4. (30 points) Figure 3 shows a three-link kinematic chain in 2D. The lengths of link A1, A2 and A3 are 11, 12 and 13, respectively. The joint angles of the chain are 02 and 03, as shown in Figure 3. For each link, we attach a local frame to the base end of that link (e.g., for link A1, the axes of frame 1, x and y are labeled). Y1 A A1 1 13 $2 A3 |03 12 Figure 3: The Three-Link Chain (a) (5 points) Determine the topology and dimension of the configuration space for this manipulator. (b) (5 points) Determine the Homogeneous coordinates v3 of the point 2 in the local frame of link A3, in terms of 11, 12, 13, 02 and 03. (c) (5 points) Determine the forward kinematics of this three-link chain. That is, calculate the homogeneous coordinates v of the point 2 in the local frame of link A, in terms of 11, 12, 13, 02 and 03. (d) (5 points) Determine the homogeneous transformations from the local frame of A3 to the local frame of A. That is, determine the transformation matrices T2 and T3 such that, 2 moves A2 from its local frame to the local frame of A1, and T3 moves A3 from its local frame to the local frame of A2. Then the transformation matrix T2 T3 moves A3 from its local frame to the local frame of A1. (e) (10 points) Show that v = T2 T3 V3.