Interval-based Computation of Generalized Aspects of Parallel Robots

The following figures show generalized aspects of several parallel robots projected onto the 2D/3D workspace. Each figure consists of sets of boxes, i.e., interval vectors, computed with a specific interval computation method. Green boxes are certified; red, blue, and black boxes are undecided (i.e., singularity condition, additional inequality constraints, or the inner test of the solution set, respectively, is not certified).

For the detail, refer the following publication:
S. Caro, D. Chablat, A. Goldsztejn, D. Ishii, and C. Jermann. A branch and prune algorithm for the computation of generalized aspects of parallel robots. Artificial Intelligence, 211:34–50, 2014. link

RPRPR (2 aspects)

RRRRR (10 aspects)

3-RPR (2 aspects)

First figure below shows the undecided boxes that cover the surface of the 3D workspace. Second and third figures show the certified CCs corresponding to the two aspects.

3-RRR (Fixed orientation, 25 aspects)

(We are still not sure about the number of aspects.)

RRRRR (Arms collisions, 10 aspects)

RRRRR (Joint limits, 12 aspects)

RRRRR (Obstacle avoidance, 26 aspects)