Maximal Singularity-free Workspace Research Articles

Synthesis and Singularity Analysis of N-UU Parallel Wrists: A Symmetric Space Approach

We report some recent advances in kinematics and singularity analysis of the mirror-symmetric N-UU parallel wrists using symmetric space theory. We show that both the finite displacement and infinitesimal singularity kinematics of a N-UU wrist are governed by the mirror symmetry property and half-angle property of the underlying motion manifold, which is a symmetric submanifold of the special Euclidean group SE(3). Our result is stronger than and may be considered a closure of Hunt's argument for instantaneous mirror symmetry in his pioneering exposition of constant velocity shaft couplings. Moreover, we show that the wrist can, to some extent, be treated as a spherical mechanism, even though dependent translation exists, and the singularity-free workspace of a N-UU wrist may be analytically derived. This leads to a straightforward optimal design for maximal singularity-free workspace.

Maximal singularity-free orientation workspace over a position region of Gough–Stewart platform

Maximizing the singularity-free workspace of parallel manipulators is highly desirable in a context of robot design. So far, no work has been found to address the maximal singularity-free orientation workspace over a position region. In practice, this type of workspace is interesting because a mechanism often works in a range of positions. This work focuses on the Gough–Stewart platform. An optimal position at which the mechanism holds the maximal singularity-free orientation workspace is determined. This optimal position lies on the line which is perpendicular to the base and passes the centroid of the base. Considering the symmetry, a parallelepiped with centre at the determined optimal position could be an interesting working position region for the Gough–Stewart platform. Two algorithms are presented to compute the maximal singularity-free orientation workspace over such an interesting position region. An example is provided for demonstration.

Determination of the maximal singularity-free workspace of 3-DOF parallel mechanisms with a constructive geometric approach

This paper proposes a novel approach to obtain the maximal singularity-free regions of planar parallel mechanisms, which is based on a constructive geometric reasoning. The proposed approach consists of two algorithms. First, the borders of the singularity-free region associated with an arbitrary starting point of the moving platform are obtained. Then, the second algorithm finds the center of the maximal singularity-free circle, which is obtained using the so-called offset curve algorithm. The procedure is applied to a 3-PRR planar parallel mechanism as an example and the obtained results illustrate graphically the effectiveness of the proposed algorithm. The proposed approach can be directly applied to obtain the maximal singularity-free circle of similar parallel mechanisms, which is not the case for other approaches proposed in the literature that are limited to a given parallel mechanism, namely, the 3-RPR planar parallel mechanism. Moreover, as the main feature of the proposed approach, it can be implemented both in a CAD system and in a computer algebra system where non-convex and reentrant curves can be considered.

Determining the maximal singularity-free circle or sphere of parallel mechanisms using interval analysis

SUMMARYThis paper proposes a systematic algorithm based on the concept of interval analysis to obtain the maximal singularity-free circle or sphere within the workspace of parallel mechanisms. As case studies the 3-RPR planar and 6-UPS parallel mechanisms are considered to illustrate the relevance of the algorithm for 2D and 3D workspaces. To this end, the main algorithm is divided into four sub-algorithms, which eases the understanding of the main approach and leads to a more effective and robust algorithm to solve the problem. The first step is introduced to obtain the constant-orientation workspace and then the singularity locus. The main purpose is to obtain the maximal singularity-free workspace for an initial guess. Eventually, the general maximal singularity-free workspace is obtained. The main contribution of the paper is the proposition of a systematic algorithm to obtain the maximal singularity-free circle/sphere in the workspace of parallel mechanisms. The combination of a maximal singularity-free circle or sphere with the workspace analysis by taking into account the stroke of actuators, as additional constraint to the latter problem, is considered. Moreover, the center point of the circle/sphere is not restrained to a prescribed point.

Six-DOF parallel manipulators with maximal singularity-free joint space or workspace

SUMMARYSingularity-free workspace is a very important criterion for the design of manipulators, especially for parallel manipulators which are well known for their limited workspace and complex singularities. This paper studies geometric parameters and dexterity measures that affect the size of a singularity-free joint space and proposes methods for the development of 6-DOF Stewart–Gough parallel manipulators that have better singularity-free joint space. With a local dexterity measure as the objective function, a systematic method is employed to search for the design with a maximal singularity-free joint space. The related workspaces are also investigated. It is shown that the workspace is not proportional to the size of the joint space and that manipulators with a larger singularity-free workspace usually have relatively poor dexterity.

GEOMETRIC SYNTHESIS OF PLANAR 3-RPR PARALLEL MECHANISMS FOR SINGULARITY-FREE WORKSPACE

This work focuses on the geometric synthesis of planar 3-RPR parallel mechanisms in order to guarantee a singularity-free workspace for a desired orientation range. The effects of the orientation angle, the minimal leg length as well as the base shape on the singularity-free workspace are analyzed using the Gauss divergence theorem. The results show that for every orientation angle, there exists an optimal minimal leg length which leads to the maximal singularity-free workspace. If the optimal minimal leg lengths are used, the equilateral triangle base yields the maximal singularity-free workspace for any orientation angle. However, for a prescribed working range of the orientation angle, the optimal minimal leg length may be different from the individual optimal minimal leg lengths. Based on the optimal minimal leg length determined for a prescribed working range of the orientation angle, a geometric synthesis procedure is proposed in order to guarantee a singularity-free workspace.

The Maximal Singularity-Free Workspace of the Gough–Stewart Platform for a Given Orientation

The maximal singularity-free workspace of parallel mechanisms is a desirable criterion in robot design. However, for a 6DOF parallel mechanism, it is very difficult to find an analytic method to determine the maximal singularity-free workspace around a prescribed point for a given orientation. Hence, a numerical algorithm is presented in this paper to compute the maximal singularity-free workspace as well as the corresponding leg length ranges of the Gough–Stewart platform. This algorithm is based on the relationship between the maximal singularity-free workspace and the singularity surface. Case studies with different orientations are performed to demonstrate the presented algorithm. The obtained results can be applied to the geometric design or parameter (leg length) setup of this type of parallel robots.

GEOMETRIC OPTIMIZATION OF PLANAR 3-RPR PARALLEL MECHANISMS

To pursue the maximal singularity-free workspace of parallel mechanisms is a very important concern for robot designers. This paper focuses on the case of planar 3-RPR parallel mechanisms. First, a relatively simple singularity equation of any point on the platform is derived. The obtained singularity equation shows that the singularity locus of any point on the platform is a circle of the same size, as long as the base and the platform are similar triangles. Furthermore, the three centres of the workspace circles lie exactly on the singularity circle. With these useful observations, the singularity-free workspace as well as the maximal leg length ranges can be determined. For a base of unit area, it is found that robots with equilateral triangle base and platform can obtain the maximal singularity-free workspace. Three case studies demonstrate this observation. Finally, a procedure for this kind of robot geometric design is provided.