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Louis Poinsot’s result, that any relative rotational motion between two frames can be realized as the motion of a moving cone rolling without slipping on a stationary cone, is stated and proved using matrix-vector algebra. Poinsot’s result is used to obtain a characterization of all attitude trajectories that are closed in the sense that the trajectory terminates at the same point that it starts from. The characterization is an extension to the continuous case of results on discrete closed sequences of rotations, and yields examples of closed attitude (orientation and angular velocity vector) trajectories that may be seen as extensions to the continuous-time case of classical theorems on discrete rotation sequences by Rodrigues, Hamilton and Donkin. One of the examples of closed attitude trajectories is used to obtain an extension of the Goodman–Robinson theorem, which reconstructs the instantaneous orientation of a rotating frame from the components of a fixed vector taken along this frame. This extension yields an explicit expression for the rotation matrix representing the instantaneous orientation of a torque-free rigid body in terms of the body components of its angular velocity.

In many scientific and commercial missions at sea, it is of crucial importance to localize one ore more targets underwater. This objective meets with difficulties due to the unavailability of GPS underwater and the high cost of the equipment classically used for target localization. For these reasons, there has been a surge of interest in the problem of range-based underwater target localization, defined as that of localizing a group of unknown (fixed/moving) targets resorting to surface vehicles (called trackers) equipped with sensors capable of measuring their distances to the targets. In this setup, the surface sensor nodes should be placed according to some optimality criterion aimed at maximizing the information available for target tracking. In addition, the number of sensor nodes should be kept to a minimum in order to avoid the use of a complex and expensive network of vehicles that must be positioned and controlled in a coordinated manner. Motivated by these considerations, in this paper we study the problem of multiple target localization using two trackers and present a constructive method to generate a family of optimal waypoints for the trackers. The use of two surface sensors is motivated by the fact that a single surface sensor does not provide enough positioning accuracy for multiple target localization simultaneously. For two and three targets, we provide a family of analytical solutions for optimal sensor node placements. Further, the proposed solutions can be extended to more than three targets under some well-defined constraints on the target configuration. The optimal sensor positions are derived by maximizing the determinants of appropriately defined Fisher information matrices associated with each of the targets. It is shown that the optimal sensor configuration thus obtained lends itself to an interesting and useful geometrical interpretation. Simulation examples illustrate the results derived.

This paper presents a solution to the problem of path following control for autonomous marine vehicles (AMVs) subject to input constraints and constant ocean current disturbances. We propose two nonlinear control strategies: the first is obtained by using a Lyapunov-based design method, while the second is developed by adopting a Model Predictive Control (MPC) framework. We show that, with the proposed control strategies, the path-following error is globally asymptotically stable (GAS). Simulations with a kinematic model of the vehicle support the theoretical results. Simulations with a realistic model of the Medusa class of AMVs show the robustness of the proposed control strategies.

We propose an integrated motion planning, geophysical navigation, and control (PNC) system for autonomous underwater vehicles (AUVs) that makes explicit use of geophysical information for both planning and navigation. The PNC system exploits the information provided by geomagnetic features observed on the sea-floor to correct the inevitable drift in the vehicle position estimates using dead-reckoning (DR). The three components of the PNC system are: a path planner that enhances the exploratory strength of the Rapidly-exploring Random Tree (RRT) algorithm with transition tests to choose new potential states, a sequential estimation algorithm, and a path following controller. The performance of the system is assessed via hardware in the loop (HIL) simulations to illustrate the significance of using magnetic data for navigation of AUVs. The results obtained are encouraging for inwater tests with an autonomous vehicle of the Medusa class aiming at the validation of the proposed combined system in real-time experiments

This paper addresses the problem of range-based autonomous underwater vehicle (AUV) localization in the presence of unknown ocean currents. In the setup adopted, the AUV is equipped with an attitude and heading reference system, a depth sensor, and an acoustic device that provides measurements of its distance to a set of stationary beacons. We consider the situation where the number of active beacons is not known in advance and may vary with time. The objective is to simultaneously localize the AUV and beacons, that is, to find their positions underwater. We start by deriving conditions under which it is possible to reconstruct the initial condition of the system under study. We consider the design model where the states evolve continuously with time, but the range measurements are only available at discrete instants of time, possibly in a nonuniform manner. For trimming maneuvers that correspond to AUV trajectories with constant linear and angular velocities expressed in the body frame, we show that if either the position of one of the beacons or the initial position of the AUV is known, then even without depth information the system is weakly observable (i.e., the set of states that are indistinguishable from a given initial configuration contains only a set of finite isolated points). If depth measurements are also available, then the system is observable even in the presence of unknown constant ocean currents. Equipped with these results, we then propose a novel observer for simultaneous AUV and beacon localization. The mathematical setup exploited borrows from minimum-energy estimation theory applied to continuous-time processes with discrete measurements, projection filters, and multiple-model estimation techniques. Convergence analysis of the resulting observer system yields conditions under which the estimation errors converge to a small neighborhood of the origin (whose size depends on the magnitude of the process and measurement noise). The results of field experiments with a robotic marine vehicle show the efficacy of the simultaneous AUV/multiple beacon localization system proposed.

The paper addresses the problems of range-based marine vehicle positioning and target localization. Vehicle positioning aims to estimate the positions of one or more vehicles from a sequence of range measurements to fixed or moving acoustic beacons with known locations. In this context, the vehicles must execute sufficiently exciting maneuvers so as to maximize the range-based information available for multiple vehicle positioning. Using an estimation theoretical setting, the vehicle trajectories are computed by maximizing the determinant of a suitably defined Fisher information matrix (FIM), subject to inter-vehicle collision avoidance and vehicle maneuvering constraints. A numerical solution is proposed for the general case. Analytical solutions are obtained in the case of one vehicle and one beacon, when the latter undergoes trajectories that are straight lines, pieces of arcs, or a combination thereof. The theoretical analysis is complemented with practical experiments that focus on the dual problem of underwater target localization. The objective is to estimate the position of a moving underwater target by using range measurements between the target and a vehicle, called a tracker, undergoing a trajectory that can be measured on-line. The experimental set-up includes a surface and an autonomous underwater vehicle of the MEDUSA*-class playing the roles of tracker and target, respectively. In the methodology adopted for system implementation the tracker runs three key algorithms simultaneously, over a sliding time window: i) tracker motion planning, ii) tracker motion control, and iii) target motion estimation based on range data acquired on-line.

We address the problem of range-based marine vehicle positioning in the presence of unknown but constant ocean currents. The goal is to estimate the position of one or more vehicles from a sequence of range measurements to fixed or moving acoustic beacons with known locations. In contrast to most range-based positioning algorithms, we address the case where the currents are unknown and seek to estimate them explicitly as well. This increases the complexity of the problem at hand and raises interesting observability issues. In particular, the vehicles must undergo sufficiently exciting maneuvers so as to maximize the range-based information available for joint current/multiple vehicle position estimation. The main contribution of the paper is the computation of vehicle trajectories for range-based vehicle positioning system in the presence of constant, unknown currents by maximizing the determinant of a suitable Fisher information matrix (FIM), subject to collision avoidance and maneuvering constraints. A numerical solution is proposed for the general set-up of multiple vehicles and beacons. Analytical solutions are obtained for the case of one vehicle and one static beacon. The efficacy of the strategies proposed for vehicle trajectory optimization is shown by numerical simulations.

We address the general problem of multiple target localization and pursuit using measurements of the ranges from the targets to a set of autonomous pursuing vehicles, referred to as trackers. We develop a general framework for targets with models exhibiting uncertainty in the initial state, process, and measurement noise. The main objective is to compute optimal motions for the trackers that maximize the range-based information available for target localization and at the same time yield good target pursuit performance. The solution proposed is rooted in an estimation-theoretical setting that involves the computation of an appropriately defined Bayesian Fisher Information Matrix (FIM). The inverse of the latter yields a posterior Cramer–Rao Lower Bound (CRLB) on the covariance of the targets’ state estimation errors that can be possibly achieved with any estimator. Using the FIM, sufficient conditions on the trackers’ motions are derived for the ideal relative geometry between the trackers and the targets for which the range information acquired is maximal. This allows for an intuitive understanding of the types of ideal tracker trajectories. To deal with realistic constraints on the trackers’ motions and the requirement that the trackers pursue the targets, we then propose a model predictive control (MPC) framework for optimal tracker motion generation with a view to maximizing the predicted range information for target localization while taking explicitly into account the trackers’ dynamics, strict constraints on the trackers’ states and inputs, and prior knowledge about the targets’ states. The efficacy of the MPC is assessed in simulation through the help of representative examples motivated by operational scenarios involving single and multiple targets and trackers.

In many of the marine applications, the availability of an accurate system to localize underwater targets using a group of surface vehicles equipped with acoustic sensors is of paramount importance. In this context, the surface vehicles must keep a well-defined formation with respect to the targets so as to maximize the range-based information available for target localization. In this paper, a target position estimation algorithm together with a simple formation control law to maintain an optimal configuration that maximizes range-related information for target localization and tracking are presented. The setup studied consists of a group of surface vehicles moving in formation and a target performing a predefined mission. The optimal formation is computed by maximizing the determinant of an appropriately defined Fisher information matrix (FIM), subject to inter-vehicle collision avoidance and vehicle manoeuvring constraints. A centralized approach for the formation control is considered, where the trajectory planning is carried out by a supervisor that computes the control actions for each agent. The formation control is complemented with a collision avoidance logic based on a discrete event law. Simulations show that this strategy leads to an interesting and well defined behavior of the formation for localization and tracking.

Single-beacon, range-based AUV localization systems work on the principle that a vehicle may find its position by maneuvering appropriately and acquiring measurements of its successive distances (ranges) to a stationary beacon deployed at a known location. This motivates the study of optimal trajectories to improve the accuracy of the vehicle's position estimate while respecting mission related criteria. In this work, the performance index used to compare different trajectories is the determinant of a properly defined Fisher Information Matrix (FIM). Assuming that heading measurements are available, the problem is studied in 2D, and a class of analytical and numerical solutions are derived. An approach to deal with the case where the initial position of the vehicle is known to lie in a region of uncertainty is also presented. Considering that depth measurements can be obtained, a 3D navigation algorithm consisting of an optimal trajectory planner and a minimum-energy estimator is proposed and its performance assessed via simulation of a practical scientific scenario.

We analyze the observability properties of the kinematic model of an autonomous underwater vehicle (AUV) moving in 3D, under the influence of ocean currents, using range and depth measurements. The results obtained shed light into the types of trajectories that an AUV may be requested to undergo in order to ensure observability, which is a crucial step in the design of single or multiple beacon positioning systems. We assume that the AUV is equipped with two sensor suites: the first computes the distance (range) of the AUV to single or multiple fixed transponders, while the second measures the vehicle's depth. In both situations, the vehicle has access to its heading angle. We further assume that the AUV undergoes maneuvers commonly known as trimming trajectories, that are naturally obtained when the inputs (thruster rpms and control surface deflections) are held constant. This is done for two main reasons: (i) the class of trajectories thus generated is sufficiently rich for a vast number of applications and (ii) from an observability-analysis standpoint they lead to mathematical tractability and allow for an intuitive physical interpretation. These facts stand in sharp contrast to common approaches adopted in the literature, where the characterization of trajectories that yield observability is only implicit and defies a simple interpretation. In the set-up adopted, the trimming trajectories are completely characterized by three variables: (a) linear body speed parallel to v parallel to; (b) flight-path angle gamma; and (c) yaw rate psi. We assume that parallel to v parallel to > 0, gamma, and psi are constant but otherwise arbitrary (within the constraints of the vehicle capabilities) and examine the observability of the resulting system with the two above mentioned sensor suites. We adopt definitions of observability and weak observability that seek inspiration from those proposed by Herman and Krener (1977) but reflect the fact that we consider specific kinds of maneuvers in 3D. We start with the single transponder case. For range measurements only, we show that in the absence of ocean currents the 3D kinematic model of an AUV undergoing trimming trajectories with nonzero flight-path angle and yaw rate is observable. In the case of non-zero but known ocean currents, identical results apply subject to the condition that the flight-path angle satisfies a current-related constraint. However, if the current is non-zero and unknown, the model is only weakly observable. The situation changes completely when both range and depth measurements are available. In this case, under the assumption that the yaw rate is different from zero, observability is obtained even when the flight-path angle is zero (vehicle moving in a horizontal plane) and there are non-zero unknown currents. These obvious advantages are lost if yaw rate is equal to zero, for in this case the model is only weakly observable. In all situations where the model is weakly observable we give a complete characterization of the sets of states that are indistinguishable from a given initial state. Finally, we show that the extended model that is obtained by considering multiple (at least two) transponders is observable in all situations if the yaw rate is different from zero. (C) 2015 International Federation of Automatic Control. Published by Elsevier Ltd. All rights reserved.

A finite sequence of rotations is closed if a sequential application of all the rotations from the sequence results in no net orientation change. A complete characterization of closed rotation sequences involving a given set of rotation axes is presented, and the set of such sequences is shown to be a smooth manifold under a nondegeneracy condition on the rotation axes. The characterization is used to derive several examples of closed rotation sequences, some of which are then shown to specialize to classical examples of such sequences provided by the Rodrigues---Hamilton theorem and the Donkin's theorem. Discrete versions of the Goodman---Robinson and Ishlinskii theorems are also derived and illustrated using the so-called Codman's paradox.

This paper addresses the observability properties of a 3D autonomous underwater vehicle (AUV) model in the presence of ocean currents, under the assumption that the vehicle can only measure its distance to a fixed transponder using an acoustic ranging device. In the set-up adopted, the AUV may undergo a wide range of maneuvers that are usually described as trimming trajectories. The latter are of paramount importance in flight dynamics and can be completely parametrized by three variables: i) linear body speed v, ii) flight-path angle γ, and iii) yaw rate ψ. We assume that v > 0, γ, and ψ are constant but otherwise arbitrary (within the constraints of the vehicle capabilities) and examine the observability of the resulting system with the output (measured) variable described above. We adopt weaker definitions of observability that are akin to those proposed by Herman and Krener (Hermann and Krener, 1977) but reflect the fact that we consider specific kinds of maneuvers in 3D. We show that in the presence of known constant ocean currents the 3D kinematic model of the AUV that corresponds to trimming trajectories with nonzero flight path angle and yaw rate is observable. In case the latter conditions fail, we give a complete characterization of the sets of states that are indistinguishable from a given initial state. We further show that in the case of unknown constant ocean currents the model is locally weakly observable for yaw rate different from zero but fails to be locally weakly observable for zero yaw rate. In both the cases we give a complete characterization of the sets of states that are indistinguishable from a given initial state.

In marine robotics, estimation of the position and orientation of an underwater agent requires lots of research efforts. Especially the realization of robot teams has opened new horizons, allowing for relative navigation based on relative range measurements between the agents. Hence, there is the need for a better understanding of optimal sensor placement related to the positions of the robots relative to each other, and for improvement of observability, based on the concrete mission scenario. In this paper, we study the tracking of a moving target by a Reference Objects (RO) capable of performing acoustic range measurements. We employ the well-known theory of the Empirical Gramians, to evaluate different scenarios and their influence on the observability properties. Emphasis will be put on the computation of a trajectory for the RO that optimizes the observability criterion. We will compare our results with others found in literature that were derived by different procedures, to proof the usability of the Empirical Gramian approach for the area of underwater robotics.

Due to an oversight during the production process, this article published without its Electronic Supplementary Material. The ESM is linked here. Springer apologizes for the error.

We analyze the observability properties of an underwater vehicle (moving in 2D) performing single beacon navigation for two specific classes of maneuvers, whereby the vehicle measures its distance to a fixed transponder located at a known position using an acoustic ranging device. We show that in the presence of known ocean currents, the system is found to be globally observable for constant relative course and constant (nonzero) relative course rate inputs in the sense of Herman and Krener. On the other hand, with unknown ocean currents the system fails to be locally weakly observable with constant relative course but we characterize the set of indistinguishable states from a given initial position and ocean current configuration. Interestingly, observability can be achieved with constant (nonzero) relative course rate in the presence of unknown, constant ocean currents.

Autonomous vehicles are becoming an essential tool in a wide range of environmental applications that include ambient data acquisition, remote sensing, and mapping of the spatial extent of pollutant spills. Among these applications, pollution source localization has drawn increasing interest due to its scientific and commercial interest and the emergence of a new breed of robotic vehicles capable of performing demanding tasks in harsh environments without human supervision. In this task, the aim is to find the location of a region that is the source of a given substance of interest (e.g. a chemical pollutant at sea or a gas leakage in air) using a group of cooperative autonomous vehicles. Motivated by fast paced advances in this challenging area, this paper surveys recent advances in searching techniques that are at the core of environmental monitoring strategies using autonomous vehicles.

International audience; Total Energy Shaping is a controller design methodology that achieves (asymptotic) stabilization of mechanical systems endowing the closed-loop system with a Lagrangian or Hamiltonian structure with a desired energy function. The success of the method relies on the possibility of solving two partial dierential equations (PDE) which identify the kinetic and potential energy functions that can be assigned to the closed-loop. Particularly troublesome is the PDE associated to the kinetic energy which is quasi-linear and inhomogeneous and the solution, that denes the desired inertia matrix, must be positive denite. This task is simplied by the inclusion of gyroscopic forces in the target dynamics, which translates into the presence of a free skew-symmetric matrix in the matching equations that reduces the number of PDE's to be solved. Recently, it has been claimed that considering a more general form for the target dynamic forces, that relax the skew-symmetry condition, further reduces the number of PDE's. The purpose of this paper is to prove that this claim is wrong.

Motivated by applications in visual servoing, we consider the state estimation problem for a class of systems described by implicit outputs and whose state lives in the special Euclidean group SE(3). We propose an observer in the group of motion SE(3) that preserves invariance and therefore takes explicitly into consideration the geometry of the problem. We discuss conditions under which the linearized state estimation error converges exponentially fast. Furthermore, we analyze the problem when the system is subject to disturbances and noises and show that the estimate converges to a neighborhood of the real solution. The size of the neighborhood increases/decreases gracefully with the bound of the disturbance and noise. We apply and illustrate these results through an application of position and attitude estimation of a rigid body using measurements from a camera attached to the rigid body.

International audience; Total energy shaping is a controller design methodology that achieves (asymptotic) stabilisation of mechanical systems endowing the closed-loop system with a Lagrangian or Hamiltonian structure with a desired energy function. The success of the method relies on the possibility of solving two partial differential equations (PDEs) which identify the kinetic and potential energy functions that can be assigned to the closed loop. Particularly troublesome is the PDE associated to the kinetic energy (KE) which is quasi-linear and non-homogeneous, and the solution that defines the desired inertia matrix must be positive definite. This task is simplified by the inclusion of gyroscopic forces in the target dynamics, which translates into the presence of a free skew-symmetric matrix in the KE matching equation that reduces the number of PDEs to be solved. Recently, it has been claimed that considering a more general form for the target dynamic forces that relax the skew-symmetry condition further reduces the number of KE PDEs. The purpose of this paper is to prove that this claim is wrong.

International audience; This paper presents the design of an observer for a class of nonlinear Lipschitz sampled-data systems. The proposed observer uses a predictor of the output between the sampling times. This predictor is re-initialized at each sampling time. Besides, unlike the conventional high gain observer, the observer introduced herein has a dynamic observation gain. Using a Lyapunov approach, we will derive an explicit relation between the bound on maximum allowable sampling period and the parameters of the observer in order to guarantee an asymptotic convergence of the observation error.

This paper considers the state estimation problem of a class of systems described by implicit outputs and whose state lives in the special Euclidean group SE(3). This type of systems are motivated by applications in dynamic vision such as the estimation of the motion of a camera from a sequence of images. We propose an observer in the group of motion SE(3) and discuss conditions under which the linearized state estimation error converges exponentially fast. We also analyze the problem when the system is subject to disturbances and noises. We show that the estimate converges to a neighborhood of the real solution. The size of the neighborhood increases/decreases gracefully with the bound of the disturbance and noise.

Range-based target localization is an important class of problems that arise in an increasing number of scientific and commercial missions at sea. Underwater target localization refers to the task of estimating the positions of fixed or moving underwater targets by using range measurements between the targets and one or more autonomous surface vehicles (ASVs), called trackers , undergoing trajectories that are known in real time. In this context, the trackers must execute sufficiently exciting maneuvers so as to maximize the range-based information available for multiple target localization. In this paper, adopting an estimation theoretical setting, we first propose a general methodology for tracker motion planning that results from maximizing the determinant of an appropriately defined Fisher information matrix (FIM) subject to inter-vehicle collision avoidance and vehicle maneuvering constraints. Then, for the single-target single-tracker problem (which is the dual problem of the classical single-beacon navigation problem), we provide a family of analytical solutions for the optimal tracker trajectories and complement the results with a practical experiment using a tracker when the target undergoes trajectories that are straight lines, pieces of arcs, or a combination thereof. In the methodology adopted for system implementation the tracker runs three key algorithms simultaneously, over a sliding time window: (i) tracker motion planning, (ii) tracker motion control, and (iii) target motion estimation based on range data acquired on-line. In order to simplify the types of trajectories that the tracker must undergo in the single target localization problem, we extend the above set-up to the case where the tracker works in cooperation with another vehicle, called companion , that can also measure ranges to the target and share this info with the tracker. The latter may have access to the position of the companion or, in some cases, only to the range between the two vehicles. We consider three different operating scenarios where the motion of the tracker is chosen so as to increase the accuracy with which the position of the target can be estimated. The scenarios reflect the situations where the motion of the companion vehicle satisfies one of three conditions: (i) the motion is not defined a priori and can also be optimized, (ii) the motion is fixed a priori and is known to the tracker (scenario in which the tracker benefits from the extra information acquired by the companion vehicle, which tracks a desired trajectory in the context of a separate, independent mission), and (iii) the motion is not known a priori and must be learned in the course of the mission. Simulation results illustrate the methodology adopted for cooperative target localization.

This paper is aimed at developing optimal motion planning for a single autonomous surface vehicle (ASV) equipped with an on-board pollutant sensor that will maximize the sensor-related information available for source seeking. The ASV uses a nonlinear diffusion model of the pollutant source to estimate the intensity/level of the pollution at the present ASV location. The rate of detection of particles depends on the relative distance between the ASV and the source. First, we use a probabilistic map of the source location built through the sensor information for a dynamic motion planning of source seeking based on an entropy reduction formulation, where an appropriately defined Fisher information matrix (FIM) is used for entropy reduction or information gain. We derive the FIM for the set-up and investigate optimal trajectories. Next, we present an online nonlinear Monte Carlo algorithm that uses the obtained sensor information about pollutant at different vehicle locations to update a probabilistic uncertainty map of pollutant source location. As the mission unfolds the ASV motion is computed by considering a moving-horizon interval of decision, which will allow for the inclusion of new information available for optimal motion planning. The proposed motion planning approach is extended to take into account external disturbances and it is able to minimize the uncertainty in the pollutant source. Finally, we provide two case studies to demonstrate efficacy of the proposed motion planning algorithm.

In the field of marine robotics, the problem of range based underwater target localization can be defined as that of localizing an unknown - fixed or moving - target from a surface vehicle called the tracker, using range information available about the target. In this set-up, the motion of the tracker (relative to the target) has a significant effect on its ability to localize the target. In general, without any prior knowledge about the target motion, this is a very challenging problem and one that requires, at the outset, a thorough observability analysis of the model adopted for localization system design. For a restricted class of target motions generated using constant linear speed and piecewise constant course rates throughout a given observation window, the target's initial position, constant speed, initial course angle, and course rate, completely characterize its motion. The conditions under which one can estimate any combinations of these four parameters using range information has not been fully studied, except for a simple isolated case where the initial target position is the only unknown parameter. In this paper, using range information, we address the observability properties of the target localization problem for a target moving along a straight line or an arc of a circumference with constant linear and angular speeds on a given observation window. In the first case there are three target parameters (initial position, linear speed, and course angle), while in the second case there are four (initial position, linear speed, angular speed, and course angle). To make the problem mathematically tractable, we consider simple kinematics models for the tracker (we assume a constant linear speed) and target with planar position and course angle as state variables. Under these conditions, for the aforementioned two cases, we investigate the observability properties of range-based target localization problem for various target parameter combinations. In particular, when the target is moving along a straight line, for most of the unknown target parameter combinations, we show that observability can be achieved by a nonzero constant tracker's course rate, which is a simple condition. In the case where the target moves along a circular path, with the knowledge of the angular speed, we derive sufficient conditions on the trackers input to achieve observability, which are more demanding and less straightforward, and do not lend themselves to a simple geometrical explanation.