Mec.E. 555 Assignment #3
Velocity Analysis Using the Jacobian
Date due: Friday, March 15, 1996, 12:00 noon
Questions?
Send a message to roger.toogood@ualberta.ca
A 3R manipulator is shown in the figure below. The shoulder joint is 0.5 m above the world XY plane. The shoulder and elbow links are 1.0 m and 0.8 m long, respectively.
This assignment concerns some kinematic analysis of this robot. You may want to consider writing a computer program to perform some of the calculations. Note that the DH parameters and A matrices for this robot are identical to the first three links of the Mitsubishi, or the elbow manipulator we discussed in class.
- Find the equations that express the forward kinematics for the robot. You can do this by inspection or by using the product of the A-matrices (or do both to check your work!).
- Find the Jacobian of the manipulator. You need to do this symbolically - by hand if necessary, although MATLAB does have a routine to determine the symbolic Jacobian of a set of equations expressed as symbolic strings.
- Find the tip velocity in Cartesian coordinates for the following conditions (ie. fill in the empty columns in the table). Verify your answers where this is relatively easy.
| Joint Positions (deg.) |
Joint Rates (deg/s) |
Tip Speed (m/s) |
| 1 | 2 | 3 | 1 | 2 | 3 | X | Y | Z |
| 0 | 0 | 0 | 1 | 0 | 0 | ? | ? | ? |
| 0 | 0 | 90 | 0 | 1 | -1 | ? | ? | ? |
| 90 | 0 | 90 | -1 | 1 | 1 | ? | ? | ? |
| 30 | 60 | -40 | .5 | -.2 | .6 | ? | ? | ? |
- Find the joint rates required to give the following desired tip velocities at the indicated robot positions. Comment on the results in the last six lines of the table.
| Joint Positions (deg.) |
Tip Speed (m/s) |
Joint Rates (deg/s) |
| 1 | 2 | 3 | X | Y | Z | 1 | 2 | 3 |
| 0 | 0 | 0 | 1 | 0 | 0 | ? | ? | ? |
| 30 | 30 | 90 | 0 | 1 | -1 | ? | ? | ? |
| 30 | 30 | -30 | 1 | 1 | 0 | ? | ? | ? |
| 30 | 10 | -10 | 1 | 1 | 0 | ? | ? | ? |
| 30 | 5 | -5 | 1 | 1 | 0 | ? | ? | ? |
| 30 | 2 | -2 | 1 | 1 | 0 | ? | ? | ? |
| 30 | 1 | -1 | 1 | 1 | 0 | ? | ? | ? |
- Determine the singular position(s) of the manipulator by examining the Jacobian.
- Consider a trajectory of the tip that starts at (X,Y,Z) = (0.8, 0, 0.6) and moves in a straight line parallel to the positive Y axis at a speed of 0.2 m/s. Compute the required joint positions and joint rates for this trajectory at intervals of 0.05m of travel. Plot these joint rates for as long as possible, and comment on the results.
THE END
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