Kinematics Primer

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Kinematics Primer

• Jyun-Ming Chen

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Contents

• General Properties of Transform
• 2D and 3D Rigid Body Transforms
• Representation
• Computation
• Conversion
• Transforms for Hierarchical Objects

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Math Primer
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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

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

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Ex 2D translation
The transform, as an operator, takes p to p,
thus changing the coordinate of p
Tr(t) p p
p
Tr(t)
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Ex 2D translation (cont)
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)
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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
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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)

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2D Kinematics

• Rigid body transform only consists of
• Tr(x,y)
• Rot(z,q)
• Computation
• 3x3 matrix is sufficient

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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

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3D Rotation Representations

• Axis-angle
• 3X3 rotation matrix
• Unit quaternion

• Learning Objectives
• Representation
• Perform rotation
• Composition
• Interpolation
• Conversion among representations

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Axis-Angle Representation

• Rot(n,q)
• n rotation axis (global)
• q rotation angle (rad. or deg.)
• follow right-handed rule
• Perform rotation
• Rodrigues formula
• Interpolation/Composition poor
• Rot(n2,q2)Rot(n1,q1) ? Rot(n3,q3)

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Rodrigues Formula
r
v
v
vR v
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Rodrigues (cont)

• http//mesh.caltech.edu/ee148/notes/rotations.pdf
• http//www.cs.berkeley.edu/ug/slide/pipeline/assi
  gnments/as5/rotation.html

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Rotation Matrix

• Meaning of three columns
• Perform rotation linear algebra
• Composition trivial
• orthogonalization might be required due to FP
  errors
• Interpolation ?

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Gram-Schmidt Orthogonalization

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

Verify!
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Quaternion

• A mathematical entity invented by Hamilton
• Definition

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Quaternion (cont)

• Operators
• Addition
• Multiplication
• Conjugate
• Length

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Unit Quaternion

• Define unit quaternion as follows to represent
  rotation
• Example
• Rot(z,90)?

Why unit? DOF point of view!
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Unit Quaternion (cont)

• Perform Rotation
• Composition
• Interpolation

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Example
p(2,1,1)
Rot(z,90)
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Example (cont)
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Example
y
x,x
z,y
z
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(No Transcript)
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Spatial Displacement

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

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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

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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

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Ex Two-Link Arm
Tip Position
T for link1 Rot(z,45) Tr(0,6) Rot(z,30) T
for link2 Rot(z,45)
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Ex Two-Link Arm
Thus, two views are equivalent The latter might
be easier to visualize.
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Ex Two-Link Arm (VRML syntax)

• Transform
• rotation 0 0 1 45
• children
• Link1
• Transform
• translation 0 0 6
• children
• Transform
• rotation 0 0 1 30
• children
• Link2

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Classes in Javax.vecmath

• Conversion Methods

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Exercises

• Study the references of Rodrigues formula
• Verify equivalence of these 2 refs
• Compute inverse Rodrigues formula