Standard Robot motion Co-ordinates, Robot Transformations, Matrix Representation of Robot,

1. Forward and Inverse Kinematics
Submitted To:
Dr. B.S Pabla
Professor
M.E Department
Submitted By:
Varinder Singh
152227
M.E(M.T)

2. Content
Introduction
Matrix Representation
Transformations
Standard Robot coordinate System
Numericals

3. Robot Kinematics: Position Analysis
INTRODUCTION
♦Forward Kinematics:
to determine where the robot’s hand is?
(If all joint variables are known)
♦Inverse Kinematics:
to calculate what each joint variable is?
(If we desire that the hand be
located at a particular point)

4. Matrix Representation
- Representation Of A Point In Space
Representation of a point in space
♦A point P in space :
3 coordinates relative to a reference frame
^^^
kcjbiaP zyx ++=

5. Representation of a vector in space
♦A Vector P in space :
3 coordinates of its tail and of its head
^^^__
kcjbiaP zyx ++=












=
w
z
y
x
P
__
Matrix Representation
-Representation of a Vector in Space
Where is Scale
factor
w

6. It can Change overall size of vector similar to
zooming function in computer graphics.
When w=1 ,
Size of components remain unchanged
When w=0,
It represent a vector whose length is infinite but it
represents the direction so called as directional
vector
Scale Factor w

7. Representation of a frame at the origin of the reference frame
♦Each Unit Vector is mutually perpendicular. :
normal, orientation, approach vector










=
zzz
yyy
xxx
aon
aon
aon
F
Matrix Representation
-Representation of a Frame at the Origin of a Fixed-
Reference Frame

8. Representation of a frame in a frame
♦Each Unit Vector is mutually perpendicular. :
normal, orientation, approach vector












=
1000
zzzz
yyyy
xxxx
Paon
Paon
Paon
F
Representation of a Frame in a Fixed Reference
Frame

9. Representation of an object in space
♦An object can be represented in space by attaching a frame
to it and representing the frame in space.












=
1000
zzzz
yyyy
xxxx
object
Paon
Paon
Paon
F
Representation of a Rigid Body

10. Homogeneous Transformation Matrices
♦A transformation matrices must be in square form.
• It is much easier to calculate the inverse of square
matrices.
• To multiply two matrices, their dimensions must match.












=
1000
zzzz
yyyy
xxxx
Paon
Paon
Paon
F

11. Transformations
A transformation is defined as making a movement
in space.
Types of Transformation are:
A pure translation
A pure rotation
A combination of translation and rotation

12. Representation of a Pure Translation
Representation of an pure translation in space












=
1000
100
010
001
z
y
x
d
d
d
T
If a frame moves in space without any change in its
orientation

13. Numerical Problem-1
A frame F has been moved 10 units along y-axis and 5
units along z-axis of reference frame. Find new location of
frame.
Answer:

14. Numerical Problem-1

15. Pure Rotation about an Axis
Coordinates of a point in a rotating frame before and after rotation.
Assumption : The frame is at the origin of the reference
frame and parallel to it.

16. Pure Rotation about an Axis

17. Combined Transformations
Combined Transformation consist of a number of
successive translations and rotations about fixed
reference frame axes.
The order of matrices written is the opposite of
the order of transformations performed.
If order of matrices changes then final position of
robot also changes

18. Numerical Problem (Forward Kinematics)-2
A point p(7,3,1) is attached to frame and subjected to
following transformations. Find coordinate of point
relative to reference frame.
1.Rotation of 90° about z-axis
2.Followed by rotation of 90 about y-axis
3.Followed by translation of [4,-3,7].
Answer: The matrix equation is given as

19. Numerical Problem-2

20. Fig. 2.13 Effects of three successive transformations
♦A number of successive translations and rotations….
Numerical Problem-2

21. Forward Kinematics and Inverse Kinematics equation
for position analysis and three types of standard robot
coordinate system are:
(a) Cartesian (gantry, rectangular) coordinates.
(b) Cylindrical coordinates.
(c) Spherical coordinates.
Forward and Inverse Kinematics Equations for Position

22. Cartesian (Gantry, Rectangular)
Coordinates
•All actuators are linear.
•A Gantry robot is a Cartesian robot and used in pick and
place applications like overhead cranes.
Cartesian Coordinates.












==
1000
100
010
001
z
y
x
cartP
R
P
P
P
TT

23. Cylindrical Coordinates
• 2 Linear translations and 1 rotation
• translation of r along the x-axis
• rotation of α about the z-axis
• translation of l along the z-axis











 −
==
1000
100
0
0
l
rSCS
rCSC
TT cylP
R ααα
ααα
,0,0))Trans(,)Rot(Trans(0,0,),,( rzllrTT cylP
R
αα ==

24. Suppose we desire to place the origin of hand frame of a
cylindrical robot at [ 3,4,7]. Calculate the joint variables of
robot.
Answer:
Numerical Problem (Inverse Kinematics)-3











 −
==
1000
100
0
0
l
rSCS
rCSC
TT cylP
R ααα
ααα r= 5 units

25. Spherical Coordinates
• 2 Linear translations and 1 rotation
• translation of r along the z-axis
• rotation of β about the y-axis
• rotation of γ along the z-axis
Spherical Coordinates.












−
⋅⋅⋅
⋅⋅−⋅
==
1000
0 βββ
γβγβγγβ
γβγβγγβ
rCCS
SrSSSCSC
CrSCSSCC
TT sphP
R
))Trans()Rot(Rot()( 0,0,,,,, γβγβ yzlrsphP
R
TT ==

26. References
Saeed B. Niku, “Introduction to Robotics – Analysis,
Control, Applications” , 2nd
Edition, John Wiley & Sons,
2016