Answer Happy

1. In this question, you will be using the following trigonometric identities: = 1 (1) cos²a + sin² a cos(x + 3) = (2) cos a cos B-sin a sin sin a cos 3 + cos a sin sin(a+ß) (3) where a, ß ER. You do not need to prove these identities. You may also use without proof the fact that the set {[oa]: GER} is exactly the set of unit vectors in R². Now for any real number a, define Ra [cosa-sina] sin a cos a (a) Prove that for all a, 3 € R, Ra Ra Ra+81 (b) Using part (a), or otherwise, prove that R, is invertible and that R¹ = Ra, for all a € R. (c) Prove that for all a E R and all x, y € R², (Rx) (Ray)=x-y (d) Suppose A is a 2 x 2 matrix such that for all x, y ER², (Ax)-(Ay)=x-y
(d) Suppose A is a 2 x 2 matrix such that for all x, y € R², (Ax)-(Ay) = x-y Must it be true that A = Ra, for some a R? Either prove this, or give a counterexample (including justification). (e) Let B= - [1] be any 2 x 2 matrix. [cos a =111 (i) Show that there are real numbers u₁1 and a such that sin a (ii) Let a € R. Use the invertibility of R to prove that there are unique U12, U22 ER such that sin a 4] <=1412 cos a sin a +U22 [ cos a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = R₂U for some a ER and some upper-triangular matrix U (iv) Suppose that B RU RAV, where a, ß ER and U and V are upper- triangular. Prove that if B is invertible, then U = V.
2. Some parts of this question refer to the attached research paper. Mili Shah (2013), Solving the Robot-World/Hand-Eye Calibration Problem Using the Kronecker Product, Journal of Mechanisms and Robotics, Volume 5, Issue 3 (2013). You do not need to add a bibliography to reference this paper, and you may refer to it in your answer as: Shah (2013). (a) A rotation matriz is a 3 x 3 matrix R such that det(R) = 1 and R¹ = R Let R and S be rotation matrices, and let t, u € R³. (i) Prove that RS is a rotation matrix. [ru r12 r13] (ii) Given R = 721 722 723 and t = 国。 y E R³, we write B1 for the [731 732 733] 4 x 4 matrix [T11 12 13 1 T21 T22 T23 y T31 132 133 Z 000 From page 1 of the attached paper, the product [R [R is a matrix of the form where R' is a rotation matrix and vER¹. Write down expressions for R and v in terms of R, S, t and u. For this question only, you do not need to give any reasons for your answer. (b) Suppose a robot is posed n times, leading to the equations n D
4 x 4 matrix T1 T2 T13 T21 722 123 731 732 733 = 0001 From page 1 of the attached paper, the product is a matrix of the form [81] where R' is a rotation matrix and vE R³ Write down expressions for R and v in terms of R, S, t and u. For this question only, you do not need to give any reasons for your answer. (b) Suppose a robot is posed n times, leading to the equations RA, RX = Ry RB, where RA,, Rx, Ry and RB, are rotation matrices, for j = 1,2,...,n. Write a few sentences to summarise the results of the attached paper by Shah on (i) the number of poses needed to obtain unique matrices Rx and Ry which satisfy these equations; and (ii) how the position errors for the method presented in the attached paper compare to the position errors for the method of Li et al, on simulated data and real-world data.