In this problem, we will see how we would understand the Adjoint operation associated with T...

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In this problem, we will see how we would understand the Adjoint operation associated with T SE(3). We learned that in class V = (ws, vs) and Vj = (@b, vb) represents the spatial twist and body twist, respectively. These twist vectors represent the velocity of the rigid body motion at time t by V = @s 0 sb (t1) T (t) = 0] = [R.R sb 0 Vb = ((0)=[]=[ T(t) sb (t) = 9 As we have seen in class, it is less intuitive than the angular velocity we learned in the rotational motion. The main reason is that the twist vector is in R6, whereas we can only visualize things as rotation or translation in three-dimensional space. In this question, let's think about visualizing the twist vectors in terms of the screw motion. Assume that ||ws|| = ||wb|| > 0 but not equal to 1. In this case, the expression for the point q and the screw pitch on the line given a twist (w, v) are RsbRT Psb (ws XP sb) ( X Psb)], 0 h 9b [RTRsb RPsb 0 (1) (2) (a) (8 points) Let's first find the corresponding screw motion for V. Let q = R a point on the line of screw motion expressed in {b}-frame, Show that 9's = ||w|| wTv ||w|| RWspsb ||ws|| Hint: ws = Rsb wb and using the Lemma Rw: RORT (b) (8 points) Now, let q's be the q, expressed in {s}-frame, which represents the point on the line expressed in {s}-frame. Show that @spsb ||ws|| hb = + Psb (c) (8 points) Now, let's compute the screw pitch h = (v)/||w|| in {b} frame. Show that T. Ws Psb S ||ws|| Note that the screw pitch is coordinate-independent quantity, which represents the rate of the translational motion vs rotational motion. (d) (8 points) Now, we have all the information on the screw motion in {s} frame: the point on the line q's, the axis direction ws/||ws||, and the screw pitch hy. From (a-c), we can find the corresponding screw axis, S', such that S' W ||ws|| -(ws)xq's |||| and the corresponding twist is V = S, M, which is , where hw + V/ = [-(ws) X -(ws) xq's + hows Computre V' to show that V' is equivalent to the spatial twist V = (Ws, Vs), Ws -|P Pb)] = Psb (ws XPsb) V = - | || || AdTsb , M = ||ws|| , which Remark: You just proved that the transformation starting from the body twist, V, to Vs preserves the screw motion. Therefore, the rigid body velocity, V and V, represents the instantaneous screw motion along the same line represented on different frames. = (e) (8 points) Lastly, We want to show that the adjoint transformation Adr,, defined as Rsb 03x3 Psb Rsb Rsb = Vs V provides the mapping between V, and Vs. Show that Vs = AdTsh Vb Ws Vs - [Ps - (. Xpa)] - [Ra] = ,Vb = (ws XPsb) RT sh (3) (4) (5) In this problem, we will see how we would understand the Adjoint operation associated with T SE(3). We learned that in class V = (ws, vs) and Vj = (@b, vb) represents the spatial twist and body twist, respectively. These twist vectors represent the velocity of the rigid body motion at time t by V = @s 0 sb (t1) T (t) = 0] = [R.R sb 0 Vb = ((0)=[]=[ T(t) sb (t) = 9 As we have seen in class, it is less intuitive than the angular velocity we learned in the rotational motion. The main reason is that the twist vector is in R6, whereas we can only visualize things as rotation or translation in three-dimensional space. In this question, let's think about visualizing the twist vectors in terms of the screw motion. Assume that ||ws|| = ||wb|| > 0 but not equal to 1. In this case, the expression for the point q and the screw pitch on the line given a twist (w, v) are RsbRT Psb (ws XP sb) ( X Psb)], 0 h 9b [RTRsb RPsb 0 (1) (2) (a) (8 points) Let's first find the corresponding screw motion for V. Let q = R a point on the line of screw motion expressed in {b}-frame, Show that 9's = ||w|| wTv ||w|| RWspsb ||ws|| Hint: ws = Rsb wb and using the Lemma Rw: RORT (b) (8 points) Now, let q's be the q, expressed in {s}-frame, which represents the point on the line expressed in {s}-frame. Show that @spsb ||ws|| hb = + Psb (c) (8 points) Now, let's compute the screw pitch h = (v)/||w|| in {b} frame. Show that T. Ws Psb S ||ws|| Note that the screw pitch is coordinate-independent quantity, which represents the rate of the translational motion vs rotational motion. (d) (8 points) Now, we have all the information on the screw motion in {s} frame: the point on the line q's, the axis direction ws/||ws||, and the screw pitch hy. From (a-c), we can find the corresponding screw axis, S', such that S' W ||ws|| -(ws)xq's |||| and the corresponding twist is V = S, M, which is , where hw + V/ = [-(ws) X -(ws) xq's + hows Computre V' to show that V' is equivalent to the spatial twist V = (Ws, Vs), Ws -|P Pb)] = Psb (ws XPsb) V = - | || || AdTsb , M = ||ws|| , which Remark: You just proved that the transformation starting from the body twist, V, to Vs preserves the screw motion. Therefore, the rigid body velocity, V and V, represents the instantaneous screw motion along the same line represented on different frames. = (e) (8 points) Lastly, We want to show that the adjoint transformation Adr,, defined as Rsb 03x3 Psb Rsb Rsb = Vs V provides the mapping between V, and Vs. Show that Vs = AdTsh Vb Ws Vs - [Ps - (. Xpa)] - [Ra] = ,Vb = (ws XPsb) RT sh (3) (4) (5)