CIS009-2, Mechatronics
Robot Coordination Systems II
David Goodwin
Department of Computer Science and Technology
University of Bedfordshire
07th February 2013
Outline
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
1 Mapping
Translation
Rotational
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
2 Operators
Translation
Rotation
3 Transformation
multiplication
Inverting
transformation equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
Mechatronics
David Goodwin
Mapping
3
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
Mapping
MAPPINGS
Mappings involving translated frames
ˆ A translated frame shifts without rotation
ˆ Translation takes place when A X k B X, A Y k
Mechatronics
David Goodwin
A
Mapping
Translation
4
Rotational
Operators
Zk Z
ˆ Mapping in this case means representing
in the form of A P
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
B
Y and
B
A
P = B P + A PBORG
B
P in {B} in {A}
MAPPINGS
Mappings involving translated frames
ˆ A translated frame shifts without rotation
ˆ Translation takes place when A X k B X, A Y k
Mechatronics
David Goodwin
A
Mapping
Translation
4
Rotational
Operators
Zk Z
ˆ Mapping in this case means representing
in the form of A P
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
B
Y and
B
A
P = B P + A PBORG
B
P in {B} in {A}
MAPPINGS
Mappings involving translated frames
ˆ A translated frame shifts without rotation
ˆ Translation takes place when A X k B X, A Y k
Mechatronics
David Goodwin
A
Mapping
Translation
4
Rotational
Operators
Zk Z
ˆ Mapping in this case means representing
in the form of A P
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
B
Y and
B
A
P = B P + A PBORG
B
P in {B} in {A}
MAPPINGS
Mappings involving translated frames
ˆ A translated frame shifts without rotation
ˆ Translation takes place when A X k B X, A Y k
Mechatronics
David Goodwin
A
Mapping
Translation
4
Rotational
Operators
Zk Z
ˆ Mapping in this case means representing
in the form of A P
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
B
Y and
B
A
P = B P + A PBORG
B
P in {B} in {A}
MAPPINGS
Mappings involving rotated frames
Mechatronics
ˆ Rotate a vector about an axis means the projection to that
David Goodwin
axis remains the same
Mapping
Translation
Rotational
5
ˆ Mapping
B
P in {B} to A P in {A} is
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
A
B
P =B
BR + P
MAPPINGS
Mappings involving rotated frames
Mechatronics
ˆ Rotate a vector about an axis means the projection to that
David Goodwin
axis remains the same
Mapping
Translation
Rotational
5
ˆ Mapping
B
P in {B} to A P in {A} is
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
A
B
P =B
BR + P
MAPPINGS
Mappings involving rotated frames
Mechatronics
ˆ Rotate a vector about an axis means the projection to that
David Goodwin
axis remains the same
Mapping
Translation
Rotational
5
ˆ Mapping
B
P in {B} to A P in {A} is
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
A
B
P =B
BR + P
MAPPINGS
Example
Mechatronics
ˆ A vector
David Goodwin
Mapping
Translation
Rotational
6
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
A
P is rotated about Z-axis by θ and is subsequently
rotated about X-axis by φ. Give rotation matrix that
accomplishes these rotations in the given order



1
0
0
cos θ − sin θ 0
R = 0 cos φ − sin φ  sin θ cos θ 0
0 sin φ cos φ
0
0
1
MAPPINGS
Example
Mechatronics
ˆ A vector
David Goodwin
Mapping
Translation
Rotational
6
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
A
P is rotated about Z-axis by θ and is subsequently
rotated about X-axis by φ. Give rotation matrix that
accomplishes these rotations in the given order



1
0
0
cos θ − sin θ 0
R = 0 cos φ − sin φ  sin θ cos θ 0
0 sin φ cos φ
0
0
1
MAPPINGS
Mappings involving general frames
Mechatronics
ˆ These mappings involve both translation and rotation
David Goodwin
Mapping
Translation
Rotational
7
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
A
B
A
P =A
B R P + PBORG
A A
A
P
R
PBORG B P
B
=
1
0 0 0
1
1
MAPPINGS
Mappings involving general frames
Mechatronics
ˆ These mappings involve both translation and rotation
David Goodwin
Mapping
Translation
Rotational
7
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
A
B
A
P =A
B R P + PBORG
A A
A
P
R
PBORG B P
B
=
1
0 0 0
1
1
MAPPINGS
Example
Mechatronics
ˆ Given {A}, {B},
David Goodwin
Mapping
Translation
Rotational
8
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
B
P and A PBORG , calculate A P , where
{B} is rotated relative to {A} about ZA -axis by 30 degrees,
translated 10 units in
XA -axis and translated 5 units in
YA -axis, and B P = 3.0 7.0 0.0
MAPPINGS
Example
Mechatronics
ˆ Given {A}, {B},
David Goodwin
Mapping
Translation
Rotational
8
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
B
P and A PBORG , calculate A P , where
{B} is rotated relative to {A} about ZA -axis by 30 degrees,
translated 10 units in
XA -axis and translated 5 units in
YA -axis, and B P = 3.0 7.0 0.0
MAPPINGS
exercises
Mechatronics
ˆ Calculate rotation matrix
David Goodwin
Mapping
Translation
Rotational
9
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
ˆ Calculate
A
P
MAPPINGS
exercises
Mechatronics
ˆ Calculate rotation matrix
David Goodwin
Mapping
Translation
Rotational
9
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
ˆ Calculate
A
P
MAPPINGS
exercises
Mechatronics
ˆ Calculate rotation matrix
David Goodwin
Mapping
Translation
Rotational
9
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
ˆ Calculate
A
P
MAPPINGS
exercises
Mechatronics
ˆ Calculate rotation matrix
David Goodwin
Mapping
Translation
Rotational
9
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
ˆ Calculate
A
P
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
10
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
Operators
OPERATORS
Mechatronics
ˆ T matrix
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
11
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
ˆ Two special cases:
ˆ Translation matrix of displacements a, b, and c along X, Y
and Z axes
OPERATORS
Mechatronics
ˆ T matrix
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
11
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
ˆ Two special cases:
ˆ Translation matrix of displacements a, b, and c along X, Y
and Z axes
OPERATORS
Mechatronics
ˆ T matrix
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
11
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
ˆ Two special cases:
ˆ Translation matrix of displacements a, b, and c along X, Y
and Z axes
OPERATORS
Mechatronics
ˆ T matrix
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
11
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
ˆ Two special cases:
ˆ Translation matrix of displacements a, b, and c along X, Y
and Z axes
OPERATORS
Mechatronics
ˆ T matrix
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
11
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
ˆ Two special cases:
ˆ Translation matrix of displacements a, b, and c along X, Y
and Z axes
OPERATORS
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
12
Rotation
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
OPERATORS
Mechatronics
ˆ Rotation
David Goodwin
Mapping
ˆ Rotation about X-axis
ˆ Rotation about Y-axis
ˆ Rotation about Z-axis
Translation
Rotational
Operators
Translation
Rotation
13
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
OPERATORS
Mechatronics
ˆ Rotation
David Goodwin
Mapping
ˆ Rotation about X-axis
ˆ Rotation about Y-axis
ˆ Rotation about Z-axis
Translation
Rotational
Operators
Translation
Rotation
13
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
OPERATORS
Mechatronics
ˆ Rotation
David Goodwin
Mapping
ˆ Rotation about X-axis
ˆ Rotation about Y-axis
ˆ Rotation about Z-axis
Translation
Rotational
Operators
Translation
Rotation
13
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
OPERATORS
Mechatronics
ˆ Rotation
David Goodwin
Mapping
ˆ Rotation about X-axis
ˆ Rotation about Y-axis
ˆ Rotation about Z-axis
Translation
Rotational
Operators
Translation
Rotation
13
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
OPERATORS
Mechatronics
ˆ Rotation
David Goodwin
Mapping
ˆ Rotation about X-axis
ˆ Rotation about Y-axis
ˆ Rotation about Z-axis
Translation
Rotational
Operators
Translation
Rotation
13
Transformation
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Transformation
Translation
Rotation
Transformation
14
multiplication
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
TRANSFORMATION
Mechatronics
ˆ Multiplication transformation
David Goodwin
ˆ Known a point in {C} and
A
BT
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
B
CT
matrix from {C} to {B} and
matrix from {B} to {A}
ˆ Find its position and orientation in {A}
Mapping
15
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
TRANSFORMATION
Mechatronics
ˆ Multiplication transformation
David Goodwin
ˆ Known a point in {C} and
A
BT
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
B
CT
matrix from {C} to {B} and
matrix from {B} to {A}
ˆ Find its position and orientation in {A}
Mapping
15
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
TRANSFORMATION
Mechatronics
ˆ Multiplication transformation
David Goodwin
ˆ Known a point in {C} and
A
BT
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
B
CT
matrix from {C} to {B} and
matrix from {B} to {A}
ˆ Find its position and orientation in {A}
Mapping
15
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
TRANSFORMATION
Mechatronics
ˆ Multiplication transformation
David Goodwin
ˆ Known a point in {C} and
A
BT
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
B
CT
matrix from {C} to {B} and
matrix from {B} to {A}
ˆ Find its position and orientation in {A}
Mapping
15
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
TRANSFORMATION
Mechatronics
David Goodwin
ˆ
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
16
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
A
CT
matrix from {C} to {A} is the multiplication of B
CT
matrix from {C} to {B} and A
T
matrix
from
{B}
to
{A}
B
TRANSFORMATION
Mechatronics
David Goodwin
ˆ
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
16
Inverting
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
A
CT
matrix from {C} to {A} is the multiplication of B
CT
matrix from {C} to {B} and A
T
matrix
from
{B}
to
{A}
B
TRANSFORMATION
Mechatronics
David Goodwin
ˆ Inverting a transformation
Mapping
ˆ Known T matrix from {B} to {A}
ˆ Find T matrix from {A} to {B}
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
17
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
TRANSFORMATION
Mechatronics
David Goodwin
ˆ Inverting a transformation
Mapping
ˆ Known T matrix from {B} to {A}
ˆ Find T matrix from {A} to {B}
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
17
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
TRANSFORMATION
Mechatronics
David Goodwin
ˆ Inverting a transformation
Mapping
ˆ Known T matrix from {B} to {A}
ˆ Find T matrix from {A} to {B}
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
17
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
TRANSFORMATION
Mechatronics
David Goodwin
ˆ Inverting a transformation
Mapping
ˆ Known T matrix from {B} to {A}
ˆ Find T matrix from {A} to {B}
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
17
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
TRANSFORMATION
Example
Mechatronics
David Goodwin
ˆ Frame {B} is rotated relative to Frame {A} about Z-axis by
Mapping
30 degrees and translated 4 units in X-axis and 3 units in
Y-axis. Find T matrix from {A} to {B}.
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
18
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
TRANSFORMATION
Example
Mechatronics
David Goodwin
ˆ Frame {B} is rotated relative to Frame {A} about Z-axis by
Mapping
30 degrees and translated 4 units in X-axis and 3 units in
Y-axis. Find T matrix from {A} to {B}.
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
18
transformation
equations
Department of
Computer Science and
Technology
University of
Bedfordshire
19
TRANSFORMATION
Transform equations
Mechatronics
David Goodwin
Mapping
ˆ Transform equation
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
19
ˆ Any unknown T matrix can then be calculated from the ones
given
Department of
Computer Science and
Technology
University of
Bedfordshire
19
TRANSFORMATION
Transform equations
Mechatronics
David Goodwin
Mapping
ˆ Transform equation
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
19
ˆ Any unknown T matrix can then be calculated from the ones
given
Department of
Computer Science and
Technology
University of
Bedfordshire
19
TRANSFORMATION
Transform equations
Mechatronics
David Goodwin
Mapping
ˆ Transform equation
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
19
ˆ Any unknown T matrix can then be calculated from the ones
given
Department of
Computer Science and
Technology
University of
Bedfordshire
19
TRANSFORMATION
Transform equations
Mechatronics
David Goodwin
Mapping
ˆ Transform equation
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
19
ˆ Any unknown T matrix can then be calculated from the ones
given
Department of
Computer Science and
Technology
University of
Bedfordshire
19
TRANSFORMATION
Transform equations
Mechatronics
David Goodwin
Mapping
ˆ Transform equation
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
19
ˆ Any unknown T matrix can then be calculated from the ones
given
Department of
Computer Science and
Technology
University of
Bedfordshire
19
TRANSFORMATION
Transform equations
Mechatronics
David Goodwin
Mapping
ˆ Transform equation
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
19
ˆ Any unknown T matrix can then be calculated from the ones
given
Department of
Computer Science and
Technology
University of
Bedfordshire
19
TRANSFORMATION
Transform equations
Mechatronics
David Goodwin
Mapping
ˆ Transform equation
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
19
ˆ Any unknown T matrix can then be calculated from the ones
given
Department of
Computer Science and
Technology
University of
Bedfordshire
19
TRANSFORMATION
Transform equations
Mechatronics
David Goodwin
Mapping
ˆ Transform equation
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformation
equations
19
ˆ Any unknown T matrix can then be calculated from the ones
given
Department of
Computer Science and
Technology
University of
Bedfordshire
19