Use software such as Word or LaTeX, scan written work, or a combination of the two.  Total: 15 points 1. Introducing the Selective Compliance Articulated Robot Arm (SCARA). This 4 degreeof freedom pick-and-place robot RRPR design is a design template used by many manufacturers (Fig. 1A). Figure 1: (A) Epson SCARA (B) Kinematic Diagram of SCARA The kinematic diagram with assigned D-H axes is shown in Fig. 1B. Frame 0 is the base frame, and frame 4 is the end effector frame. For simplicity, we are going to set many of the offsets that exist in a real-world robot to be zero. For instance, the offset from the base to the elbow, d1 = 0, and the offset from the last revolute joint to the end effector, d4 = 0 The D-H parameters are: Link ri ai di ?i 1 L1 0 0 ?1 2 L2 p 0 ?2 3 0 0 d3 0 4 0 0 0 ?4 The corresponding forward kinematics matrix is given here (although you will have to redo some of this work for part (e). 1 of 2 UBC BMEG 400F 2021W2 Assignment 2 ? ? ? ? c12c4 + s12s4 -c12s4 + s12c4 0 L1c1 + L2c12 s12c4 - c12s4 -s12s4 - c12c4 0 L1s1 + L2s12 0 0 -1 -d3 0 0 0 1 ? ? ? ? (a) (1 pt) Look at the translation component of the forward kinematics. Why does this look familiar? Can you figure out these terms just by looking at the structure of the manipulator? (b) (1 pt) Based on this intuition, what is the shape of the reachable workspace of the end effector? (assuming L1 > L2) (c) (1 pt) Based on this intuition, write down the inverse kinematics equations of the robot. i.e. given (x, y, z) return possible solutions for the three joint variables. (?4 does not matter since it does not affect end effector position). (d) (2 pt) Find the linear velocity Jacobian (3  4) of the robot. (e) (3 pts) Find the angular velocity Jacobian (3  4) of the robot. (f) (1 pt) Consider only the first three joint variables (including ?4 will make the Jacobian non-square and it does not affect position anyway), what are the singularities of the robot? There are two singularities that are not obvious from |J| = 0. What are they? 2. Write a program that makes the end effector of this robot move in a cubic polynomial trajectory from initial position p0 to final position pf over time tf , using the algorithm discussed in class.  The program should take p0, pf and tf as its inputs, and return a set of joint variables qi(t), desired end effector positions p(t), and actual end effector positions x(t) as outputs, where t are equally spaced time intervals ?t from 0 to tf , e.g. ?t = 0.01 s.  Remember that you have to first drive the robot differentially from its zero configuration to the starting point p0. Alternatively, you can use inverse kinematics to start the robot at p0.  MATLAB or Python preferred. Please clear it with the instructor if you plan to use another language. (a) (3 pts) Include the source code for the file. You can add them as pages in the solution pdf. (b) (1 pt) Plot two examples of the robot moving from an initial position to a final position. Specifically, plot the desired polynomial trajectory as a dashed line and the actual trajectory as a solid line. You dont have to animate the joints and links of the robot (although that would be fabulous!). (c) (1 pt) Would this program work if you try to input any two points within the workspace of the robot? In what conditions would it fail? (d) (1 pt) Increase the sampling frequency (decrease ?t). What happens to the desired trajectory vs. actual trajectory?