Exercise 3.1 In terms of the is, ýs, żs coordinates of a fixed space frame (s), the frame {a) has its Xa-axis pointing in the direction (0,0,1) and its ya-axis pointing in the direction (-1,0,0), and the frame {b) has its xb-axis pointing in the direction (1, 0.0) and its Уь-axis pointing in the direction (0,0,-1) (a) Draw by hand the three frames, at different locations so that they are easy to see. (b) Write down the rotation matrices Rsa and Rsb. (c) Given Rsb, how do you calculate Rs without using a matrix inverse? Write down Rc, and verify its correctness using your drawing. (d) Given Rsa and Rsb, how do you calculate Rab (again without using ma- trix inverses)? Compute the answer and verify its correctness using your drawing (e) Let R -Rsb be considered as a transformation operator consisting of a rotation about x by-90°. Calculate Rı = RsaR, and think of Rsa as a representation of an orientation, R as a rotation of Rsa, and Ri as the new orientation after the rotation has been performed. Does the new orientation R1 correspond to a rotation of Rsa by -90° about the world- fixed Xs-axis or about the body-fixed Xa-axis? Now calculate R2 RRsa Does the new orientation R2 correspond to a rotation of Rsa by-900 about the world-fixed is-axis or about the body-fixed Xa-axis? (f) Use Rsb to change the representation of the point pb - (1, 2, 3) (which is in {b} coordinates) to {s) coordinates (g) Choose a point p represented by p (1,2,3) in [s coordinates. Calculate p, Rsbps and p', _ RTbps. For each operation, should the result be interpreted as changing coordinates (from the {s) frame to {b]) without moving the point p or as moving the location of the point without changing the reference frame of the representation (h) An angular velocity w is represented in {s) as w, = (3,2, 1). What is its representation wa in {a)? (i) By hand, calculate the matrix logarithm Įwja of Rsa. (You may verify your answer with software.) Extract the unit angular velocity and rotation amount θ. Redraw the fixed frame {s) and in it draw w (j) Calculate the matrix exponential corresponding to the exponential coor- dinates of rotation ώθ-(1,2,0). Draw the corresponding frame relative to Ist, as well as the rotation axis