In the previous problem, we prove that the transformation between the body and spatial twists preserve...

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In the previous problem, we prove that the transformation between the body and spatial twists preserve the screw motion. When we say the twist represents the velocity of the rigid motion in SE(3), it is very natural to ask if the speed of motion is preserved in between the different representations. This is an excellent question from our ECE569 student during the class. However, we may need to define what we mean by the speed carefully. In the rotational motion, given the angular velocities ws and wb, the speed was the magnitude of the angular velocities, so = ||ws|| = ww = w RT Rsbw = w w = ||wv|| sb Wb the angular speed was preserved. Now, the question for the twist is such that what is the speed of the screw motion. It contains two things, rotational speed and translational speed. The rotational speed can be understood as ||wb|| and ||ws|| but what would be the translational speed? Would it be ||v6|| and ||vs||? The problem is that vs does not have a direct meaning of the velocity of the origin, whereas v is the velocity of the origin expressed in {b}-frame. However, we have the quantity screw pitch which represents the ratio of the translational speed and the rotational speed. (a) (10 points) We want to check that the screw pitch for the body and spatial twists are the same. Show that s hs = = ||wb|| 2 ||wb||2 where vs = psb - (ws xpsb), v = Rpsb, ws = Rwb = (b) (10 points) As we proved in (a), the ||wb|| ||ws|| and hs hb, which indicates that the speed of rigid body velocity is preserved between two representations.. It is interesting to check if the adjoint operation associated with Tsb, AdTsb, will preserve the magnitude of ||V|| and ||V||. Or equivalently would Ad AdTsb 16x6? 1) Compute Ad AdTsb and sb 2) find Psb R which makes Ad 3) find Psb R which makes Ad Isb AdT 166 AdT = 166. sb Remark: Therefore, the magnitude of the twist does not necessarily be preserved, and there are no meaning of computing ||V||, but the ||w|| and h are preserved through this adjoint operation. (c) (10 points) Let T = (R1, p) = SE(3) and T2 = (R2, p2) E SE(3). Show that Ad Ad = Ad (6) In the previous problem, we prove that the transformation between the body and spatial twists preserve the screw motion. When we say the twist represents the velocity of the rigid motion in SE(3), it is very natural to ask if the speed of motion is preserved in between the different representations. This is an excellent question from our ECE569 student during the class. However, we may need to define what we mean by the speed carefully. In the rotational motion, given the angular velocities ws and wb, the speed was the magnitude of the angular velocities, so = ||ws|| = ww = w RT Rsbw = w w = ||wv|| sb Wb the angular speed was preserved. Now, the question for the twist is such that what is the speed of the screw motion. It contains two things, rotational speed and translational speed. The rotational speed can be understood as ||wb|| and ||ws|| but what would be the translational speed? Would it be ||v6|| and ||vs||? The problem is that vs does not have a direct meaning of the velocity of the origin, whereas v is the velocity of the origin expressed in {b}-frame. However, we have the quantity screw pitch which represents the ratio of the translational speed and the rotational speed. (a) (10 points) We want to check that the screw pitch for the body and spatial twists are the same. Show that s hs = = ||wb|| 2 ||wb||2 where vs = psb - (ws xpsb), v = Rpsb, ws = Rwb = (b) (10 points) As we proved in (a), the ||wb|| ||ws|| and hs hb, which indicates that the speed of rigid body velocity is preserved between two representations.. It is interesting to check if the adjoint operation associated with Tsb, AdTsb, will preserve the magnitude of ||V|| and ||V||. Or equivalently would Ad AdTsb 16x6? 1) Compute Ad AdTsb and sb 2) find Psb R which makes Ad 3) find Psb R which makes Ad Isb AdT 166 AdT = 166. sb Remark: Therefore, the magnitude of the twist does not necessarily be preserved, and there are no meaning of computing ||V||, but the ||w|| and h are preserved through this adjoint operation. (c) (10 points) Let T = (R1, p) = SE(3) and T2 = (R2, p2) E SE(3). Show that Ad Ad = Ad (6)