3-D Kinematics

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3-D Kinematics
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Position and Orientation of a Rigid Body
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Position and Orientation of a Rigid Body

• The position of origin O with respect to O-xyz
  is expressed by the relation

• The component of each unit vector are the
  direction cosines of the axes of frame O-xyz

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Rotation Matrix

• Orientation can be described by rotation matrix

• R is orthogonal matrix

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Elementary Rotations
Rotation by an angle about axis z
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Elementary Rotations

• Rotation by an angle about axis y

• Rotation by an angle about axis x

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Representation of a Vector
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Representation of a Vector

• Representation of p w.r.t O-xyz

• Representation of p w.r.t O-xyz

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Rotation of a Vector
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Equivalent Geometrical Meaningsof Rotation Matrix
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Composition of Rotation Matrices

• Let Rij denote the rotation matrix of Frame i
  with respect to Frame j
• Post-multiplication interpretation
• Refer to current frame
• Pre-multiplication interpretation
• Refer to fixed frame

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Euler Angles

• Minimal representation of orientation
• Three parameters are sufficient
• Euler Angles
• Two successive rotations are not made about
  parallel axes
• How many kinds of Euler angles are there?

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ZYZ Angles

• The rotation described by ZYZ angles is

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ZYZ Angles
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ZYZ Angles

• The rotation matrix is

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ZYZ Angles

• Inverse problem determine the Euler angles
  corresponding to a given rotation matrix
• Solution 1 theta is in the range (0, pi)

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ZYZ Angles
y1 x1 y-1 x1 y1 x-1 y-1 x-1
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ZYZ Angles

• Solution 1 theta is in the range (0, pi)

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ZYZ Angles

• Solution 1 theta is in the range (0, pi)

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ZYZ Angles

• Solution 2 theta is in the range (-pi, 0)

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ZYZ Angles

• Solution 2 theta is in the range (-pi, 0)

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ZYZ Angles

• Solution 2 theta is in the range (-pi, 0)

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ZYZ Angles

• What will happen if sin(theta) 0?
• Matlab eul2tr, tr2eul

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Roll-Pitch-Yaw Angles

• Originate from (aero)nautical field

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Roll-Pitch-Yaw Angles
MATLAB QUATDEMO
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Roll-Pitch-Yaw Angles

• The rotation matrix is

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Roll-Pitch-Yaw Angles

• Inverse problem determine the Euler angles
  corresponding to a given rotation matrix
• Solution 1 theta is in the range (-pi/2, pi/2)

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Roll-Pitch-Yaw Angles

• Solution 2 theta is in the range (pi/2, 3pi/2)

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Roll-Pitch-Yaw Angles

• What will happen if cos(theta) 0?
• Matlab rpy2tr, tr2rpy

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Angle and Axis

• Non-minimal representation four parameters
• The unit vector of a rotation axis w.r.t O-xyz
• The angle theta about the axis
• Matlab quatdemo

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Angle and Axis

• Align r with z
• Rotate by theta about z
• Realign with the initial direction of r

Attention always refer to the fixed frame
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Angle and Axis

• The resulting rotation matrix is

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Angle and Axis

• The inverse problem
• Remember the three component of r is not
  independent

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Angle and Axis

• Problems
• solution is not unique
• r is arbitrary when theta 0

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Unit Quaternion

• Unit quaternion is defined as

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Unit Quaternion

• Inverse problem
• Matlabquaternion, plot, quaternion.t