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

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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 41 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​=Rsa​R, 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′=Rsb​ps​ and p′′=RsbT​ps​. 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^.