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This assignment concerns the simple 2R robot shown in the figure below. The dynamic model has been considerably simplified by considering the mass of each link as concentrated at the center of gravity at the midpoint of each link. Note that the robot is in a vertical plane with gravity acting downward. The link masses are 2m and m, respectively, and both have length l.
The figure shows a plot of a function s(t*) versus t*, a normalized time obtained using
where T_seg is the total real time (in seconds) required for the move. This function must have the following properties:
| t* = 0 | t* = 1 | |
|---|---|---|
| position, s | 0 | 1 |
| velocity, s dot | 0 | 0 |
| acceleration, s ddot | 0 | 0 |
You will have to determine what kind of function to use for this. I suggest a polynomial of some kind, in which case you must find the coefficients in order to satisfy the end conditions stated in the table.
Once you know what the motion function s(t*) is, you can compute the position, velocity, and acceleration of each joint (separately) using the following relations:
On the left are the desired kinematic parameters for calculating the joint torques. The T_seg appears on the right hand side because the derivatives of s are with respect to t*.
Once you have all that in place, you can compute the required joint torques to move the robot from the initial position (30,-60) to the final position (80, -30) (all angles in degrees) in a period of 3 seconds. You will need the values m = 20kg and l=0.75m.
As this motion is being computed, store the torque values and then plot them versus actual time.
How much does the maximum required torque change if we try to accomplish this move in 1 second instead of 3?
If you are feeling ambitious, you can separate the total torque on each joint into inertia, coriolis, centrifugal, and gravity terms and plot each of these separately.