Reliable Robot Localization

3Trajectories under Differential Constraints

3.1. Introduction

As mentioned in Chapter 1, interval-based methods effortlessly handle nonlinear situations while ensuring guaranteed results. We have seen that we can also deal with trajectories by means of tubes presented in Chapter 2. However, the differential problem remains to be addressed.

3.1.1. The differential problem

In fact, the robotic example provided in section 2.4 does not represent the range of state estimation problems we usually encounter, especially in mobile robotics. Indeed, a convincing approach must be able to assess observations through time while ensuring the evolution of the state. And sometimes, as for the kidnapped robot problem, the situation is even more complicated as initial conditions are not known. Hence, there is a need to fully deal with ordinary differential equations (ODEs) in the most generic way.

Formally, we consider the problem of guaranteed integration of dynamical systems (see, for example, (Konečný et al. 2016)) of the form

[3.1]

One of the main motivations of guaranteed integration methods is to develop reliable cyber-physical systems such as Acumen¹ (Taha et al. 2015) for dynamical systems simulation and verification.

3.1.2. Attempts with set-membership methods

Our problem corresponds to the area of interval integration (Moore 1979, Berz 1996). Set-membership methods allow the computation ...