Assignment 2: Kinematics
Contact: asathyam@cs.umd.edu February 2020
1 DH Parameters and Forward Kinematics
1. [20 points] The UR-6 is a robot arm with six degrees of freedom, all of which are rotary, as shown in Figure 1.
Figure 1: UR-6 with necessary dimensions marked.
a. Draw a diagram showing your assignment of coordinate frames to this robot arm. Choose the positive direction of the z-axes to be in the direction of
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the (blue/ light gray) end caps for the joint. The base is at the bottom. The lengths marked on the figure should make the frame assignment easier.
b. Construct the table of DH parameters for this robot arm. There should be 6 rows in the table, not counting the first row which consists of column la- bels. NOTE: please use the letters a, d, ¦Á, and ¦È for the DH parameters, not the numerical values shown in the figure.
c. Write the A-matrices for each joint.
2 Forward Kinematics Visualization
2. [10 points] Consider the 3-link robot arm shown in Fig.2.
Figure 2: 3-link planar manipulator.
a. Using the equations of motion which relate the end-effector position (cartesian coordinates x and y) to the joint angles, derive the equation for the robot workspace. This is the reachable workspace, i.e. workspace as defined conventionally which is the set of all point reachable by the robot in at least one pose. The workspace equation should be reduced in its simplest form. Do not restrict the motion of the robot to the first quadrant. Your proof should be a formal (i.e. purely analytical) derivation and should not use Matlab or any other computational software. Note that, there may be multiple workspaces depending on the relationship between the magnitude of the link lengths. If so, derive all possible workspaces analytically. Use drawings to elucidate your
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answer.
b. Calculate the expression for the area of all possible robot workspace(s).
3 Inverse Kinematics
3. [10 points] Given a desired position of the end-effector, how many solutions are there to the inverse kinematics of the three-link planar arm shown in Fig.2? If the orientation of the end-effector is also specified, how many solutions can be calculated? Use the geometric approach to find them.
4. [15 points] [Use any programming language] Write a computer program to compute the inverse kinematic equations for the elbow manipulator using Equations (3.75) – (3.80) in Spong et. al. Include procedures for identifying sin- gular configurations and choosing a particular solution when the configuration is singular.
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