Two satellites are circling the Earth, as shown in Figure 2 Frames (1) and {2} are...

 

 

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Two satellites are circling the Earth, as shown in Figure 2 Frames (1) and {2} are rigidly attached to the satellites in such a way that their x axis always point toward the Earth. Satellite 1 moves at a constant speed V1, while satellite 2 moves at a constant speed v2. To simplify the matters, ignore the rotation of the Earth about its own axis. The fixed frame {0} is located at the center of the Earth. The Figure 2 shows the position of two satellites at t = 0. To help you better understand, the origin of each satellite at time t = 0 expressed in {0}-frame are poi(t = 0) = (0, R 3/2, R/2) and poz(t = 0) = (0, R,0). (a) (12 points) We want to find the trajectory of each satellite observed from the earth, namely, Toi(t) and To2(t), for any given time t. Let's try it! In this subquestion, we want to find To1 (0) and To2(0), the coordinates of each satellites at time t = 0. Given the information in the Figure 2, we can think the configuration of To1 (0) and To2(0) as a rigidly transformed coordinates from the 14x4 SE(3), a frame aligned with {0}. Now, think about the sequential body rotations To1 (0) 13x3 (d) TRot(x,a) TRot(y,B)Td T02 (0) = 13x3 (d) TRot(x,a2) TRot(y,B) Tdz where Rot(x, a) is the rotation along body x axis about a angle. Similarly, Rot(y, B) is the rotation along the new body y axis about angle. Also Ta, represents the translation along d R vector in the new body cooridnates (after rotating a and ). Find a1, a2, B1, B2 [-n, pi) and d, d2 R. (Hint: You can check your answer is correct by looking in to the origin of the frame of Toi (0) and To(0), which should be equal to por (0) and po2(0)) (b) (6 points) Now, we want to calculate the trajectory of each satellite Toi(t), To2(t) SE (3). By knowing that, after time t, the satellite will be turned along each axis shown in Figure 2 Find the screw axis S, S R6 for this rigid body transformation expressed in {0} frame. (Hint: you need to find S = (w, v) where either ||w|| = 1 or ||v|| = 1). (c) (12 points) Based on the instantaneous constant speed for each satellite, we can find the angle at time t = 0, 0(t)= vit, 0(t) = R v2t R {0} Zo yo 30 X V1 R Satellite 1 {1} R Figure 2: Picture from "Modern Robotics" $ {2} Satellite 2 Now, find the corresponding rigid body transformation by computing T' (t) = 01(t) and T(t) = e202(t) You need to answer all elements in T(t) and T(t) in terms of functions of 0(t), 0(t) (just use 0(t) and 0(t) as the variables). Show your work. (d) (4 points) Now, we have everything ready to find the coordinates of two satellites at any given time t. Write the expression for Toi(t) and To2(t), using To1 (0), To2(0), T, T. (e) (4 points) Finally, we want to find the coordinates of the satellite 1 observed from the satellite 2. What is the expression for T21(t) using Toi(t) and To2(t)? Write the expressions in terms of Toi(t), To2(t). Show your work. Two satellites are circling the Earth, as shown in Figure 2 Frames (1) and {2} are rigidly attached to the satellites in such a way that their x axis always point toward the Earth. Satellite 1 moves at a constant speed V1, while satellite 2 moves at a constant speed v2. To simplify the matters, ignore the rotation of the Earth about its own axis. The fixed frame {0} is located at the center of the Earth. The Figure 2 shows the position of two satellites at t = 0. To help you better understand, the origin of each satellite at time t = 0 expressed in {0}-frame are poi(t = 0) = (0, R 3/2, R/2) and poz(t = 0) = (0, R,0). (a) (12 points) We want to find the trajectory of each satellite observed from the earth, namely, Toi(t) and To2(t), for any given time t. Let's try it! In this subquestion, we want to find To1 (0) and To2(0), the coordinates of each satellites at time t = 0. Given the information in the Figure 2, we can think the configuration of To1 (0) and To2(0) as a rigidly transformed coordinates from the 14x4 SE(3), a frame aligned with {0}. Now, think about the sequential body rotations To1 (0) 13x3 (d) TRot(x,a) TRot(y,B)Td T02 (0) = 13x3 (d) TRot(x,a2) TRot(y,B) Tdz where Rot(x, a) is the rotation along body x axis about a angle. Similarly, Rot(y, B) is the rotation along the new body y axis about angle. Also Ta, represents the translation along d R vector in the new body cooridnates (after rotating a and ). Find a1, a2, B1, B2 [-n, pi) and d, d2 R. (Hint: You can check your answer is correct by looking in to the origin of the frame of Toi (0) and To(0), which should be equal to por (0) and po2(0)) (b) (6 points) Now, we want to calculate the trajectory of each satellite Toi(t), To2(t) SE (3). By knowing that, after time t, the satellite will be turned along each axis shown in Figure 2 Find the screw axis S, S R6 for this rigid body transformation expressed in {0} frame. (Hint: you need to find S = (w, v) where either ||w|| = 1 or ||v|| = 1). (c) (12 points) Based on the instantaneous constant speed for each satellite, we can find the angle at time t = 0, 0(t)= vit, 0(t) = R v2t R {0} Zo yo 30 X V1 R Satellite 1 {1} R Figure 2: Picture from "Modern Robotics" $ {2} Satellite 2 Now, find the corresponding rigid body transformation by computing T' (t) = 01(t) and T(t) = e202(t) You need to answer all elements in T(t) and T(t) in terms of functions of 0(t), 0(t) (just use 0(t) and 0(t) as the variables). Show your work. (d) (4 points) Now, we have everything ready to find the coordinates of two satellites at any given time t. Write the expression for Toi(t) and To2(t), using To1 (0), To2(0), T, T. (e) (4 points) Finally, we want to find the coordinates of the satellite 1 observed from the satellite 2. What is the expression for T21(t) using Toi(t) and To2(t)? Write the expressions in terms of Toi(t), To2(t). Show your work.

Answer rating: 100% (QA)

a To find T010 and T020 we need to determine the rotation angles alpha1 beta1 alpha2 beta2 ... View the full answer