Solving the Forward Kinematics of a Gough-Type Parallel Manipulator with Interval Analysis

Abstract

We consider in this paper a Gough-type parallel robot and we present an efficient algorithm based on interval analysis that allows us to solve the forward kinematics, i.e., to determine all the possible poses of the platform for given joint coordinates. This algorithm is numerically robust as numerical round-off errors are taken into account; the provided solutions are either exact in the sense that it will be possible to refine them up to an arbitrary accuracy or they are flagged only as a "possible" solution as either the numerical accuracy of the computation does not allow us to guarantee them or the robot is in a singular configuration. It allows us to take into account physical and technological constraints on the robot (for example, limited motion of the passive joints). Another advantage is that, assuming realistic constraints on the velocity of the robot, it is competitive in term of computation time with a real-time algorithm such as the Newton scheme, while being safer.

Injectivity analysis using interval analysis: Application to structural identifiability

“…To perform this verification, we propose to generalize the interval Newton method by defining an extension of the unicity operators [9]. This new result will be exploited in the next section in order to show that a function f :…”

Singularity-Invariant Leg Rearrangements in Stewart–Gough Platforms

“…It started to receive a lot of attention twenty years ago with the advent of molecular magnetic resonance experiments to obtain distances between atoms in rigid molecules [24], and more recently in Robotics to obtain intrinsic formulations of different kinematics problems, thus avoiding the introduction of arbitrary reference frames [77,98], or the problems derived from the tangent half-angle substitution [59,85]. Interval analysis and linear relaxation techniques have also been used for general position analysis solvers, giving box approximation solutions for general serial and parallel manipulators [69,71,80,97].…”

Exact interval propagation for the efficient solution of position analysis problems on planar linkages

“…In recent years there has been a growing interest in the use of interval methods in Kinematics, specially for the task of finding the feasible configurations of complex linkages-the so-called position analysis problem-which, when performed analytically, may involve a large number of equations or high-degree polynomials [1][2][3]. When the solution space is continuous, interval methods can provide a discrete approximation of such space as a set of enclosing boxes, which can, at least in principle, be refined as much as desired [4].…”

“…When the box cannot be further reduced, it is split, usually into two halves (or branches), each of which is treated recursively by the same process. The pruning method may be a general interval propagation algorithm applied to the kinematic equations of the linkage [1,[4][5][6][7], or a more specific method suited to the problem [2,3,8].…”