Question

Exercise 3.16 In terms of the Xs, ŷs, Zs coordinates of a fixed space frame {s}, frame {a} has its xg-axis pointing in the direction (0,0,1) and its y -axis pointing in the direction (-1,0,0), and frame {b} has its Xb-axis pointing in the direction (1,0,0) and its ýb-axis pointing in the direction (0,0,-1). The origin of {a} is at (3,0,0) in {s} and the origin of {b} is at (0,2,0) is {s}. (a) Draw by hand a diagram showing {a} and {b} relative to {s}. (b) Write down the rotation matrices Rga and Rgb and the transformation matrices Tsa and Tgb. (c) Given T , how do you calculate without using a matrix inverse? Write T and verify its correctness using your drawing. (d) Given Tga and Tgb, how do you calculate Tab (again without using ma- trix inverses)? Compute the answer and verify its correctness using your drawing (e) Let T = To be considered as a transformation operator consisting of a rotation about by -90° and a translation along ý by 2 units. Calculate T =T, T. Does T correspond to a rotation and translation about x, and ỹ,, respectively (a world-fixed transformation of Ta), or a rotation and translation about Xa and ga, respectively (a body-fixed transformation of Tsa)? Now calculate T2 = TTg. Does T2 correspond to a body-fixed or world-fixed transformation of Tsa? (f) Use Tg to change the representation of the point po = (1,2,3) in {b} coordinates to {S} coordinates. (g) Choose a point p represented by P, = (1,2,3) in {s} coordinates. Calcu- late p = Tsp, and p = T6 ps. For each operation, should the result be interpreted as changing coordinates (from the {s} frame to {b}) with- out moving the point p, or as moving the location of the point without changing the reference frame of the representation?

İ need e f and g part thank you for attention

Answer 1

е) е) Тѕь – т. — Т. o o от 2 о - оо To o o Т-Го - о з Тоо - 10 1 o o o o o o 1 Т - Тат = Tasb- -ro o - 1 1 o 1 оо 1 ооо 7 y coordinate changes from 3 to 1. лио листа Xa remains unchanged. So, body-fined transformation,

(1) T₂ = TTsa = Tub Tsa o to 2. 10 -1 6 37 y coordindes changes feron To 0 1 0 To o 1 1 ² =ña remains unchanged Tsa - So coworld-fined transformation of ( Ps= Tsb Pb So, Pot (1,5, (1) pl= Tsbps = 1 Moving location of the point ? without changing reforece frame af representahon P"= Tub Ps Changing coordinates from &se to aby frame, without moving point po