Page 1 of 7 ME5286 Robotics Spring 2015 Quiz 1 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 your blue book. 2. This quiz has 7 pages (including this cover page) and contains 2 problems. There are 5 parts to problem # 1 and 3 parts to problem # 2. 3. 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:
Page 2 of 7 PROBLEM 1 (18 points) RoboSimian, designed by NASA s Jet Propulsion Laboratory, is a simian-inspired limbed robot that will participate in the DARPA Robotics Challenge. RoboSimian has four general purpose limbs and hands, capable of both mobility and manipulation, to achieve passively stable stances; establish multi-point anchored connections to supports such as ladders, railings, and stair treads; and brace itself during forceful manipulation operations. Figure 1.1 show RoboSimian in various poses. Leg Arm Head Figure 1.1: RoboSimian shown in a wheeled pose (Left Figure) and a legged walking pose (Right Figure) A. (4 points) You are given the base coordinate frame as well as the rotational axis for each joint in one arm of RoboSimian as depicted in Figure 1.2. Clearly draw and label the remaining link coordinate frames on Figure 1.2. B. (4 points) The elements of the A matrices from the (i-1) th to the i th joint can be calculated by using the Denavit- Hartenburg variables; fill in the table of θ, d, a, and α for each of the A matrices. Fill in the appropriate geometric dimensions, from Figure 1.2, with the assumption that there are no offsets in or out of the page.
Page 3 of 7 C. (4 points) At one instant in time, the joints move through their zero position (the zero pose of Fig 1.2). Compute A matrices, (AA 0 1, AA 1 2, AA 2 3, AA 3 4 ), using the Denavit-Hartenburg variables from part B. D. (3 points) Find the homogenous transformation matrix ( TT 0 3 ). E. (3 points) Compute the end effectors position [XEE YEE ZEE ]when the joint angles are θ = [ 30 45 30 60 0 EEEE 45 30 ] and TT 4 is: 0.61 0.707 0.35 0.035 TT EEEE 0.5 0 0.866 0.61 4 = 0.61 0.707 0.35 0.45 0 0 0 1 Table 1.1: D-H Table for RoboSimian Joint θ d a α 1 2 3 4 5 6 7 EE
Page 4 of 7 Figure 1.2: RoboSimian right arm (front view) at the zero pose with all joint variable equal to zero. Assume there are no offsets.
Page 5 of 7 PROBLEM 2 (12 points) A custom robotic arm named GREG has been developed by a research lab at the University of Minnesota. This arm has 3 rotational servo joints and an end effector. A CAD model of this robot is pictured in Figure 2.1. Figure 2.1: The GREG arm pictured in a non-zero pose. The joint configuration of the GREG arm is RRR. The base coordinate frame (XX 0, YY 0, ZZ 0 ) is situated at the intersection of the 1 st and 2 nd joint axis as pictured in Figure 2.2. The first joint rotates about joint axis JJ 1 which is coincident with the base Z 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 axis are always parallel. The joint lengths for the upper and lower arms are LL 1 = 100mmmm and LL 2 = 100mm. The end-effector coordinate frame (XX EE, YY EE, ZZ EE ) is located at the end of the second arm. A DH table with the relevant information is provided in Figure 2.3.
Page 6 of 7 Figure 2.2: The GREG arm in its zero pose (straight up). Table 2.1: The DH table for the GREG robot in the zero position. Joint i θ (degrees) d (mm) a (mm) α (degrees) 1 θθ 1 0 0-90 2 θθ 2 90 0 100 0 3 θθ 3 0 100-90
Page 7 of 7 Figure 2.3: Two different viewpoints of the GREG arm in a joint configuration such that the origin of the end effector coordinate frame (XX EE, YY EE, ZZ EE ) is at the desired position (XX dd, YY dd, ZZ dd ). Note: This is no longer in the zero pose. It is desired to move the joints of the GREG arm such that the origin of the end-effector coordinate frame is located at a desired position (XX dd, YY dd, ZZ dd ) in 3 dimensional space. For a specific task this desired position is (XX dd, YY dd, ZZ dd ) = (60,40,160). These coordinates are relative to the base coordinate frame. A. (2 points) Determine the degrees of freedom for the GREG arm in terms of position and orientation. B. (6 points) Using either transformation matrices or geometric methods, determine the joint angles (θθ 1, θθ 2, θθ 3 ) such that the end effector is moved to the desired position (XX dd, YY dd, ZZ dd ). In other words, solve for the inverse kinematics to achieve this position. Note: There may be multiple solutions; you only need to provide one. C. (4 points) Write a formula to solve for the outer limits of the reachable workspace for the GREG arm.