Robots, computer algebra and eight connected components
Abstract
Answering connectivity queries in semi-algebraic sets is a long-standing and
challenging computational issue with applications in robotics, in particular
for the analysis of kinematic singularities. One task there is to compute the
number of connected components of the complementary of the singularities of
the kinematic map. Another task is to design a continuous path joining two
given points lying in the same connected component of such a set. In this
paper, we push forward the current capabilities of computer algebra to obtain
computer-aided proofs of the analysis of the kinematic singularities of
various robots used in industry.
We first show how to combine mathematical reasoning with easy symbolic
computations to study the kinematic singularities of an infinite family
(depending on paramaters) modelled by the UR-series produced by the company
``Universal Robots''. Next, we compute roadmaps (which are curves used to
answer connectivity queries) for this family of robots. We design an algorithm
for ``solving'' positive dimensional polynomial system depending on
parameters. The meaning of solving here means partitioning the parameter's
space into semi-algebraic components over which the number of connected
components of the semi-algebraic set defined by the input system is invariant.
Practical experiments confirm our computer-aided proof and show that such an
algorithm can already be used to analyze the kinematic
singularities of the UR-series family. The number of connected components of
the complementary of the kinematic singularities of generic robots in this
family is $8$.
Domains
Symbolic Computation [cs.SC]Origin | Files produced by the author(s) |
---|