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A simulation program was written to test the parallel manipulation algorithm. A typical run of the program is shown in the 3 screenshots below. They illustrate a manipulation task with 3 parts moving simultaneously along 3 independent trajectories. The minimum number of rotations/step is therefore 6 (it is twice the number of parts). To better condition the solving step, the program will assume a total of 25 centers of rotation, i.e., allow for 25-6=19 degrees of redundancy. SVD is used to invert a 6x25 matrix at every step. In the snapshots, the centers of rotation are rendered with green dots.

Below, the user has placed the first part at the upper left corner of the workspace (purple disk) and created a rectangular path for it.

19_23_49.GIF (8780 bytes)

Next, a second part is placed initially at the lower left corner of the workspace (yellow disk), and is also prescribed a rectangular path.

19_23_57.GIF (9537 bytes)

The simulation is allowed to run for a few steps. Both parts advance along their intended paths (it works, it works!!!); midway along these paths, a third part is added to the experiment (the cyan disk), and a self-intersecting polygonal path is specified.

19_24_42.GIF (12571 bytes)

The simulation go on, with the three parts moving simultaneously, as shown below (note that in these frames, the force field lines which result from the sum of all rotations are shown as a grid of red vectors):


19_25_08.gif (15030 bytes) 19_25_41.gif (14431 bytes)
19_26_23.gif (13275 bytes) 19_26_54.gif (13634 bytes)

An .avi animation of the whole process is available here.


2000 Dan S. Reznik, <>