cis009-2, mechatronics - robot coordination systems ii · cis009-2, mechatronics robot coordination...

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