This book considers the localization of autonomous robots evolving in unstructured underwater environments. The common thread of this work is the use of temporal constraints and inter-temporal measurements that open new opportunities of localization, which are underexploited so far. In addition, these robotic problems have motivated the study of academic research topics in the fields of interval analysis and constraint programming. Our goal was to develop new reliable tools to fit our requirements. While their application to underwater experiments appears to be efficient, their use in other fields, such as automatic and control, collision avoidance, path planning, terrestrial localization or spatial trajectory evaluations, could also be considered.

Our primary motivation is the localization of mobile robots in environments characterized by the paucity of relevant information. A SLAM approach would allow a concurrent localization of the vehicle and mapping of the area, without prior knowledge on the environment or the use of positioning systems. However, the previously existing approaches do not provide a comfortable fit: with poor measurements, unknown observation functions or strong positioning uncertainties. In addition, they do not provide guaranteed results that may be expected for safety reasons.

Our strategy consisted of taking a temporal approach to solving a spatial problem. Indeed, time references tie together state estimations and environment observations that are uncertain spatial values due to errors arising from the sensors. The matching of these values implies further uncertainties, and we proposed handling them in a bi-temporal space representing inter-temporal configurations. This view differs from usual methods that only consider uncertainties in the spatial space without performing estimations of temporal references.

Chapter 6 detailed this temporal SLAM and provided illustrations on actual underwater experiments. Our aim was to address this problem with a constraint programming approach coupled with interval analysis. This decision was motivated by the simplicity and genericity allowed by this paradigm. By representing a problem with constraints and sets of feasible solutions, we can perfectly deal with complicated situations or poor datasets. Furthermore, the developed algorithms do not require sophisticated settings; in this book, only two parameters have been introduced: the time discretization of a tube *δ* and the precision *ε* of set-inversion algorithms (SIVIA).

To achieve our goal, though, three primary constraints were studied. All of them apply to trajectories by means of new operators designed to reduce the domains of feasible dynamical solutions: so-called *tubes*. The first study of was the object of earlier work, but a reliable *contractor* was still missing. Definitions and proofs of the new contractor were provided in Chapter 3, together with robotic applications. We also discussed the limits of the approach when it comes to overcoming unwanted pessimism on the trajectory evaluations.

The second constraint of interest, was fully investigated for the first time in this work. It allows the consideration of strong time uncertainties: a topic that has been poorly studied with set-membership methods and in the state estimation community. Chapter 4 presented 𝒞_{eval}, which aims to consider any uncertainty about the evaluation of a trajectory at a given time. Its application to robotic problems, such as the correction of a drifting clock, is the first step towards new problem-solving techniques, allowing resolutions from a temporal aspect. Chapter 6 typically illustrated an original application that was made possible by this contractor.

Our SLAM problem has also triggered the development of a so-called inter-temporal implication constraint denoted by **z**(*t*_{1}) = **z**(*t*_{2}). Its implementation required the study of a new zero verification test that has been successfully applied in robotics to prove the existence of loops along uncertain robot trajectories. Indeed, Chapter 5 demonstrated the efficiency of the topological degree theory when coupled with function evaluations in a bounded-error context. In our robotic applications, we discussed the optimality of the approach. These results have a strong impact on the effectiveness of the contractor, which was detailed in Chapter 6 and illustrated in the context of a new bathymetric SLAM approach.

In a nutshell, this work is rooted in robotic problems, which gradually leads to the development of new academic tools. It brought new advances in the field of constraint programming by proposing a declarative way to deal with dynamical systems. A reliable contractor programming framework is now at hand, allowing us to build solvers for dynamical systems. This set of tools has been illustrated in this book with realistic robotic applications.

Chapters 3, 4, 5 and 6 are subject to publications in robotic journals. The last one is still a work in progress. We plan new experiments with accurate datasets in order to clearly assess the relevancy of our approach for underwater navigation.

The list of published papers related to this book is summarized as follows:

- – Rohou, S., Jaulin, L., Mihaylova, L., Le Bars, F., and Veres, S. M. (2017). Guaranteed computation of robot trajectories.
*Robotics and Autonomous Systems*, 93:76–84. - – Rohou, S., Franek, P., Aubry, C., and Jaulin, L. (2018a). Proving the existence of loops in robot trajectories.
*The International Journal of Robotics Research*, 37(12):1500–1516. - – Rohou, S., Jaulin, L., Mihaylova, L., Le Bars, F., and Veres, S. M. (2018b). Reliable non-linear state estimation involving time uncertainties.
*Automatica*, 93:379–388.

A significant contribution of this work is the development, by the first author, of the new open source library *Tubex*, which is freely available at:

- – http://www.simon-rohou.fr/research/tubex-lib.

This project collects all the elementary tools presented in this book. In turn, the reader will be able to process the simulated examples and build its own solvers for the resolution of more dedicated dynamical problems.

By now the reader should be able to address problems related to dynamical systems by breaking them down into a set of elementary constraints involving vectors, trajectories or sets. Then, he or she could appropriately define initial domains of variables and use the presented contractors to approximate solution sets. However, building such solver requires technical skills to efficiently handle intervals, tubes, contractors and related settings. This knowledge is not at hand for many users who could benefit from this constraint programming approach. There is then a need to provide an extra abstraction level in the design of solvers.

The continuation of this work will be to design a dedicated language to define problems. This would involve syntax definitions and an appropriate semantic to maintain the ability to deal with a wide range of problems. A close link must be investigated between this language and its automatic implementation under the form of contractors. For example, tubes should be automatically instantiated with an appropriate time discretization *δ*. Similarly, while we explained that the order of contractor calls has no impact on the final result, a smart scheduling could be processed to speed up the computations.

In addition, this would also be a good opportunity to push the limits of our approach by coupling it with further tools. We could merge several methods of guaranteed integration, such as those presented in the introduction of Chapter 3, which would be helpful in avoiding some overestimation of trajectory sets. Furthermore, the complicated task of managing hybrid constraints with our approach could be dealt by merging it with some *Eulerian* approaches such as (Le Mézo *et al*. 2018).

All of these optimizations can be hidden from the user who would only focus on its problem definition. This opening is the subject of the new *Contredo* consortium^{1} that will gather several academic and industrial partners focusing on this topic over the next three years.

- 1 From the French National Research Agency (ANR). ANR Programme: (DS0702) 2016. Project ID: ANR-16-CE33-0024. Project coordinator: Pr. Gilles Trombettoni.