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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 5
Velocity Analysis (Ch. 6):
Jacobianby
S.K. SahaSep. 19, 2017 (Tu)@JRL301 (Rob. Tech.)
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Course Project: Mobile Platform
• Build a two-wheeled driven mobile robot
platform
• Perform kinematic analysis, i.e., given a path of
the platform what are the wheel motions
required [SK Saha]
• Attach a camera for some application
(navigation?) [S Dutta Roy]
• Apply some form of control [IN Kar]
• Implement in some hardware! [Kolin Paul]
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Velocity Analysis: Jacobian
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
twist of 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
Check from position
expression also!
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Example: Singularity of 2-link RR Arm
12212211
12212211
cacaca
sasasaJ 2 = 0 or
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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.
Jacobian in
Statics (Ch. 7)
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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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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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Two Jacobian Matrices
• From
Statics
00
0
2221
21
a acθa
sθa
J
• From
Kinematics
12212211
12212211
cacaca
sasasaJ
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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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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.
Dynamics (Ch. 8)
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Euler-Lagrange Formulation
ii
q
L
iq
L
dt
d
L (Lagrangian) = T – U;
T: Kinetic energy; U: Potential energy;
qi: Generalized coordinate;
i : Generalized force.
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Generalized Coordinates
• Coordinates that specify the configuration (position and
orientation) generalized coordinates
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Kinetic and Potential Energies
• Kinetic Energy
• Potential Energy
n
i
Tii
mU
1
gc
n
iii
Tii
Tii
mn
ii
TT
12
1
1
ωIωcc
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Euler-Lagrange Equation
Kinetic energy
Velocity constraint:
Euler-Lagrange:
1;
2c c TT m
;c i x
Lmx;
x
ef
mx f
External
force,
Mass, m
i
j
c
cfReaction,
0U
21
2L(= T -U) mx
( )
d Lmx;
dt x0
L
x
f
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Example: One-DOF Arm (EL)2
2 21 1( ) ;
2 2 12
( )2 2
a maT m
2
a aU mg c
mgas
Lma
L
dt
d
2
1;
3
1)( 2
mgasma3
1 2
2
1
)1(2
2 ca
mg6
maU-TL
2
Please
write!
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Simulation of One-link Arm
RoboAnalyzer
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Simulation of One-link Arm using
MATLAB
2 1( sin )
22mga
ma
Hence, the state-space form is given by
1 2
2
2 1( sin )
22
y y
y mgama
