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  3. exercise 31 in terms of the xs ys coordinates...

Question: exercise 31 in terms of the xs ys coordinates...

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Exercise 3.1 In terms of the Xs, ys, ż coordinates of a fixed space frame {s) the frame {a} has its Xa-axis pointing in the direction (0,0,1) and its ỷa-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) (e) Let R-Rsb be considered as a transformation operator consisting of a rotation about x by-90°. Calculate R1 = RsaR, and think of Rsa as a representation of an orientation, R as a rotation of Rsa, and Ri the new orientation after the rotation has been performed. Does the new orientation Ri 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 -90° about the world-fixed Ks-axis or about the body-fixed Xa-axis? as (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 Ps = (1, 2, 3) in {s) coordinates. Calculate p Rabp, and p-Rp. 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 (aj? (i) By hand, calculate the matrix logarithm [we of Rsa. (You may verify your and rotation answer with software.) Extract the unit angular velocity amount θ. Redraw the fixed frame {s) and in it draw a. (i) Calculate the matrix exponential corresponding to the exponential coor- dinates of rotation (1,2,0). Draw the corresponding frame relative to [s}, as well as the rotation axis .01 ブtelohonohue oiling tr gtohalfon om above by ao xs .b

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