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This paper presents a new approach to bounded-error state estimation involving time uncertainties. For a given bounded observation of a continuous-time non-linear system, it is assumed that neither the values of the observed data nor their acquisition instants are known exactly. For systems described by state-space equations, we prove theoretically and demonstrate by simulations that the proposed constraint propagation approach enables the computation of bounding sets for the systems’ state vectors that are consistent with the uncertain measurements. The bounding property of the method is guaranteed even if the system is strongly non-linear. Compared with other existing constraint propagation approaches, the originality of the method stems from our definition and use of bounding tubes which enable to enclose the set of all feasible trajectories inside sets. This method makes it possible to build specific operators for the propagation of time uncertainties through the whole trajectory. The efficiency of the approach is illustrated on two examples: the dynamic localization of a mobile robot and the correction of a drifting clock. [ABSTRACT FROM AUTHOR]

In this paper, we present a new method for bracketing ( i.e. , characterizing from inside and from outside) all solutions of an ordinary differential equation in the case where the initial time is inside an interval and the initial state is inside a box. The principle of the approach is to cast the problem into bracketing the largest positive invariant set which is included inside a given set X . Although there exists an efficient algorithm to solve this problem when X is bounded, we need to adapt it to deal with cases where X is unbounded. [ABSTRACT FROM AUTHOR]

The stability analysis of nonlinear continuous systems often requires manual calculation, which can become time-consuming when dealing with complex systems. Some works use positive invariant sets to discuss stability. These sets can be numerically approximated using Interval analysis but the computational complexity is exponential. In this paper, we propose a computational tractable numerical but guaranteed method based on Interval analysis to verify the robust positive invariance of ellipsoids to automatize the study of n-dimensional nonlinear systems' stability. This method relies on a fast enclosure of a state integration by an Euler method. Interval analysis guarantees the results of the developed algorithms. Several examples show the effectiveness of the proposed approaches on n -dimensional non-asymptotical continuous systems subject to bounded perturbation. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new approach for improving significantly existing guaranteed integration methods for state equations with uncertain initial conditions. We first find a tube that encloses the solution of the differential equation assuming that the initial state is known. Then, using Lie symmetries, we inflate the tube in order to contain the uncertainty associated with the initial state. The method is shown to be efficient on examples coming from reachability analysis and robotics. [ABSTRACT FROM AUTHOR]

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 a new notion of separators which is a pair of complementary contractors and presents the corresponding algebra. Using 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. [Copyright &y& Elsevier]

Abstract: An interval problem is a problem where the unknown variables take interval values. Such a problem can be defined by interval constraints, such as “the interval ”. Interval problems often appear when we want to analyze the behavior of an interval solver. To solve interval problems, we propose to transform the constraints on intervals into constraints on their bounds. For instance, the previous interval constraint can be transformed into the following bound constraints “ and ”. Classical interval solvers can then be used to solve the resulting bound constraints. The procedure which transforms interval constraints into equivalent bound constraints can be facilitated by using symbolic interval arithmetic. While classical intervals can be defined as a pair of two real numbers, symbolic intervals can be defined as a pair of two symbolic expressions. An arithmetic similar to classical interval arithmetic can be defined for symbolic intervals. The approach will be illustrated on several applications. [Copyright &y& Elsevier]

Abstract: This paper proposes a new observer for estimating the state vector of a nonlinear system. This observer, which is robust with respect to outliers, assumes that the measurement errors, as well as the number of outliers that could occur within a given time window, are bounded. The principle of the approach is to use interval analysis to deal properly with the nonlinearities involved in the system (without any linearization or approximation) and to propagate through the time, in a forward and backward manner, the assumptions made about outliers. A test case related to the localization and control of an underwater robot is also proposed to illustrate the efficiency of the approach. [Copyright &y& Elsevier]

This paper presents a first study on the application of interval analysis and consistency techniques to state estimation of continuous-time systems described by nonlinear ordinary differential equations. The approach is presented in a bounded-error context and the resulting methodology is illustrated by an example. [Copyright &y& Elsevier]

This paper proposes a new concept of thick gradual sets (TGSs), which is based on the notions of thick sets (TSs) and gradual sets (GSs). A TS is an uncertain set, which is represented by a pair of crisp sets (CSs). These CSs represent the upper and lower bounds of the TS. Therefore, a TS can be considered as an interval of CSs. A GS is a CS, which is parameterized by a degree of pertinence and aims to increase the specificity of CSs. Furthermore, a TGS is an interval of GSs, i.e., a pair of lower and upper GSs. In situations when the constraint of monotonicity (consistency) is guaranteed, a GS becomes a type-1 fuzzy set (T1FS) and a TGS can be regarded as a thick fuzzy set (TFS). Moreover, a TFS, which is composed of lower and upper T1FS bounds, can be interpreted as a type-2 fuzzy set (T2FS). According to the TGS representation, this new approach offers an original concept for interpreting, manipulating, and computing some uncertain quantities that cannot be represented by GSs, T1FSs, and/or T2FSs. The potential applications of the TGS concept has been validated using application examples in the frameworks of solving fuzzy systems of equations and uncertain fuzzy regression and through a real-world application where the trajectory of an underwater robot is uncertain and cannot be precisely known because of disturbances induced by the environment. The proposed approach makes it possible to compute the uncertain zone explored by the underwater robot. [ABSTRACT FROM AUTHOR]

In the paper, we present a new interval-based set inversion algorithm which takes into account the continuity of the problem. In the case where the set Y to be inverted has some volume, we show that inverting the boundary ∂ Y of Y is sufficient to reconstruct the preimage X = f − 1 (Y). The inversion of ∂ Y separates the domain of f into two regions : one inside X and one outside. To detect which part is inside or outside, we show that we can retro-propagate the information coming from Y at negligible cost. The efficiency of the approach is illustrated on a localization problem. [ABSTRACT FROM AUTHOR]

Obstacle avoidance is a major challenge for underwater robot navigation, especially in cluttered environments where sonars are subject to reverberation and noise. Optic flow cues are widely used for aerial drone navigation, as they can be measured with broadly available sensors such as monocular cameras. Few studies have explored the possibility of using optic flow cues for underwater robot navigation, although this is more difficult due to imaging conditions and natural scene characteristics. In this study, we investigate the use of optic flow divergence for real-time collision avoidance onboard a Remotely Operated underwater Vehicle (ROV) equipped with monocular cameras. The measured optic flow divergence was first used to trigger an autonomous emergency braking response, then to estimate the relative distance of the underwater vehicle from the detected obstacle using an Extended Kalman Filter. This relative distance was used to maintain a predefined safe distance from the obstacle. Tests were first carried out in a pool, then in a harbour, in a lake and in the sea. Our findings show that this minimalistic method is effective under a wide range of visibility and turbidity conditions. • Measure of optic flow divergence underwater in real-time on a Remotely Operated underwater Vehicle (ROV), performed in a pool, a lake, a harbor, and in the sea. • The use of optic flow divergence to trigger an autonomous emergency braking response. • The use of optic flow divergence to estimate the relative distance of the underwater vehicle from the detected obstacle using an Extended Kalman Filter. [ABSTRACT FROM AUTHOR]

Abstract: This paper proposes an original set-membership approach for loop detection of mobile robots in the situation where proprioceptive sensors only are available. To detect loops, the new concepts of the-plane (which is a two dimensional space with time coordinates) are introduced. Intervals of functions (or tubes) are then used to represent uncertain trajectories and tests are provided in order to eliminate parts of the -plane that do not correspond to any loop. An experiment with an actual underwater robot is proposed in order to illustrate the principle and the efficiency of the approach. [Copyright &y& Elsevier]

Abstract: This paper deals with offline nonlinear state estimation where measurements are available only when some given equality conditions are satisfied. For this type of problems, which are often met in robot localization when sonar or radar are involved, the data are qualified as fleeting because the measurements are available only at some given unknown dates. In this paper, the first approach able to deal with nonlinear estimation with fleeting data is presented. An illustration related to offline robot localization with a laser rangefinder will be given. [Copyright &y& Elsevier]

Abstract: This paper presents a new numerical algorithm based on interval analysis able to prove that a differentiable function is injective. This algorithm also performs a partition of the domain in subsets where, for all , the cardinality of is constant. In the context of parameter estimation, we show how this algorithm provides an efficient and numerical method to study the structural identifiability of parametric models. [Copyright &y& Elsevier]

This paper deals with feedback controller synthesis for timed event graphs, where the number of initial tokens and time delays are only known to belong to intervals. We discuss here the existence and the computation of a robust controller set for uncertain systems that can be described by parametric models, the unknown parameters of which are assumed to vary between known bounds. Each controller is computed in order to guarantee that the closed-loop system behavior is greater than the lower bound of a reference model set and is lower than the upper bound of this set. The synthesis presented here is mainly based on dioid, interval analysis and residuation theory. [Copyright &y& Elsevier]