Robotic engineering HCCS robotic vision questions

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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 π›Όπ‘—βˆ’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 π‘₯ βˆ’ 𝑦 plane in this case. The modified D-H parameter table: Link (𝑗) 1 π›Όπ‘—βˆ’1 π‘Žπ‘—βˆ’1 𝑑𝑗 πœƒπ‘— 2 3 π’‹βˆ’πŸ 𝝃𝒋 (πœ½π’‹ , 𝒅𝒋 , 𝒂𝒋 , πœΆπ’‹ ) = 𝑹𝒙 (πœΆπ’‹βˆ’πŸ )⨁𝑻𝒙 (π’‚π’‹βˆ’πŸ ) βŠ• 𝑻𝒛 (𝒅𝒋 ) βŠ• 𝑹 (πœ½π’‹ ) 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 πœƒπ‘— βˆ’ S πœƒπ‘— 0 π‘Žπ‘—βˆ’1 S πœƒ C𝛼 C πœƒπ‘— C π›Όπ‘—βˆ’1 βˆ’S π›Όπ‘—βˆ’1 βˆ’S π›Όπ‘—βˆ’1 𝑑𝑗 π‘—βˆ’1 ) 𝑇𝑗 = ( 𝑗 π‘—βˆ’1 S πœƒπ‘— 𝑆 π›Όπ‘—βˆ’1 C πœƒπ‘— S π›Όπ‘—βˆ’1 C π›Όπ‘—βˆ’1 C π›Όπ‘—βˆ’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 π‘ž1 and π‘ž2 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
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MENG 5308 Robotic Vision and Control
Spring Semester 2024

University of Texas at Tyler
Mechanical Engineering

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Dr. Chung Hyun Goh

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MENG 5308 Robotic Vision and Control
Spring Semester 2024

Dr. Chung Hyun Goh

University of Texas at Tyler
Mechanical Engineering

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MENG 5308 Robotic Vision and Control
Spring Semester 2024

Dr. Chung Hyun Goh

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