robotkinematics-16092vsdfva sdaf7173439.ppt

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

2. 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)

3. Matrix Representation
-Representation Of A Point In Space
Representation of a point in space
A point Pin space :
3 coordinates relative to a reference frame^^^
kcjbiaP zyx 

4. Representation of a vector in space
A Vector Pin 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
factorw

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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:

13. Numerical Problem-1

14. 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.

15. Pure Rotation about an Axis

16. 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

17. 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

18. Numerical Problem-2

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

20. 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 KinematicsEquations for Position

21. 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

22. Cylindrical Coordinates
•2 Linear translations and 1 rotation
•translation of r along the x-axis
•rotation of about the z-axis
•translation of l along the z-axis 












1000
100
0
0
l
rSCS
rCSC
TT cylP
R

 ,0,0))Trans(,)Rot(Trans(0,0,),,( rzllrTT
cylP
R


23. 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


r= 5 units

24. Spherical Coordinates
•2 Linear translations and 1 rotation
• translation of r along the z-axis
• rotation of about the y-axis
• rotation of along the z-axis
Spherical Coordinates.















1000
0 


rCCS
SrSSSCSC
CrSCSSCC
TT sphP
R ))Trans()Rot(Rot()( 0,0,,,,,  yzlrsphP
R
TT 