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Fundamentals of RoboticsFundamentals of RoboticsLinking perception to actionLinking perception to action

2. Motion of Rigid Bodies2. Motion of Rigid Bodies

南台科技大學南台科技大學電機工程系電機工程系

謝銘原謝銘原

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Chapter 2. Motion of Rigid BodiesChapter 2. Motion of Rigid Bodies

2.2 Cartesian Coordinate Systems2.2 Cartesian Coordinate Systems 2.3 Projective Coordinate Systems2.3 Projective Coordinate Systems 2.4 Translational Motions2.4 Translational Motions 2.5 Rotational Motions2.5 Rotational Motions 2.6 Composite Motions2.6 Composite Motions

Homogeneous Transformation Differential Homogeneous Transformation Successive Elementary Transformation Successive Elementary Rotations Euler Angles Equivalent Axis and Angle of Rotation

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2.2 Cartesian Coordinate Systems2.2 Cartesian Coordinate Systems

Two references for describing the motions of a rigid body Two references for describing the motions of a rigid body –– The time reference (for velocity and acceleration, t ) The spatial reference (for position and orientation, X, Y, Z )

Cartesian Coordinate System – Cartesian Coordinate System – OO00 – X – X00YY00ZZ00

1 0 0

0 1 0

0 0 1

r x i y j z k R r

x

y

z

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2.3 Projective Coordinate Systems2.3 Projective Coordinate Systems

Geometric operationsGeometric operations Translation, Rotation, Scaling, Projection

Typical projection –Typical projection – Display of 3D objects onto a 2D screen Visual perception (onto an image plane)

Projective coordinates Projective coordinates ((kk – homogeneous coordinate– homogeneous coordinate))

, , ,t

X Y Z k

, ,

, ,

t

t

P x y z

Q kx ky kz

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2.4 Translational Motions2.4 Translational Motions

2.4.1 Linear Displacement2.4.1 Linear Displacement

The relative distance (displacement) Between the origin of frame 0 and

the origin of frame 1

1 2 1 1

1 1

( ) ( )

( ) ( , , )tx y z

O t O t T

O t t t t

0 1 0

1 1 1O O T

01T

0 1 01

0 0 0 1 1 1

( ) ( )

( , , ) ( , , ) ( , , )t tx y z

P t P t T

x y z x y z t t t

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Translational transformationTranslational transformation

The Cartesian coordinatesThe Cartesian coordinates

The equivalent projective coordinatesThe equivalent projective coordinates

0 1 01

0 0 0 1 1 1

( ) ( )

( , , ) ( , , ) ( , , )t tx y z

P t P t T

x y z x y z t t t

0 1

0 1

0 1

0 101

1 0 0

0 1 0

0 0 1

1 10 0 0 1

x

y

z

tx x

ty y

z z

P M

t

P

Homogeneous motion transformation matrix

01T

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2.4.2 Linear Velocity and Acceleration2.4.2 Linear Velocity and Acceleration

The displacement vectorThe displacement vector

The corresponding linear velocity and accelerationThe corresponding linear velocity and acceleration

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2.5 Rotational Motion2.5 Rotational Motion

Rotational motion –Rotational motion – The rotation of a rigid body about any straight line (a rotation

axis) Force, or torque

2.5.1 Circular displacement2.5.1 Circular displacement

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Rotation matrixRotation matrix

Rotation matrix, Rotation matrix, RR – – inverseinverse

Initial configurationInitial configuration

After rotationAfter rotation

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Example 2.4Example 2.4

Rotational transformation Rotational transformation from frame1 to frame0

Inverse (from frame0 to frame1)

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The homogeneous motion transformation matrixThe homogeneous motion transformation matrix

Rotational motionRotational motion

Translational motionTranslational motion

0 1

0 1

0 1

0 101

1 0 0

0 1 0

0 0 1

1 10 0 0 1

x

y

z

tx x

ty y

z z

P M

t

P

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2.5.2 Circular Velocity and Acceleration2.5.2 Circular Velocity and Acceleration

Circular velocityCircular velocity caused by a rigid body’s angular velocity

Circular accelerationCircular acceleration Vector is parallel to the rotation axis and its norm is equal to

at antangential acceleration vector centrifugal acceleration vector

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2.6 Composite Motions2.6 Composite Motions

Any complex motion can be treated as the combination of Any complex motion can be treated as the combination of translational and rotational motions.translational and rotational motions.

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2.6.1 Homogeneous Transformation2.6.1 Homogeneous Transformation

Homogeneous TransformationHomogeneous Transformation

with

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2.6.2 Differential2.6.2 Differential Homogeneous TransformationHomogeneous Transformation

Homogeneous TransformationHomogeneous Transformation

By differentiating,By differentiating,

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In a matrix formIn a matrix form

Homogeneous TransformationHomogeneous Transformation

Differential Homogeneous TransformationDifferential Homogeneous Transformation

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2.6.3 Successive Elementary Translations2.6.3 Successive Elementary Translations

Three successive translationsThree successive translations

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2.6.4 Successive Elementary Rotations2.6.4 Successive Elementary Rotations

Three successive rotations of frame 1 to frame 0Three successive rotations of frame 1 to frame 0

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The equivalent projective coordinatesThe equivalent projective coordinates

In robotics, the above equation describes the forward kinematics of a spherical joint having three degrees of freedom.

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Useful expressionUseful expression

Imagine now that Imagine now that x, x, y,y, and and zz can undergo instantaneous variations can undergo instantaneous variations

with respect to time.with respect to time.

The property of skew-symmetric matrixThe property of skew-symmetric matrix

We can derive the following equalities:We can derive the following equalities:

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2.6.5 Euler Angles2.6.5 Euler Angles

The three successive rotationsThe three successive rotations 1st elementary rotation can choose X, Y, or Z axes as its rotation

axis. 2nd, 3rd, both have two axes to choose from. Total, 3*2*2 = 12 possible combinations

These sets are commonly called Euler Angles.

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Example 2.5 Example 2.5 ZYZ Euler AnglesZYZ Euler Angles

1st – about Z axis, 1st – about Z axis, rotation angle rotation angle

2nd – about Y axis, 2nd – about Y axis, rotation angle rotation angle

3rd – about Z axis, 3rd – about Z axis, rotation angle rotation angle

are called are called ZYZ Euler AnglesZYZ Euler Angles

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2.6.6 Equivalent Axis and Angle of rotation2.6.6 Equivalent Axis and Angle of rotation

Euler anglesEuler angles The set of minimum angles which fully determine a frame’s

rotation matrix with respect to another frame. Each set has three angles which make three successive elementary

rotations.

Thus,the orientation of a frame, with respect to another frame,depends on three independent motion parameterseven through the rotation matrix is a 3*3 matrix with 9 elements inside.

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In roboticsIn robotics

It is necessary It is necessary to interpolate the orientation of a frame from its initial orientation to to interpolate the orientation of a frame from its initial orientation to an actual orientationan actual orientationso that so that a physical rigid body (e.g. the end-effector)a physical rigid body (e.g. the end-effector)can smoothly execute the rotational motion in real space.can smoothly execute the rotational motion in real space.

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Equivalent axis of rotationEquivalent axis of rotation

Now, imagine that there is am intermediate frame Now, imagine that there is am intermediate frame ii which has the which has the ZZ axis that coincides with the equivalent rotation axis axis that coincides with the equivalent rotation axis rr. .

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RotationRotation

If r is the equivalent axis of rotation for the rotational motion between If r is the equivalent axis of rotation for the rotational motion between frame 1 and frame 0:frame 1 and frame 0:

The orientation of The orientation of frame iframe i with respect to with respect to frame 0frame 0,,

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The solutions for The solutions for rr and and

To derive,To derive,

If If = 0, R = 0, Rzz = I = I 3*33*3

given