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Kinematics Primer

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Transforms are usually not commutable. TaTb p TbTa p (in general) Rigid body transform: ... not commutable. rot(n,q) tr(t) p tr(t) rot(n,q) p ... – PowerPoint PPT presentation

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Title: Kinematics Primer


1
Kinematics (???) Primer
  • Jyun-Ming Chen
  • Fall 2009

2
Contents
  • General Properties of Transform
  • 2D and 3D Rigid Body Transforms
  • DOF (degree of freedom)
  • Representation
  • Computation
  • Conversion
  • Transforms for Hierarchical Objects

3
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
  • Explain these concepts via 2D translation
  • Verify that the same holds for rotation, 3D,

4
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)
5
Ex 2D translation (move frame)
The transform moves the xy-frame to xy-frame
and the point is placed with the same local
coordinate. To determine the corresponding
position of p in xy-frame
Tr(t)
6
Properties of Transform
  • 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)

Rotation axis n passes thru origin
7
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)

8
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

9
Ex Two-Link Arm (2D)
  • Configuration
  • Link 1 Box (6,1) bend 45 deg
  • Link 2 Box (8,1) bend 30 deg
  • Goals
  • Draw it
  • find tip position

10
Ex Two-Link Arm
Tip Position
T for link1 Rot(z,45) Tr(0,6) Rot(z,30) T
for link2 Rot(z,45)
11
Ex Two-Link Arm
Thus, two views are equivalent The latter might
be easier to visualize.
12
2D Kinematics
  • Rigid body transform only consists of
  • Tr(x,y)
  • Rot(z,q)
  • Computation
  • 3x3 matrix is sufficient to realize Tr and Rot

13
3D Kinematics
  • Consists of two parts
  • 3D 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

14
DOF (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)

15
DOF of
  • A particle in
  • R2
  • R3
  • A rigid body in
  • R2
  • R3

3
2
6
3
2
  • a line in R2
  • a line in R3
  • a plane in R3

4 (Plucker coordinates)
3
16
3D Rotation Representations
  • Euler angles
  • Axis-angle
  • 3X3 rotation matrix
  • Unit quaternion
  • Learning Objectives
  • Representation (uniqueness)
  • Perform rotation
  • Composition
  • Interpolation
  • Conversion among representations

17
Euler Angles
  • Roll, pitch, yaw
  • Gimbal lock reduced DOF due to overlapping axes

Ref http//www.fho-emden.de/hoffmann/gimbal09082
002.pdf
18
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)

19
Rodrigues Formula
r unit vector (axis)
r
This is the cross-product matrix rL v r ? v
v
v
vR v
References http//mesh.caltech.edu/ee148/notes/ro
tations.pdf http//www.cs.berkeley.edu/ug/slide/p
ipeline/assignments/as5/rotation.html
20
Rotation Matrix (xy plane)
21
Rotation Matrix
  • Rodrigues formula
  • About any axis
  • About y axis

22
Rotation Matrix
  • Three columns of R the transformed bases
  • Perform rotation
  • x Rx

23
Rotation Matrix (example)
z
x
z
y
x
y
24
Rotation Matrix (cont)
  • Composition trivial
  • orthogonalization might be required due to FP
    errors
  • Interpolation ?

25
Gram-Schmidt Orthogonalization
  • If 3x3 rotation matrix no longer orthonormal,
    metric properties might change!

Verify!
26
Transformation Matrices
v Hu
Rotation (about principal axes)
Translation
Scaling
27
Translation (math)
28
Frame
Object coordinates
Recall how we did the spotlight beam!
29
Quaternion
  • A mathematical entity invented by Hamilton
  • Definition

30
Quaternion (cont)
  • Operators
  • Addition
  • Multiplication
  • Conjugate
  • Length

31
Unit Quaternion
  • Define unit quaternion as follows to represent
    rotation
  • Example
  • Rot(z,90)?
  • q and q represent the same rotation

Why unit? DOF point of view!
32
Example
x
y
z
Rot (90, 0,0,1) OR Rot (-90,0,0,-1)
33
Unit Quaternion (cont)
  • Perform Rotation
  • Composition
  • Interpolation
  • Linear
  • Spherical linear (slerp, more later)

34
Example
p(2,1,1)
Rot(z,90)
35
Example (cont)
36
Example
y
x,x
z,y
z
37
Matrix Conversion
38
Matrix Conversion (cont)
Find largest qi2 solve the rest
39
Spherical Linear Interpolation
unit sphere in R4
The computed rotation quaternion rotates about a
fixed axis at constant speed
References http//www.gamedev.net/reference/artic
les/article1095.asp http//www.diku.dk/research-gr
oups/image/teaching/Studentprojects/Quaternion/ ht
tp//www.sjbrown.co.uk/quaternions.html http//www
.theory.org/software/qfa/writeup/node12.html
40
Spatial Displacement
  • Any displacement can be decomposed into a
    rotation followed by a translation
  • Matrix
  • Quaternion
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