On the determination of cusp points of 3-RPR parallel manipulators
Introduction
Because at a singularity a parallel manipulator loses its stiffness, it is of primary importance to be able to characterize these special configurations. This is, however, a very challenging task for a general parallel manipulator. Planar parallel manipulators have received a lot of attention [1], [2], [3], [4], [5] because of their relative simplicity with respect to their spatial counterparts. Moreover, studying the former may help understand the latter. Planar manipulators with three extensible leg-rods, referred to as 3-RPR manipulators, have often been studied. Such manipulators may have up to six assembly modes and their direct kinematics can be written in a polynomial of degree six [1], [2]. Moreover, they may have singularities (configurations where two direct kinematic solutions coincide). It was first pointed out that to move from one assembly mode to another, the manipulator should cross a singularity [3], [4]. Later, Innocenti and Parenti-Castelli [5] showed, using numerical experiments, that this statement is not true in general. In fact, this statement is only true under some special geometric conditions, such as similar base and mobile platforms [6], [7]. More recently, Macho et al. [8] proposed a method to plan non-singular assembly-mode changing trajectories. McAree [6] pointed out that for a 3-RPR parallel manipulator, if a point with triple direct kinematic solutions exists in the joint space, then the non-singular change of assembly mode is possible. This result holds under some assumptions on the topology of the singularities [9]. For other mechanisms than 3-RPR manipulator, it is also interesting to note that encircling a cusp point is not the only way to execute a non-singular change of assembly mode [10]. A condition for three direct kinematic solutions to coincide was established in [6]. This condition was then exploited in [11] to derive a univariate polynomial of degree 96. A factored expression was obtained, one of the factors being a 24th-degree polynomial. The authors observed on many examples that the cusp points were each time defined by the 24th-degree polynomial, the remaining factors always defining spurious solutions only. However, they did not attempt to certify that the 24th-degree polynomial was really the only valid factor. Moreover, they never found more than 8 cusp points.
The purpose of this paper is to propose a rigorous methodology to determine the cusp points of any 3-RPR manipulator and to certify that all cusp points are found. This methodology uses the notion of discriminant varieties and resorts to Gröbner bases to solve the systems of equations. Using symbolic computation, it is verified that the cusp points are really defined by a 24th-degree polynomial. For any given 3-RPR manipulator geometry, the maximum number of cusp points in sectional sections of the joint space is determined and the results are certified. In particular, a robot with 10 cusp points in a cross section of its joint space is found for the first time.
The following section introduces the manipulators studied and recalls the main known results. Section 3 describes the algebraic tools used. Last section presents the methodology that is proposed to determine the cusp points using the algebraic tools.
Section snippets
Robot studied and modeling
A general 3-RPR planar parallel manipulator is shown in Fig. 1. This manipulator has three extensible leg-rods actuated with prismatic joints. This manipulator can move its moving platform B1B2B3 in the plane. The vector of the three leg-rod lengths is L ≡ (ρ1, ρ2, ρ3) (Fig. 1). The geometric parameters of the manipulators are the three sides of the moving platform d1, d2, d3 and the position of the base revolute joint centers A1, A2 and A3. The reference frame is centered at A1 and the x-axis
Algebraic tools
This section recalls some mathematic definitions and their implementation in Maple software. The parametric systems we consider are supposed to have a finite number of complex solutions for almost all parameter values. This property is checked by computing a Gröbner basis of the system at hand (see ([16], page 274) and ([17], Theorem 2) for more details).
New modeling and method
In this section and the subsequent ones, we use the equation system and the notations induced by the GSR Model presented in Section 2.1.2. In this model, the position of the platform is given by 4 variables: B1x, B1y the coordinates of the point B1 and αx, αy, the cosine and sine of the angle α respectively.
The configurations of a 3-RPR manipulator may be classified into three categories: the regular configurations, the singular configurations and the cuspidal configurations. The description of
Conclusion
This paper shows that efficient algebraic tools can be applied to analyse in a certified way important kinematic features of parallel manipulators such as the determination of cusp points. These points are known to play an important role in planning non-singular assembly mode changing motions.
A new method was introduced, which is able to characterize all the cusp points for the 3-RPR manipulators. It allowed us to determine the number of cusp points for all the slices of the joint space. For
Acknowledgement
We would like to thank there viewers for their useful remarks, which we have all taken into account. This work has been supported by the Agence Nationale de Recherche (ANR SiRoPa).
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