CSci 5551: Introduction to Intelligent Robotic Systems Spring 2022
Homework 3
Due: 11:59 PM (CST), Thursday Aprl 14, 2022
The points assigned to each problem are in the square braces. Do not just write the final result. Present your work in detail explaining your methodology at every step. For the Denavit-Hartenberg (DH) convention questions (Q 1, 2, and 3a), please show the frames clearly and all associated measurements. If link lengths are not shown, use variables ai to indicate the length of link i, if needed. Remember that phrase “derive the forward kinematic equations” means showing both the orientation (in the form of a rotation matrix R) and position of the end-effector (in the form of a vector) with respect to the base frame.
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1. [10+10=20 points] For both problems below, please perform matrix multiplication at the end and come up with the final homogeneous transformation matrix. Do not just leave the matrices in multiplicative form.
(a) For the two-link Cartesian robot shown in Figure (1a), please derive the forward kinematic equations using the DH convention.
(b) For the three-link robot shown in Figure (1b), please derive the forward kine- matic equations using the Denavit-Hartenberg (DH) convention.
2. [20 points] The arm with three degrees of freedom shown in Figure 2 is a 3R robot, except that joint 1’s Z axis is not parallel to the other two. Instead, there is a twist of
90 degrees in magnitude between axes 1 and 2. Find D-H table parameters for this robot, and find the transform T13, with F1 being coordinate frame for the first joint.
Figure 2: A 3R, 3-DOF robot arm.
3. [20+20=40 points]:
(a) Consider the PUMA 260 manipulator shown in Figure 3. Derive the complete set of forward kinematic equations, by establishing appropriate DH coordinate frames, constructing a table of link parameters, forming the A-matrices, etc. You do not need to multiply the matrices at the end. Please use the given dimensions, where necessary, in your solutions.
(b) Solve the inverse position kinematics problem for the PUMA 260 robot. Use any method you deem appropriate.
4. [20 points] A kinematic model of the human arm (see Figure 4) co-locates joints (1-3) at the shoulder creating spherical joint; joint (4), a pin joint axis – shown coming out of the page – at the elbow; joint (5) along the forearm (axial roll), and then joints (6,7) at the wrist for yaw and then pitch. Starting with the z0 axis coming out at you from the page, and the x0 axis to your left, as shown (i.e., θ1 rotation lowers/raises the arm from the side), neatly draw and label all the z and x axes and make a complete DH table. Show and label all non-zero displacements on the diagram (on Figure 4) and include them in your table. Give the tool transform to the tip of the hand frame as shown. Points will be given for clarity.
Figure 4: The human arm with joints.
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