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EE5106/MCH5209/ME5402 NATIONAL UNIVERSITY OF SINGAPORE FACULTY OF ENGINEERING EXAMINATION FOR (Semester II: 2002/2003) EE5106/MCH5209/ME5402 – ADVANCED ROBOTICS April / May 2003 - Time Allowed: 2 ½ Hours INSTRUCTIONS TO CANDIDATES: 1. This examination paper contains four (4) questions and comprises seven (7) pages. 2. Answer all four (4) questions. 3. All questions carry equal marks. 4. This is an open-book examination. EE5106/MCH5209/ME5402 Advanced Robotics/Page 2 Q.1 (i) Figure 1.1 is the schematic diagram of a robot with one translational joint (along the yu axis) followed by three rotational joints. The first rotational joint following the translational joint has an axis of rotation parallel to zu. The 2nd and 3rd rotational joint axes are parallel to each other and to the xy plane of Frame U. The joint limits of travel are indicated in the figure. Note the direction of the positive motion for each joint axis. a) How many frames are needed to completely describe the kinematics of this robot? (2 marks) b) Define all needed frames according to the Denavit-Hartenberg convention such that the position and orientation of adjacent frames are defined by the four kinematic parameters (, r, d, ): Ti Rot ( z , i )Trans (0, 0, ri )Trans (d i , 0, 0) Rot ( x, i ) i 1 (7 marks) c) Provide the values for all the kinematic parameters and indicate which parameter is the joint coordinate (variable) for each link. That is, fill in the following table. Link r d 1 2 etc (6 marks) d) Indicate the range of possible motions for each joint coordinate (qi) of the robot, where qi is the kinematic parameter that is variable and defined according to your frame assignment. (4 marks) XE 1m 3m 90 to 180 ZE 60 to 300 ZU 1 to 10 m 2m 0 to 360 (no joint limit) XU YU Figure 1.1 EE5106/MCH5209/ME5402 Advanced Robotics/Page 3 Q.1 (ii) Frame B is attached to a moving body. Frames A and C are fixed to the ground with ATC known, as shown in Figure 1.2. The body B is initially at a known position and orientation with respect to Frame A, and defined by the 4 4 homogenous transformation matrix ATB. The body undergoes the following motion in the indicated sequence: 1st: Rotation about X axis of Frame A by 30 2nd: Rotation about Y axis of Frame C by 40 3rd: Translation along Frame B (itself) by (1,2,3) m. Determine the new position and orientation of Frame B is Frame A, ATB, after the three motions above. You need not evaluate nor simplify the expression for ATB. (6 marks) B C A Figure 1.2 EE5106/MCH5209/ME5402 Advanced Robotics/Page 4 Q.2 (i) Figure 2.1 is the schematic diagram of a robot with three rotational joints whose axes of rotation are all parallel to the Z axis of Frame 0. The joint configuration of the robot is indicated by (q1, q2, q3) whose zero positions and positive directions of motions are shown in Figure 2.1. The position of the end-effector expressed in Frame 0 is (x,y), and the orientation of the end effector is desribed by the angle between the end-effector (last link) and the positive x axis of Frame 0. a) Does the robot have singularities with respect to the task (x, y, )? If so, determine the joint coordinates that correspond to each singular configuration. (5 marks) b) Does the robot have singularities with respect to the task (x, y)? This means that the robot is programmed to position its end-effector at (x, y) and its orientation is not important. If there are singularities, determine the joint coordinates that correspond to each singular configuration. (5 marks) c) Does the robot have singularities with respect to the task (x)? This means that the robot is programmed to position its end-effector to satisfy the desired x- coordinate position only. The y-coordinate position and orientation of the end-effector are not important. If there are singularities, determine the joint coordinates that correspond to each singular configuration. (5 marks) q3 (x,y) q2 Y0 q1 Z0 X0 Figure 2.1 EE5106/MCH5209/ME5402 Advanced Robotics/Page 5 Q.2 (ii) Figure 2.2 shows a schematic diagram of a robot with n rotational joints hanging on a ceiling and carrying a load of mass m kg. At a certain robot configuration, the following are known: Centre of gravity of the load is at coordinates (a, b, c) m with respect to Frame E, where Frame E is the frame attached to the end-effector. The position and orientation of the end-effector in Frame 0 is given by 3 1 vector 0pE and 3 3 rotation matrix 0RE. The n 1 robot joint configuration q. The 6 n manipulator Jacobian relating the velocity of Frame E with respect to Frame 0, 0JE. Determine the n joint actuation torques needed to maintain the load at the robot configuration q. Assume that the mass of the robot is negligible compared to the load. (10 marks) Y0 Centre of X0 gravity of load Z0 Robot links End Effector Frame E Figure 2.2 EE5106/MCH5209/ME5402 Advanced Robotics/Page 6 Q.3 Figure 3.1 shows a two link (RP) robot in the vertical plane under the influence of gravity. Assume that I 1 is the moment of inertia of link 1 about its rotating axis, m2 is the equivalent lumped mass of link 2, m3 is the mass of the payload, l1 is the distance between joint 1 and the starting point of link 2, 1 is the driving torque at the joint 1, and f 2 is the applied force for link 2. (i) Sketch a set of generalized coordinates, and derive the positions and velocities of m2 and m3 with respect to a base coordinate system. (5 marks) (ii) Find the kinetic energy K and the potential energy V (q) of the robot, and derive equations of motion using Langrangian formulation. (10 marks) l2 m3 m2 l1 Figure 3.1 (iii) Assuming the drive chain of the second link is flexible as shown in Figure 3.2, where m0 is the equivalent mass of the rotor of the motor, k is the sping constant, and the rests of the system remain the same as in Figure 3.1. Derived the corresponding equations of the motion using any method you are comfortable with. l2 m3 m2 l1 k m0 Figure 3.2 (10 marks) EE5106/MCH5209/ME5402 Advanced Robotics/Page 7 Q.4 Figure 4 is a schematic drawing of a single link robot, where m and are the motor and load torques, I m and I the inertias of the motor and load, bm and b the viscous damping coefficients at the motor and load bearings, m and the joint angles of the motor and load respectively, and is the ratio of an ideal belt transmission mechanism with no slippage. (i) If the inertial load varies between 5 and 9 kgm2 , the rotor inertia is I m 0.01kgm2 , the viscous damping coefficients are bm 0.01kgm2 s 1 and b 0.02kgm2 s , the transmission ratio 10 , and the system posses unmodelled resonances at 8.0, 12.0 and 20.0 radians/second, design a PD controller with feedback from such that the system is never underdamped, will not excite the resonances, but is as stiff as possible. (7 marks) (ii) Briefly discuss the advantages and disadvantages of the scheme of independent joint control such as the one in part (i). Explain the situation where there is a need for integral control. (5 marks) (iii) Explain why most industrial robot control systems work well though independent PID control has been used. (5 marks) (iv) Assume that all the parameters ( I m , I , bm , b and ) are unknown. Design an adaptive controller that can guarantee the stability of the system, and briefly explain the main ideas. (8 marks) I m Im b m bm END OF PAPER