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