n n
n
v Z Z P Z Z P
q
vv J q
1
21 1 2 2 n n
n
Z Z Z
q
J q
The Jacobian Explicit Form
![Page 36: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/36.jpg)
1
21 1 1 1 1 2 2 2 2 2
ˆ ˆn n
n
v Z Z P Z Z P
q
vv J q
1
21 1 2 2 n n
n
Z Z Z
q
J q
The Jacobian Explicit Form
![Page 37: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/37.jpg)
vxyz
xxq
qxq
qxq
qPP P P
nn
F
HGGI
KJJ
. . .
1
12
2
1 2
P P Pv
n
x x xJq q q
v
w
JJJ
Matrix (direct differentiation)vJ
The Jacobian Explicit Form
![Page 38: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/38.jpg)
Vector Representation Jxq
xq
xq
Z Z Z
P P P
n
n n
FHGG
IKJJ
1 2
1 1 2 2
. . .
In {0}
0
0
1
0
2
0
10
1 20
20
Jxq
xq
xq
Z Z Z
P P P
n
n n
F
HGG
I
KJJ
. . .
The Jacobian Explicit Form
![Page 39: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/39.jpg)
J in Frame {0}
01
0
2
0 0
1 10
2 20 0
J qx
qx
qx
R Z R Z R Z
P Pn
P
n n
F
HGG
I
KJJ
( ) ( ) ( )
.( . ) .( . ) .( . )
0iZ 0 ;i
i iR Z001
iiZ Z
![Page 40: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/40.jpg)
Stanford Scheinman Arm
![Page 41: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/41.jpg)
J F
HGG
I
KJJ1 13Z P 2 23Z P 3Z
2Z1Z 4Z 5Z 6Z
0 0 0
0
Z6
Z0 X6X0
d3d2
Y6Y0
![Page 42: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/42.jpg)
0 0 0
01 2 3
0 0 0 0 01 2 4 5 6
0 0 0
0
P P Px x xJ q q q
Z Z Z Z Z
c d s s d c c d c ss d c s d s c d s s
s d cs c s c c s s c c c c s s s s c s c
c s s s c s c c s c c s c s s s s cc s s s c s c c
L
N
MMMMMMM
O
Q
P1 2 1 2 3 1 2 3 1 2
1 2 1 2 3 1 2 3 1 2
2 3 2
1 1 2 1 2 4 1 4 1 2 4 5 1 4 5 1 2 5
1 1 2 1 2 4 1 4 1 2 4 5 1 4 5 1 2 5
2 2 4 2 4 5 5 2
0 0 00 0 0
0 0 0 00 00 01 0 0
PPPPPP
Stanford Scheinman Arm Jacobian
![Page 43: cs225a-2020 · Apr 7: Introduction & Course Review May 7: Project Proposal (Presentation) Apr 9: Simulation and Graphics May 12: Individual Group Meetings Apr 10: Software Setup (Special](https://reader033.vdocument.in/reader033/viewer/2022050313/5f755c1ada074805700a71c1/html5/thumbnails/43.jpg)
Jx
Velocity/Force Duality
T FJ