1)   A frame {B} is located initially coincident with {A}. We rotate {B} about  by θ degrees, and  then we rotate the resulting frame about  , by ϕ degrees. Give the rotation matrix that will change the descriptions of vectors from   to   .                                                     [3 points]

2)   In the diagram shown below, find the homogenous transformations between each pair of

frames ( 1(0) , 2(0) , 2(1) ).

[6 points] 

3)   Find homogeneous transformation matrix representing a rotation by angle α about the current x-axis, followed by translation of b units along current x-axis, followed by translation of d units along current z-axis, followed by rotation of angle θ about current z-axis.                      [4 points]

4)   (a) Using Z-Y-X (α- β-γ) Euler angle convention, write a MATLAB program to calculate the rotation matrix  when the user enters the Euler angles. Test for two examples:

i)     α = 10 deg, β = 20 deg, γ = 30 deg

ii)    α = 30 deg, β = 90 deg, γ = -55 deg

For case (i), demonstrate the six constraints for orthonormal rotation matrices. Also demonstrate that   =  − 1  =  .

(b) Write a MATLAB program to calculate the Euler angles when the user enters the rotation     matrix  (the inverse problem). Calculate both possible solutions. Demonstrate this inverse    solution for the two cases from part (a). Use a circular check to verify your results – enter Euler angles and compute rotation matrix using the code above. Then use the rotation matrix and     recalculate Euler angles. You will get two answers, one should match the original input and the

other set can be re-run in the code in part (a) to get the same rotation matrix.           [15 points] 

5)   A velocity vector is given by

  = [20]