Interval Analysis for Certified Numerical Solution of Problems in Robotics

Abstract

Interval analysis is a relatively new mathematical tool that allows one to deal with problems that may have to be solved numerically with a computer. Examples of such problems are system solving and global optimization, but numerous other problems may be addressed as well. This approach has the following general advantages: (a) it allows to find solutions of a problem only within some finite domain which make sense as soon as the unknowns in the problem are physical parameters; (b) numerical computer round-off errors are taken into account so that the solutions are guaranteed; (c) it allows one to take into account the uncertainties that are inherent to a physical system. Properties (a) and (c) are of special interest in robotics problems, in which many of the variables are parameters that are measured (i.e., known only up to some bounded errors) while the modeling of the robot is based on parameters that are submitted to uncertainties (e.g., because of manufacturing tolerances). Taking into account these uncertainties is essential for many robotics applications such as medical or space robotics for which safety is a crucial issue. A further inherent property of interval analysis that is of interest for robotics problems is that this approach allows one to deal with the uncertainties that are unavoidable in robotics. Although the basic principles of interval analysis are easy to understand and to implement, this approach will be efficient only if the right heuristics are used and if the problem at hand is formulated appropriately. In this paper we will emphasize various robotics problems that have been solved with interval analysis, many of which are currently beyond the reach of other mathematical approaches.

Variable symmetry breaking in numerical constraint problems

“…The 3-cyclic roots problem consists in finding all (x 1 , x 2 , x 3 ) ∈ R 3 satisfying (x 1 + x 2 + x 3 = 0) ∧ (x 1 x 2 + x 2 x 3 + x 3 x 1 = 0) ∧ (x 1 x 2 x 3 = 1). This problem has six variable symmetries (including identity): , 3], [1,3,2], [2,1,3], [2,3,1], [3,1,2], [3, 2, 1]},…”

“…As a consequence, their application has raised interest in fields as diverse as neurophysiology and economics [2], biochemistry, crystallography, robotics [3] and, more generally, in those related to global optimization [4]. Symmetries naturally occur in many of these applications, and it is advisable to exploit them in order to reduce the search space and, thus, to increase the efficiency of the solvers.…”

Uncertainties in structural dynamics: overview and comparative analysis of methods

“…Nevertheless, uncertainties are sometimes epistemic due to imprecise or incomplete information, and that is why non-probabilistic approaches were developed [9]. Among the first developments was the interval theory, usually applied when the focus lies in identifying the bounds of the interval in which the output of interest varies, such as to problems in mechanics [10,11] and robotics [12]. The fuzzy set theory is another nonprobabilistic approach that uses the fuzzy logic which is based on human reasoning rather than rigid calculations [13,14].…”

Sensitivity analysis of parallel manipulators using an interval linearization method

“…This information is useful for the evaluation of the pose error of the end-effector and for the tolerance synthesis. The three main methods used for the sensitivity analysis to variations in geometric parameters and joint clearances of serial and parallel robots are: (i) the probabilistic methods [1]; (ii) the linearization methods [2] and (iii) the interval analysis methods [3].…”