Translation, Rotation, and Transformation

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Translation, Rotation, and Transformation
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Translations(Simple, Linear, Commutative)
X Y Z
1 0 0 Dx
0 1 0 Dy
0 0 1 Dz
0 0 0 1
Dy
Dx
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Rotations Differ from Translations

• Rotations are non-Euclidean
• like travelling on a globe vs. a grid
• Rotations are not commutative
• x-rotate, y-rotate is not equal y-rotate,
  x-rotate etc.
• Rotations are non-linear

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Basic Rotation about Z-axis
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Rotation Parameterization

• Represent rotation space in Euclidean R3
• e.g. Euler angles
• Pros
• three parameters for three DOFs
• Cons
• singularities, potentially poor interpolation

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Euler angles (f,?,?)

• An Euler angle is a rotation about a single
  Cartesian axis
• Create multi-DOF rotations by concatenating
  Eulers
• R R? R? Rf
• 3 DOFs can be obtained by concatenating

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X-Convention

• Most commonly used
• The rotation given by Euler angles (f,?,?), where
  the first rotation is by an angle f about the
  z-axis, the second is by an angle ? about the
  x-axis, and the third is by an angle ? about the
  z-axis (again).
• R R? R? Rf

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Yaw-Pitch-Roll Convention
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Singularities

• More than one sets of parameters can create the
  same rotation matrix.
• Gimbal lock - two or more axes align, results in
  loss of rotational DOFs
• For Yaw-Pitch-Roll Convention

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Rotation Axis Angle

• Eulers Rotation Theorem
• all rotations can be expressed as axis/angle

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Rotation Matrix
V (1-Cosq) C Cosq S Sinq
For given axis U(unit length) u1, u2, u3T and
rotation angle q
u1u 2 V u3 S
u12 V C
u1u3 V u 2S
R
u 2u1 V u3 S
u 2u3 V u1 S
u 22 V C
u32 V C
u3u1 V u2 S
u3u 2 V u1 S
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Solution of Axis and Angle
Sinq ½(R32-R23)2(R13-R31)2(R21-R12)2(1/2)
Cosq (TraceR-1) / 2
q Atan2(Sinq, Cosq) -p lt q lt p
y
u1 (R32-R23) / (2 Sinq)
All
Sine
u2 (R13-R31) / (2 Sinq)
x
Cosine
Tan
u3 (R21-R12) / (2 Sinq)
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TransformationAP ATB BP
B
BP
A
AP
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Example
0 1 0 0
1 0 0 5
0 0 -1 0
0 0 0 1
ATB
zA
5
xB
yA
yB
xA
zB
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Cube of Sides 2
0 -1 0 2
-S45 0 S45 2
-C45 0 -C45 2
0 0 0 1
ATB
yB
xB
yA
xA
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Multiple Transformations
AP ATB BTC CTDDP
Also DTA (ATD)T
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EOMs

• Newton SFma, SMIa

I
L
SMIa - m g L Sinq I a m g L Sinq I a
0
q
mg
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EOMs

• Lagrangian L K P (kinetic and potential
  energy)

L
q
mg
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EOMs