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3-D Kinematics. Position and Orientation of a Rigid Body. 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’-x’y’z’. Rotation Matrix.
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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’-x’y’z’
Rotation Matrix • Orientation can be described by rotation matrix • R is orthogonal matrix
Elementary Rotations Rotation by an angle about axis z
Elementary Rotations • Rotation by an angle about axis y • Rotation by an angle about axis x
Representation of a Vector • Representation of p w.r.t O-xyz • Representation of p w.r.t O-x’y’z’
Composition of Rotation Matrices • Let Rijdenote 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
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?
ZYZ Angles • The rotation described by ZYZ angles is
ZYZ Angles • The rotation matrix is
ZYZ Angles • Inverse problem: determine the Euler angles corresponding to a given rotation matrix • Solution 1: theta is in the range (0, pi)
ZYZ Angles y=1 x=1; y=-1 x=1; y=1 x=-1; y=-1 x=-1;
ZYZ Angles • Solution 1: theta is in the range (0, pi)
ZYZ Angles • Solution 1: theta is in the range (0, pi)
ZYZ Angles • Solution 2: theta is in the range (-pi, 0)
ZYZ Angles • Solution 2: theta is in the range (-pi, 0)
ZYZ Angles • Solution 2: theta is in the range (-pi, 0)
ZYZ Angles • What will happen if sin(theta) = 0? • Matlab: eul2tr, tr2eul
Roll-Pitch-Yaw Angles • Originate from (aero)nautical field
Roll-Pitch-Yaw Angles MATLAB: QUATDEMO
Roll-Pitch-Yaw Angles • The rotation matrix is
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)
Roll-Pitch-Yaw Angles • Solution 2: theta is in the range (pi/2, 3pi/2)
Roll-Pitch-Yaw Angles • What will happen if cos(theta) = 0? • Matlab: rpy2tr, tr2rpy
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
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
Angle and Axis • The resulting rotation matrix is
Angle and Axis • The inverse problem • Remember: the three component of r is not independent
Angle and Axis • Problems: • solution is not unique • r is arbitrary when theta = 0
Unit Quaternion • Unit quaternion is defined as
Unit Quaternion • Inverse problem: • Matlab:quaternion, plot, quaternion.t