16-11-2012, 01:52 PM
Cusp Points in the Parameter Space of Degenerate 3-RPR Planar Parallel Manipulators
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ABSTRACT
This paper investigates the conditions in the design parameter space for the existence
and distribution of the cusp locus for planar parallel manipulators. Cusp points make
possible nonsingular assembly-mode changing motion, which increases the maximum
singularity-free workspace. An accurate algorithm for the determination is proposed
amending some imprecisions done by previous existing algorithms. This is combined with
methods of cylindric algebraic decomposition, Gro¨bner bases, and discriminant varieties
in order to partition the parameter space into cells with constant number of cusp points.
These algorithms will allow us to classify a family of degenerate 3-RPR manipulators.
[DOI: 10.1115/1.4006921]
Introduction
In the past, singularities were believed to physically separate
the different assembly modes, meaning that for fixed joint values
one could not find a path going from one assembly mode to
another without crossing a singular configuration. So the interest
relied on considering the widest connected nonsingular domain,
called aspect. Innocenti and Parenti-Castelli pointed out in
Ref. [1] that nonsingular changes of assembly mode are possible,
and McAree and Daniel showed in Ref. [2] that such changes are
possible when triple roots of the forward kinematic problem
(FKP) exist. In Ref. [3] Zein et al. showed that for the case of
3-RPR manipulators a nonsingular change of assembly mode can
be accomplished by encircling a cusp point, and Husty recently
proved in Ref. [4] that the generic 3-RPR parallel manipulators
without joint limits always have two aspects.
From the algebraic point of view, the locus of cusp points can
be described by means of symbolic equations. In order to avoid
long symbolic-algebraic manipulations, these equations are usually
solved by numerical approximation at an early stage, which
may lead to small deviations that can be propagated along the process.
However, there exist efficient symbolic-algebraic techniques
that may leave the use of numerical methods to the last step. In
particular, we will apply Gro¨bner bases [5] in order to adopt a
more suitable equivalent system defining the same solution points.
Discussion on the Joint Space
We now extend our improved method to partition a parameter
space with regard to the associated cusp locus. So, we want to discuss
the solutions of a parametric system. Among the numerous
possible ways of solving parametric systems, we focus on the use
of DV [6] for two main reasons: it provides a formal decomposition
of the parameter space through an exactly known algebraic
variety (no approximation), and it has been successfully used in
similar problems [20].
Higher-Dimensional Discussion by Means of a CAD
By construction we know that over any connected open region
not intersecting the DV the system has a constant number of real
roots, for whatever chosen parameters. But if we want to discuss
larger parameter spaces, then the open regions will no longer be
as simple as one-dimensional intervals. So the goal of this section
is to provide an accurate description of the regions with constant
number of solutions. For this, we will use the CAD [7,23].
Conclusions
This paper has introduced both an efficient method for the computation
of the cuspidal configurations of a mechanism, and a reliable
algorithm that partitions a given parameter space into open
regions with constant number of associated cusp points.
The first one is based on a symbolic-algebraic approach able to
describe the roots of exact multiplicity 3 and a certified numerical
algorithm that isolates among them the real (i.e., not complex)
ones. This symbolic-numeric approach is more efficient than other
previously existing methods, which mainly relied on the approximation
of roots of multiplicity at least 3 after reducing the initial
system to a simpler one and projecting it onto the q-space.
This new method is combined with some algebraic tools such
as the DV and the CAD in order to analyze a two-dimensional parameter
space with respect to the associated number of cusp
points. This second algorithm provides a partition of the parameter
space into cells with constant number of cusp points, which is
certified for whatever values are picked inside each open cell but
not on their borders. Cell borders are further analyzed by Algorithm
4 based on Conjecture 1, which still remains unproved.