代写TRC4800/MEC4456 Robotics PC 5: Velocity, Jacobian and Statics调试SPSS

Department of Mechanical and Aerospace Engineering

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

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 ofthe 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 ̂(z)0 . 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:

Figure 3: Schematic of a 4-Dof RRPR mechanism

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  &  4w4 .

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  &   0w4 .

c.    Find the Jacobian of the end-effector in the tool frame,i.e. J4  (includeonly linear velocity terms).

d.   Find the Jacobian of the end-effector in the base frame,i.e. J0  (includeonly linear velocity terms).

e.   Find the joint torques required to maintain a static force vector  0F  =  [fx   fy  0]T, 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:

Note: You answers should reflect use of these transformation matrices, or otherwise correct 4(0)T generated by the matrices above. Do not reassign frames and use a different set of matrices.

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  &  0w5 .

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  &   5w5 .

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:

Where c12−4  = cos(θ1  + θ2  − θ4) and s12−4  = sin(θ1  + θ2  − θ4)