- Exercise 3 1 In Terms Of The X S Y S Z S Coordinates Of A Fixed Space Frame S The Frame A Has Its X A Axis Poi 1 (769.65 KiB) Viewed 42 times
Exercise 3.1 In terms of the x^s,y^s,z^s coordinates of a fixed space frame {s}, the frame {a} has its x^a-axis pointing in the direction (0,0,1) and its y^a-axis pointing in the direction (−1,0,0), and the frame {b} has its x^b-axis pointing in the direction (1,0,0) and its y^b-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 Rsb−1 without using a matrix inverse? Write down Rsb−1 and verify its correctness using your drawing. (d) Given Rsa and Rsb, how do you calculate Rab (again without using matrix 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 R1=RsaR, and think of Rsa as a representation of an orientation, R as a rotation of Rsa, and R1 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 worldfixed x^s-axis or about the body-fixed x^a-axis? Now calculate R2=RRsa. Does the new orientation R2 correspond to a rotation of Rsa by −90∘ about the world-fixed x^s-axis or about the body-fixed x^a-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 ps=(1,2,3) in {s} coordinates. Calculate p′=Rsbps and p′′=RsbTps. 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 ωs=(3,2,1). What is its representation ωa in {a} ? (i) By hand, calculate the matrix logarithm [ω^]θ of Rsa. (You may verify your answer with software.) Extract the unit angular velocity ω^ and rotation amount θ. Redraw the fixed frame {s} and in it draw ω^. (j) Calculate the matrix exponential corresponding to the exponential coordinates of rotation ω^θ=(1,2,0). Draw the corresponding frame relative to {s}, as well as the rotation axis ω^.