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Kinematics ( 運動學) PrimerPowerPoint Presentation

Kinematics ( 運動學) Primer

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Contents

- General Properties of Transform
- 2D and 3D Rigid Body Transforms
- DOF (degree of freedom)
- Representation
- Computation
- Conversion
- …

- Transforms for Hierarchical Objects

Explain these concepts via 2D translation

Verify that the same holds for rotation, 3D, …

Kinematic Modeling- Two interpretations of transform
- “Global”:
- An operator that “displaces” a point (or set of points) to desired location

- “Local”:
- specify where objects are placed in WCS by moving the local frame

- “Global”:

p

x

Ex: 2D translation (move point)The transform, as an operator, takes

p to p’, thus changing the coordinate

of p:

Tr(t) p = p’

p’

Tr(t)

y’

y

p’

p

x’

x

Ex: 2D translation (move frame)The transform moves the xy-frame to

x’y’-frame and the point is placed

with the same local coordinate.

To determine the corresponding position of p’ in xy-frame:

Tr(t)

Transforms are usually not commutable

TaTb p TbTa p (in general)

Rigid body transform:

the ones preserving the shape

Two types:

rotation rot(n,q)

translation tr(t)

Properties of TransformRotation axis n passes thru origin

Rigid Body Transform

- transforming a point/object
- rot(n,q) p; tr(t) p

- not commutable
- rot(n,q) tr(t) p tr(t) rot(n,q) p

- two interpretations (local vs. global axes, see next pages)

Hierarchical Objects

- For modeling articulated objects
- Robots, mechanism, …

- Goals:
- Draw it
- Given the configuration, able to compute the (global) coordinate of every point on body

Link 1: Box (6,1); bend 45 deg

Link 2: Box (8,1); bend 30 deg

Goals:

Draw it

find tip position

y

x

y

x

Ex: Two-Link Arm (2D)Rot(z,45)

Tr(0,6)

Rot(z,30)

Ex: Two-Link ArmTip Position:

T for link1:

Rot(z,45) Tr(0,6) Rot(z,30)

T for link2:

Rot(z,45)

y”

y’”

y’

x”

x”’

x’

Rot(z”,30)

Tip pos:(0’”,8’”)

Tr(0,6’)

Rot(z,45)

Ex: Two-Link ArmThus, two views are equivalent

The latter might be easier to

visualize.

Rigid body transform only consists of

Tr(x,y)

Rot(z,q)

Computation:

3x3 matrix is sufficient to realize Tr and Rot

2D Kinematics3D rotation

3D translation

The same as 2D

3D rotation is more complicated than 2D rotation (restricted to z-axis)

Next, we will discuss the treatment for spatial (3D) rotation

3D KinematicsDOF (degree of freedom)

- … of a system of moving bodies is the number of independent variables required to specify the configuration
- … is closely related to kinematic representation
- (# of independent variables) = (# of total variables) ̶ (# of equality constraints)

R2

R3

A rigid body in

R2

R3

DOF of …3

2

6

3

a line in R2

a line in R3

a plane in R3

2

4 (Plucker coordinates)

3

Axis-angle

3X3 rotation matrix

Unit quaternion

Learning Objectives

Representation (uniqueness)

Perform rotation

Composition

Interpolation

Conversion among representations

…

3D Rotation RepresentationsGimbal lock: reduced DOF due to overlapping axes

Euler AnglesRef: http://www.fho-emden.de/~hoffmann/gimbal09082002.pdf

Axis-Angle Representation

- Rot(n,q)
- n: rotation axis (global)
- q: rotation angle (rad. or deg.)
- follow right-handed rule

- Rot(n,q)=Rot (-n,-q)
- Problem with null rotation: rot(n,0), any n
- Perform rotation
- Rodrigues formula (next page)

- Interpolation/Composition: poor
- Rot(n2,q2)Rot(n1,q1) =?= Rot(n3,q3)

Rodrigues Formula

r : unit vector (axis)

r

This is the “cross-product matrix”

rL v = r v

v

v’

v’=R v

References:

http://mesh.caltech.edu/ee148/notes/rotations.pdf

http://www.cs.berkeley.edu/~ug/slide/pipeline/assignments/as5/rotation.html

Rotation Matrix (xy plane)

orthogonalization might be required due to FP errors

Interpolation: ?

Rotation Matrix (cont)Gram-Schmidt Orthogonalization

- If 3x3 rotation matrix no longer orthonormal, metric properties might change!

Verify!

Unit Quaternion

- Define unit quaternion as follows to represent rotation
- Example
- Rot(z,90°)

- q and –q represent the same rotation
- Null rotation: q = [1,0,0,0]

Why “unit”?

DOF point of view!

Unit Quaternion (cont)

- Perform Rotation
- Composition
- Interpolation
- Linear
- Spherical linear (slerp, more later)

Matrix Conversion (cont)

Find largest qi2; solve the rest

r

Spherical Linear Interpolationunit sphere in R4

The computed rotation quaternion rotates about a fixed axis at constant speed

References:

http://www.gamedev.net/reference/articles/article1095.asp

http://www.diku.dk/research-groups/image/teaching/Studentprojects/Quaternion/

http://www.sjbrown.co.uk/quaternions.html

http://www.theory.org/software/qfa/writeup/node12.html

Spatial Displacement

- Any displacement can be decomposed into a rotation followed by a translation
- Matrix
- Quaternion

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