Singularity-free Zones Research Articles

Forward Kinematics Solution Distribution and Analytic Singularity-Free Workspace of Linear-Actuated Symmetrical Spherical Parallel Manipulators

This paper presents a new kinematics model for linear-actuated symmetrical spherical parallel manipulators (LASSPMs) which are commonly used considering their symmetrical kinematics and dynamics properties. The model has significant advantages in solving the forward kinematic equations, and in analytically obtaining singularity loci and the singularity-free workspace. The Cayley formula, including the three Rodriguez–Hamilton parameters from a general rotation matrix, is provided and used in describing the rotation motion and geometric constraints of LASSPMs. Analytical solutions of the forward kinematic equations are obtained. Then singularity loci are derived, and represented in a new coordinate system with the three Rodriguez–Hamilton parameters assigned in three perpendicular directions. Limb-actuation singularity loci are illustrated and forward kinematics (FK) solution distribution in the singularity-free zones is discussed. Based on this analysis, unique forward kinematic solutions of LASSPMs can be determined. By using Cayley formula, analytical workspace boundaries are expressed, based on a given mechanism structure and input actuation limits. The singularity-free workspace is demonstrated in the proposed coordinate system. The work gives a systematic method in modeling kinematics, singularity and workspace analysis which provides new optimization design index and a simpler kinematics model for dynamics and control of LASSPMs.

Singularity-free workspace analysis of general 6-UPS parallel mechanisms via convex optimization

This paper investigates the singularity-free zones in the workspace of general 6-UPS parallel mechanisms. The emphasis is placed on obtaining the maximum volume ellipsoid or sphere in the singularity-free subregions, while taking into account the actuator strokes. The proposed algorithms are based on the convex optimization and have several advantages over the existing methods such as Lagrange multipliers approach, which makes it appropriate for other applicable optimization problems. For determining the maximal singularity-free zones, in order to find the maximum volume ellipsoid, a judicious iterative procedure, referred to as Improved Lower Bound Semidefinite Programming, is proposed. Additionally, different reformulations of the problem are proposed in order to solve other interesting problems in the robotics community such as obtaining the maximum volume sphere. Furthermore, an approach based on the sum of squares method is proposed to solve this problem for a general Gough–Stewart platform with any arbitrary geometric parameter which is conducive to a polynomial optimization problem. The computational time for the proposed algorithms are considerably low compared to other methods proposed in the literature for obtaining the singularity-free workspace which opens an avenue to use them as systematic algorithms for on-line purposes.

Eliminating singularity of a parallel driving mechanism of axisymmetric vectoring exhaust nozzle

The singularity is the inherent characteristics of parallel manipulators. At near-singular configurations, the parallel manipulator cannot resist externally applied force/torque along certain directions. The axisymmetric vectoring exhaust nozzle is driven by the 3-SPS + 3-PRS parallel manipulator to change its exit area A9 and make the universal vector of its divergent section. Preventing the 3-SPS + 3-PRS parallel manipulator from falling into the singular configuration is very important for the maneuverability and safety of the jet-thrust aircraft equipping with the axisymmetric vectoring exhaust nozzle. In this paper, a methodology to eliminate the singularity of the 3-SPS + 3-PRS parallel manipulator is presented. At first, with the aid of the configuration homotopy-tracing algorithm, the configuration curves relative to the input parameters are figured out. It is found that the singularity-free zone corresponding to the input parameter exists between the left and the right extreme singular positions. Based on the extended equation algorithm, the curves of singular points going with the input parameters are drawn. By selecting the suitable initial working point and letting the input parameters locate within the singularity-free zones of input parameters determined by these curves, the singularity can be eliminated in the design stage. The method to eliminate the singularity presented in this paper is simple, efficient, and easy to be implemented directly through inspecting the lengths of the input parameters.

Choosing Actuation Scheme for Optimal Performance of 3-DOF Planar Parallel Manipulators

This paper describes how the actuation scheme affects the size and shape of the workspace of 3RPR and 3PRR, which are the two popular planar parallel manipulators. This study helps in achieving the required singularity-free zones for a specific task by adopting a suitable actuation scheme without resorting to changing the architecture of the manipulator. In addition, the performance index GCI for each actuation scheme has been evaluated and compared with each other. Finally, the GCI for each actuation scheme has been optimized subjected to the geometric constraints for the 3PRR planar parallel manipulator.

A new approach to determine the maximal singularity-free zone of 3-RPR planar parallel manipulator

SUMMARYThis paper presents a new algorithm to find the singularity-free cylindrical workspace of parallel manipulators. Because of the limited workspace of parallel manipulators cluttered with different types of singularities, a simple and robust technique to determine continuous singularity-free zones in the workspace of parallel manipulators is required. Here, the largest singularity-free cylinder within the workspace for any prescribed orientation ranging around a reference orientation angle of moving platform is determined. To this end, Particle Swarm Optimization is utilized to find the closest point on the singularity surface to the axis of the cylinder. By implementing the algorithm on 3-RPR planar parallel manipulator, the results show that this algorithm improves the efficiency and leads to significantly larger singularity-free workspace than those reported earlier.

Determination of Singularity-Free Zones in the Workspace of Planar Parallel Mechanisms with Revolute Actuators

The singularity-free workspace of parallel mechanisms is important in the trajectory planning and the kinematic design of the mechanisms. The planar parallel mechanism with revolute actuators, which the expression for the singularity locus and the algorithms for determining the singularity-free zones involve both the Cartesian and the joint variables, has been studied in the paper. Algorithms are proposed here to simplify the determination of the singularity-free zones in Cartesian space. The methodology and procedures for determining the singularity-free zones of planar parallel mechanisms with revolute actuators are introduced. The numerical results show that the simplified algorithms and the techniques are correct and efficient in determining singularity-free zones in the workspace of planar parallel mechanisms.

Increase of singularity-free zones in the workspace of parallel manipulators using mechanisms of variable structure

This paper is focused on the study of singularity of planar parallel manipulators taking into account the force transmission, i.e. study of singularity of planar manipulator by introducing the force transmission factor. Thus the singularity zones in the workspace of the manipulator are defined not only by kinematic criterions from the theoretical perfect model of the manipulator but also by the quality of force transmission. For this purpose, the pressure angle is used as an indicator of force transmission. The optimal control of the pressure angle for a given trajectory of the manipulator is realized by means of legs with variable structure. The suggested procedure to determination of the optimal structure of the planar parallel manipulator 3-RPR is illustrated by two numerical simulations.

Determination of the maximal singularity-free zones in the six-dimensional workspace of the general Gough–Stewart platform

Abstract This paper addresses the singularity-free workspace analysis of the general Gough–Stewart platform. Unlike in serial mechanisms, where there exist only boundary singularities, the closed-loop nature of parallel mechanisms generates complex singularities inside the workspace, which makes the workspace analysis and the trajectory planning of parallel mechanisms a very difficult problem. Therefore, it is highly desirable to develop algorithms that can locate singularity-free zones inside the workspace. The analytical expression of the six-dimensional singularity locus of the general Gough–Stewart platform was previously obtained by the authors [H.D. Li, C.M. Gosselin, M.J. Richard, B. Mayer St-Onge, Analytic form of the six-dimensional singularity locus of the general Gough–Stewart platform, in: Proceedings of the 2004 ASME International Design Engineering Technical Conference, September 28–October 2, Salt Lake City, USA, 2004]. In this paper, first, the singularity equation is briefly recalled. Then, based on the known singularity locus equation, three procedures are presented to determine singularity-free zones in the 3-D and 6-D workspace, respectively. The zones obtained are the maximal singularity-free spheres or hyper-spheres for a given centre configuration within a given workspace. The determination of these singularity-free zones does not involve any discretization of the workspace and the results obtained are guaranteed. Finally, numerical examples are given to graphically illustrate the procedures. The procedures can be applied to analyze other similar parallel mechanisms with known singularity equations.

Determination of Singularity-Free Zones in the Workspace of Planar 3-P̱RR Parallel Mechanisms

This paper presents an algorithm for the determination of singularity-free zones in the workspace of the planar 3-P̱RR mechanism. The mathematical derivation of the algorithm is first given. Numerical examples are then included to demonstrate the application of the proposed approach.

Determination of maximal singularity-free zones in the workspace of planar three-degree-of-freedom parallel mechanisms

In this paper, a new procedure is introduced to determine maximal singularity-free zones of the 3- R P R parallel mechanisms. The procedure does not involve a discretization of the workspace and the zones obtained are guaranteed to be free from singularities. Numerical examples are given in order to graphically illustrate the effectiveness of the procedure. The proposed approach can be applied to analyze other similar parallel mechanisms.