Singularity-free Regions Research Articles

Classification of 3-Degree-of-Freedom 3-UPU Translational Parallel Mechanisms Based on Constraint Singularity Loci Using Gröbner Cover

Abstract A 3-degree-of-freedom (DOF) 3-UPU translational parallel mechanism (TPM) is one of the typical TPMs. Despite comprehensive studies on 3-UPU TPMs in which the joint axes on the base and the moving platform are coplanar, only a few 3-UPU TPMs with skewed base and moving platform have been proposed, and the impact of link parameters on constraint singularity loci of such TPMs has not been systematically investigated. The advances in computing comprehensive Gröbner system (CGS) or Gröbner cover of parametric polynomial systems provide an efficient tool for solving this problem. This paper presents a systematic classification of 3-UPU TPMs with skewed base and moving platform based on constraint singularity loci. First, the constraint singularity equation of a 3-UPU TPM is derived. Using Gröbner cover, the 3-UPU TPMs are classified into 12 types. Finally, a novel 3-UPU TPM is proposed. Reconfiguration analysis shows that unlike most existing 3-UPU TPMs which can transit from a 3-DOF translational mode to two or more 3-DOF operation modes, the proposed 3-UPU TPM can only transit from a 3-DOF translational mode to one general 3-DOF operation mode. The singularity locus divides the workspace of this 3-UPU TPM into two constraint singularity-free regions. As a by-product, a 3-UPU parallel mechanism that the moving platform can undergo 3-DOF translation and 1-DOF infinitesimal rotation is revealed. This work provides a solid foundation for the design of 3-UPU TPMs and a starting point for the classification of 3-UPU parallel mechanisms.

Parallel robots modelling and optimization

Parallel robots are mechanical systems with numerous applications in manufacturing, medicine, space industry, etc. Such robots are characterized by closed kinematic chains ensuring robustness and maneuverability. The design of a fully functional parallel robot is a time-consuming task as an engineer needs to enumerate and compare many variants in order to find one with the best characteristics. Typical characteristics are the area of the workspace, the number of singularity-free regions, dexterity index. The design of a parallel robot is thus a multi-objective constraint optimization problem. We propose to use interval analysis to reliably compute characteristics and multiobjective optimization methods to find the Pareto-optimal set of robot parameters. The proposed approach is illustrated on a set of several realistic parallel robotic systems.

Singularity-free spheres in the position and orientation workspaces of Stewart platform manipulators

This paper presents the computation of singularity-free regions in the position and orientation workspaces of Stewart platform manipulators (SPMs). Notwithstanding several previous publications in this regard, certain issues persist, such as the unavailability of analytical solutions and the consequent lack of accurate estimates of the numbers of solutions in various cases; the improper use of an incompatible metric while finding such regions inside SO(3) or SE(3). In this paper, the singularity-free regions have been envisaged as spheres in either R3 or SO(3). An analytical formulation for identifying such singularity-free spheres (SFSs) is presented. Several algebraic methods are explored to solve the resulting set of polynomial equations, leading to valuable insights into the numbers of (finite, complex) solutions to this problem, in both the position and the orientation workspaces. A physically meaningful and mathematically sound metric is used while computing the SFS in the orientation workspace. The formulation and computational algorithms have been demonstrated by identifying the SFSs (in R3 as well as SO(3)) for multiple architectures of the SPM through implementations in the C language.

Comparison of 3-[PP]S Parallel Manipulators based on their Singularity Free Orientation Workspace, Parasitic Motions and Complexity

The choice of a parallel manipulator for a particular application is a challenging task due to the existence of numerous designs that generate the same motion type. Hence, this paper attempts to compare a class of 3 degree-of-freedom parallel manipulators belonging to the 3-[PP]S family, for which each leg is confined to move in a plane with the first two joints generating a motion equivalent to two coplanar translations. Specifically, the 3-[PP]S parallel manipulators in which the planes are arranged in a Y-pattern are considered and are compared with respect to their singularity-free orientation workspace, parasitic motions within that workspace and complexity indices. Moreover, the comparison is done between the 3-[PP]S-Y family of manipulators for their different operation modes, with different actuation schemes and for different sets of design parameters. Two indices are used to quantify the singularity-free workspace and the parasitic motions of the manipulators at hand. Additionally, the non-dominated manipulator architectures with regard to their singularity-free regions and parasitic motions are highlighted thanks to a Pareto front. Finally, the Pareto-optimal solutions are ranked in ascending order of complexity.

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.

Enlarging operational workspaces in parallel manipulators by connecting working modes. Application to the 3RSS robot

SUMMARYThe aim of this paper is to describe a general methodology for enlarging the workspace within which a parallel manipulator can move in a controllable way. The basis for obtaining this consists in superimposing all the singularity-free regions associated with the various different robot working modes. These can be connected because such transitions do not imply a loss of control of the manipulator. This enlarged operational workspace is associated with a certain assembly mode. In addition, the strategy to be used for path planning in this kind of workspace is presented.

Forward displacement analysis of a quadratic 4-DOF 3T1R parallel manipulator

A quadratic parallel manipulator refers to a parallel manipulator with a quadratic characteristic polynomial. This paper revisits the forward displacement analysis (FDA) and the Type II singularity analysis of a quadratic 4-DOF 3T1R (SCARA) parallel manipulator: the Quadrupteron. It will be proved that there exists a one-to-one correspondence between the two formulas, each producing one solution to the FDA, and the two singularity-free regions. Therefore, a unique solution to the FDA can be obtained in a straightforward way for such a parallel manipulator if the singularity-free region in which it works is specified. The Type II singularity analysis in the joint space will also be investigated in order to identify the conditions on the inputs to keep the Quadrupteron working in the same singularity-free region in its Cartesian workspace.

Forward Displacement Analysis of a Quadratic Planar Parallel Manipulator: 3-RP̱R Parallel Manipulator With Similar Triangular Platforms 1

A quadratic parallel manipulator refers to a parallel manipulator with a quadratic characteristic polynomial. This paper revisits the forward displacement analysis (FDA) of a quadratic parallel manipulator: 3-RP̱R planar parallel manipulator with similar triangular platforms. Although it has been revealed numerically elsewhere that for this parallel manipulator, the four solutions to the FDA fall, respectively, into its four singularity-free regions (in its workspace), it is unclear if there exists a one-to-one correspondence between the four formulas, each producing one solution to the FDA, and the four singularity-free regions. Using an algebraic approach, this paper will prove that such a one-to-one correspondence exists. Therefore, a unique solution to the FDA can be obtained in a straightforward way for such a parallel manipulator if the singularity-free region in which it works is specified.

Traversing the singularity hypersurface by applying the input disturbances to 6-SPS parallel manipulator

The singular points of a 6-SPS Stewart platform are distributed on the multi-dimensional singularity hypersurface in the task-space, which divides the workspace of the manipulator into several singularity-free regions. Because of the motion uncertainty at singular points, while the manipulator traverses this kind of hypersurface from one singularity-free region to another, its motion cannot be predetermined. In this paper, a detailed approach for the manipulator to traverse the singularity hypersurface with its non-persistent configuration is presented. First, the singular point transfer disturbance and the pose disturbance, which make the perturbed singular point transfer horizontally and vertically, respectively, are constructed. Through applying these disturbances into the input parameters within the maximum loss control domain, the perturbed persistent configuration is transformed into its corresponding non-persistent one. Under the action of the disturbances, the manipulator can traverse the singularity hypersurface from one singularity-free region to another with a desired configuration.

Uniqueness Domains and Regions of Feasible Paths for Cuspidal Manipulators

This paper addresses the kinematics of cuspidal manipulators, i.e., nonredundant manipulators which can change posture without meeting a singularity. It focuses on the uniqueness domains and on the regions of feasible paths in the workspace. For cuspidal manipulators, the uniqueness domains are not the singularity-free regions of the joint space. It is shown that additional surfaces, called characteristic surfaces, separate the inverse kinematic solutions in the joint space. The uniqueness domains define the regions of feasible paths in the workspace. Joint limits are taken into account in this paper. This paper should help the reader better understand the kinematics of cuspidal manipulators.

Singularity-Free Regions for Solutions of Systems of Nonlinear Complex Differential Equations

Singularity-Free Regions for Solutions of Systems of Nonlinear Complex Differential Equations

Singularity-free regions for solutions of systems of nonlinear complex differential equations

where for each k, k = 1, * n fk is regular in z for | zj <R and entire in yl, * , Yn. A region S is said to be singularity-free for a solution (yi, , yn) if each of yi, * * *, yn is regular-analytic in S. In this paper conditions are obtained under which the system (1.1) has solutions satisfying given initial conditions which are regular in a region which contains a set which is star-like with respect to the origin. The principal result, Theorem 2.1, compares the growth of the norms of such solutions along each ray emanating from the origin with the growth of solutions of associated real differential equations. Analogous results are obtained for the scalar equation

Asymptotic almost periodicity of solutions of a system of differential equations

1. K. M. Das, Singularity-free regions for solutions of 2nd order non-linear dzfferential equations, J. Math. Mech. 13 (1964), 73-84. 2. P. Hartman, Ordinary differential equations, Wiley, New York, 1964. 3. E. L. Ince, Ordtnary differential equations, Dover, New York, 1926. 4. V. Lakshmikantham, Upper and lower bounds of the norm of solutions of differ ential equations, Proc. Amer. Math. Soc. 13 (1962), 615-616.