cis009-2, mechatronics - robot coordination systems ii · cis009-2, mechatronics robot coordination...
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CIS009-2, MechatronicsRobot Coordination Systems II
David Goodwin
Department of Computer Science and TechnologyUniversity of Bedfordshire
07th February 2013
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
19
Mechatronics
David Goodwin
3Mapping
Translation
Rotational
Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
Mapping
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
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
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
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
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
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
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
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
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
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
]
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
]
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
]
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
]
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
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
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
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
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
10Operators
Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
Operators
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
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
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
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
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
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
12Translation
Rotation
Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
OPERATORS
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
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
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
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
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
19
Mechatronics
David Goodwin
Mapping
Translation
Rotational
Operators
Translation
Rotation
14Transformation
multiplication
Inverting
transformationequations
Department ofComputer Science and
TechnologyUniversity ofBedfordshire
Transformation
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}.
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}.
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
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
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
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
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
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
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
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