Robot Motion Analysis Transformations

1. Robot Motion Analysis
Presented By
Deepam Goyal
Department of Mechanical Engineering
National Institute of Technical Teachers Training & Research
Chandigarh – 160 019
July, 2015

2. Historical Perspective
 Robot Definitions
 Basic Fundamentals of Robots
 Types of Kinematics
 Transformations
 Geometric Interpretation of Rotation Matrix
 Inverse transformations
 Homogeneous transformations
 Inverse Homogeneous transformations
 Inverse Kinematics of a Two Link Manipulator
 Important Terminology
 References
Contents

3. Historical Perspective
• The word robot was first used in 1921 by Czech playwright
Karel Capek in his satirical drama titled Rossum’s Universal
Robots
– Derived from Czech word robota which literally means
‘forced labour’.
• The term robotics was coined by American author and
professor of biochemistry at Boston University, Isaac
Asimov in his short story titled Runaround.

4. Robot Definitions
• Webster dictionary
– An automatic apparatus or device
that performs functions ordinarily
ascribed to humans or operates with
what appears to be almost human
intelligence
• Robotic Institute of America (RIA)
– A robot is a reprogrammable,
multifunctional manipulator
designed to move material, parts,
tools or specialized devices through
variable programmed motions for
the performance of a variety of
tasks.

5. Basic Components of a Robot System
• Manipulator
– Series of rigid members, called links, connected by joints.
• Actuators
– Provide power to the manipulator.
• Sensory Devices
– To monitor position, speed, acceleration, torque etc.
• Controller
– Provides the intelligence to make the manipulator perform
in a certain manner.
• Power Conversion Unit
– Takes signal from controller and converts it into meaningful
power level so that actuators can move.

6. Robot Configurations
• Cartesian (3P)
– Three Degrees of freedom (DOF) are linear & at right angles
to each other.
– Rectangular Workspace.
• Cylindrical (R2P)
– Two DOFs are linear and one DOF is rotational.
– Cylindrical Workspace.
• Spherical (2RP)
– One DOF is linear and two DOFs are rotational.
– Spherical Workspace.
• Articulated (3R)
– All three DOFs are rotational.
– Irregular Workspace.

7. Contd..

8. Types of Joints
Joints Motions Degree of Freedom
Revolute joint Rotary motion One
Prismatic joint Sliding motion One
Cylindrical joint One sliding & one rotary motion Two
Planar joint Two sliding & one rotary motion Three
Screw pair One translatory & one rotary motion Two
Spherical joint Three rotary motion three

9. Types of Kinematics

10. The transformation matrix is 4X4 matrix which consists of sub-
matrices :
 Rotation Matrix
 Translation or Position Vector
 Perspective Transformation
 Scaling/ Stretching
Transformation

11. Geometric Interpretation of Rotation Matrix
• Rotation about X-axis
• Rotation about Y-axis
• Rotation about Z-axis

12. Inverse transformations

13. Problem
Example : A mobile body reference frame OABC is rotated 60 about OY-
axis of the fixed base reference system OXYZ. If and are
the coordinates with respect to OXYZ plane, what are the
corresponding coordinates of p and q with respect to OABC frame.
T
xyzp )6,4,2(
T
xyzq )7,5,3(

14. Homogeneous Transformations
Components of Homogeneous Transformations

15. Contd..

16. Problem
Example: Determine the homogeneous transformation
matrix to represent the following sequence of
operations.
a) Rotation of about OX-axis
b) Translation of 4 units along OX-axis
c) Translation of -6 units along OC-axis
d) Rotation of about OB-axis
3

6


17. Inverse Homogeneous Transformation
















1000
100
)cossin(0cossin
)sincos(0sincos
),(1
z
yx
yx
p
pp
pp
zH




18. Revolute and Prismatic
Joints Combined
1
X
Y
S
(x , y) )
x
y
(tanθ 1

)y(xS 22

(i)
(ii)
Inverse Kinematics of a One Link Manipulator

19. (x , y)
2
1
l2
l1
Given: l1, l2 , x , y
Find: 1, 2
Redundancy:
A unique solution to this problem does not exist. Notice,
that using the “givens” two solutions are possible.
Sometimes no solution is possible.
(x , y)
Inverse Kinematics of a Two Link Manipulator

20.    
 







 








21
2
2
2
1
22
1
2
221
2
2
2
1
21121121
2
2
2
1
21121
2
21
2
2
2
1
2
121121
2
21
2
2
2
1
2
1
2222
2
yx
cosθ
c2
)(sins)(cc2
)(sins2)(sins)(cc2)(cc
yx)2((1)
ll
ll
llll
llll
llllllll
The Algebraic Solution
21
21211
21211
1221
11
θθθ)(
sinsy)(
ccx)(
)θcos(θc
cosθc








iii
llii
lli
Only
Unknown

21. )c(s)s(c
cscss
sinsy
)()c(c
ccc
ccx
2211221
12221211
21211
2212211
21221211
21211
lll
lll
ll
slsll
sslll
ll








We know what 2 is from the
previous slide. We need to solve for
1 . Now we have two equations and
two unknowns (sin 1 and cos 1 )
 














22
22221
1
221
2
2
2
1122
221
221122
221
221
221
221
1
yx
x)c(y
s
)c2(sx
)c(
1
)c(s)s(
)c(
)(x
y
)c(
)(x
c
slll
llllsl
ll
lll
ll
sls
ll
sls
(Substituting for c1 and simplifying
many times)
(This is the law of cosines and can be
replaced by x2+ y2)








 
22
222211
1
yx
x)c(y
sinθ
slll

22. The Geometric Solution
l2
l1
2
1

(x , y)
Using the Law of Cosines:







 







21
2
2
2
1
22
1
21
2
2
2
1
22
21
2
2
2
1
22
222
2
cosθ
2
)cos(θ
)cos(θ)θ180cos(
)θ180cos(2)(
cos2
ll
llyx
ll
llyx
llllyx
Cabbac
2
2
22
2
Using the Law of Sines:















x
y
2tanα
αθθ
yx
)sin(θ
yx
)θsin(180θsin
sinsin
1
11
22
2
22
2
2
1
l
c
C
b
B
















 
x
y
2tan
yx
)sin(θ
sinθ 1
22
221
1
l
Redundant since 2 could be in the
first or fourth quadrant.
Redundancy caused since 2 has two
possible values

23. Degeneracy :
The robot looses a degree of freedom and thus cannot
perform as desired.
‫٭‬ When the robot’s joints reach their physical limits, and as a
result, cannot move any further.
‫٭‬ In the middle point of its workspace if the z-axes of two similar
joints becomes collinear.
Dexterity :
The volume of points where one can position the robot
as desired, but not orientate it.
Important Terminology

24. Problems
Example 1: Determine the homogeneous transformation matrix to
represent the following sequence of operations.
a) Translation of 4 units along OX-axis
b) Rotation of OX-axis
c) Translation of -6 units along OC-axis
d) Rotation of about OB-axis
3

6


25. References
• Groover, M.P., Emory W. Zimmers JR. “CAD/CAM:Computer-
Aided Design and Manufacturing”. 25. New Delhi: Prentice
Hall of India Private Limited, 2002. 324-332. Print.
• Hegde, Ganesh S. "Robot Motion Analysis." A Textbook on
Industrial Robotics. Second ed. New Delhi: U Science, 2009.
25-114. Print.
• Niku, Saeed B. "Robotic Kinematics : Position
Analysis." Introduction to Robotics: Analysis, Systems,
Applications. 1st ed. New Delhi: PHI Learning Private
Limited, 2009. 29-90. Print.

26. Email ID:-
bkdeepamgoyal@gmail.com