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PROPRIETARY MATERIAL. © 2014, 2008 The McGraw-Hill Companies, Inc. All rights reserved. No part of this PowerPoint slide may be displayed,
reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers
and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this PowerPoint slide, you are using it without
permission.
Lecture 2 (III-LT1)
Robot Kinematics (Ch. 5)by
S.K. SahaAug. 07, 2015 (F)@JRL301 (Robotics Tech.)
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2
F
p΄
o
p
U
MOM
V
P
W
O
X
Z
Y
Pose Position + Rotation
Translation: 3
Rotation: 3
Total: 6
A moving body Pose or Configuration
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3
xp
zp
ypF][p . . . (5.8)
[ ] , [ ] , and [ ]
1 0 00 1 0
00 1F F F
x y z
. . . (5.10)
Position Description
p = px x + py y + pz z . . . (5.9)
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4
Orientation Description
1. Direction cosine representation
2. Fixed-axes rotations
3. Euler angles representation
4. Single- and double-axes rotations
5. Euler parameters representation
I will illustrate the first TWO only
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5
u = ux x + uy y + uz z
. . . (5.11a)
v = vx x + vy y + vz z
. . . (5.11b)
w = wx x + wy y + wz z
. . . (5.11c)
Direction Cosine Representation
Refer to Fig. 5.12
p = puu + pvv + pww
. . . (5.12)
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6
p = (puux + pvvx + pwwx)x + (puuy + pvvy + pwwy)y
+ (puuz + pvvz + pwwz)z . . . (5.13)
px = uxpu + vxpv + wxpw . . . (5.14a)
py = uypu + vypv + wypw . . . (5.14b)
pz = uzpu + vzpv + wzpw . . . (5.14c)
Substitute eqs. (5.11a-c) into eq. (5.12)
[p]F = Q [p]M . . . (5.15)
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[p]F = Q [p]M . . . (5.15)
xwxvxu
zwzvzu
ywyvyuQpp
TTT
TTT
TTT,][,][x
wx
vx
u
zw
zv
zu
yw
yv
yu
F
up
wp
vp
xp
zp
yp
M
.. . (5.16)
uTu = vTv = wTw = 1, and
uTv(vTu) = uTw(wTu) = vTw(wTv) = 0 … (5.17)
Q is called Orthogonal
Orientation description 1
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u v = w, v w = u, and w u = v . . . (5.18)
QTQ = QQT = 1 ; det (Q) = 1; Q1 = QT . . . (5.19)
Due to orthogonality
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9
[ ,
[ ] ,
[ ]
0
0
001
u]
v
w
F
F
F
CαSα
SαCα
. . . (5.20)
Example 5.6 Rotations [Elementary] (Fig. 5.13a)In
terp
reta
tion 1
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10
100
0
0
CS
SC
ZQ . . . (5.21)
CS
SC
CS
SC
XY
0
0
001
;
0
010
0
. . . (5.22)
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Non-commutative Property: An Illustration
Fig. 5.20 Successive rotation of a box about Z and Y-axes
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Non-commutative Property (contd.)
Fig. 5.21 Successive rotation of a box about Y and Z-axes
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PROPRIETARY MATERIAL. © 2014, 2008 The McGraw-Hill Companies, Inc. All rights reserved. No part of this PowerPoint slide may be displayed,
reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers
and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this PowerPoint slide, you are using it without
permission.
Lecture 3
Robot Kinematics (Ch. 5)by
S.K. SahaAug. 10, 2015 (M)@JRL301 (Rob. Tech.)
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14
Recap
• Orientation representations
– Non-commutative
• Direction cosines: Has disadv. of 9 param.
• Fixed-axes (RPY) rotations (12 sets)
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15
Homogeneous Transformation
F
p΄
o
p
U
MOM
V
P
W
O
X
Z
Y
Task: Point P is known in moving frame M. Find P in fixed frame F.
Fig. 5.23 Two coordinate frames
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p = o + p . . . (5.45)
[p]F = [o]F + Q[p’]M . . . (5.46)
1
][
1
][
1
][T
F MF poQp
0. . . (5.47)
MF ][][ pTp . . . (5.48)
Homogenous Transformation
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17
TTT 1 or T1 TT . . . (5.49)
1
][T
TT1
0
oQQT F . . . (5.50)
1000
1100
2010
0001
T
. . . (5.51)
Example 5.10 Pure Translation
T: Homogenous transformation matrix (4 4)
Fig. 5.24 (a)
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18
. . . (5.52)
Example 5.11 Pure Rotation
30 30 0 0
30 30 0 0
0 0 1 0
0 0 0 1
3 10 0
2 2
1 30 0
2 2
0 0 1 0
0 0 0 1
T
o o
o o
C S
S C
Fig. 5.24 (b)
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19
Like rotation matrices homogeneous transformation
matrices are non-commutative, i. e.,
Non-commutative Property
TATB TBTA
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Denavit and Hartenberg (DH) Parameters
• Serial chain
- Two links connected
by revolute or
prismatic joint
• Four parameters– Joint offset (b)
– Joint angle ()
– Link length (a)
– Twist angle ()
Fig. 5.27 Serial manipulator
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• Joint axis i: Link i-1 + link i
• Link i: Fixed to frame i+1 (Saha) / frame i (Craig)
DH Variables
bi and i
[Screw@Z]
Constants
ai and i
[Screw@X]
Saha XiXi+1@Zi ZiZi+1@Xi+1
Craig Xi-1Xi@Zi ZiZi+1@Xi
Z’’’i
Zi+1
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• bi (Joint offset): Length of the intersections of the
common normals on the joint axis Zi, i.e., Oi and
Oi. It is the relative position of links i 1 and i.
This is measured as the distance between Xi
and Xi + 1 along Zi.
Z’’’i
Zi+1
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• i (Joint angle): Angle between the orthogonal projections of
the common normals, Xi and Xi + 1, to a plane normal to the
joint axes Zi. Rotation is positive when it is made counter
clockwise. It is the relative angle between links i 1 and i.
This is measured as the angle between Xi and Xi + 1 about Zi.
Z’’’i
Zi+1
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• ai (Link length): Length between the O’i and Oi
+1. This is measured as the distance between
the common normals to axes Zi and Zi + 1 along
Xi + 1.
Z’’’i
Zi+1
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25
• i (Twist angle): Angle between the orthogonal
projections of joint axes, Zi and Zi+1 onto a plane
normal to the common normal. This is measured as
the angle between the axes, Zi and Zi + 1, about axis Xi
+ 1 to be taken positive when rotation is made counter
clockwise.
Z’’’i
Zi+1
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Revolute Joint
Fig. 5.28
• DH@Z (Variable)– Joint offset (b)
– Joint angle ()
• DH@X (Const.)– Link length (a)
– Twist angle ()
Z’’’i
Zi+1
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Tb =
1000
100
0010
0001
ib. . . (5.60a)
T =
1000
0100
00
00
ii
ii
CθSθ
θSCθ. . . (5.60b)
Mathematically• Translation along Zi
• Rotation about Zi
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1000
00
00
0001
ii
ii
CαSα
αSCαT = . . . (5.60d)
Ta =
1000
0100
0010
001 ia
. . . (5.60c)
• Translation along Xi+1
• Rotation about Xi+1
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Ti = TbTTaT . . . (5.61a)
Ti =
1000
0 iii
iiiiiii
iiiiiii
bCαSα
SθaSαCθCαCθSθ
CaSαSθCαSθCθ
. . . (5.61b)
• Total transformation from Frame i to Frame i+1
Rotation
Matrix
Po
sitio
n
Do it yourself!
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Spherical-type Arm
• DH-parameters
Link bi i ai i
1 0 1 (JV) 0 /2
2 b2 2 (JV) 0 /2
3 b3
(JV)
0 0 0
Fill-up the DH parameters
Fig. 5.32 A spherical arm
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31
PUMA 560
i Variable
DH
Constant
DH
bi i ai i
1 0 1 0 -/2
2 0 2 a2 0
3 B3 3 a3 -/2
4 b4 4 0 /2
5 0 5 0 -/2
6 0 6 0 0
Fig. 5.35 PUMA 560 and its frames
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PROPRIETARY MATERIAL. © 2014, 2008 The McGraw-Hill Companies, Inc. All rights reserved. No part of this PowerPoint slide may be displayed,
reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers
and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this PowerPoint slide, you are using it without
permission.
Lecture 4 (SIT Sem. Rm.)
Forward and Inverse Kinematics
(Ch. 6)by
S.K. SahaAug. 12, 2015 (W)@JRL301(Robotics Tech.)
@ McGraw-Hill Education
33
Forward and Inverse Kinematics
Inverse: 1st soln.
.
Inverse: nth soln.
Forward: One soln.S
olv
e
Non
-lin. e
qns.
Multip
ly
+ A
dd
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34
Three-link Planar Arm
Ti =
1000
0100
0
0
iiii
iiii
SθaCθSθ
CaSθCθ
• DH-parameters
, for i=1,2,3
Link bi i ai i
1 0 1 (JV) a1 0
2 0 2 (JV) a2 0
3 0 3 (JV) a3 0
• Frame transformations
(Homogeneous)
Fill-up the DH
parameters
Fill-up with the elements
Fig. 5.29 A three-link planar arm
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DH Parameters of Articulated Arm
Link bi i ai i
1 0 1 (JV) 0 π/2
2 0 2 (JV) a2 0
3 0 3 (JV) a3 0
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36
Matrices for Articulated Arm1 1
1 1
10 1 0 0
0 0 0 1
c 0 s 0
s 0 c 0
T
2 2 2 2
2 2 2 2
2
c s 0 a c
s c 0 a s
0 0 1 0
0 0 0 1
T
3 3 3 3
3 3 3 3
3
c s 0 a c
s c 0 a s
0 0 1 0
0 0 0 1
T
1000
sasa0cs
)cac(ascsscs
)cac(acssc-cc
233222323
2332211231231
2332211231231
)(T … (6.11)
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Inverse Kinematics
• Unlike Forward Kinematics, general solutions
are not possible.
• Several architectures are to be solved
differently.
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38
Two-link Arm
X22a2
a1
X1
X3
Y2
Y1 Y3
1px
py
12211
12211
sasap
cacap
y
x
21
2
2
2
1
22
22 aa
aappc
yx
2
22 1 cs
2 = atan2 (s2, c2)
Δ
psap)ca(as
xy 22221
1
22
221
2
2
2
1 2 yx ppcaaaaΔ
Δ
psap)ca(ac
yx 22221
1
1 = atan2 (s1, c1)
1
2
RoboA
naly
zer
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39
Inverse Kinematics of 3-DOF RRR Arm
321 θθθφ
123312211 cacacapx
123312211 sasasapy
122113 cacac φ apw xx
122113 sasas φ apw yy
… (6.18a)
… (6.18b)
… (6.18c)
… (6.19a)
… (6.19b)
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40
w2x + w2
y = a12+ a2
2 + 2 a1a2c2
21
2
2
2
1
22
22 aa
aawwc 21
2
22 1 cs
2 = atan2 (s2, c2) . . . (6.21)
2121221 ssa)ccaa(wx
2121221y sca)sca(aw
Δ
wsaw)ca(as
xy 22221
1
Δ
wsaw)ca(ac
yx 22221
1
22
221
2
2
2
1 2 yx wwcaaaaΔ
1 = atan2 (s1, c1) . . . (6.23c)
3 = - 1 2 . . . (6.24)
… (6.22a)
… (6.22b)
… (6.20a)
… (6.20b,c)
… (6.23a,b)
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Numerical Example
3 5
2 2
3 31
2 2T
10 3
2
10
2
0 0 1 0
0 0 0 1
• An RRR planar arm (Example 6.15). Input
where = 60o, and a1 = a2 = 2 units, and a3 = 1 unit.
Rotation
Matrix
Origin
of end-
effector
frame
4.23
1.86
0
Do it yourself Verify using RoboAnalyzer
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42
Using eqs. (6.13b-c), c2 = 0.866, and s2 = 0.5,
Next, from eqs. (6.16a-b), s1 = 0, and c1= 0.866.
Finally, from eq. (6.17) ,
Therefore …(6.30b)
The positive values of s2 was used in evaluating 2 = 30o.
The use of negative value would result in :
…(6.30c)
2 = 30o
1 = 0o.
3 = 30o.
1 = 0o 2 = 30o, and 3 = 30
1 = 30o 2 = -30o, and 3 = 60o
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43
Watch
• Forward and Inverse Kinematics: Watch 3/3 of
IGNOU Lectures [29min]
https://www.youtube.com/watch?v=duKD8cvtBTI
• For more clarity: Watch 12 of Addis Ababa
Lectures [77 min]
[https://www.youtube.com/watch?v=NXWzk1toze4
• Robotics (13 of Addis Ababa Lectures): Inverse
Kinematics [82 min]
https://www.youtube.com/watch?v=ulP3YiJLiEM
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Velocity Analysis
1 2where andJ j j j n
i
i
i ie
, if Joint is revolutei
ej
e aprismatic isointJif, i
iei
i
ae
0j
et Jθ
1
twistof end - effector : ; Joint rates : e
e
e
n
ωt θ
v
Jacobian maps joint rates into end-effector’s velocities. It
depends on the manipulator configuration.
nee1e aeaeae
eeeJ
n2
n221
1
. . (6.86)
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Jacobian of a 2-link Planar Arm
ee 2211 aeaeJ
1 1 2 12 2 12
1 1 2 12 2 12
Hence, Ja s a s a s
a c a c a c
1 2where [0 0 1]e eT
1 1 2
1 1 2 12 1 1 2 12[ 0]
a a a
e
Ta c a c a s a s
2 2
2 12 2 12[ 0]
a a
e
Ta c a s
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46
Example: Singularity of 2-link RR Arm
12212211
12212211
cacaca
sasasaJ 2 = 0 or
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47
PROPRIETARY MATERIAL. © 2014, 2008 The McGraw-Hill Companies, Inc. All rights reserved. No part of this PowerPoint slide may be displayed,
reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers
and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this PowerPoint slide, you are using it without
permission.
Lecture 4 (SIT Sem. Rm.)
Forward and Inverse Kinematics
(Ch. 6)by
S.K. SahaAug. 12, 2015 (W)@JRL301(Robotics Tech.)
@ McGraw-Hill Education
48
PROPRIETARY MATERIAL. © 2014, 2008 The McGraw-Hill Companies, Inc. All rights reserved. No part of this PowerPoint slide may be displayed,
reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers
and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this PowerPoint slide, you are using it without
permission.
Statics and Manipulator
Design (Ch. 7)
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49
Principle of Virtual Work
• Relation between two virtual displacements
(Can be derived from velocity expression)
θτxw TTe
θJx
θτθJw TTe
TT
e τJw
e
TwJτ
… (7.28)
… (7.29)
… (7.32)
… (7.31)
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50
Example: 2-link RR Planar Arm
1 1 1 01 1
1 2 2 1 2
[ ] [ ] e n
T
x y
τ
a f sθ (a a cθ )f
y
T faτ 2212222 ][][ ne
fJτT
2
1
τ
ττ
0
y
x
f
f
f
00
0
2
22121
a
acθasθaT
J
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51
Two Jacobian Matrices
• From
Statics
00
0
2221
21
a acθa
sθa
J
• From
Kinematics
12212211
12212211
cacaca
sasasaJ
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52
Jacobian from Statics in Frame 1
00
00
0
100
0
0
100
0
0
][
12212211
12212211
2221
21
22
22
11
11
1
cθacθ acθa
sθ asθ asθa
a acθa
sθa
cs
sc
cs
sc
J
… (7.34)
• Without the last row, it is the same as
the one from kinematics Should be!
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53
Manipulator Design
• High investment in robot usage low
technological level of mechanical structure
• Functional Requirements
• Kinetostatic Measures
• Structural Design and Dynamics
• Economics
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54
Functional Requirements of a
Robot
• Payload
• Mobility
• Configuration
• Speed, Accuracy and Repeatability
• Actuators and Sensors
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• Dexterity
• Manipulability
• Nonredundant manipulator square
Jacobian
Dexterity and Manipulability
det( )Jdw
det( )T
mw JJ
det( )mw J d mw w
… (7.44)
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57
Motor Selection (Thumb Rule)
• Rapid movement with high torques (>
3.5 kW): Hydraulic actuator
• < 1.5 kW (no fire hazard): Electric
motors
• 1-5 kW: Availability or cost will
determine the choice
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Simple Calculation
2 m robot arm to lift 25 kg mass at 10
rpm
• Force = 25 x 9.81 = 245.25 N
• Torque = 245.25 x 2 = 490.5 Nm
• Speed = 2 x 10/60 = 1.047 rad/sec
• Power = Torque x Speed = 0.513 kW
• Simple but sufficient for approximation
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59
Practical Application
Subscript l for load; m for motor;
G = l/m (< 1); : Motor + Gear box efficiency
Trapezoidal Trajectory
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Accelerations & Torques
Ang. accn. during t1:
Ang. accn. during t3:
Ang. accn. during t2: Zero (Const. Vel.)
Torque during t1: T1 =
Torque during t2: T2 =
Torque during t3: T3 =
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63
Final Selection
• Peak speed and peak torque
requirements , where TPeak is max of
(magnitudes) T1, T2, and T3
• Use individual torque and RMS values
+ Performance curves provided by the
manufacturer.
• Check heat generation + natural
frequency of the drive.
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64
Dynamics and Control
Measures
1
2n r
• Rule of Thumb
n
r
: closed-loop natural frequency
: lowest structural resonant frequency
… (7.51)
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65
Manipulator Stiffness
2
1 2
1 1 1
ek k k ek
equivalent stiffness
gear ratio … (7.48)
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66
Link Material Selection
• Mild (low carbon) steel:
Sy = 350 Mpa; Su = 420 Mpa
• High alloyed steel
Sy = 1750-1900 Mpa; Su = 2000-2300
Mpa
• Aluminum
• Sy = 150-500 Mpa; Su = 165-580 Mpa
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67
Driver Selection
• Driver of a DC motor: A hardware unit
which generates the necessary current
to energize the windings of the motor
• Commercial motors come with
matching drive systems
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68
Summary
• Forward Kinematics
• Inverse kinematics
– A spatial 6-DOF wrist-portioned has 8
solutions
• Velocity and Jacobian
• Mechanical Design