{2} 2 v1 21 1{1} Top View {1} 2 R = 2 {2} 45 45 L= 1 {0} H=2 Zo L=1 L = 1 {0} xo L=1 Figure 3: Picture from "Modern Robotics" Textbook Two toy cars are moving on a round table, as shown in Figure 3. Car 1 moves at a constant speed v along the circumference of the table, while car two moves at a constant speed v2 along a radius; the position of the two vehicles at t = 0 are shown on the left of Figure 3. The top view of the initial position is shown on the right of Figure 3. The table is not moving and rigidly attached to the ground and the {0} frame is the fixed frame. Don't get confused from the top-view. {0} frame is not on the table. Hashtag-smiley face. The goal is to find the coordinates of each car at time t. In other words, we want to know T01(t) and T02(t) as a function of time t. Let's do it! (a) (6 points) First, Find the initial configuration To1(0) and T02 (0) and write down in homogeneous representation in R44. (b) (6 points) For this simple motion, finding a constant screw axis for each car observed in the {0} frame might be possible. Car 1 is moving on the circular path on the table, and car 2 is following a straight line along the radius of the circular table. Find the screw axis for each rigid transformation, and the angle of rotation OR distance traveled as function of time t. Write the screw axis S = R6 for carl and the screw axis S2 for car 2. Also write the angle 01(t) represents how much it turned along S and distance traveled algons s2, 02(t). We simply use the same symbol 0 for car 2 but it is actually in meter unit not radians. (c) (12 points) Based on your answer in (a), find the rigid body transformation for Car 1 and Car 2. In other words, compute T(t) = e(t) and T(t) = e202(t) (d) (6 points) Now we are ready! Find the homogeneous representation of each car as a function of time t. Write down To1(t), T02(t) = R4 (Hint: T(t) and T(t) are the rigid body transformation which brings your To1(0) and T02 (0) to the current configuration at time t)