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CIS009-2, MechatronicsRobot Coordination Systems II
David Goodwin
Department of Computer Science and TechnologyUniversity of Bedfordshire
07th February 2013
![Page 2: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/2.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
Outline
1 MappingTranslationRotational
2 OperatorsTranslationRotation
3 TransformationmultiplicationInvertingtransformation equations
![Page 3: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/3.jpg)
19
Mechatronics
David Goodwin
3Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
Mapping
![Page 4: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/4.jpg)
19
Mechatronics
David Goodwin
Mapping
4Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
MAPPINGSMappings involving translated frames
� A translated frame shifts without rotation� Translation takes place when AX ‖ BX, AY ‖ BY and
AZ ‖ BZ� Mapping in this case means representing BP in {B} in {A}in the form of AP
AP = BP + APBORG
![Page 5: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/5.jpg)
19
Mechatronics
David Goodwin
Mapping
4Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
MAPPINGSMappings involving translated frames
� A translated frame shifts without rotation� Translation takes place when AX ‖ BX, AY ‖ BY and
AZ ‖ BZ� Mapping in this case means representing BP in {B} in {A}in the form of AP
AP = BP + APBORG
![Page 6: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/6.jpg)
19
Mechatronics
David Goodwin
Mapping
4Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
MAPPINGSMappings involving translated frames
� A translated frame shifts without rotation� Translation takes place when AX ‖ BX, AY ‖ BY and
AZ ‖ BZ� Mapping in this case means representing BP in {B} in {A}in the form of AP
AP = BP + APBORG
![Page 7: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/7.jpg)
19
Mechatronics
David Goodwin
Mapping
4Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
MAPPINGSMappings involving translated frames
� A translated frame shifts without rotation� Translation takes place when AX ‖ BX, AY ‖ BY and
AZ ‖ BZ� Mapping in this case means representing BP in {B} in {A}in the form of AP
AP = BP + APBORG
![Page 8: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/8.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
5Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
MAPPINGSMappings involving rotated frames
� Rotate a vector about an axis means the projection to thataxis remains the same
� Mapping BP in {B} to AP in {A} is
AP = BBR+ BP
![Page 9: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/9.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
5Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
MAPPINGSMappings involving rotated frames
� Rotate a vector about an axis means the projection to thataxis remains the same
� Mapping BP in {B} to AP in {A} is
AP = BBR+ BP
![Page 10: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/10.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
5Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
MAPPINGSMappings involving rotated frames
� Rotate a vector about an axis means the projection to thataxis remains the same
� Mapping BP in {B} to AP in {A} is
AP = BBR+ BP
![Page 11: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/11.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
6Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
MAPPINGSExample
� A vector AP is rotated about Z-axis by θ and is subsequentlyrotated about X-axis by φ. Give rotation matrix thataccomplishes these rotations in the given order
R =
1 0 00 cosφ − sinφ0 sinφ cosφ
cos θ − sin θ 0sin θ cos θ 00 0 1
![Page 12: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/12.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
6Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
MAPPINGSExample
� A vector AP is rotated about Z-axis by θ and is subsequentlyrotated about X-axis by φ. Give rotation matrix thataccomplishes these rotations in the given order
R =
1 0 00 cosφ − sinφ0 sinφ cosφ
cos θ − sin θ 0sin θ cos θ 00 0 1
![Page 13: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/13.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
7Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
MAPPINGSMappings involving general frames
� These mappings involve both translation and rotation
AP = ABR
BP + APBORG[AP1
]=
[ABR
APBORG
0 0 0 1
] [BP1
]
![Page 14: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/14.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
7Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
MAPPINGSMappings involving general frames
� These mappings involve both translation and rotation
AP = ABR
BP + APBORG[AP1
]=
[ABR
APBORG
0 0 0 1
] [BP1
]
![Page 15: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/15.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
8Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
MAPPINGSExample
� Given {A}, {B}, BP and APBORG, calculateAP , where
{B} is rotated relative to {A} about ZA-axis by 30 degrees,translated 10 units in XA-axis and translated 5 units inYA-axis, and
BP =[3.0 7.0 0.0
]
![Page 16: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/16.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
8Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
MAPPINGSExample
� Given {A}, {B}, BP and APBORG, calculateAP , where
{B} is rotated relative to {A} about ZA-axis by 30 degrees,translated 10 units in XA-axis and translated 5 units inYA-axis, and
BP =[3.0 7.0 0.0
]
![Page 17: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/17.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
9Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
MAPPINGSexercises
� Calculate rotation matrix
� Calculate AP
![Page 18: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/18.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
9Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
MAPPINGSexercises
� Calculate rotation matrix
� Calculate AP
![Page 19: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/19.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
9Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
MAPPINGSexercises
� Calculate rotation matrix
� Calculate AP
![Page 20: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/20.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
9Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
MAPPINGSexercises
� Calculate rotation matrix
� Calculate AP
![Page 21: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/21.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
10Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
Operators
![Page 22: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/22.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
11Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
OPERATORS
� T matrix
� Two special cases:
� Translation matrix of displacements a, b, and c along X, Yand Z axes
![Page 23: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/23.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
11Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
OPERATORS
� T matrix
� Two special cases:
� Translation matrix of displacements a, b, and c along X, Yand Z axes
![Page 24: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/24.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
11Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
OPERATORS
� T matrix
� Two special cases:
� Translation matrix of displacements a, b, and c along X, Yand Z axes
![Page 25: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/25.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
11Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
OPERATORS
� T matrix
� Two special cases:
� Translation matrix of displacements a, b, and c along X, Yand Z axes
![Page 26: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/26.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
11Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
OPERATORS
� T matrix
� Two special cases:
� Translation matrix of displacements a, b, and c along X, Yand Z axes
![Page 27: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/27.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
12Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
OPERATORS
![Page 28: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/28.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
13Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
OPERATORS
� Rotation
� Rotation about X-axis� Rotation about Y-axis� Rotation about Z-axis
![Page 29: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/29.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
13Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
OPERATORS
� Rotation
� Rotation about X-axis� Rotation about Y-axis� Rotation about Z-axis
![Page 30: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/30.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
13Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
OPERATORS
� Rotation
� Rotation about X-axis� Rotation about Y-axis� Rotation about Z-axis
![Page 31: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/31.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
13Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
OPERATORS
� Rotation
� Rotation about X-axis� Rotation about Y-axis� Rotation about Z-axis
![Page 32: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/32.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
13Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
OPERATORS
� Rotation
� Rotation about X-axis� Rotation about Y-axis� Rotation about Z-axis
![Page 33: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/33.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
14Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
Transformation
![Page 34: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/34.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
15multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
TRANSFORMATION
� Multiplication transformation
� Known a point in {C} and BCT matrix from {C} to {B} and
ABT matrix from {B} to {A}
� Find its position and orientation in {A}
![Page 35: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/35.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
15multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
TRANSFORMATION
� Multiplication transformation
� Known a point in {C} and BCT matrix from {C} to {B} and
ABT matrix from {B} to {A}
� Find its position and orientation in {A}
![Page 36: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/36.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
15multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
TRANSFORMATION
� Multiplication transformation
� Known a point in {C} and BCT matrix from {C} to {B} and
ABT matrix from {B} to {A}
� Find its position and orientation in {A}
![Page 37: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/37.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
15multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
TRANSFORMATION
� Multiplication transformation
� Known a point in {C} and BCT matrix from {C} to {B} and
ABT matrix from {B} to {A}
� Find its position and orientation in {A}
![Page 38: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/38.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
16multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
TRANSFORMATION
�ACT matrix from {C} to {A} is the multiplication of B
CTmatrix from {C} to {B} and A
BT matrix from {B} to {A}
![Page 39: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/39.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
16multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
TRANSFORMATION
�ACT matrix from {C} to {A} is the multiplication of B
CTmatrix from {C} to {B} and A
BT matrix from {B} to {A}
![Page 40: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/40.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
17Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
TRANSFORMATION
� Inverting a transformation
� Known T matrix from {B} to {A}� Find T matrix from {A} to {B}
![Page 41: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/41.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
17Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
TRANSFORMATION
� Inverting a transformation
� Known T matrix from {B} to {A}� Find T matrix from {A} to {B}
![Page 42: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/42.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
17Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
TRANSFORMATION
� Inverting a transformation
� Known T matrix from {B} to {A}� Find T matrix from {A} to {B}
![Page 43: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/43.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
17Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
TRANSFORMATION
� Inverting a transformation
� Known T matrix from {B} to {A}� Find T matrix from {A} to {B}
![Page 44: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/44.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
18Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
TRANSFORMATIONExample
� Frame {B} is rotated relative to Frame {A} about Z-axis by30 degrees and translated 4 units in X-axis and 3 units inY-axis. Find T matrix from {A} to {B}.
![Page 45: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/45.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
18Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
TRANSFORMATIONExample
� Frame {B} is rotated relative to Frame {A} about Z-axis by30 degrees and translated 4 units in X-axis and 3 units inY-axis. Find T matrix from {A} to {B}.
![Page 46: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/46.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
19transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
TRANSFORMATIONTransform equations
� Transform equation
� Any unknown T matrix can then be calculated from the onesgiven
![Page 47: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/47.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
19transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
TRANSFORMATIONTransform equations
� Transform equation
� Any unknown T matrix can then be calculated from the onesgiven
![Page 48: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/48.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
19transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
TRANSFORMATIONTransform equations
� Transform equation
� Any unknown T matrix can then be calculated from the onesgiven
![Page 49: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/49.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
19transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
TRANSFORMATIONTransform equations
� Transform equation
� Any unknown T matrix can then be calculated from the onesgiven
![Page 50: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/50.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
19transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
TRANSFORMATIONTransform equations
� Transform equation
� Any unknown T matrix can then be calculated from the onesgiven
![Page 51: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/51.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
19transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
TRANSFORMATIONTransform equations
� Transform equation
� Any unknown T matrix can then be calculated from the onesgiven
![Page 52: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/52.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
19transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
TRANSFORMATIONTransform equations
� Transform equation
� Any unknown T matrix can then be calculated from the onesgiven
![Page 53: CIS009-2, Mechatronics - Robot Coordination Systems II · CIS009-2, Mechatronics Robot Coordination Systems II David Goodwin Department of Computer Science and Technology University](https://reader030.vdocuments.us/reader030/viewer/2022040817/5e60ec208e88eb1fb45c168b/html5/thumbnails/53.jpg)
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
19transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
TRANSFORMATIONTransform equations
� Transform equation
� Any unknown T matrix can then be calculated from the onesgiven