5Looped Trajectories: From Detections to Proofs

5.1. Introduction

5.1.1. The difference between detection and verification

In this chapter, we present a reliable method to detect and verify the existence of loops along the uncertain trajectory of a robot, based on proprioceptive measurements only, using a bounded-error approach. In a reliable context, a distinction must be made between the detection and the verification of a loop. Considering a set of feasible trajectories, some of them may cross themselves at some point; this will lead to a detection. In addition, when we verify that all the feasible trajectories are looped, we can speak about a loop proof since a loop occurs regardless of the considered uncertainties. Figure 5.1 provides an illustration of this distinction.

Section 5.2 focuses on loop detections, presenting the concepts introduced in Aubry et al. (2013). Our contribution is the object of section 5.3, in which we propose a complementary tool for verification purposes. Such a problem is not trivial, even in two-dimensional contexts. Our approach is to rely on the topological degree theory (Fonseca and Gangbo 1995) to verify zeros in uncertain contexts. This study was the object of Rohou, Franek, Aubry and Jaulin (2018).


Figure 5.1. Only one loop can be verified in this set of trajectories, while at least two feasible loops are detected. Indeed, there exist ...

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