physics question

MENG 5340 Robotic Vision and Control Ch09 & Ch15 University of Texas at Tyler Mechanical Engineering HW #2 Name: Problem 1. Consider the single-link robot arm shown in figure below, consisting of a rigid link coupled through a gear train to a DC-motor. Let 𝜃𝑚 and 𝜃𝑙 denote the angles of the motor shaft and the link, respectively. 𝐽𝑚 and 𝐽𝑙 represent the rotational inertias of the motor and the link, respectively. 𝜏𝑚 and 𝜏𝑙 are the motor torque and the disturbance torque due to the link motion. 𝐾𝑎 and 𝐾𝑚 are the transconductance of the amplifier and the motor torque constant. 𝑢 and 𝑖𝑚 are the input (applied) control voltage and control current on a motor driver, respectively. In the figure, 𝑀 is the total mass of the link and 𝑙 is the distance from the joint axis to the link center of mass, 𝐺 is the gear ratio. 𝐵𝑚 denotes viscous friction (or damping) coefficient of the motor. a. (40 pts) Find the kinetic energy (𝐸𝑘 ) and potential energy (𝐸𝑝 ) of the system as a function of 𝜃𝑙 and 𝜃̇𝑙 . b. (30 pts) Establish the equation of the motion on the link of the robot arm by applying the Euler-Lagrangian equation (6) with the results of part (a). c. (30 pts) Determine the complete expression for the dynamics of this system. Hint: the gear ratio is defined as 𝐺= 𝜃𝑚 𝜃𝑙 (1) The total rotational inertia is evaluated by 𝐽 = 𝐽𝑚 + 𝐽𝑙 𝐺2 (2) and the total damping (or viscous friction) coefficient is computed by 𝐵 = 𝐺𝐵𝑚 + 𝐵𝑙 (3) The disturbance torque due to the link motion may be calculated by 𝜏𝑙 = 𝑢 − 𝐵𝜃̇𝑙 (4) The Lagrangian ℒ is given by ℒ = 𝐸𝑘 − 𝐸𝑝 (5) For dynamic force analysis, the Euler-Lagrangian equation is evaluated by 𝜕ℒ 𝑑 𝜕ℒ − ( ) = 𝜏𝑙 (6) 𝜕𝜃 𝑑𝑡 𝜕𝜃̇ Dr. Chung Hyun Goh Page 1 MENG 5340 Robotic Vision and Control Ch09 & Ch15 University of Texas at Tyler Mechanical Engineering Problem 2. We create a model of a central perspective camera with a focal length of 8 mm of the lens. Assume that the world point is located at 𝑃 = [1, 1, 5] 𝑚. a. Find the corresponding nonhomogeneous image-plane coordinates (𝑥, 𝑦). b. Convert the nonhomogeneous image-plane coordinates (𝑥, 𝑦) into the pixel coordinates (𝑢, 𝑣). For part (b), we assume that the pixels are 10 𝜇𝑚 square and the camera resolution is 1024 × 1024 𝑝𝑖𝑥𝑒𝑙𝑠 with its principal point at image-plane coordinate (512, 512). c. Find the pixel coordinates relative to the principal point. d. Determine the image Jacobian matrix. Dr. Chung Hyun Goh Page 2