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Lecture 5 (SIT Sem. Rm.)
by
S.K. Saha
Aug. 17, 2015 (M)@JRL301(Robotics Tech.)
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.
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Principle of Virtual Work
T
w e x
 τ θ
T
… (7.28)
• Relation between two virtual displacements
(Can be derived from velocity expression)
x  Jθ
T
w e Jθ
… (7.29)
w J  τ … (7.31)
T
e
 τ θ
T
τ  J we
T
T
… (7.32)
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Example: 2-link RR Planar Arm
From FBD
τ1  [e ] [n01 ]1
T
1 1
 a1 f x sθ2  (a2  a1cθ2 )f y
τ 2  [e ] [n12 ]2  a2 f y
T
2 2
τJ f
T
 τ1  T a1sθ 2 a1cθ2  a2
τ  J 
0
a
τ
2

 2
 fx 
0
f 
f

y


0
 0 
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Two Jacobian Matrices
• From
Statics
 a1 sθ 2

J  a1cθ 2  a2

0
0

a2 
0 
• From
 a1 s1  a 2 s12
Kinematics J  
 a1 c1  a 2 c12
 a 2 s12 

a 2 c12 
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Jacobian from Statics in Frame 1
0
c1  s1 0 c 2  s 2 0  a1sθ 2
[J ]1   s1 c1 0  s 2 c 2 0 a1cθ2  a2 a2 
 0
0
1  0
0
1 
0
0 
 a1sθ1  a2 sθ12  a2 sθ12 
  a1cθ1  a2 cθ12
a2 cθ12 

0
0 
… (7.34)
• Without the last row, it is the same as
the one from kinematics  Should be!
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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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Functional Requirements of a
Robot
• Payload
• Mobility
• Configuration
• Speed, Accuracy and Repeatability
• Actuators and Sensors
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bmin  b  bmax, for 0o    360o
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Dexterity and Manipulability
• Dexterity  wd  det(J )
• Manipulability  wm 
… (7.44)
T
det(JJ )
• Non-redundant manipulator  square
Jacobian
wm  det(J)
wd  wm
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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
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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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Practical Application
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Trapezoidal
Trajectory
Subscript l for load; m for motor;
G = l/m (< 1); : Motor + Gear box efficiency
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Accelerations & Torques
Ang. accn. during t1:
Ang. accn. during t2: Zero (Const. Vel.)
Ang. accn. during t3:
Torque during t1: T1 =
Torque during t2: T2 =
Torque during t3: T3 =
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RMS Value
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Motor Performance
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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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Dynamics and Control
Measures
• Rule of Thumb
1
n  r
2
… (7.51)
n
: closed-loop natural frequency
r
: lowest structural resonant frequency
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Manipulator Stiffness
1
1
1
 2 
ke  k1 k2
… (7.48)
ke  equivalent stiffness
  gear ratio
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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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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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Summary
• Statics in robotics
• Manipulator design
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Lecture 6 (SIT Sem. Rm.)
by
S.K. Saha
Aug. 24, 2015 (M)@JRL301(Robotics Tech.)
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.
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Outline
• Definition
• Euler-Lagrange Formulation
– Generalized coordinates
– Kinetic and potential energy
– Equations of Motion
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Euler-Lagrange Formulation


d  L  L


i
dt   q  q
i
 i
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
n
n 1
T   T    m cT c  ωT I ω 
i
i i i
i i i

2
i 1
i 1
• Potential Energy
n
T
U   m c g
i i
i 1
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Euler-Lagrange Equation
External
force, f e
j
f
Mass,
m
c
Reaction, f c
i
Kinetic energy
1
T  mcT c; U  0
2
1
2
L(=
T
-U)

mx
c

x
i
;
Velocity constraint:
2
L
d L
L
 mx;
(
)  mx;
0
x
dt x
x
Euler-Lagrange:
mx  f
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Example: One-DOF Arm (EL)
2
1 a Please
1
ma
T  m(  ) 2 
 2;
2 2 write!
2 12
a a
U  mg (  c )
2 2
ma2  2
a
L  T -U 
  mg (1  c )
6
2
d L
1
1
2  L
( )  ma ;
  mgas
dt 
3

2
1 2  1
ma   mgas  
3
2
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Simulation of One-link Arm using MATLAB and MuPAD
(contd…)
RoboAnalyzer
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Simulation of One-link Arm using
MATLAB
2
1

(  mga sin  )
2
ma
2
Hence, the state-space form is given by
y1  y2
2
1
y2 
(  mga sin  )
2
ma
2
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Mobile Robots
• Non-holonomic systems
– Necessary and sufficient no. of variables
defining a pose exceeds the number of
actuators
• Holonomic
– Necessary and sufficient no. of variables
defining a pose is same as the no.
independent actuators
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Summary
• Euler-Lagrange equation was shown
– Generalized coordinates, generalized
forces were defined
• Demonstration with MATLAB and
RoboAnalyzer
• Mobile Robot Dynamics
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Lecture 7 (SIT Sem. Rm.)
by
S.K. Saha
Aug. 26, 2015 (W)@JRL301(Robotics Tech.)
Mobile Robot Dynamics
[Ref: Dynamics and Design of Nonholonomic Robotic
Mechanical Systems, Ph. D thesis, McGill Univ., Canada, 1991]
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Two-wheeled System
Hand calculations on white board using
Euler-Lagrange equation
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Kinematic & Dynamic Models
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Circular Path
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Joint Torques (- 1; .. 2)
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Three-DOF 3-Wheeled Mobile
Robot
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Three-DOF 4-Wheeled
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Joint Torques@j (- 1&4; .. 2&3)
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Joint Torques@i (-1&2; .. 3&4
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Three-DOF 6-Wheeled
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Isotropic 3-DOF 4-Wheeled
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Summary
• Mobile robots are nonholonomic systems
• Dynamic model for a 2-wheeled system
• Several mobile robots with omnidirectional
wheels are shown
• Isotropic design was emphasized
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Thank You
saha@mech.iitd.ac.in
http://sksaha.com