w6_PC5_Velocity_2022.pdf

Monash University 2022 TRC4800/MEC4456 Department of Mechanical and Aerospace Engineering PC 5 TRC4800/MEC4456 Robotics PC 5: Velocity, Jacobian and Statics Objective: To apply vector propagation to solve direct ve...

Monash University 2022 TRC4800/MEC4456 Department of Mechanical and Aerospace Engineering PC 5 TRC4800/MEC4456 Robotics PC 5: Velocity, Jacobian and Statics Objective: To apply vector propagation to solve direct velocity problems, and to understand the functions of the Jacobian in robotic manipulators. Problem 1. Derive the point velocity and angular velocity, represented in 𝐹0 (frame 0) of the end-effector of the three-link manipulator shown in Figure 1 by using the time derivative of the point position. Figure 1: A 3R Planar Arm Problem 2. Derive the point velocity and angular velocity, represented in 𝔽4 (frame 4) of the end-effector of the three-link manipulator shown in Figure 2 using the velocity propagation method. Assume the length of the last link is L3. Note 𝔽4 is located at the tip of the end- effector, with the same orientation as 𝔽3. Figure 2: A 3R non-planar arm Monash University 2022 TRC4800/MEC4456 Department of Mechanical and Aerospace Engineering PC 5 Problem 3. Derive the Jacobian for the manipulators mapping the joint velocity inputs to the point velocity of the end-effector tips for both the 3 link manipulators from Figure 1 and 2. Represent the Jacobian in a. 𝔽0 attached to the base. b. 𝔽3 attached to the 3rd link. The angular velocity of the end-effector is not required. Problem 4. What happens to the rank of a square Jacobian matrix under singularity configuration? Under that assumption, would singularities in the force domain exist at the same configuration as singularities in the position domain. Explain the physical meaning of a singularity in the force domain. Problem 5. A simplified model of a personnel lifter mechanism is shown in Figure 3 that has a working platform as the end-effector, which for safety purposes is always oriented in the direction of 𝑧̂0. Joints 1, 2 and 4 are revolute with the first two coincident and the third joint prismatic, offset from joint 2 by distance L with variable extension d. You can assume a load in any direction on the platform translates to a point force in the same direction coincident at the fourth joint. Hence find equations for the torques, 𝜏1 , 𝜏2 and prismatic joint force 𝐹3 such that the mechanism can support any force 0𝐹 = [𝑓𝑥 𝑓𝑦 𝑓𝑧 ]𝑇 , applied to the platform. What are the singularities of this system and what does it physically mean? Assume the robot is at home position in Figure 3, therefore an offset is needed in row 2 of the DH table. Note: If you are using the velocity propagation method, you may need to use the following equations for a prismatic joint: 𝑖+1 𝑖+1 𝑖 𝜔𝑖+1 = 𝑖 𝑅 𝜔𝑖 𝑖+1 𝑣𝑖+1 = 𝑖+1 𝑖𝑅 ( 𝑖𝑣𝑖 + 𝑖𝜔𝑖 × 𝑖𝑃𝑖+1 ) + 𝑑̂𝑖+1 𝑖+1 𝑍̂𝑖+1 Figure 3: Schematic of a 4-Dof RRPR mechanism Monash University 2022 TRC4800/MEC4456 Department of Mechanical and Aerospace Engineering PC 5 Problem 6. For the robot shown in Figure 4. a. Using the propagation method, find the velocity (linear and angular) of the end- effector in the tool frame, i.e. find 4𝑣4 & 4𝜔4. b. Using the results you obtained in (a), find the velocity (linear and angular) of the end- effector in the base frame in its simplest form, i.e. find 0𝑣4 & 0𝜔4. c. Find the Jacobian of the end-effector in the tool frame, i.e. 𝐽4 (include only linear velocity terms). d. Find the Jacobian of the end-effector in the base frame, i.e. 𝐽0 (include only linear velocity terms). 0 0]𝑇 , e. Find the joint torques required to maintain a static force vector 𝐹 = [𝑓𝑥 𝑓𝑦 represent the result as a matrix equation. Figure 4: Planar Robot The transformation matrices between the base and tool frame for the robot in Figure 4 are as follows: 𝑐1 −𝑠1 0 0 1 0 0 𝐿1 0 𝑠1 𝑐1 0 0 1 0 0 −1 −𝑑2 1𝑇 = [ ] 𝑇=[ ] 0 0 1 0 2 0 1 0 0 0 0 0 1 0 0 0 1 𝑠3 𝑐3 0 0 1 0 0 𝐿3 2 0 0 1 0 3 0 1 0 0 3𝑇 =[ ] 𝑇=[ ] 𝑐3 −𝑠3 0 0 4 0 0 1 0 0 0 0 1 0 0 0 1 Note: You answers should reflect use of these transformation matrices, or otherwise correct 0 4𝑇 generated by the matrices above. Do not reassign frames and use a different set of matrices. Monash University 2022 TRC4800/MEC4456 Department of Mechanical and Aerospace Engineering PC 5 Problem 7. For the robot shown in Figure 5. a. Using the time differentiation method, find the velocity (linear and angular) of the end-effector in the base frame, i.e. find 0𝑣5 & 0𝜔5. b. Using the results you obtained in (a), find the velocity (linear and angular) of the end- effector in the tool frame in its simplest form, i.e. find 5𝑣5 & 5𝜔5. c. Find the Jacobian of the end-effector in the base frame, i.e. 𝐽0 (include both terms for linear and angular velocity). d. Find the Jacobian of the end-effector in the tool frame, i.e. 𝐽5 (include both terms for linear and angular velocity). e. Find the singularities (if any) of the robot, giving a physical interpretation of them. f. Find the symbolic equation which represents the workspace boundary of the robot. Figure 5: Orthopaedic Robot The transformation matrices between the base and tool frame for the robot in Figure 5 are as follows: 𝑐1 −𝑠1 0 0 𝑐2 −𝑠2 0 𝐿1 1 0 0 𝐿2 0 𝑠1 𝑐1 0 0 1 𝑠 𝑐2 0 0 2 0 −1 0 0 1𝑇 = [ ] 2𝑇 = [ 2 ] 3𝑇 = [ ] 0 0 1 0 0 0 1 𝑑2 0 0 −1 −𝑑 3 0 0 0 1 0 0 0 1 0 0 0 1 𝑐4 −𝑠4 0 0 1 0 0 0 3 𝑠4 𝑐4 0 0 4 0 1 0 0 4𝑇 = [ ] 𝑇=[ ] 0 0 1 0 5 0 0 1 𝐿4 0 0 0 1 0 0 0 1 𝑐12−4 𝑠12−4 0 𝐿1 𝑐1 + 𝐿2 𝑐12 0 𝑠12−4 −𝑐12−4 0 𝐿1 𝑠1 + 𝐿2 𝑠12 5𝑇 = [ ] 0 0 −1 𝑑2 − 𝑑3 − 𝐿4 0 0 0 1 Where 𝑐12−4 = cos(𝜃1 + 𝜃2 − 𝜃4 ) and 𝑠12−4 = sin(𝜃1 + 𝜃2 − 𝜃4 )