Description

Problem 1
The relative orientation of two frames of reference are given as follows. Frame A forms the base/reference frame while Frame B is obtained by taking Frame A and rotating it by the following three incremental relative rotations:
R[z,pi/3] followed by R[x,pi/3] and finally R[z,pi/4]. For the above sets of orientations determine:
a) The alternate relative rotations representation called Z-Y-Z Euler Angles (obtained by three successive relative rotations first about z -axis by the angle phi, then around y-axis by theta, and finally rotated about z by psi – see section 2.5.1 in Spong, Hutchinson and Vidyasagar)
b) The Roll-Pitch-Yaw Angle representation (note these are absolute angles – see section 2.5.2 in Spong,
Hutchinson and Vidyasagar)
c) The Axis/Angle representation (see section 2.5.3 in Spong, Hutchinson and Vidyasagar) In particular, find the singularities and multiple solutions of these representations if any.

Problem 2
The initial and final positions of corners of a unit cube are given in the inertial coordinate frame as:
 xi0 0 1 1 0 0 1 1 0
 0  
The initial coordinates of each vertex is given by:  yi =0 0 1 1 0 0 1 1 .
 zi0 0 0 0 0 1 1 1 1  
The final coordinates of each vertex is:
xif  0.5000 0.9698 1.0805 0.6107 1.3758 1.8456 1.9563 1.4865 
yif = 
 0 -0.8660 -0.6160 0.2500 0.4330 -0.4330 -0.1830 0.6830
zif   0 -0.1710 -1.1329 -0.9619 0.2133 0.0423 -0.9196 -0.7486 
Determine the homogenous transformation for the displacement.

Problem 3
Prove that the following matrix is a rotation matrix

Problem 4
Find the values of the missing elements to complete the 3 3× rotation matrix representation of the location of a body fixed frame {M} with respect to an inertial frame {F}.

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Problem 5
For the figure shown below, find:
(i) the 4 4× homogeneous transformation matrices, i−1Ai for i=1, 2, 3, 4 and
(ii) the 4×4 homogeneous transformation matrices 0Ai for i=1, 2, 3, 4.

Problem 6
Rodriques’ formula for the rotation matrix during rotation of a rigid body about the unit vector u=[ux, uy, uz]T

Using symbolic calculations only verify that it satisfies all the properties of a rotation matrix.

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