Aerospace and Mechanical Engineering Me 469: Introduction to Robotics Homework 4 Solutions

1. Hopefully, every group accomplished at least some work this week on project 2. 2. (a) You can do this problem in two ways. The first way is attaching frames to each link and determining the Denavit–Hartenberg parameters, as illustrated in Figure 1. Referring to the figure, the link parameters are: i α i−1 a i−1 d i θ i 1 0 0 0 θ 1 2 0 l 1 0 θ 2 3 0 l 2 0 θ 3 Using equation 3.6 or your Mathematica function, gives the transformation 0 3 T =    cos(θ 1 + θ 2 + θ 3) − sin(θ 1 + θ 2 + θ 3) 0 cos(θ 1) l 1 + cos(θ 1 + θ 2) l 2 sin(θ 1 + θ 2 + θ 3) cos(θ 1 + θ 2 + θ 3) 0 sin(θ 1) l 1 + sin(θ 1 + θ 2) l 2 0 0 1 0 0 0 0 1    Note that frame 0 is the same as frame S; however, the tool frame T is not frame 3. To get the overall transformation S T T , we need to multiply 0 3 T by 3 T T , which is pure displacement in the x–direction: 3 T T =     1 0 0 l 3 0 1 0 0 0 0 1 l 3 0 0 0 1    . Evaluating the matrix product gives S T T =    cos(θ1 + θ2 + θ3) − sin(θ1 + θ2 + θ3) 0 cos(θ1) l1 + cos(θ1 + θ2) l2 + cos(θ1 + θ2 + θ3) l3 sin(θ1 + θ2 + θ3) cos(θ1 + θ2 + θ3) 0 sin(θ1) l1 + sin(θ1 + θ2) l2 + sin(θ1 + θ2 + θ3) l3 0 0 1 0 0 0 0 1   . The (x, y) displacement of the end effector (the origin of the tool frame) is given by the upper two terms of the last column, and inspecting the rotation matrix component of T