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Set computation methods have been widely used to compute reachable sets, design invariant sets and estimate system state for dynamic systems. The wrapping effect of such set computation methods plays an essential role in the accuracy of their solutions. This paper studies the wrapping effect of existing interval, zonotopic and polytopic set computation methods and proposes novel approaches to reduce the wrapping effect for these set computation methods based on the task of computing the dynamic evolution of a nonlinear uncertain discrete-time system with a set as the initial state. The proposed novel approaches include the partition of a polytopic set via Delaunay triangulation and also the representation of a polytopic set by the union of small zonotopes for the following set propagation. The proposed novel approaches with the reduced wrapping effect has been further applied to state estimation of a nonlinear uncertain discrete-time system with improved accuracy. Similar to bisection for interval and zonotopic sets, Delaunay triangulation has been introduced as a set partition tool for polytopic sets, which has opened new research directions in terms of novel set partition, set representation and set propagation for reducing the wrapping effect of set computation. [ABSTRACT FROM AUTHOR]

Stability contractors, based on interval analysis, were introduced in recent work as a tool to verify stability domains for nonlinear dynamic systems. These contractors rely on the property that -- in case of provable asymptotic stability -- a finitely large domain in a multi-dimensional state space is mapped into its interior after a certain integration time for continuous-time processes or after a certain number of discretization steps in a discrete-time setting. However, a disadvantage of the use of axis-aligned interval boxes in such computations is the omnipresent wrapping effect. As shown in this contribution, the replacement of classical interval representations by ellipsoidal domain enclosures reduces this undesirable effect. It also helps to find suitable ratios for the edge lengths if interval-based domain representations are investigated. Moreover, ellipsoidal domains naturally represent the possible regions of attraction of asymptotically stable equilibrium points that can be analyzed with the help of quadratic Lyapunov functions, for which stability criteria can be cast into linear matrix inequality (LMI) constraints. For that reason, this paper further presents possible interfaces of ellipsoidal enclosure techniques with LMI approaches. This combination aims at the maximization of those domains that can be proven to be stable for a discrete-time range-only localization algorithm in robotics. There, an Extended Kalman Filter (EKF) is applied to a system for which the dynamics are characterized by a discrete-time integrator disturbance model with additive Gaussian noise. In this scenario, the measurement equations correspond to the distances between the object to be localized and beacons with known positions. [ABSTRACT FROM AUTHOR]

In this paper, we propose to study the stability of a navigation method that allows a robot to move in an unstructured environment without compass by measuring a scalar function' which only depends on the position. The principle is to ask the robot to roll along an isovalue of '. Using an interval method, we prove the stability of our closed loop system in the special case where' is linear. [ABSTRACT FROM AUTHOR]

The Unscented Kalman Filter (UKF) is widely used for the state, disturbance, and parameter estimation of nonlinear dynamic systems, for which both process and measurement uncertainties are represented in a probabilistic form. Although the UKF can often be shown to be more reliable for nonlinear processes than the linearization-based Extended Kalman Filter (EKF) due to the enhanced approximation capabilities of its underlying probability distribution, it is not a priori obvious whether its strategy for selecting sigma points is sufficiently accurate to handle nonlinearities in the system dynamics and output equations. Such inaccuracies may arise for sufficiently strong nonlinearities in combination with large state, disturbance, and parameter covariances. Then, computationally more demanding approaches such as particle filters or the representation of (multi-modal) probability densities with the help of (Gaussian) mixture representations are possible ways to resolve this issue. To detect cases in a systematic manner that are not reliably handled by a standard EKF or UKF, this paper proposes the computation of outer bounds for state domains that are compatible with a certain percentage of confidence under the assumption of normally distributed states with the help of a set-based ellipsoidal calculus. The practical applicability of this approach is demonstrated for the estimation of state variables and parameters for the nonlinear dynamics of an unmanned surface vessel (USV). [ABSTRACT FROM AUTHOR]

The two-part Special Issue "Algorithms for Reliable Estimation, Identification and Control" deals with the optimization of feedforward and feedback controllers with respect to predefined performance criteria as well as the state and parameter estimation for systems with uncertainty. Finally, bounds on the system parameters to be estimated are determined with an algorithm for set inversion via interval analysis. Possible applications of related optimization algorithms can be found not only in the frame of a control and estimator synthesis, but also in the field of reliable modeling and model-based analysis of measured data. The associated stability proof is carried out through control Lyapunov functions, where the enclosure approach for invariant sets exploits an algorithm for a set-based inversion of nonlinear mappings. [Extracted from the article]

A reliable quantification of the worst-case influence of model uncertainty and external disturbances is crucial for the localization of vessels in marine applications. This is especially true if uncertain GPS-based position measurements are used to update predicted vessel locations that are obtained from the evaluation of a ship's state equation. To reflect real-life working conditions, these state equations need to account for uncertainty in the system model, such as imperfect actuation and external disturbances due to effects such as wind and currents. As an application scenario, the GPS-based localization of autonomous DDboat robots is considered in this paper. Using experimental data, the efficiency of an ellipsoidal approach, which exploits a bounded-error representation of disturbances and uncertainties, is demonstrated. [ABSTRACT FROM AUTHOR]

Recent advances in computational power, algorithms, and sensors allow robots to perform complex and dangerous tasks, such as autonomous missions in space or underwater. Given the high operational costs, simulations are run beforehand to predict the possible outcomes of a mission. However, this approach is limited as it is based on parameter space discretization and therefore cannot be considered a proof of feasibility. To overcome this limitation, set-membership methods based on interval analysis, guaranteed integration, and contractor programming have proven their efficiency. Guaranteed integration algorithms can predict the possible trajectories of a system initialized in a given set in the form of tubes of trajectories. The contractor programming consists in removing the trajectories violating predefined constraints from a system's tube of possible trajectories. Our contribution consists in merging both approaches to allow for the usage of differential constraints in a contractor programming framework. We illustrate our method through examples related to robotics. We also released an open-source implementation of our algorithm in a unified library for tubes, allowing one to combine it with other constraints and increase the number of possible applications. [ABSTRACT FROM AUTHOR]

For linear time-invariant dynamic systems with exactly known coefficients of their system matrices for which measurements with bounded errors are available at discrete time instants, an optimal polygonal state estimation scheme was recently published. This scheme allows for tightly enclosing all possible state trajectories in presence of uncertain, but bounded, system inputs which may be varying arbitrarily within in their bounds. Moreover, this approach is also capable of accounting for uncertainty related to the measurement time instants. However, the drawback of this polygonal technique is its rapidly increasing complexity for larger system dimensions. For that reason, the polygonal state enclosures are replaced by a computationally less expensive, but nearly optimal, ellipsoidal enclosure technique in this paper. Numerical simulations for representative benchmark examples focusing both on applications with precisely known and uncertain parameters conclude this contribution. [ABSTRACT FROM AUTHOR]

This article aims at proposing the concept of thick fuzzy sets (TFSs). A TFS is based on the joint use of thick sets (TSs) and α-cuts concepts. A TFS is represented by a family of nested TSs. A TS is an uncertain set, which is represented by a pair of crisp sets (CSs). These CSs characterize the upper and lower bounds of the TS. Therefore, a TS can be regarded as an interval of CSs. In this framework, as a type-1 fuzzy set (T1FS) is regarded as a family of nested CSs, a TFS can be represented by a family of nested TSs. Furthermore, according to the vertical dimension α, a TFS can be regarded as an interval with T1FS bounds. The potentialities of the TFS concept have been validated using application examples where a real-world application for modeling the zone explored by an underwater robot is given. [ABSTRACT FROM AUTHOR]

A wide variety of approaches for set-valued simulation, parameter identification, state estimation as well as reachability, observability and stability analysis for nonlinear discrete-time systems involve the propagation of ellipsoids via nonlinear functions. It is well known that the corresponding image sets usually possess a complex shape and may even be nonconvex despite the convexity of the input data. For that reason, domain splitting procedures are often employed which help to reduce the phenomenon of overestimation that can be traced back to the well-known dependency and wrapping effects of interval analysis. In this paper, we propose a simple, yet efficient scheme for simultaneously determining outer and inner ellipsoidal range enclosures of the solution for the evaluation of multi-dimensional functions if the input domains are themselves described by ellipsoids. The Hausdorff distance between the computed enclosure and the exact solution set reduces at least linearly when decreasing the size of the input domains. In addition to algebraic function evaluations, the proposed technique is--for the first time, to our knowledge--employed for quantifying worst-case errors when extended Kalman filter-like, linearization-based techniques are used for forecasting confidence ellipsoids in a stochastic setting. [ABSTRACT FROM AUTHOR]

For a mobile robot to operate in its environment it is crucial to determine its position with respect to an external reference frame using noisy sensor readings. A scenario in which the robot is moved to another position during its operation without being told, known as the kidnapped robot problem, complicates global localisation. In addition to that, sensor malfunction and external influences of the environment can cause unexpected errors, called outliers, that negatively affect the localisation process. This paper proposes a method based on the fusion of a particle filter with bounded-error localisation, which is able to deal with outliers in the measurement data. The application of our algorithm to solve the kidnapped robot problem using simulated data shows an improvement over conventional probabilistic filtering methods. [ABSTRACT FROM AUTHOR]

In this paper, we present a new method for bracketing (i.e. characterising from inside and from outside) backward reach set of the target region T of a continuous-time dynamical system. The principle of the method is to formalise the problem as a constraint network, where the variables are the trajectories (or paths) of the system. The resolution is made possible by using mazes which is a set of paths that contain all solutions of the problem. As a result, we will be able to derive a method able to compute a backward reach set for a huge class of systems without any knowledge of a parametric Lyapunov function and without assuming any linearity for our system. The method will be illustrated in several examples. [ABSTRACT FROM AUTHOR]

This paper presents the development made around the SeaBot, a new low-cost profiling float design for shallow water. We introduce a simplified dynamical model of the float and propose a state feedback depth controller coupled with an Extended Kalman Filter (EKF) to estimate model parameters. We show experimental results of the depth control that validate the model and the controller. We finally propose a loop design method to build low-cost floats by highlighting key design choices along with design rules. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of platooning control for a fleet of heterogeneous sailboats. The platooning maintains a constant time headway (CTH) between sailboats following a circular path, a complex problem for sailboats due to the influence of wind direction. First, the desired acceleration based on the CTH and the sailboat velocity needed to converge to the platooning is defined. Second, a control of sailboat orientation to manage the sailboat acceleration is proposed. The proposed platooning strategy adapts to the specific characteristics of sailboats, which are different from other motorized marine vehicles. Two tack strategies can be used for the method: the first is to regulate the sailboat velocity; the second is to go front of the wind while staying in a short corridor. The desired acceleration for fulfilling the platooning has been derived and validated. The simulation results demonstrate the effectiveness of the proposed approach in comparison with an optimal receding horizon control algorithm. [ABSTRACT FROM AUTHOR]

When implementing a non-continuous controller for a cyber-physical system, it may happen that the evolution function of the closed-loop system is not anymore piecewise continuous along the trajectory, mainly due to if statements inside the control algorithm. As a consequence, an unwanted chattering effect may occur. This behavior is often difficult to observe even in simulation. We propose here a set-membership method based on interval analysis to detect different types of discontinuities. One of them is the sliding surface where the state trajectory jumps indefinitely between two distinct behaviors. As an application, we consider the validation of a sailboat controller. We show that our approach is able to detect and explain some unwanted sliding effects that may be observed in rare and specific situations on our actual sailboat robots. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of platooning control for a fleet of sailboats. A quadrilateral path is proposed to avoid going into the wind. A tack strategy is defined to go front the wind, stay in a short corridor and regulate the sailboat acceleration, which is a challenging task in comparison to autonomous underwater vehicles (AUVs) or surface vehicles that can use motors for such regulation. Desired acceleration which guarantees to reach the platooning has been derived and validated. A method of regulating the sail angle using a proportional command is proposed to control the sailboat acceleration. Simulation results demonstrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

In this paper we present a reliable method to verify the existence of loops along the uncertain trajectory of a robot, based on proprioceptive measurements only, within a bounded-error context. The loop closure detection is one of the key points in simultaneous localization and mapping (SLAM) methods, especially in homogeneous environments with difficult scenes recognitions. The proposed approach is generic and could be coupled with conventional SLAM algorithms to reliably reduce their computing burden, thus improving the localization and mapping processes in the most challenging environments such as unexplored underwater extents. To prove that a robot performed a loop whatever the uncertainties in its evolution, we employ the notion of topological degree that originates in the field of differential topology. We show that a verification tool based on the topological degree is an optimal method for proving robot loops. This is demonstrated both on datasets from real missions involving autonomous underwater vehicles and by a mathematical discussion. [ABSTRACT FROM AUTHOR]

This paper deals with the simultaneous localization and mapping problem (SLAM) for a robot. The robot has to build a map of its environment while localizing itself using a partially built map. It is assumed that (i) the map is made of point landmarks, (ii) the landmarks are indistinguishable, (iii) the only exteroceptive measurements correspond to the distance between the robot and the landmarks. This paper shows that SLAM can be cast into a constraint network the variables of which being trajectories, digraphs and subsets of $\mathbb {R}^{n}.$ Then, we show how constraint propagation can be extended to deal with such generalized constraint networks. As a result, due to the redundancy of measurements of SLAM, we demonstrate that a constraint-based approach provides an efficient backtrack-free algorithm able to solve our SLAM problem in a guaranteed way. [ABSTRACT FROM AUTHOR]

The minimal model of glucose-insulin dynamics is currently being used in several diabetes-related applications, such as investigating the glucose metabolism, and in the developments of model predictive controllers and fault detection techniques for automatic blood glucose control (i.e., artificial pancreas). Different approaches have been proposed to identify this model, but none of them is capable of providing guaranteed robust enclosures for its parameters, something very desired in applications such as the artificial pancreas, where robustness is paramount. This paper presents a novel approach for guaranteed set-membership parameter estimation of the minimal model based on the well-renowned Set Inversion via Interval Analysis (SIVIA) algorithm. Because the computational complexity of this algorithm is the main barrier for its applicability, an efficient vectorial implementation of SIVIA was employed. Clinical data from a standard intravenous glucose tolerance test were used to prove the validity of the presented approach. Finally, Modal Interval Analysis was used to reduce the numerical overestimation due to the dependency problem of interval arithmetic and significantly speeding up the computations. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper addresses the common issue of locating an Underwater Vehicle. Usually, the positioning of a vehicle is based on the propagation of electromagnetic waves, using systems such as the Global Positioning System. However, water, and particularly salty water, makes the use of electromagnetic waves impractical due to the attenuation caused by the high conductivity of the medium. Hence, the most reliable way to transmit information underwater is by using sound waves and many technologies have emerged to solve the positioning problem in this way. Technologies such as Long BaseLine (LBL), Short BaseLine (SBL) and Ultra-Short BaseLine (USBL) are the most frequently used underwater. These technologies are based on the use of multiple emitters in the case of LBL and SBL, or multiple receivers in the case of USBL. This paper describes a way of finding a vehicle location, on-board, based on the measurement at the vehicle by a single receiver of the phase of an acoustic sine wave transmitted from a single emitter that is at a fixed and known location. The method also uses other proprioceptive measurements: vehicle's velocity and heading. An algorithm based on contractors and bisections scatters the solution space searching for all possible solutions (positions in this case) to the set of equations. Moreover, this paper introduces the Time Constraint Satisfaction Problem. Indeed, the proposed algorithm does not compute the solutions from measurements at a single point in time, rather it uses a set of measurements taken over a time window and stored in a buffer. As a result, the location is not only known at the latest instant but the past locations can be tracked back over the length of the chosen time window. [ABSTRACT FROM AUTHOR]

This paper proposes a set-membership approach to characterize the kernel of an interval-valued function. In the context of a bounded-error estimation, this formulation makes it possible to embed all uncertainties of the problem inside the interval function and thus to avoid bisections with respect to all these uncertainties. To illustrate the principle of the approach, two testcases taken from robotics will be presented. The first testcase deals with the characterization of all loops of a mobile robot from proprioceptive measurements only. The second testcase is the localization of a robot from range-only measurements. [ABSTRACT FROM AUTHOR]

In this paper, we consider the resolution of non-linear constraint satisfaction problems where the variables of the systems are trajectories (functions from $${\mathbb{R}}$$ to $${\mathbb{R}^{n}}$$ ). We introduce the notion of tubes as intervals of functions, for which the lower and upper bounds are trajectories with respect to the inclusion. We then define basic operators and prove propositions verified by tubes. We show the possibility to build contractors on tubes and propagate constraints to solve problems involving time-dependant functions as the unknown variables. We show that the approach is particularly powerful when inter-temporal equations (e.g. delays) are involved. Finally, in order to illustrate the principle and efficiency of the approach, several test cases are provided. [ABSTRACT FROM AUTHOR]