Introduction to the algebra of separators with application to path planning

Abstract

Contractor algebra is a numerical tool based on interval analysis which makes it possible to solve many nonlinear problems arising in robotics, such as identification, path planning or robust control. This paper presents the new notion of separators which is a pair of complementary contractors and presents the corresponding algebra. Using the separator algebra inside a paver will allow us to get an inner and an outer approximation of the solution set in a much simpler way than using any other interval approach. A path planning problem will then be considered in order to illustrate the principle of the approach.

A boundary approach for set inversion

“…At Steps 2 and 5, the contractions are minimal since the functions f k are contractible. When we want to find an inner and an outer approximations, we have to run the contractor given by Algorithm 2 twice: once for both Y and once for its complementary Y [16]. This is clearly not optimal and it is what we want to avoid in this paper.…”

“…Section VI concludes the paper. Figure 1 illustrates the behavior of a typical contractorbased algorithm [16] to characterize a set X (magenta). The algorithm uses two types of contractors: a contractor C X for X and a contractor C X for its complementary X.…”

Characterizing Sliding Surfaces of Cyber-Physical Systems

“…Since we need to compute with sets defined by non-linear inequalities that are connected with intersection, union, complementary operators, we decided to use separators [15] instead of contractors [5] (which do not allow the use of complementary operators).…”

Minkowski Operations of Sets with Application to Robot Localization

“…Other operations on separators such as the complement or the projection can also be considered [6]. Example 2.…”

“…Interval methods are generally used to solve equations or optimization problems [5] but can also been used to solve set-membership problems where the sets are represented by subpavings [7]. The efficiency of interval algorithms can be improved by the use of contractors [2] or (separators [6] which correspond to pairs of contractors). This paper deals with localization of a robot with sonar rangefinders in a unstructured environment.…”