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
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
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