ME5286 Robotics Spring 2017 Quiz 2

Transcription

1 Page 1 of 5 ME5286 Robotics Spring 2017 Quiz 2 Total Points: 30 You are responsible for following these instructions. Please take a minute and read them completely. 1. Put your name on this page, any other page you write on, and on your blue book cover. 2. This quiz has 5 pages (including this cover page) and contains 2 problems. There are 4 parts to problem # 1, and 2 parts to problem # This quiz is open book and open notes. You may use a calculator. You may NOT use any device that is capable of wireless communication (including cell phones, laptops, etc.). 4. To get full credit, your response must have a single, correct solution reported with appropriate units. Partial credit is awarded, so be sure to show your work. 5. If you believe a problem statement is missing a necessary parameter, make an assumption and carry on. Be sure to specify the exact nature of your assumption. 6. If you get stuck and cannot derive the solution to one part that you will need for a subsequent part, assume an answer and carry on. 7. If you can t get an answer, or you believe your answer is incorrect, and cannot find the problem in the time available, write a brief explanation of what you think is wrong, why you don t believe your answer is correct, and how you would continue to find the correct solution. Name: Student ID:

2 Page 2 of 5 PROBLEM 1 (16 points) A three jointed robot is shown in Figure 1.1. The configuration of this robot is RRR. The base coordinate frame {XX 0, YY 0, ZZ 0 } is situated at the intersection of the 0 th and 1 st joint axes as pictured in Figure 1.1. The first joint rotates about joint axis JJ 1 which is coincident with the base ZZ axis ZZ 0. The second joint rotates about joint axis JJ 2 which is coincident with the base Y axis YY 0. Finally the third joint rotates about joint axis JJ 3 which is parallel with the base Y axis YY 0. Note: the JJ 2 and JJ 3 axes are always parallel. Figure 1.1: The RRR arm in its zero pose (straight up). The end-effector coordinate frame {XX ee, YY ee, ZZ ee } is located at the end of the second link. A DH table with the relevant information is provided in Table 1.1. Table 1.1: The D-H table for the RRR robot. Joint i θ (degrees) d (mm) a (mm) α (degrees) 1 θθ θθ LL θθ 3 0 LL 2-90

3 Page 3 of 5 The AA matrices are: and c 1 0 ss 1 0 c 2 ss 2 0 LL 1 cc 2 c 3 0 ss 3 LL 2 cc 2 AA 1 ss 0 = 1 0 cc 1 0, AA 2 ss = 2 cc 2 0 LL 1 ss 2, AA 3 ss = 3 0 cc 3 LL 2 ss c 23 cc 1 ss 1 ss 23 cc 1 cc 1 (LL 2 cc 23 + LL 1 cc 2 ) TT 3 0 = cc 23 ss 1 cc 1 ss 23 ss 1 ss 1 (LL 2 cc 23 + LL 1 cc 2 ) ss 23 0 cc 23 LL 2 ss 23 LL 1 ss 2 Part A (1 Point) Suppose a robot has 8 joints. What would be the size of the Jacobian matrix for this robot? (What does m and n equal in m x n?) Part B (2 Points) Explain what the columns and rows of the Jacobian signify. Part C (4 Points) Given the D-H Table (Table 1.1) clearly draw the coordinate frames that move with each joint for the three rotational joint axes on Figure 1.1. Part D (9 Points) Calculate the symbolic Jacobian for the robot shown in Figure 1.1 using the D-H parameters of Table 1.1.

4 Page 4 of 5 PROBLEM 2 (14 points) The UR5 is a lightweight 6 jointed robot arm which can work in collaborative environments with humans. The UR5 is primarily utilized in manufacturing tasks. The UR5 s D-H table is given in Table 2.1 and the joint coordinates are shown in Figure 2.1. The robot passes through θθ = [30, 45, 80, 216, 70, 0] dddddd resulting in the following Jacobian matrix and transformation matrix: Part A (7 Points) JJ UUUU5 = TT UUUU5 = A sensor records the velocity at the end-effector (EE) at VV EEEE = [ ] mm/ss and ωω EEEE = [ ] rrrrrr/ss. Calculate the joint velocities at this instance in time. Report your answer in rrrrrr/ss. Part B (7 Points) Consider the robot colliding with an object at the joint angles, θθ, and Jacobian, JJ UUUU5, described above. This causes a reaction force and torque at the end-effector. The measured force and torque are FF EEEE = [ ] TT NN and ττ EEEE = [ ] TT NNNN respectively. Calculate the torques that must be generated by each joint in the arm when applying the given force and torque vectors, FF EEEE & ττ EEEE, and the Jacobian, JJ UUUU5. Report your answer in NNNN.

5 Page 5 of 5 Table 2.1: D-H Table for a UR5 robot. Joint θ d [m] a [m] α [rad] 1 θθ π/2 2 θθ θθ θθ π/2 5 θθ π/2 6 θθ EE Figure 2.1: A top down view of the UR5 robot is shown for the zero configuration of each joint, i.e. at [0, 0, 0, 0, 0, 0] deg. The robot base frame is {XX 0, YY 0, ZZ 0 } and is provided on this image. The location of the origin of each joint coordinate frame is shown on the robot and the joint coordinate frames are shown in the corresponding boxes. Each joint coordinate frame is a top down view of the robot at each joint.