1. 1

2.  The study of the position and orientation of a
robot hand with respect to a reference
coordinate system, given the joint variables and
the arm parameters, OR
 The analytical study of the geometry of motion
of a robot arm with respect to a reference
coordinate system.
 Without regard the forces of moments that
cause the robot motion.
 It is the first step towards robotic control.
2

3.  What you are given:
 The length of each link
 The angle of each joint
 What you can find:
 The position of any point (i.e. it’s (x, y, z) coordinates
3

4.  Between two frames, the is a kinematic
relationship either a translation, rotation or
both. The relationship can be describe by a
transformation matrix.
{D}
z2
z0 {C} Translation
y2 and rotation
{B} {A}
Rotation Translation
y0
x2
Rereference
x0 frame Note: {D} = Frame D
4

5. pu cos( ) sin( ) px A
Puv RPxy
B
pv sin( ) cos( ) py
A y
Puv B RPxy v
{A}
A Puv
Pxy B QPuv
{B}
Pxy
u
A A 1 A
Q
B BR BRT B
A R
x
A
Note: B R describes the rotations of {B} w.r.t. {A}
5

6. px ix iu i x jv ix k w pu
Pxyz py jy i u j y jv jy k w pv RPuvw
pz k z iu k z jv kz kw pw
z
A
Puvw B RPxyz
Puvw
Pxyz
A
Pxyz QP
B uvw y
A A 1 A
BQ BR BRT B
A R
x
6

7.  Rotation transformation matrices
Rotation about x-axis by degrees - Yaw
1 0 0 z
Rx ( ) 0 Cos Sin
Roll
0 Sin Cos
Rotation about y-axis by degrees - Pitch Pitch
Cos 0 Sin y
Ry ( ) 0 1 0
Sin 0 Cos x Yaw
Rotation about z-axis by degrees - Roll
Cos Sin 0
Rz ( ) Sin Cos 0
0 0 1
7

8.  Roll-pitch-yaw angles (Z-Y-X Euler angle-Relative axis)
It provides a method to decompose a complex rotation into
three consecutive fundamental rotations; roll, pitch, and yaw.
Use post multiplication rule.
Ruvw (mobile
) Rz ( ) Ry ( ) Rx ( )
Cos Sin 0 Cos 0 Sin 1 0 0
Sin Cos 0 0 1 0 0 Cos Sin
0 0 1 Sin 0 Cos 0 Sin Cos
Ruvw is mobile with respect to the Rxyz
8

9.  Yaw-pitch-roll angles (X-Y-Z fixed angle)
Representation in yaw-pitch-roll angles allows complex rotation
to be decomposed into a sequence of yaw, pitch and roll about the
x, y and z axis.
Use pre-multiplication rule.
Ruvw ( fixed) RZ ( ) RY ( ) RX ( )
Cos Sin 0 Cos 0 Sin 1 0 0
Sin Cos 0 0 1 0 0 Cos Sin
0 0 1 Sin 0 Cos 0 Sin Cos
Conclusion: Ruvw ( fixed) Ruvw (mobile
)
9

10.  Z-Y-Z Euler angle
Read the Z-Y-Z Euler angles on page 30 (M. Zhihong)
10

11.  Find the position of point P=[10 10] with
respect to the global axis after it is
transformed/rotated by [pi/3]
 Find the position of point P=[10 10 10] with
 respect to the global axis after it is transformed
by [pi/4; pi/3; pi/6]
11

12.  Homogeneous transformations
•Transforms and translates.
•The homogenous transformation matrix below is used to
transform and translate. R is a 3x3 rotation matrix and P is a
3x1translation/position vector.
R P
H
0 0 0 1
Three fundamental rotation matrices of roll, pitch and yaw in
the homogeneous coordinate system:
C S 0 0 C 0 S 0 1 0 0 0
S C 0 0 0 1 0 0 0 C S 0
Hz( ) Hy( ) Hx( )
0 0 1 0 S 0 C 0 0 S C 0
0 0 0 1 0 0 0 1 0 0 0 1
12

13.  Homogeneous transformations
Three fundamental rotation matrices of roll, pitch and yaw Hrpy
in the homogeneous coordinate system:
R
CC CS S SC CS C SS Px
SC SS S CC SS C CS Py
H rpy
S C S C C Pz
0 0 0 1
A point B’ can be found from the following relationship:
B' H rpy B
13

14.  Homogeneous transformations
O
Translation without rotation
Y
1 0 0 Px
0 1 0 Py
N H
P 0 0 1 Pz
X A 0 0 0 1
Z
Rotation without translation
Y
O nx ox ax 0
N ny oy ay 0
H
nz oz az 0
X
0 0 0 1
Z
14 A

15.  Example 1: Find a point B’ in {B} w.r.t to the
reference frame {A} if the origin of {B} is (5,5,5) .
Given B=(1,2,3). Given 0; 0; 0.
B’
(5,5,5)
B(1,2,3)
15

16.  Solution
1 0 0 5
0 1 0 5
H
0 0 1 5
0 0 0 1
B' HB
1 0 0 5 1 6
0 1 0 5 2 7
B'
0 0 1 5 3 8
0 0 0 1 1 1
16

17.  Example 2: Find a point B’ in {N} w.r.t to the
reference frame {M} if the origin of {N} is (3,5,4) .
Given B=(3,2,1). {N} is rotated by ; 0; .
2
{N}
B’
(3,5,4)
{M}
B(3,2,1)
17

18. Solution:
1 0 0 3 3 0
0 0 1 5 2 6
B'
0 1 0 4 1 6
0 0 0 1 1 1
18

19.  Example 3: Find a point P’ in {N} w.r.t to the
reference frame {M} if the origin of {N} is (3,5,4) .
Given B=(3,2,1). {N} is rotated by ; ; .
3 2
{N}
B’
(3,5,4)
{M}
B (3,2,1)
B’ =[ 0.7679
4.8660
1.0000
1.0000]
19