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

Abstract

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.

Review article: Performance measures of parallel kinematics manipulators

“…-Interval analysis method (Kaloorazi et al, 2014) which can handle almost any constraints and any number of DOF.…”

“…This type of singularity leads to so-called self-motion or sometimes called the Borel-Bricard motion. Karger and Husty (1998) and Karger (2008) studied all self-motions in a Stewart-Gough PKM. Wohlhart (2003Wohlhart ( , 2010 termed this type of singularity as architectural shakiness and pointed out that it means more than mere singularity; it means "architectural mobility".…”

Collision-free workspace of parallel mechanisms based on an interval analysis approach

“…The mathematical framework used in this paper is based on interval analysis. [14][15][16] Interval analysis is a reliable method to evaluate functions and is used frequently in the field of robotics. 17 An interval variable of [x] = [x, x] is a set of all real numbers from the lower bound, x, to the upper bound, x.…”

“…Most of them dealt with solving the Forward Kinematic Problem (FKP) 18 and obtaining the workspace. 16,19,20 Merlet, in ref. [21], investigated the workspace of PMs.…”

“…[21], investigated the workspace of PMs. Several papers have been published in the context of obtaining the singularity-free workspace using geometrical approach, 22 and using interval analysis 16 in which the novelty is about demonstrating the state of the art of applying interval analysis in solving the foregoing problems. In ref.…”