In this article, we propose a new algorithm for solving the forward dynamics of multibody systems consisting of rigid bodies connected in arbitrary topologies by localized joints and/or soft links, possibly actuated or not. The simulation is based on the implicit time integration of the Lagrangian model of these systems, where the soft links are modeled by Cosserat rods parameterized by assumed strain modes. This choice imposes a predictor–corrector structure on the approach, and requires computing both the residual vector and the Jacobian of the residual vector of the dynamics constrained by the time integrator. These additional calculations are handled here with a new Newton–Euler recursive inverse dynamics algorithm and its linearized tangent version. The approach is illustrated with numerical examples from the Cosserat rod literature and from recent robotic applications.

This article explores the relationship between optimal control and Cosserat beam theory from the perspective of solving the forward and inverse dynamics (and statics as a subcase) of continuous manipulators and snake-like bioinspired locomotors. By invoking the principle of minimum potential energy and the Gauss principle of least constraint, it is shown that the quasi-static and dynamic evolutions of these robots are the solutions of optimal control problems in the space variable, which can be solved at each step (of loading or time) of a simulation with the shooting method. In addition to offering an alternative viewpoint on several simulation approaches proposed in the recent past, the optimal control viewpoint allows us to improve some of them while providing a better understanding of their numerical properties. The approach and its properties are illustrated through a set of numerical examples validated against a reference simulator.

Cosserat rod theory proved efficient modeling performances in robotics, especially in the context of continuum robots, in the past decade. The implementation of such theory is far from being unique and straightforward. We consider the illustrative example of multisegment, general routing tendon actuated continuum robots in their nominal static operating regime. This article details two main approaches based on Cosserat rod modeling, namely, the Newtonian and Lagrangian approaches. We provide a walk-through guide regarding theoretical derivations and numerical implementation of both approaches, together with a proof of equivalence. This comparative study is supplemented with novel contributions and extensions of each approach and in-depth discussion of their performances and applicability, as well as highlighting their special features.

This paper reports methods to compute the equilibrium stances of a new snake-like robot designed to stabilize its head on a free water surface. To adjust rapidly the stability of the robot, this bio-inspired robot can rotate independently each body-shell, and modify the level of immersion of each module. To predict the stable stance accessible by this additional degree of freedom, a model is developed to compute the equilibrium configurations of the robot from a given parametrization of the body shape. Then, an algorithm is introduced to compute a sequence of controlled body deformations, such that the head configuration relatively to the water surface remains unchanged. Finally, we explore in simulation stances and quasi-static gaits, and investigate to what extent the buoyancy and the body deformations can be used to stabilize the head of the snake-like robot.

In this article, we propose a new dynamic model of Cosserat beams in view of its application to continuum and soft robotics manipulation and locomotion. In contrast to usual approaches, it is based on the nonlinear parameterization of the beam shape by its strain fields and their reduction on a functional basis of strain modes. While remaining geometrically exact, the approach provides us with a minimal set of ordinary differential equations in the usual Lagrange matrix form that can be exploited for analysis and control design. Inspired from rigid robotics, the calculation of the matrices of this Lagrangian model is performed with a new reduced inverse Newton-Euler algorithm. To assess the approach, this Lagrangian model is compared against a well-validated finite element method through several benches of nonlinear structural statics and dynamics.

In the 50s, biologists discovered that some electric hsh is capable of discriminating the pose as well as the electric and geometric properties of surrounding objects by navigating and measuring the distortions of a self-generated electric held. In this article, we address the challenging issue of ellipsoidal objects pose and size estimation for underwater robots equipped with artihcial electric sense. Unlike current methods, the approach can estimate both the position and size in parallel with a single straight trajectory. Neither multipolarization nor reactive self-alignment control are necessary to locate the object. The approach is a purely model-based heuristic that selects the best ellipsoid parameters among a set of potential candidates. It is based on a set of four electric measurements recorded at several positions along the robot trajectory along which the displacement is measured. The efhciency of the method is assessed over numerous experiments with different objects, several positions, and orientations, and two different kinds of water (fresh and salt water). Despite some model simplifications and experimental errors, location and size estimation errors are on average below 1 cm and 15%, respectively, while offering promising perspectives for real-time computation.

This paper deals with the geometrical properties of an ellipsoidal object (aspect ratio, volume, orientation) estimation with an underwater sensor inspired by the uncommon sense of the electric fish. The proposed method first locates the object independently of its geometrical properties thanks to the MUSIC (MUltiple SIgnal Classification) algorithm and then, estimates the geometrical properties using an optimization method and the object's electrical response model. The simulation results show the relevance of the method.

By manipulation of electric fields, it is possible to detect the presence of foreign objects underwater. The presented work builds upon a previous result, in which was developed a neural network-based methodology allowing to address this detection problem for spherical objects. Hereafter, we show that the approach generalizes to the case of continuous walls. The technique relies on a neural model of the forward map (from scene configuration to electric measures). Exploiting this model, together with collected electric measures, it becomes possible to detect and infer the relative distance and orientation of a planar wall. In addition, we show that relying on a single forward model, only descriptive of the presence of a single wall, it is possible to address the same problem in presence of a combination of walls forming a corner or a corridor. Closing the motion control loop with information obtained using the proposed approach, it becomes possible to regulate position of a system at a fixed distance and orientation from a wall, with applications to the exploration and monitoring of flooded pipelines, or to surface quality monitoring of ships' hulls (in relation to biofouling). Data collected experimentally are used together with analytical models and numerical simulations to illustrate efficacy of the approach.

Navigation of cluttered underwater environments remains to this day a challenging task in mobile robotics. Applying an electric field to a mobile robot's direct environment and measuring perturbations of this field, one is able to detect the presence of foreign objects in close proximity of the system. In addition, one is also able to infer a range of information relative to the detected objects, such as their position or electrical characteristics. Extracting such information from available measures typically requires a model (analytical, numerical or heuristic) descriptive of the relationship from geometry of the scene to measures performed (typically referred to as forward model), or of the inverse relationship (inverse model). In the following, we directly extract one such model from experimental data, and capture a forward model using a neural formalism. Then, using an iterative procedure, we are able to estimate the position of a detected object and assess the degree of confidence one can place on this estimate. Merit of the approach is illustrated using experimental data for a spherical object.

Some fish species use electric sense to navigate efficiently in the turbid waters of confined spaces. This paper presents a first attempt to use this sense to control a group of nonholonomic rigid underwater vehicles navigating in a cooperative way. A leader whose motion is unknown to the others serves as an active agent for its passive neighbor, which perceives the leader's electric field via current measurements and moves in order to follow a trajectory relative to it. Then, this passive agent, becomes in its turn the leader for the next agent and so on. Sufficient conditions of convergence of the control law are derived for electric current servoing. This is achieved without the explicit knowledge of the location of the agents. Some limits on the possible motion of the leader along with the importance of the choice of controlled outputs are demonstrated. Switching between different group configurations by following a virtual agent is also described. Simulation and experimental results illustrate the theoretical study.

Thanks to an electro-sensible skin, some species of fish can feel the surrounding electric field generated by them-self or other fish. Known under the name of “electric-sense”, this ability allows these fish to navigate in confined surroundings. Based on a bio-inspired electric sensor, this article presents how this electric sense can be used for the navigation in formation of several underwater vehicles. The formation considered is a file, each vehicle is assumed to follow its predecessor at a given distance. In confined environment, the file formation is interesting since fish can follow the same safe path. Being based on the servoing of the electric measurements, these laws do not require the knowledge of the location of the agents. The underwater vehicle studied have non holonomic properties, their forward velocity has no lateral component. Depending on the choice of the controlled outputs (combination of electric measures) we will see that path followed by the follower agents can be different and a methodology to choose the output will be defined in order that all the agents follow the leader path in presence of curved motion of the leader. The influence of the number of electrodes is discussed. Simulation results illustrate the proposed approach.

This paper shows how a sensor inspired by an electric fish could be used to help navigate in confined environments. Exploiting the morphology of the sensor, the physics of electric interactions, as well as taking inspiration from passive electrolocation in real fish, a set of reactive control laws encoding simple behaviors, such as avoiding any electrically contrasted object, or seeking a set of objects while avoiding others according to their electric properties, is proposed. These reflex behaviors are illustrated on simulations and experiments carried out on a setup dedicated to the study of electric sense. The approach does not require a model of the environment and is quite cheap to implement.

Thanks to an electro-sensible skin, some species of fish can feel the perturbations of a self generated electric field caused by their surroundings variations. Known under the name of “electric-sense”, this ability allows these fish to communicate and navigate in confined surroundings wetted by turbid waters where vision and sonar cannot work. Based on a bio-inspired electric sensor recently proposed in [1], this article presents a first attempt to use electric sense for the navigation in formation of a set of rigid underwater vehicles. The navigation strategy combines some behaviours observed in electric fish as well as a follower-leader strategy well known from multi-robot navigation. Being based one the servoing of the electric measurements, these laws do not require the knowledge of the location of the agents. Sufficient convergence conditions of the resulting control laws are given. Moreover, some limits on the possible motion of the leader are exhibited and the importance of the choice of controlled outputs is discussed too. Finally, simulation results illustrate the feasibility of the approach.

This article proposes a solution to the problem of the navigation of underwater robots in confined unstructured environments wetted by turbid waters. The solution is based on a new sensor bio-inspired from electric fish. Exploiting the morphology of the sensor as well as taking inspiration from passive electro-location in real fish, the solution turns out to be a sensory-motor loop encoding a simple behavior relevant to exploration missions. This behavior consists in seeking conductive objects while avoiding insulating ones. The solution is illustrated on experiments. It is robust and works even in very unstructured scenes. It does not require any model and is quite cheap to implement.

Electric fish feel the perturbations of a self-generated electric field through their electro-receptive skin. This sense allows them to navigate and reconstruct their environment in conditions where vision and sonar cannot work. In this article, we use a sensor bio-inspired from this active sense to address the problem of small objects reconstruction and electrolocation. Based on a Kalman filter, any small object in the surrounding of the motion controlled sensor can be encapsulated into an equivalent sphere whose location is well estimated by the filter. Experimental results illustrate the feasibility of the approach.

Morphology, perception and locomotion are three key features highly inter-dependent in robotics. This paper gives an overview of an underwater modular robotic platform equipped with a bio-inspired electric sense. The platform is reconfigurable in the sense that it can split into independent rigid modules and vice-versa. Composed of 9 modules, the longer entity can swim like an eel over long distances, while once detached, each of its modules is efficient for small displacements with a high accuracy. Challenges are to mechanically ensure the morphology changes and to do it automatically. Electric sense is used to guide the modules during docking phases and to navigate in unknown scenes. Several aspects of the design of the robot are described and a particular attention is paid to the inter-module docking system. The feasibility of the design is assessed through experiments.

This paper reports the first results from a program of work aimed at developing a swimming robot equipped with electric sense. After having presented the principles of a bioinspired electric sensor that is now working, we will build the models for electrolocation of objects that are suited to this kind of sensor. The produced models are in a compact analytical form in order to be tractable on the onboard computers of the future robot. These models are tested by comparing them with numerical simulations based on the boundary elements method. The results demonstrate the feasibility of the approach and its compatibility with online objects electrolocation, i.e., another parallel program of ours.

The paper deals with the modeling of a fish-like robot equipped with the electric sense, suited to study sensorimotor loops. The proposed multi-physics model merges a swimming dynamic model of a fish-like robot with an electric model of an embedded electrolocation sensor. Based on a TCP-IP and threaded framework, the resulting simulator works in real time. After presenting the modeling aspects of this work, this article focuses on two numerical studies. In the first, the interactions between body deformations and perception variables are studied and a current correction process is proposed. In the second study, an electric exteroceptive feedback loop based on a direct current measurement method is designed and tested for obstacle avoidance.

This paper deals with a planar biped. The aim of this paper is the estimation, during the imbalance phases of a walking cyclic gait, of its absolute orientation by only using the measurement of the actuated joint variables. The main contribution is the experimental evaluation of an original finite-time convergent-posture observer.

The aim of this paper is to perform the navigation of an underwater robot equipped with a sensor using the electric sense. The robot navigates in an unbounded environment in presence of spheres. This sensor is inspired of some species of electric fish. A model of this sensor composed of n spherical electrodes is established. The variations of the current due to the presence of the sphere is related to the model of Rasnow. Unscented Kalman Filter is used to localize the robot with respect to the sphere and to estimate the size of the sphere. We show that bio-inspired motions improve the detection of the spheres. We illustrate the efficiency of the method in two cases: a two electrodes sensor and a four electrodes sensor.

Experimental validation of absolute orientation estimation solutions is displayed for the dynamical stable five-link biped robot Rabbit during a walking gait. The objective is to prove the technical feasibility of posture online software estimation in order to remove sensors. Finally, this paper presents the first experimental results of walking biped robot posture estimation.

A step-by-step observer based on second order sliding mode approach is proposed to determine the absolute orientation of a biped robot during a walking cyclic gait. The gait are composed of single support phases and impacts. The biped is underactuated in single support because there is no motor in the ankles. The observer and the control law converge in finite-time. The originality is both: firstly, the observer is based on second-order sliding mode approach and is original in biped robot context Secondly, the estimation of all state variables are derived from only actuated joint variables, which induces a minimization of the sensors number. Numerical tests are proposed to show the robustness of the observer coupled with the control.