Robotic engineering HCCS robotic vision questions

MENG 5308 Robotics Vision and Control Ch 07 Robotics University of Texas at Tyler Mechanical Engineering Quiz #4 Take-home (120 minutes) Name: Starting time: , Ending time: Problem #1 Figure shows a three-link nonplanar arm. Because all three joints are revolute, this manipulator is sometimes called an RRR (or 3R) mechanism. There is a twist of 90 degrees in magnitude between axes 1 and 2. a. (20 pts) Fill out the table for representing the modified Denavit-Hartenberg (D-H) parameters. b. (40 pts) Calculate the homogeneous transformation matrix for each link, i.e., 0𝑇1, 1𝑇2, 2𝑇3 . Hint: use 𝛼𝑗−1 = 0 and 𝑑𝑗 = 0 in this case. c. (40 pts) Find the total homogeneous transformation of the robot at {3} coordinate with respect to the {0} coordinate, i.e., 0𝑇3 . Note that the 𝑧-direction of each link is upward, perpendicular to the 𝑥 − 𝑦 plane in this case. The modified D-H parameter table: Link (𝑗) 1 𝛼𝑗−1 𝑎𝑗−1 𝑑𝑗 𝜃𝑗 2 3 𝒋−𝟏 𝝃𝒋 (𝜽𝒋 , 𝒅𝒋 , 𝒂𝒋 , 𝜶𝒋 ) = 𝑹𝒙 (𝜶𝒋−𝟏 )⨁𝑻𝒙 (𝒂𝒋−𝟏 ) ⊕ 𝑻𝒛 (𝒅𝒋 ) ⊕ 𝑹 (𝜽𝒋 ) 𝒛 Dr. Chung Hyun Goh Page 1 MENG 5308 Robotics Vision and Control Ch 07 Robotics University of Texas at Tyler Mechanical Engineering Note that in the modified D-H parameters, the link coordinate frames are attached to the near (proximal), rather than the far (distal) end of each link. The homogeneous transformation matrix using the modified D-H parameters is represented by C 𝜃𝑗 − S 𝜃𝑗 0 𝑎𝑗−1 S 𝜃 C𝛼 C 𝜃𝑗 C 𝛼𝑗−1 −S 𝛼𝑗−1 −S 𝛼𝑗−1 𝑑𝑗 𝑗−1 ) 𝑇𝑗 = ( 𝑗 𝑗−1 S 𝜃𝑗 𝑆 𝛼𝑗−1 C 𝜃𝑗 S 𝛼𝑗−1 C 𝛼𝑗−1 C 𝛼𝑗−1 𝑑𝑗 0 0 0 1 Dr. Chung Hyun Goh Page 2 MENG 5308 Robotics Vision and Control Ch 07 Robotics University of Texas at Tyler Mechanical Engineering Problem #2 Consider 2-dimensional (planar) robotic arm shown in figure below. Derive the formulae of joint angles 𝑞1 and 𝑞2 using Pythagorean identities and cosine law and apply trigonometric addition formulae. Show all the steps to get full credits. Dr. Chung Hyun Goh Page 3