Lounis Adouane
email: lounis.adouane@uca.fr

Toward Fully Autonomous Vehicle Navigation: From Behavioral to Hybrid Multi-Controller Architectures *

This paper makes the focus on the way to increase gradually the autonomy of single vehicle as well as multi-vehicle systems (MVS) to achieve autonomous navigation in complex environments (e.g., cluttered, uncertain and/or dynamic). Its main objective is to give an overview of the developed generic control architectures (mainly decision/action aspects), and their different components, in order to enhance the safety, flexibility and the reliability of autonomous navigation. Furthermore, among the main ideas developed in this paper are related to the potentiality of using multi-controller architectures and their hybrid nature (continuous/discrete). Indeed, using this kind of control allows us to break the complexity of the overall tasks to be carried out while using bottom-up construction. This implies the development of appropriate reliable elementary controllers (obstacle avoidance, target reaching/tracking, formation maintaining, etc.), but also the proposition of specific mechanisms to manage the controllers' interactions, which ensures the respect of different constraints and enhancing metrics/criteria linked to the overall control reliability. Although the developed concepts/methods/architectures could be applied for different domains (such as service robotics or agriculture), the transportation-related purposes remains the privileged target.

Two main examples of task achievement will be highlighted in this paper: appropriate flexible and reliable navigation based on sequential target reaching, and dynamic multi-vehicle navigation in formation.

I. INTRODUCTION

In recent years, autonomous UGVs (Unmanned Ground Vehicles) are used in a multitude of tasks/domains, for instance: area surveillance [START_REF] Memon | Fully autonomous flammable gases (Methane-Gas) sensing and surveillance robot[END_REF], mapping of unknown environments [START_REF] Fernández-Madrigal | Simultaneous Localization and Mapping for Mobile Robots: Introduction and Methods[END_REF], human search and rescue [START_REF] Simpkins | Rescue robotics [on the shelf[END_REF], exploration [START_REF] Lozenguez | Punctual versus continuous auction coordination for multi-robot and multi-task topological navigation[END_REF], military [START_REF] Springer | Military Robots and Drones: A Reference Handbook[END_REF], agriculture [START_REF] Lu | Automated truck platoon control and field test[END_REF], service robotics [START_REF] Güttler | Towards a future robotic home environment: A survey[END_REF] or transportation [START_REF] Vilca | A novel safe and flexible control strategy based on target reaching for the navigation of urban vehicles[END_REF]. Although the big interest of scientific as well as industrial community on these UGVs, the transportation domain remains among the most challenging and strategic topic knowing its high potential impact for both futuristic smart cities and manufacture of the future. Indeed, this transportation can touch people (private car or public transport) as well as goods transportation (in warehouses or ports, for instance).

In terms of autonomous cars, among the last main events which highly contribute to draw the interest of the international communauty let us cite the DARPA 1 Grand Challenges *This work was supported by the French National Research Agency (ANR) and ADEME (Agence de l'Environnement et de la Maîtrise de l'Energie) through: research programs Investissements d'avenir of RobotEx Equipment of Excellence (ANR-10-EQPX-44), the IMoBS 3 Laboratory of Excellence (ANR-10-LABX-16-01) and Businova Evolution project. 1 Defense Advanced Research Projects Agency of USA.

that happened respectively in 2004 and 2007 [START_REF]Grand Challenge[END_REF] [10] [START_REF] Buehler | The DARPA urban challenge: Autonomous vehicles in city traffic[END_REF].

The important rate of success of these challenges opens a large spectrum to define the actual possibilities of the driverless car for mid-and long-term prospects. Autonomous mobile robotics reach a milestone and automotive industries as well as new companies increase their interest for these new applications [START_REF] Adouane | Autonomous Vehicle Navigation: From Behavioral to Hybrid Multi-Controller Architectures[END_REF]Chapter 1]. In recent years, the development of fully autonomous vehicle in the transportation field has received even more attention from different countries [START_REF] Burns | Sustainable mobility: A vision of our transport future[END_REF].

One of the pioneering and important work which has drawn a lot of public attention to autonomous cars have been happen respectively: in 2010 with the Google driverless car [START_REF] Thrun | IEEE Spectrum, How Google's Self-Driving Car Works[END_REF] (cf. Figure 1(a)) and in 2013 with the BRAiVE (cf. Figure 1(b)) 2 car from Parma University (Italy). These two vehicles allow fully autonomous driving in different contexts (rural / free-way / urban). Nowadays several important automotive manufacturers like Mercedes, GM, Ford, Nissan/Renault, Daimler, PSA, Audi or BMW announce to sell a driverless car at the mid-term horizon (less than 10 years), but before that, important challenges must be resolved in terms of perception-decision-action aspects [START_REF] Adouane | Autonomous Vehicle Navigation: From Behavioral to Hybrid Multi-Controller Architectures[END_REF]Chapter 1]. This paper will make the focus on the decision-action phases.

It is important to notice that the driverless car is not only synonym of a car as we commonly known but with the addition just of the automation of its displacement functions. In fact, in parallel with the developments of this area by automotive industries and certain laboratories, another generation of UGVs like EZ10 (cf. Figure 1(c)) or VIPALAB (cf. Figure 1(d)) aim also to autonomously transport passengers but in a more restricted area like midtown or inside big companies, amusement parks, airports, etc. which need autonomous shuttles between their different areas. This specific autonomous navigation function is among the most important applications started at Institut Pascal since 2000. Although the environment of navigation is generally delimited and the dynamic of UGV evolution is not the same as for the Google car for instance, nevertheless an important part of the autonomous navigation issues are shared. Indeed, this kind of vehicles must, like the Google car, navigate autonomously while taking into account the different events (e.g., traffic light, obstructing objects, etc.) which could be much dense and dynamic in midtown for instance.

While laboratories and innovative companies (like Google) developed a driverless car from scratch, the automotive manufacturers proceed with incremental developments using ADAS (Advanced Driver Assistance Systems). Several sophisticated and reliable ADAS have already entered the market, such as Automatic Parking, Adaptive Cruise Control, Lane Keeping Assistance or Collision avoidance system [START_REF]Advanced Driver Assistance Systems[END_REF]. This incremental (or bottom-up approach) is close in certain manner to what we claim in this paper though the use of multi-controller architectures (MCA) and their hybrid nature (continuous/discrete) (cf. section II). Indeed, among the main ideas developed in this paper are related to the potentiality of using MCA [START_REF] Adouane | Autonomous Vehicle Navigation: From Behavioral to Hybrid Multi-Controller Architectures[END_REF] to achieve fully autonomous navigation in complex environments (e.g., cluttered or/and dynamic). This implies first, impel to components' standardization (input/output) but also the development of appropriate reliable elementary controllers (e.g., obstacle avoidance, target reaching/tracking, formation maintaining, etc.), but also the proposition of specific mechanisms to manage the controllers' in-teractions, which ensures the respect of different constraints related to the overall control structure. This paper makes thus the focus on the way to increase gradually the autonomy of single vehicle as well as multi-vehicle systems (MVS) to achieve fully autonomous navigation. Its main objective is to give an overview of the developed generic control architectures based on MCA (decision/action aspects), and their different components, in order to enhance the safety, flexibility and the reliability of autonomous navigation. Although the developed concepts/methods/architectures could be applied for different domains (such as service robotics or agriculture), this paper will make the focus on transportationrelated application.

The rest of the paper is organized as follows. Section II presents: the genesis of using MCA, the adopted overall navigation framework and also an overview of the main elementary components composing MCAs. Sections III and IV are devoted to highlight the design of appropriate MCAs to achieve respectively two complex tasks: flexible and reliable navigation based on sequential target reaching, and dynamic multi-vehicle navigation in formation. This paper ends with some conclusions and further work.

II. FROM BEHAVIORAL TO MULTI-CONTROLLER ARCHITECTURES

There exist in animals innate behaviors which could be qualified as atomic (or elementary) in the sense that they are not reducible to simpler behaviors (directly observable). In general, all animals' motor actions (coordination of a set of muscle activities) are included in this category. These behaviors are the building blocks with which the behavior of a higher level can be built and described [START_REF] Arbib | Perceptual structures and distributed motor control[END_REF] [START_REF] Anderson | Animal behavior as a paradigm for developing robot autonomy[END_REF]. Further, autonomous mobile robots can have to perform several tasks, for instance going to a specific target (location) in the environment while avoiding obstacles and in certain cases while maintaining a formation (as targeted notably in the work given in [START_REF] Vilca | A novel safe and flexible control strategy based on target reaching for the navigation of urban vehicles[END_REF]) and so on. In addition, these subtasks must also be achieved generally while guaranteeing multi-objective criteria to obtain for instance reliable and smooth robot navigation. All these sub-tasks and several criteria increase considerably the complexity to attain efficient autonomous robot navigation. To address this complexity (in terms of task definition and multi-objective criteria), the control architectures can be elaborated in a modular and bottom-up way as introduced in [START_REF] Brooks | A robust layered control system for a mobile robot[END_REF] and so-called behavioral architectures [START_REF] Arkin | Behavior-Based Robotics[END_REF]. Behavioral control architectures are based on the concept that a robot can achieve a global complex task while using only the coordination of several elementary behaviors [START_REF] Arbib | Perceptual structures and distributed motor control[END_REF] [18] [START_REF] Anderson | Animal behavior as a paradigm for developing robot autonomy[END_REF]. To tackle this complexity, behavioral control architecture decomposes the global control into a set of elementary behaviors/controllers (e.g., attraction to a target, obstacle avoidance, trajectory following, etc.) to better master the overall robot behavior. Indeed, each behavior can be tested either individually or collectively with other behaviors. The goal is to verify the reliability and the efficiency of the corresponding behavior to achieve each determined sub-task [START_REF] Adouane | Hybrid behavioral control architecture for the cooperation of minimalist mobile robots[END_REF] [START_REF] Adouane | Behavioral and distributed control architecture of control for minimalist mobile robots[END_REF]. Nevertheless, several challenges remain to be addressed before obtaining an effective and reliable multi-controller architecture 3 . Among the main objectives of the works highlighted in this paper is to lead to stable and reliable multi-controller architectures while maintaining a high level of flexibility, necessary to tend toward fully autonomous vehicles in any complex situation.

It is presented in what follows first, the adopted overall navigation framework (cf. section II-A), secondly an overview of the used obstacle avoidance technique, based on limit-cycles, will be given (cf. section II-B). This latter is an important component for any autonomous navigation in complex environment. Third, an overview of the main elementary components composing MCAs will be highlighted (cf. section II-C).

A. Overall navigation framework definition

Autonomous vehicle navigation aims, in the proposed generic framework, to lead the vehicle from its initial configuration, to a final configuration while avoiding any obstacle (which could have different shapes, cf. Figure 2). This navigation could be done even with reactive control (while acting online according to the vehicle's local perception) or with cognitive control (while following an already planned trajectory) [START_REF] Adouane | Reactive versus Cognitive Vehicle Navigation based on Optimal Local and Global PELC[END_REF]. The desired vehicle's movement needs to be safe and smooth along all its displacement. One supposes in the setup that the vehicle and the final target to reach are surrounded by circle shapes with a radius R R and R T respectively (cf. Figure 2). For the obstacles/walls, it is supposed that they can be surround by appropriate ellipses (cf. Figure 2). Each ellipse is characterized by: (h, k) the coordinates of the ellipse's center w.r.t. global reference frame, (a, b) which correspond respectively to the major and minor elliptic semi-axes and Ω gives the ellipse orientation.

The Surrounded Ellipse's (SE) parameters (h, k, a, b and Ω) can be obtained by the vehicle either offline (using for 3 The term multi-controller will replace in what follows, the term behavioral because it has been wildly investigated in our works the use of automatic control theory to confirm among others the reliability of each controller as well as the overall multi-controller architecture [START_REF] Adouane | Autonomous Vehicle Navigation: From Behavioral to Hybrid Multi-Controller Architectures[END_REF].

instance a road map of the static environment) or online using for example a camera positioned in the environment [START_REF] Benzerrouk | Multi lyapunov function theorem applied to a mobile robot tracking a trajectory in presence of obstacles[END_REF] or the vehicle's telemetric sensors [START_REF] Vilca | Robust on-line obstacle detection using data range for reactive navigation[END_REF]. Among the challenging aspects when the vehicle navigates in fully reactive way (thus with discovering online its environment), is to update smoothly and efficiently the ellipses' parameters as the vehicle discovers the entire shape of the obstacles. In order to perform this important perceptive functionality, an appropriate weighted least square method, on the range data given by telemetric sensors, has been used in [START_REF] Vilca | On-line obstacle detection using data range for reactive obstacle avoidance[END_REF]. An extension of this last approach while using Extended Kalman Filter and an appropriate sub-optimal heuristic method has been developed in [START_REF] Vilca | Robust on-line obstacle detection using data range for reactive navigation[END_REF] and [START_REF] Vilca | Reactive navigation of mobile robot using elliptic trajectories and effective on-line obstacle detection, Gyroscopy and Navigation[END_REF].

B. Safe obstacle avoidance as an important component

Before to make the focus on multi-controller architectures, let us introduce in short the used "obstacle avoidance" controller. This kind of function is always an important primitive to performe autonomous vehicle navigation in complex environments. Thus, special attention should be taken for its development [START_REF] Minguez | Handbook of Robotics[END_REF]. A multitude of methods exist in the literature to deal with obstacle avoidance, among them those based on: artificial potential field [START_REF] Khatib | Real-time obstacle avoidance for manipulators and mobile robots[END_REF], Voronoï diagrams [START_REF] Aurenhammer | Voronoi diagrams -a survey of a fundamental geometric data structure[END_REF], visibility graphs [START_REF] Latombe | Robot Motion Planning[END_REF], navigation functions [START_REF] Rimon | Exact robot navigation using artficial potential flelds[END_REF], Rapidly-exploring Random Tree (RRT) [START_REF] Lavalle | Rapidly-exploring random trees: A new tool for path planning[END_REF] or Deformable Virtual Zone (DVZ) [START_REF] Zapata | DVZ-based collision avoidance control of non-holonomic mobile manipulators[END_REF]. Each of them have their interests and drawbacks [START_REF] Minguez | Handbook of Robotics[END_REF] [START_REF] Adouane | Reactive versus Cognitive Vehicle Navigation based on Optimal Local and Global PELC[END_REF]. It is used mainly in our works limit-cycles approach [START_REF] Kim | A real-time limit-cycle navigation method for fast mobile robots and its application to robot soccer[END_REF] [35] [START_REF] Adouane | Orbital obstacle avoidance algorithm for reliable and on-line mobile robot navigation[END_REF][37] [START_REF] Soltan | Ode-based obstacle avoidance and trajectory planning for unmanned surface vessels[END_REF] [26] [START_REF] Adouane | Reactive versus Cognitive Vehicle Navigation based on Optimal Local and Global PELC[END_REF]. The navigation methodologies based on limit-cycles have been used in the literature to perform mainly intuitive and efficient obstacle avoidance behavior. They are defined according to a circular [START_REF] Adouane | Orbital obstacle avoidance algorithm for reliable and on-line mobile robot navigation[END_REF], elliptic [START_REF] Adouane | Mobile robot navigation in cluttered environment using reactive elliptic trajectories[END_REF] or parallel to ellipse [START_REF] Adouane | Reactive versus Cognitive Vehicle Navigation based on Optimal Local and Global PELC[END_REF] periodic orbits. These periodic orbits can guarantee, if they are well-dimensioned (far enough from any obstacle) and accurately followed, to avoid any obstructing obstacle. It is to be noted that the used overall strategy for obstacle avoidance remains almost the same while using Elliptic Limit-Cycles (ELC) or Parallel ELC (PELC). Without being exhaustive and to avoid too complex developments it is given below the formulation of ELC.

1) ELC mathematical formulation:

The differential equations of ELC are given by:

ẋs = r(Bys + 0.5Cxs) + µxs(1 -Ax 2 s -By 2 s -Cxsys) (1) ẏs = -r(Axs + 0.5Cys) + µys(1 -Ax 2 s -By 2 s -Cxsys) (2)
with (x s , y s ) corresponds to the position of the vehicle according to the center of the ellipse; r = ±1 according to the avoidance direction (clockwise (+) or counter-clockwise (-) respectively) (cf. Figure 3); µ ∈ R + a positive constant value which allows us to modulate the convergence of the ELC. The convergence is as slow as µ is smaller, which allows us also to obtain smoother ELC (Figure 5 highlights the influence of µ in the case of PELC). The variables A, B and C are given by: A =(sin(Ω)/b lc ) 2 + (cos(Ω)/a lc ) 2 (3) where a lc and b lc characterize respectively the major and minor elliptic semi-axes and Ω gives the ellipse orientation.

B =(cos(Ω)/b lc ) 2 + (sin(Ω)/a lc ) 2 (4)
C =(1/a 2 lc -1/b 2 lc ) sin(2Ω) (5)
Figure 3 shows that the ellipse of a major axis = 2a lc = 4 and of minor axis = 2b lc = 2 is a periodic orbit. Figure 3(a) and 3(b) show the shape of equations ( 1) and ( 2) when r = 1 and -1 respectively. They show the direction of trajectories (clockwise or counter-clockwise) according to (x s , y s ) axes. The trajectories from all points (x s , y s ) of X, Y reference frame, including inside the ellipse, move towards the ellipse. This can be demonstrated mathematically while using Lyapunov synthesis.

It is to be noted that the case of Parallel Elliptic Limit-Cycle (PELC) addressed in [START_REF] Adouane | Reactive versus Cognitive Vehicle Navigation based on Optimal Local and Global PELC[END_REF] lead to much complex formulation of equations ( 1) and ( 2) since it is targeted to have orbit which is always parallel to the ellipse (constant distance w.r.t. the ellipse border) which surround the obstacle. This leads to equations of 8 th order. It is also important to notice that these limit-cycles could be used as local path planner (to avoid the closest obstacle) or global planner. As global planner, the aim is to find the optimal (or sub-optimal) global path leading the vehicle from its initial configuration until a final configuration while avoiding all the hindering obstacles (cf. Figure 4 for an example of global planning using PELC).

2) References frame linked to the task achievement: For simple and efficient description of vehicle navigation in any kind of environment, it is presented in what follows a specific reference frame assigned for each obstacle / wall / (or any object which could obstruct the vehicle's movement) inside the considered environment (or at least for each element inside the vehicle's field of view). These specific reference frames will guide the vehicle behaviors and allows the vehicle to evaluate the success of the current achieved subtask (e.g., wall following, obstacle avoidance, etc.) [START_REF] Adouane | Reactive versus Cognitive Vehicle Navigation based on Optimal Local and Global PELC[END_REF]. Each elementary reference frame will orient thus locally the achievement of the vehicle navigation toward its final objective. A kind of analogy could be established with robot manipulator modeling. In fact, when we would like to control the movement of a robot end-effector (w.r.t. its base), it is assigned for each articulation an appropriate reference frame while using dedicated conventions [START_REF] Denavit | A kinematic notation for lower-pair mechanisms based on matrices[END_REF] [START_REF] Khalil | Modeling, identification and control of robots[END_REF]. These local Example of simple simulation highlighting the use of several PELC to reach the final assigned target. PELC paths given by "+" symbols mean invalid PELCs. The others elementary PELC in dotted colored lines show valid PELCs. The arrows given in solid lines correspond to the final PELCs' configurations. The numbering correspond to the successive computed PELCs in order to obtain the global optimized path based on PELC [START_REF] Adouane | Reactive versus Cognitive Vehicle Navigation based on Optimal Local and Global PELC[END_REF].

reference frames are mainly used to express simply the local elementary articulations' movements (translation / rotations) in order to obtain the desired final end-effector movement. The context of vehicle navigation is obviously different but the proposed reference frames will similarly lead to make a reasoning on the efficiency of the vehicle movements in order to reach its final objective.

The reference frame (R OT ) attributed to each obstacle will allow to evaluate the obstacle avoidance sub-task achievement while knowing the relative vehicle's localization according to it. R OT is obtained with a simple geometric construction and has the following features [START_REF] Adouane | Reactive versus Cognitive Vehicle Navigation based on Optimal Local and Global PELC[END_REF] (cf. Figure 5):

• X OT axis connects the center of the obstacle (x O , y O )
to the center of the final Target (x f , y f ). This axis is oriented toward this target. • the Y OT axis is defined by two points PT 1 and PT 2 , which correspond to the tangent points between the two straight lines coming from the final target (x f , y f ) and the Parallel Ellipse of Influence (PEI). Y OT axis is oriented while following trigonometric convention.

To guide the vehicle's future movements it is important to define its localization w.r.t. R OT . One needs, therefore, to make a transformation from the global reference frame X G Y G to the local reference frame X OT Y OT [START_REF] Adouane | Reactive versus Cognitive Vehicle Navigation based on Optimal Local and Global PELC[END_REF]. Once the transformation is obtained, it is enough for instance to check the sign of the vehicle's localization according to the axis X OT to assign the vehicle appropriate behavior. For instance, if the sign of x RO is negative, the vehicle must follows the defined limit-cycle (to avoid the obstacle) and if positive the vehicle can consider that the obstacle is not a hindering obstacle and can go thus straight toward its final target (cf. Figure 5). At the condition obviously that there is no other constrained obstacle; if not, the process will be reiterated. In previous work given in [START_REF] Adouane | Orbital obstacle avoidance algorithm for reliable and on-line mobile robot navigation[END_REF] the sign of y RO (ordinate of the robot in R OT ) has been used to determine the right direction to avoid the obstacle. If y RO ≥ 0 then apply clockwise limitcycle direction else apply counter-clockwise direction. This simple rule allows to reduce the length of robot trajectory in the case where the obstacle is surrounded by a circle. In the work given in [START_REF] Adouane | Reactive versus Cognitive Vehicle Navigation based on Optimal Local and Global PELC[END_REF] the presented reference frame has been served as an important component to determine the best direction and shape of local planning using PELC and global planning using an optimal sequence of PELC to reach the final target.

After introducing the global idea to obtain the limit-cycles which should be followed by the vehicles to avoid obstacles, let us highlight the main structures characterizing MCAs and their main components to achieve autonomous vehicle navigation.

C. Multi-controller architectures' main structures

The main aim of multi-controller architectures is to have a bottom-up construction of the principal functionalities of autonomous vehicle navigation, while maintaining a high level of flexibility and reliability of the achieved complex tasks [START_REF] Adouane | Autonomous Vehicle Navigation: From Behavioral to Hybrid Multi-Controller Architectures[END_REF]. It will be presented, in what follows, two simple multicontroller architectures (cf. Figure 6) which could serve as basic structures to highlight the main components to perform flexible and reliable navigation in cluttered environments.

Before highlighting the main difference between the two control structures depicted in Figure 6, it is important to give the definition of the robot's elementary behavior (controller). In what follows each robot's controller is constituted by a dedicated set-point and stable control law blocks which allow us to achieve safely and reliably the desired robot's behavior. As shown in Figures 6(a) and 6(b) the two structures 1 and 2, are different in terms of the used control laws. In fact, structure 1 has two distinct control laws whereas structure 2 has only one common control law shared by the two setpoint blocks. It is interesting to notice that structure 1 is the less restrictive architecture, in the sense that both controllers can have completely different set-points and control laws definition. It is enough to have two already stable elementary controllers to integrate them in this structure, without any harmonization of the used set-points or control laws. The possible drawback of this kind of multi-controller architecture corresponds to its difficulty in having a simple analysis of the overall control architecture stability (since it could use non-uniform control laws) [START_REF] Adouane | Autonomous Vehicle Navigation: From Behavioral to Hybrid Multi-Controller Architectures[END_REF]Chapter 3].

The two simple multi-controller architectures depicted in Figure 6 allow us to manage the interaction between different elementary blocks. The main features of each block composing these architectures are detailed below.

1) Sensor information block: While using robot's sensors and any already known data on the environment (using a road-map for instance), this block is in charge of detecting / localizing / characterizing any important features in the environment. Mainly this block, in the case of the basic architectures given in Figure 6, must provide the list of all perceived obstacles and the relative final target localization w.r.t. the robot (cf. Figure 2).

2) Controllers' coordination block: The coordination of the several controllers constituting a multi-controller architecture is among the most important aspect to master in order to obtain coherent and reliable architecture. Generally once the stability of each elementary controller is proved, they are gathered in specific multi-controller architectures to perform complex tasks. The objective is to ensure in addition, the stability and the smoothness of the overall control. This can be done mainly if the coordination between the elementary controllers is mastered [12, Chapter 3].

There exist two major principles of controller coordination: action selection and fusion of actions. Even if fusion of actions process gives very interesting robot behaviors, as it has been shown in [START_REF] Adouane | Behavioral and distributed control architecture of control for minimalist mobile robots[END_REF] (using a kind of schema motor principle); in [START_REF] Dafflon | Adaptive autonomous navigation using reactive multi-agent system for control laws merging[END_REF] (using Multi-Agent System) or in [START_REF] Boufera | Mobile robot navigation using fuzzy limit-cycles in cluttered environment[END_REF] (using Fuzzy Logic principles), nevertheless the stability of the overall control architecture remains very complex, even impossible to demonstrate. However, control architectures based on the action selection process are relatively much easier to demonstrate even when switches between conttollers occur [START_REF] Branicky | Multiple Lyapunov functions and other analysis tools for switched and hybrid systems[END_REF] [44] [START_REF] Liberzon | Switching in Systems and Control[END_REF]. However, the challenge is, in addition to the overall stability, to guarantee control smoothness. In fact, during mainly the phase of switching between controllers, the robot's set-points or the control could be subject to jerking/discontinuities/oscillations, the objective of the Contrtollers' coordination block is therefore to avoid (or at least minimize) these drawbacks to obtain finally reliable and smooth robot navigation [START_REF] Adouane | Hybrid and safe control architecture for mobile robot navigation[END_REF] [23] [47] [START_REF] Adouane | Towards smooth and stable reactive mobile robot navigation using on-line control set-points[END_REF]. More details on the hybrid behavior (continuous/discrete) of the overall multi-controller architectures in order to guarantee at the same time, the overall control stability and the smoothness of the switch between controllers is emphasized in [START_REF] Adouane | Autonomous Vehicle Navigation: From Behavioral to Hybrid Multi-Controller Architectures[END_REF]Chapter 3].

3) Set-point blocks and their homogeneous definition: Set-point blocks (cf. Figure 6), which have as input the perceptions P i coming from the sensor information block, are responsible to give for each dedicated controller, the appropriate set-points for its operation. The design of multicontroller architecture aims to decompose the overall complex task into a multitude of sub-tasks to achieve (e.g., target reaching, wall following, obstacle avoidance, etc.) (cf. section II). According to these elementary sub-tasks, performed in a reactive or cognitive way [START_REF] Adouane | Reactive versus Cognitive Vehicle Navigation based on Optimal Local and Global PELC[END_REF], it has been noted in general that the robot must follow/track a path/trajectory or reach/track a specific target. This section aims to propose a homogeneous set-points definition for the multitude of the robot's navigation sub-tasks in order to simplify the design of control architectures.

It will be described in sections III and IV the use of static and/or dynamic targets to lead to a much more flexible way to define the robot's sub-tasks. It is promoted therefore in what follows the use of target set-points, defined by a pose (x T , y T , θ T ) and a velocity v T . The following subsections (II-C.3.a to II-C.3.c) will highlight the fact that this set-point formulation is generic enough to define an important number of the robot's behaviors. It is to be noted that, once the setpoints are defined, at each sample time, it is important to have reliable control laws to reach/track these assigned setpoints. To do that, one of the reliable control laws defined in section II-C.4 will be used to stabilize the errors to zero. This corresponds thus to the scheme of control given by structure 2 (cf. Figure 6(b)).

a) Target tracking set-points based on global planned path: The first identified case corresponds to the one where a global path is already defined using for instance a PELC [START_REF] Adouane | Reactive versus Cognitive Vehicle Navigation based on Optimal Local and Global PELC[END_REF]. In fact, in certain situations (e.g., static environment) it is enough for the robot to follow the path as accurately as possible without modifying its initial planning. In that situation, a Frenet reference frame is used [START_REF] Samson | Control of chained systems: Application to path following and time-varying point-stabilization of mobile robots[END_REF] to extract the robot's set-points. The target set-point, at each sample time, When the environment is not very well known or dynamic, it is better to navigate reactively. In that situation, the current PELC takes as initial configuration, and at each sample time, the current robot configuration. The target set-point is given by (cf. Figure 7

Obstacle i O m Y m X m O G X G Y G Y F X F (Target) T  O F (x T, y T ) PELC (a) Frenet refrence frame Obstacle i PELC O G X G Y G Target (x T, y T ) T  Y m X m R S O m (b) Eucledian distance R S Target i  i T Y m X m O m Follower i Leader robot (main Target) Static target O G X G Y G (c) Cartesian configuration

(b)):

• A position (x T , y T ) corresponding to the intersection between the circle (which has as origin the origin of the reference frame X m Y m and as radius R S ) and the planned PELC.

• An orientation θ T corresponding to the tangent to the PELC w.r.t. X G Y G reference frame at the intersection point (x T , y T ). If R S = 0, the robot has to apply only an orientation control. Indeed, since the robot is already on the current computed PELC, the robot has only to control its heading w.r.t. θ T . This simple control has been used in [START_REF] Adouane | Orbital obstacle avoidance algorithm for reliable and on-line mobile robot navigation[END_REF] and [START_REF] Adouane | Mobile robot navigation in cluttered environment using reactive elliptic trajectories[END_REF]. • A velocity v T which could be constant or variable indifferently. c) General target reaching/tracking set-points: The last identified case (cf. Figure 7(c)) corresponds to the general situation where the robot must reach/track a static/dynamic target (x T , y T , θ T , v T ). The sub-tasks which can deal with this kind of target definition correspond to all the cases where the set-points are not restricted to evolve inside a specific path. For instance, let us cite:

• For a static target, the set-points could correspond to the final robot destination as given in Figure 2. They could also correspond to an appropriate waypoints in the environment through which the robot must cross sequentially [START_REF] Vilca | A novel safe and flexible control strategy based on target reaching for the navigation of urban vehicles[END_REF] (section III will give an example showing this navigation strategy). • For a dynamic target, this kind of target set-point can serve for the Follower robot (as depicted in Figure 7(c)) to track a secondary target referenced w.r.t. the Leader. Section IV, focused on multi-robot systems, will highlight better this kind of target set-point definition. 4) Uniform used control law: Before presenting the used control law, it is important to know the robot's model. The used robot corresponds to a tricycle vehicle [START_REF] Luca | Feedback control of a nonholonomic car-like robot[END_REF] modeled according to the well-known kinematics model given by equation 6.

   ẋ = v cos(θ) ẏ = v sin(θ) θ = v tan(γ)/l b (6)
where (x, y, θ) is the posture (configuration state) of the vehicle at the point O m (origin of the local reference frame X m Y m linked to the vehicle (cf. Figure 8)), γ is the orientation of the equivalent front wheel (cf. Figure 8), v is the linear velocity of the vehicle at O m and l b is the vehicle's wheelbase. v and γ are the two control inputs of the vehicle (cf. equations 10 and 11 respectively). According to Figure 8, w b corresponds to the track width of the vehicle and I cc the instantaneous center of curvature of the vehicle trajectory.

The radius of curvature r c is given by:

r c = l b / tan(γ) (7)
and c c = 1/r c is the curvature of the vehicle trajectory. The used control law [START_REF] Vilca | A novel safe and flexible control strategy based on target reaching for the navigation of urban vehicles[END_REF] aims to drive the vehicle toward specific targets (static or dynamic) in the environment. At each sample time the tracked target is defined by a posture (x T , y T , θ T ) and a velocity v T (this velocity could be = 0 if the target is static). As it was shown in subsection II-C.3, different vehicles behaviors are described with a uniform way where the vehicle has to reach/follow/track a specific target set-points. In order that this paper becomes at maximum self-contained, let us give in summary the main elements of synthesis, using a Lyapunov formulation [START_REF] Khalil | Frequency domain analysis of feedback systems[END_REF], the used control law [START_REF] Vilca | A novel safe and flexible control strategy based on target reaching for the navigation of urban vehicles[END_REF]. The adopted Lyapunov function V is given by equation ( 8) (cf. Figure 8):

V = 1 2 K d d 2 + 1 2 K l d 2 l + K o [1 -cos(e θ )] = 1 2 K d d 2 + 1 2 K l d 2 sin 2 (e RT ) + K o [1 -cos(e θ )] (8)
where the initial values of e RT and e θ must satisfy the following initial conditions:

e RT ∈ ] -π/2, π/2[ and e θ ∈ ] -π/2, π/2[ (9)
The Lyapunov function ( 8) is therefore a function of three parameters which depend on: the distance d between the target and vehicle's position; the distance d l from the vehicle to the target line (line that passes through the target position with an orientation equal to the target orientation), this term is related to the Line of Sight and Flight of the target [START_REF] Siouris | Missile Guidance and Control Systems[END_REF]; and the orientation error e θ between the vehicle and the target.

The desired linear velocity v and the front wheel orientation γ of the vehicle which allows to asymptotically stabilize the error vector (e x , e y , e θ , (v -v T )) toward zero (permitting therefore to have V < 0) are given by:

v = v T cos(e θ ) + v b (10) γ = arctan(l b c c ) (11)
where v b and c c are defined by:

v b =K x [K d e x + K l d sin(e RT ) sin(e θ ) + K o sin(e θ )c c ] (12)
with:

c c = 1 r c T cos(e θ ) +
K = (K d , K l , K o , K x , K θ , K RT (13)
) is a vector of positive constants defined by the designer. Accurate analysis of this stable and efficient control law is given in [START_REF] Vilca | An overall control strategy based on target reaching for the navigation of a urban electric vehicle[END_REF] and [START_REF] Vilca | A novel safe and flexible control strategy based on target reaching for the navigation of urban vehicles[END_REF].

Knowing the used common control law permitting to reach any static or dynamic target with stable way, let us present in what follows, how to perform different navigation sub-tasks using appropriate multi-controller architectures.

III. FLEXIBLE AND RELIABLE NAVIGATION BASED ON SEQUENTIAL TARGET REACHING

This section emphasizes in summary the fact that it is not absolutely mandatory (as commonly admitted and broadly used in the literature) to have a predetermined trajectory to be followed by a robot to perform reliable and safe navigation in an urban and/or cluttered environment [START_REF] Adouane | Autonomous Vehicle Navigation: From Behavioral to Hybrid Multi-Controller Architectures[END_REF]Chapter 1]. It is presented in what follows the idea to use only a set of waypoints, appropriately disposed in the environment, to perform such navigation. The use of only a discrete number of waypoints in the environment will allow us even more flexibility of the vehicle's movements, since it is allowed to perform more maneuvers between waypoints, while remaining obviously safe (non-collision of the vehicle w.r.t. the road limits or any obstructing obstacle). Hence, navigation using only waypoints allows us to avoid any path/trajectory planning which could be timeconsuming and complex, mainly in cluttered and dynamic environments. Moreover, this kind of navigation does not require knowledge of the pose of the closest point to the followed trajectory (w.r.t. the robot configuration) and/or the value of the curvature at this point [START_REF] Gu | On-road motion planning for autonomous vehicles[END_REF]. Consequently, the navigation problem is simplified to a waypoint-reaching problem, i.e., the vehicle is guided by waypoints instead of following a specific fixed path [START_REF] Vilca | A novel safe and flexible control strategy based on target reaching for the navigation of urban vehicles[END_REF]. Moreover, it is important to notice that if the successive waypoints are closer to each other, then the vehicle tends to perform a path-following navigation. The proposed technique tends therefore to gather the different navigation techniques. In addition, the use of only waypoints to control the vehicle instead of a fixed trajectory, allows the robot to carry out local operations (to avoid such obstacle) while maintaining overall stability of the used hybrid multi-controller architectures [START_REF] Adouane | Autonomous Vehicle Navigation: From Behavioral to Hybrid Multi-Controller Architectures[END_REF]Chapter 3].

A. Problem statement and Task modeling

The proposed navigation strategy uses a sequence of N sorted waypoints appropriately disposed in the environment. The aim of this sequence is to guarantee safe and flexible robot navigation. Each waypoint T j = (x Tj , y Tj , θ Tj , v Tj ) corresponds to a specific key configuration in the environment (cf. Figure 9). T j is characterized by:

• A position (x Tj , y Tj ).

• An orientation θ Tj such as: θ Tj = arctan (y Tj+1 -y Tj )/(x Tj+1 -x Tj )

where: (x Tj+1 , y Tj+1 ) corresponds to the position of the next target T j+1 . T j orientation is therefore always oriented toward the waypoint T j+1 .

w R w R T j T j+1 T j-1 L j-1 L j L j+1 θTj d j+1 v Edis Eangle d j+ 2 Y Tj X Tj T j+2 L j+2 T j θTj Edis Eangle Y Tj X Tj d j d j+1 d j
Fig. 9. Description of waypoints assignment.

• A velocity v Tj . It is important to mention that to perform the proposed navigation by reaching sequential waypoints (targets), it is mandatory to reach each target (except the final waypoint) with a velocity v Tj = 0 to not have a jerky vehicle movement, at the starting and the arrival phase for each waypoint. The overall vehicle navigation becomes therefore smoother without oscillations in terms of linear velocity.

Different methods to obtain the appropriate set of waypoints (target set-points ((x Tj , y Tj , θ Tj , v Tj ) | j=1...N )) are presented in [START_REF] Vilca | Optimal multi-criteria waypoint selection for autonomous vehicle navigation in structured environment[END_REF]. They are based either on a heuristic method or on optimal multi-criteria optimization.

To define the robot's navigation strategy between the successive waypoints (cf. subsection III-B), an orthogonal reference frame X Tj Y Tj (cf. Figure 9) is attributed to each waypoint, where:

• the X Tj axis connects the position of T j to the following waypoint T j+1 and oriented toward T j+1 , and • the Y Tj axis is perpendicular to X Tj and is oriented while following trigonometric convention.

This reference frame will be used in subsection III-B to perform the target assignment process. In addition, to ensure safe robot navigation between successive waypoints, each waypoint is assigned upper error bounds defined by E dis and E angle (cf. Figure 9). They correspond respectively to the maximal distance d and angle e θ errors between the robot and the target when it crosses the axis Y Tj . Further, E dis and E angle correspond to a kind of maximal error tolerance when the robot reaches the target T j . This tolerance is notably related to the inaccuracies of the robot localization and/or to the performance of the used control law. The maximum authorized values of E dis and E angle allow us to keep reliable robot navigation toward the target T j (cf. Figure 9) while guaranteeing the appropriate robot configuration to reach the next target T j+1 [START_REF] Vilca | A novel safe and flexible control strategy based on target reaching for the navigation of urban vehicles[END_REF], and so on.

B. Proposed MCA and Experimental validation

To perform the navigation based on sequential target reaching, the multi-controller architecture depicted in Figure 10 is used. This architecture is composed of several blocks: • The "Waypoint determination" block (dashed green box in Figure 10) obtains the set of appropriate waypoints configuration [START_REF] Vilca | Optimal multi-criteria waypoint selection for autonomous vehicle navigation in structured environment[END_REF]. • The "Obstacle avoidance" block is activated when an obstacle obstructs the robot's movement toward its current assigned waypoint. The used obstacle avoidance is based on limit-cycle technique (as given in subsection II-B) and allows us to avoid locally any obstructing obstacle. • The "Control law" block ensures asymptotic stability to reach the current assigned waypoint T j (x Tj , y Tj , θ Tj , v Tj ), as given in subsection II-C. [START_REF] Lozenguez | Punctual versus continuous auction coordination for multi-robot and multi-task topological navigation[END_REF], where the control law can perform either static as well as dynamic target reaching/tracking. • The "Target assignment" block lets us obtain, at each sample time, the current waypoint (target) to reach. This block is detailed in what follows. Sequential target assignment: The strategy to assign, at each sample time, the waypoint to reach by the vehicle is shown in Algorithm 1. The stable and reliable control law has to reach each assigned waypoint while ensuring that the vehicle's trajectory is always within the road boundaries.

The error conditions, E dis and E angle , are used to switch to the next waypoint when the vehicle's position is inside a circle given by the center (x Tj , y Tj ) and a radius E dis . Hence, the current waypoint index is updated with the next waypoint and the vehicle has to thereafter adapt its movement according to this new target. If the vehicle does not satisfy the distance and orientation error conditions (the errors d and e θ > than E dis and E angle respectively) when crossing the Y Tj axis (cf. Figure 9), then the vehicle must nevertheless switch to the next waypoint. Obviously, this situation should not occur if the environment is accurately modeled/identified and the control law well settled. Despite all these aspects, if this situation happens, then the value of the maximal distance and angular errors can be used to decide if the vehicle could or not continue its navigation. This fault detection/diagnosis Algorithm 1: Sequential target assignment Require: Vehicle pose, current target Tj and a set of N sorted waypoints Ensure: Reaching Tj while guaranteeing to the vehicle to be in the best configuration to reach after the next waypoint Tj+1. 1: if ( (d ≤ E dis and e θ ≤ E angle ) or (x T j ≥ 0) ) {x T j is the coordinate of the vehicle in the local Target frame XT j YT j (cf. Figure 9)} then Keep going to waypoint Tj 5: end if is not addressed in this paper.

It is also interesting to mention that the definition of Y Tj axis, as in section II-B, guides the task achievement. In fact, it is used in the cited section to perform elementary obstacle avoidance. This axis is used here as a mean to decide when to switch to the next waypoint.

Experimental validation: The presented scenario was built to show different situations in urban environment, such as multi-vehicle navigating in platoon (based on Leader-Follower formulation (cf. section IV)), static and dynamic target-reaching and obstacle-avoidance situation. The experiment was done using VIPALAB vehicles (cf. Figure 1(d)) in PAVIN platform (Plate-forme d'Auvergne pour Véhicules INtelligents). A metric map of PAVIN [START_REF] Ip | Data.Sets[END_REF] is used to plan the optimal configuration of geo-referenced waypoints with optimal configurations [START_REF] Vilca | Optimal multi-criteria waypoint selection for autonomous vehicle navigation in structured environment[END_REF]. In this experiment, each vehicle uses a combination of RTK-GPS and gyrometer to estimate its current position and orientation at a sample time of T s = 0.1 s. The vehicles have a range sensor (LIDAR) with a maximum detected range equal to 10 m. Moreover, the vehicles communicate by Wi-Fi, enabling the transmission of the Leader's pose data.

The presented experiment highlights the performance of the proposed control law and target assignment strategy using waypoint selection. The Leader vehicle has to successively reach static waypoints. Moreover, the proposed control law (cf. section II-C.4) was implemented in another vehicle (Follower) which takes the first vehicle (Leader) as a dynamic target to track at a curvilinear distance equal to 5 m (behind the Leader). The tracking of the dynamic target allows us to apply the proposed control law to multi-vehicle systems where the dynamic set-point is given by the leader and the desired geometric formation shape [START_REF] Vilca | Adaptive leader-follower formation in cluttered environment using dynamic target reconfiguration[END_REF]. The configuration of the dynamic target is sent by the Leader to the Follower via Wi-Fi. This experiment can be found online. 4 Furthermore, to exhibit the flexibility of the proposed navigation strategy, an obstacle is placed between the waypoints. Therefore, the vehicle can perform different maneuvers between waypoints, in this case obstacle avoidance, without the use of any trajectory replanning method. The obstacle avoidance is activated as soon as the vehicle detects at least one obstacle which can hinder the future vehicle movements toward the current assigned waypoint. It can be seen in Figure 12(a) that the Leader accurately reaches the successive assigned static waypoints and the Follower also accurately tracks the dynamic target (Leader). Moreover, the Follower trajectory using the proposed control law is close to the Leader trajectory (cf. Figure 12(a)). Figure 12(b) focuses on the vehicles' trajectories when the obstacle avoidance is activated. The Leader detects the obstacle between the waypoints and it applies local obstacle avoidance based on limit-cycles (cf. section II-B). The Follower also avoids the obstacle since it accurately tracks the Leader trajectory. It can be noted that the proposed navigation strategy allows flexible and smooth movements between the waypoints and performance of different behaviors, such as obstacle avoidance, emergency stop or waypoint reassignment.

IV. DYNAMIC MULTI-VEHICLE NAVIGATION IN

FORMATION

The navigation in formation is addressed in this section while using the Leader-follower approach. This approach has been adopted and applied on VIPALAB vehicles for dynamic reconfiguration of a fleet of vehicles [START_REF] Vilca | Adaptive leader-follower formation in cluttered environment using dynamic target reconfiguration[END_REF]. Each follower tracks the instantaneous state (pose and velocity) of its assigned virtual targets, given w.r.t. the Leader dynamic.

A. Problem statement and Task modeling

The Leader-follower approach enables us to maintain a rigid geometric shape (e.g., a triangle in Figure 13). The formation is defined w.r.t. the Cartesian frame (local frame of the leader), as follows:

• A Leader (UGV L in Fig. 13); its pose (x L , y L , θ L ) and its linear velocity v L determine the dynamic of the formation. • The formation structure is defined with as many nodes as necessary to obtain the desired formation shape. Each node i is a virtual dynamic target (T di ). The formation is defined as

F = {f i , i = 1 • • • N },
where f i are the coordinates (h i , l i ) T of the dynamic target T di w.r.t. the leader local reference frame.

The position and orientation of each node (virtual target) are computed from the leader configuration. The leader position determines the node positions according to the formation shape. The instantaneous center of curvature I cc L of the formation is determined by the leader according to its movements (cf. Figure 13). I cc L allows us to compute the desired orientation of the nodes according to the formation 13. Maintaining a triangular formation by defining set-points according to a mobile reference frame linked to the Leader (Leader-follower approach).

L v i T v L Icc j T v m X m Y i l i h i    , , L L L x y    , , i i i T T T x y  i  L UGV Virtual dynamic Target for the UGV i i d T j U G V Desired formation shape i UGV j d T L  G X G Y i U G V i v j v L c r T i c r   , j j j h l  f Fig.
shape. The leader turns around I cc L (positioned perpendicularly to its rear wheels), then the other target set-points T di must also turn around I cc L to maintain a rigid formation. Thus, the target velocity v Ti must be tangent to the circle which has I cc L as center and the distance between T di and I cc L as radius r c T i .

The idea behind this strategy is to eliminate the dependency of each vehicle to a global reference frame. A straightforward transformation can be applied to obtain the set-point w.r.t. a local reference frame attached to the leader. The polar coordinates (r i , Φ i ) can also be used by applying a straightforward transformation. An important advantage of the used Leader-follower approach is that it does not depend here on any reference trajectory and the formation is fully defined by the instantaneous dynamic of the leader. Furthermore, the presented approach is more reactive in the sense that it takes at each sample time only the current configuration and velocity of the Leader, instead of using the trajectory of the Leader as a reference for the formation [START_REF] Chen | Smooth formation navigation of multiple mobile robots for avoiding moving obstacles[END_REF], [START_REF] Shames | Safe formation control with obstacle avoidance[END_REF].

An important consideration to take into account to achieve the presented formation strategy, is that the followers have to know, as accurately as possible, the leader state (pose and velocity). It is assumed in what follows that the Leader sends its state by stable Wi-Fi communication without latency. However, cameras and/or LIDAR sensors embedded in each follower, can be used to estimate the leader state [START_REF] Das | A vision-based formation control framework[END_REF] [61] [START_REF] Vilca | A novel safe and flexible control strategy based on target reaching for the navigation of urban vehicles[END_REF]. In the sequel, f i is given in a global Cartesian frame to homogenize the notation of the equations [START_REF] Vilca | Adaptive leader-follower formation in cluttered environment using dynamic target reconfiguration[END_REF]. The pose of the virtual target T di w.r.t. the leader pose in the global reference frame can be written as (cf. Figure 13):

   x Ti = x L + h i cos(θ L ) -l i sin(θ L ) y Ti = y L + h i sin(θ L ) + l i cos(θ L ) θ Ti = θ L + β i (15)
where (x L , y L , θ L ) is the current pose of the leader and β i is the T di orientation w.r.t. the leader pose. It is given by:

β i = arctan (h i /(r c L -l i )) (16)
where r c L is the radius of curvature of the leader. Differentiating equation ( 15), the velocities of each T di are given thus by:

v Ti = (v L -l i ω L ) 2 + (h i ω L ) 2 (17)
ω Ti =ω L + βi (18)
where v L and ω L are respectively the linear and angular velocities of the leader, βi is computed as:

βi = -h i ṙc L / (r c L -l i ) 2 + (h i ) 2 . ( 19
)
One can note from equation ( 19) that when βi is equal to zero, the formation has a constant radius of curvature r c L and the angular velocities of the virtual targets are equal to the angular velocity of the leader (ω Ti = ω L ) [START_REF] Brooks | A robust layered control system for a mobile robot[END_REF].

Dynamic and smooth formation reconfiguration: It is presented summarily in what follows a new Strategy for Formation Reconfiguration (SFR) [START_REF] Vilca | Adaptive leader-follower formation in cluttered environment using dynamic target reconfiguration[END_REF] based on suitable smooth switches between different virtual target configurations. It is considered in the following a deterministic target assignment, and a label H i of the virtual target T di is assigned to UGV i at the beginning of the experiments. This label is kept by each UGV in the reconfiguration phase. It is important to notice that the new virtual targets (defined on the new formation shape) must be ahead of the UGVs to guarantee the stability of the overall system (the vehicle must not go back to reach the new virtual target). If this condition is not satisfied, then the former formation will be adapted by increasing smoothly and contentiously the longitudinal coordinates h i until all UGVs are positioned in the right configuration. The error between the coordinates of the former and the new formation e fi (e hi , e li ) is defined as:

e fi = f n i -f f i (20)
where

f f i (h f i , l f i ) and f n i (h n i , l n i )
are respectively the coordinates of the former formation and the new desired formation (cf. Figures 13).

The reconfiguration process between the different formation shapes is given by:

f i = h i = h n i -e hi e -kr(t-tr) , l i = l n i ; if e hi < 0 h i = h n i , l i = l n i ; if e hi ≥ 0 (21
) where f i (h i , l i ) are the coordinates of the current virtual target T di to be tracked by the follower UGV i . e hi is the longitudinal coordinate of e fi that enables to detect if the virtual target is ahead of its corresponding follower (e hi ≥ 0). The adaptation function when e hi < 0 (virtual target behind to follower i ) is set as proportional to the error between formation shapes, where k r is a real positive constant designed according to the dynamic of the leader and t r > 0 is the initial time for the reconfiguration process. Multi-controller architecture embedded in each robot in the formation.

B. Proposed MCA and Experimental validation

This section presents the proposed multi-controller architecture (cf. Figure 14) to obtain safe and smooth robot navigation in formation. A basic structure of type 2 (cf. Figure 6(b)) is used with notably the addition of a Formation Parameters block which determines the desired multirobot configurations. An overview of the different blocks composing this architecture is briefly presented below while emphasizing the new blocks/features.

• "Perceptions & communication" block: as emphasized in section II-C.1, this block is in charge of all the local and/or global robot perceptions. Furthermore, knowing that several robots have to coordinate their movements, it is important to have reliable and low-latency communication between the robots. • "Hierarchical action selection" block: it aims to manage the switches between the two elementary controllers, Obstacle avoidance and Dynamic target reaching blocks, according to the formation parameters and environment perception. It activates the Obstacle avoidance controller as soon as it detects at least one obstacle which can hinder the robot's future movement toward its assigned dynamic virtual target. • In terms of set-point blocks, they are harmonized as motivated in section II-C.3. Indeed, they are always defined according to an appropriate target pose (x T , y T , θ T ) and a linear velocity v T .

-"Obstacle avoidance set-point" block: This well detailed block, in section II-B, allows to each elementary robot to avoid reactively, and in a safe and reliable way, any obstructing obstacle. -"Dynamic target reaching set-point" block: These set-points are defined according to the assigned formation shape (e.g., triangle, line, etc.). All the robots (except the Leader) have to track their assigned dynamic target (given according to the desired formation).

• "Control law" block: A stable and generic common control law for target tracking, as given in section II-C.4 is used. Experimental validation: An experimentation was made using 3 VIPALABs. The objective is to validate the pro-posed strategies based on the Leader-follower approach and reconfiguration mechanism. Figure 15(a) shows the sequence of the MVS evolution, from the beginning with an initial triangular formation to a linear one, when the Leader detects an obstacle, and once the last follower detects the end of the obstacle, the formation returns to the triangular formation in a smooth way. Figure 15(b) shows the trajectories of the 3 VIPALABs, and it attests on the safety and the smoothness of the performed navigation in formation and its reconfiguration. This experiment can be found online. 5 More details about the proposed strategy which deals with the navigation in formation of a group of autonomous vehicles are given in [START_REF] Vilca | Adaptive leader-follower formation in cluttered environment using dynamic target reconfiguration[END_REF] and [START_REF] Vilca | Safe and flexible hybrid control architecture for the navigation in formation of a group of vehicles[END_REF]Chapter 5]. These references highlight also the way to guarantee that the assigned target's set-points are always attainable by the Followers, which allows thus to guarantee the reliability of the overall multivehicle navigation in formation.

V. CONCLUSION AND FURTHER WORK

This paper emphasizes the use of homogeneous Multi-Controller Architectures (MCAs) in order to increase gradually the autonomy of single vehicle as well as multivehicle systems to achieve complex navigation in cluttered and/or dynamic environments. Indeed, the obtainment of fully autonomous vehicle could be reached only if the overall used control architecture (perception-decision-action) can guarantee the safety and reliability of the navigation while allowing to maintain a high level of flexibility and adaptability to achieve different assigned tasks. This paper made a focus on the decision/action aspects, while emphasizing the importance to have generic and bottom-up developments of the MCA. Three closely related elements have been highlighted concerning MCA: (i) Importance of flexible and reliable obstacle avoidance controller (ii) Homogenization and standardization of task modeling: in terms of set-points definition and their corresponding stable control law; appropriate reference frames for task modeling/achievement. These specific reference frames guide for instance the vehicle's behaviors and allows us to evaluate the success of the current achieved sub-task (e.g., obstacle avoidance, targets reaching/tracking, etc.). Therefore, each elementary reference frame orientates locally the vehicle navigation. (iii) Navigation sub-tasks definition while using only appropriate attraction/tracking toward static/dynamic targets. This has been highlighted with two examples: flexible and reliable navigation based on sequential target reaching, and dynamic multi-vehicle navigation and formation reconfiguration. As main future work to enhance the features of MCAs, several works have been initiated to prove the stability and reliability of the overall MCA even in the presence of uncertainties (due for instance to the perception or to the vehicle modeling).

Fig. 1 .

 1 Fig. 1. Different autonomous vehicles with several use cases.

Fig. 2 .

 2 Fig. 2.Vehicle's pose and its elementary perceptions for performing reactive navigation. The parameters (h, k, a, b and Ω) characterize the SE i and the EI i (cf. section II-B).

Fig. 3 .

 3 Fig. 3. Shape possibilities for the used Elliptic Limit-Cycles (ELC).

  Fig.[START_REF] Lozenguez | Punctual versus continuous auction coordination for multi-robot and multi-task topological navigation[END_REF].Example of simple simulation highlighting the use of several PELC to reach the final assigned target. PELC paths given by "+" symbols mean invalid PELCs. The others elementary PELC in dotted colored lines show valid PELCs. The arrows given in solid lines correspond to the final PELCs' configurations. The numbering correspond to the successive computed PELCs in order to obtain the global optimized path based on PELC[START_REF] Adouane | Reactive versus Cognitive Vehicle Navigation based on Optimal Local and Global PELC[END_REF].

Fig. 5 .

 5 Fig.5. Reference frame linked to each PELC and different PELC shapes according to the value of µ. α and ϕ charaxterize the refernce frame linked to each obstcale[START_REF] Adouane | Reactive versus Cognitive Vehicle Navigation based on Optimal Local and Global PELC[END_REF].

  Fig.6. The two main used multi-controller structures.

Fig. 7 .

 7 Fig. 7. Set-points definition based on (a) global planned path, (b) local planned path, (c) general static/dynamic target.

Fig. 8 .

 8 Fig. 8. Vehicle and target configuration in Global and Local reference frames. Control variables according to Lyapunov synthesis are also shown.

Fig. 10 .

 10 Fig. 10. Proposed multi-controller architecture to perform autonomous vehicle navigation based on sequential target reaching [8].

2 :

 2 Switch from the current target Tj to the next sequential waypoint Tj+1 3: else 4:

Fig. 11 .

 11 Fig. 11. Some images from the performed navigation in urban environment.

Fig. 12 .

 12 Fig. 12. Vehicle trajectories obtained using GPS and a set of waypoints positioned in the environment.

  Fig. 14.Multi-controller architecture embedded in each robot in the formation.

http://www.braive.vislab.it/, consulted in May 2017.

https://goo.gl/gVgsIa


B Hichri

L Adouane
email: lounis.adouane@univ-bpclermont.fr

J-C Fauroux

Y Mezouar
email: youcef.mezouar@univ-bpclermont.fr

I Doroftei
email: idorofte@mail.tuiasi.ro

Cooperative Lifting and Transport by a Group of Mobile Robots

Keywords: Cooperative mobile robots, Payload co-manipulation and transport, Force closure grasping, Static stability margin, Control architecture

This paper addresses cooperative manipulation and transportation of any payload shape, by assembling a group of simple mobile robots (denoted m-bots) into a modular poly-robot (p-bot). The focus is made in this paper on the chosen methodology to obtain sub-optimal positioning of the robots around the payload to lift it and to transport it while maintaining a geometric multi-robot formation. This appropriate positioning is obtained by combining the constraint to ensure Force Closure Grasping (FCG) for stable and safe lifting of the payload and the maximization of the Static Stability Margin (SSM) during the transport. A predefined control law is then used to track a virtual structure in which each elementary robot has to keep the desired position relative to the payload. Simulation results for an object of any shape, described by a parametric curve, are presented. Additional 3D simulation results with a multi-body dynamic software validate our proposal.

Introduction

In recent years, many researches were oriented to survey and design collaborative mobile robotic systems [START_REF] Siegwart | Introduction to Autonomous Mobile Robots[END_REF][START_REF] Souma | Mobile Robotics, Moving Intelligence[END_REF] gathering different engineering and science disciplines. This blend between those disciplines allows the design of autonomous systems able to interact with the environment without human mediation and also to achieve diverse complex tasks or infeasible by humans, such as exploring dangerous and/or unreachable areas [START_REF] Wilcox | Athlete: A cargo handling and manipulation robot for the moon[END_REF] or navigation in formation for a group of autonomous robots [START_REF] Benzerrouk | Navigation of multi-robot formation in unstructured environment using dynamical virtual structures[END_REF]. Autonomous mobile robots have the ability for sensing and reacting in the environment by acquiring additional abilities. They can also collaborate when a task needs more than one robot, such as heavy objects co-manipulation or transport [START_REF] Ijspeert | Collaboration Through the Exploitation of Local Interactions in Autonomous Collective Robotics: The Stick Pulling Experiment[END_REF][START_REF] Wilcox | Athlete: A cargo handling and manipulation robot for the moon[END_REF][START_REF] Dorigo | Swarmanoid: "a novel concept for the study of heterogeneous robotic swarms[END_REF][START_REF] Adouane | Hybrid behavioral control architecture for the cooperation of minimalist mobile robots[END_REF]. The aim of our research is to co-manipulate and to transport objects using a group of mobile robots. We aim to design an innovative architecture for payload transport on structured environment. Collaborative robots behaviors may be also interesting for transporting tasks with mobile robots. Many robotic examples can be mentioned such as in [START_REF] Yamashita | Cooperative Manipulation of Objects by Multiple Mobile Robots with Tools[END_REF][START_REF] Dorigo | Swarmanoid: "a novel concept for the study of heterogeneous robotic swarms[END_REF][START_REF] Bay | Design of the army-ant cooperative lifting robot[END_REF][START_REF] Abou-Samah | Optimal configuration selection for a cooperating system of mobile manipulators[END_REF][START_REF] Kernbach | Symbiotic Robot Organisms: REPLICATOR and SYMBRION Projects[END_REF]. Our goal in the C 3 Bots project (Collaborative Cross and Carry mobile roBots) is to design several mobile robots with a simple mechanical architecture called m-bots that will be able to autonomously comanipulate and transport objects of any shape by connecting together. The resulting poly-robot system, called p-bot, will be able to solve the so-called removal-mantask to transport object of any shape and mass repartition. Reconfiguring the p-bot by adjusting the number of m-bots allows to manipulate heavy objects with any shape, particularly if they are wider than a single m-bot. During the manipulation, the grasping task [START_REF] Sahbani | An overview of 3D object grasp synthesis algorithms[END_REF][START_REF] El-Khoury | 3D objects grasps synthesis: a survey[END_REF] is a crucial phase for payload lifting and if it fails the whole task cannot be achieved.

To ensure the co-manipulation task, he group of m-bots must succeed to ensure the payload Force Closure Grasping (FCG) [START_REF] Sahbani | An overview of 3D object grasp synthesis algorithms[END_REF][START_REF] Ding | Computing 3-D optimal form-closure grasps[END_REF][START_REF] Li | A new algorithm for three-finger force-closure grasp of polygonal objects[END_REF][START_REF] Roa | Finding locally optimum force-closure grasps[END_REF][START_REF] El-Khoury | 3D objects grasps synthesis: a survey[END_REF][START_REF] Liu | Qualitative test and force optimization of 3-D frictional form closure grasps using linear programming[END_REF] until putting it on their top platform. FCG refers to Newton laws which allows to ensure the payload immobility [START_REF] El-Khoury | 3D objects grasps synthesis: a survey[END_REF]. In the aim of ensuring object stability, which is the goal of any used grasping strategy, several methods have been developed using various approaches. Avoiding too large forces allows to reduce the power for the manipulator's actuation and the deformation of the manipulated object. A grasp is considered stable when a miniature disturbance on the position of the manipulated object or contact force, generates a restoring wrench that brings the system back to a stable configuration [START_REF] Sahbani | An overview of 3D object grasp synthesis algorithms[END_REF]. In [START_REF] Nguyen | Constructing Stable Grasps in 3D[END_REF], Nguyen presents an algorithm for stable grasps construction and he proved the possibility of making stable all 3D force closure grasps. According to [START_REF] Sahbani | An overview of 3D object grasp synthesis algorithms[END_REF][START_REF] El-Khoury | 3D objects grasps synthesis: a survey[END_REF], a grasping strategy should ensure stability, task compatibility and adaptability to novel objects. Analytical and empirical approaches were developed in different literatures to ensure a stable grasping. The former approach choose the manipulator configuration and contact positions with kinematical and dynamical formulation whereas empirical approaches use learning to achieve a grasp depending on the task and on the geometry of the object. Diverse analytical methods were developed to find a force closure grasp [START_REF] Ding | Computing 3-D optimal form-closure grasps[END_REF][START_REF] Li | A new algorithm for three-finger force-closure grasp of polygonal objects[END_REF][START_REF] Liu | Qualitative test and force optimization of 3-D frictional form closure grasps using linear programming[END_REF]. The latter approach avoids the complexity of computation by attempting to mimic human strategies for grasping. Datagloves and map human hand were used by researchers for empirical approaches to learn the different joint angles [START_REF] Fischer | Learning techniques in a dataglove based telemanipulation system for the DLR hand[END_REF][START_REF] Ekvall | Interactive grasp learning based on humain demonstration[END_REF], hand preshape [START_REF] Kyota | Detection and evaluation of grasping positions for autonomous agents[END_REF]. Vision based approach is also used to demonstrate grasping skills. A robot can track an operator hand for several times to collect sufficient data [START_REF] Aarno | Early Reactive Grasping with Second Order 3D Feature Relations[END_REF][START_REF] Hueser | Learning of demonstrated grasping skills by stereoscopic tracking of human head configuration[END_REF].

Payload stability during movement is evaluated according to developed metrics in literature. In the late sixties, stability margin metrics were developed and classified mainly in two categories: static [START_REF] Mcghee | On the stability properties of quadruped creeping gaits[END_REF]- [START_REF] Orin | Interactive compute-control of a six-legged robot vehicle with optimization of stability, terrain adaptibility and energy[END_REF] and dynamic [START_REF] Papadopoulos | A new measure of tipover stability margin for mobile manipulators[END_REF]- [START_REF] Grand | Stability and Traction Optimization of a Reconfigurable Wheel-Legged Robot[END_REF] stability margins. We consider the Static Stability Margin (SSM) since our system evolves at low speed in a structured environment. This margin was defined by McGhee and Frank [START_REF] Mcghee | On the stability properties of quadruped creeping gaits[END_REF] as follows: "static stability margin is the shortest distance from the vertical pro-jection of the centre of gravity to any point on the boundary of the support pattern". Considering the payload lifting and transport using mobile robots, stability is also ensured by coordinating the group of transporting robots which means multi-robot control problem. The multi-robot navigation in formation is the main research area linked to the phase of payload transportation. A multitude of control architecture to deal with this task were proposed in the literature [START_REF] Benzerrouk | Navigation of multi-robot formation in unstructured environment using dynamical virtual structures[END_REF][START_REF] Adouane | Architectures de controle comportementales et reactives pour la cooperation d'un groupe de robots mobiles[END_REF][START_REF] Van Den Broek | Formation control of unicycle mobile robots: a virtual structure approach[END_REF]. A multi-robot system control can be either centralized or distributed. The control problem is discussed to provide a suitable control strategy for this task. Formation control can be classified according to recent literature, [START_REF] Benzerrouk | Navigation of multi-robot formation in unstructured environment using dynamical virtual structures[END_REF][START_REF] Van Den Broek | Formation control of unicycle mobile robots: a virtual structure approach[END_REF], into three main approaches: the behaviorbased approach, the leader-follower approach and the virtual structure approach.

This paper presents an algorithm allowing to determine an optimal positioning of m-bots around a general payload in order to maximize the Static Stability Margin (SSM) and to ensure Force Closure Grasping (FCG). A centralized control will be used for its higher calculation performances to calculate different desired positions according to a payload of any shape. For targets reaching and payload transport, the groups of robots will act according to centralized control approach. A predefined control law is then used to track a virtual structure in which each elementary robot has to keep the desired position relative to the payload. This paper is organized as follow: in Section 2 the paradigm of C 3 Bots project is introduced and the general problem is presented for co-manipulation and transport using multi-robot system; Section 3 will present the robots positioning according to both criteria SSM and FCG computation and the multi-robot transport strategy. Simulation results for an object of any shape, described by a parametric curve, and 3D simulations with a multi-body dynamic software are also presented. Finally Section 4 provides a conclusion and future works.

Paradigm and problem statement

The paradigm of C 3 Bots project is to co-manipulate and transport a common payload through collaboration between several similar elementary robots (see Fig. 1). Wheeled robots were selected for their versatility on various terrains and good efficiency on regular grounds compared to legs and tracks. The C 3 Bots transport strategy takes inspiration from Army Ants [START_REF] Bay | Design of the army-ant cooperative lifting robot[END_REF] by laying the payload on top of robot's bodies, and from the structure given in [START_REF] Abou-Samah | Optimal configuration selection for a cooperating system of mobile manipulators[END_REF], that has a rotative arm on top of it. The concept of modularity was also kept and each m-bot is built from two parts: a mobile platform and a manipulation mechanism [START_REF] Hichri | Design of Collaborative, Cross and Carry Mobile RoBots (C3Bots)[END_REF]. The mobile platform is a single-axle Khepera robot and the manipulator is fixed on a rotary platform that lets the robot turn freely on itself when the object lays on the transporting platform. The manipulator has a parallelogram structure to bring the payload from the ground to the m-bot top platform with a circular trajectory [START_REF] Hichri | Lifting Mechanism for Payload Transport by Collaborative Mobile Robots[END_REF].

The resulting p-bot system (cf. Fig. 1(a), Fig. 1(c)) is thus allowed to translate along any direction and rotate around any point in the ground plane. Before starting the transport task, the m-bots have to achieve the co-manipulation process using the mechanism presented in [START_REF] Hichri | Design of Collaborative, Cross and Carry Mobile RoBots (C3Bots)[END_REF] and detailed in [START_REF] Hichri | Lifting Mechanism for Payload Transport by Collaborative Mobile Robots[END_REF]. Its role is to hold firmly the payload and to ensure FCG [START_REF] Roa | Finding locally optimum force-closure grasps[END_REF] to lift the object by applying a sufficient normal force f m,p,n (see Fig. 1(d) and Fig. 4) with:

⃗ z m +1 f tpm µ 2 f npm f ngm f tgm µ 1 ⃗ x m G i ⃗ y m ⃗ x m ⃗ z m C m , g (µ g ) C m+1, g (µ g ) f m+1, g , n C m+1, p (µ p ) f m+1, g ,t f m , g ,n f m , g ,t f m , p , t f m+1, p , t f m , p , n C m , p (µ p ) f m+1, p , n ⃗ x m+1 G m +1
f m,p,n ∈ [0, f max ] = [0, µ p µ g m r g] (1)
The value of f max is obtained while applying the well known resultant of the force/moment for the all system (First and Second principle of Newton). We obtained thus a simple formulation of f max while taking into account: µ p the payloadend-effector friction coefficient; µ g the wheel-ground friction coefficient; m m is the robot mass and g is the gravity.

The minimum number m min of m-bots that have to be used to lift and transport the payload is obtained according to the following equation:

m min ∑ m=1 f m,p,t = M pl g (2)
Where f m,p,t is the generated vertical tangential force to lift the object (Fig. 1(d)).

Cooperative mobile robot manipulation and transport

The proposed overall cooperative manipulation and transport strategy, for any payload shape, by a group of m-bots is presented in Figure 2. This figure gives the most important steps to be achieved during this cooperative task. The details of the chosen criteria for cooperative manipulation and transportation are given respectively in sub-sections 3.1 and 3.2.

Step 1 (cf. Fig. 2) presents the first phase of the task and which consists on payload detection and estimation of its mass and gravity center position. Step 2 consists on determining the minimum number mmin of m-bots that could be used to ensure the payload lifting and transport with relative to (2). Step 3 presents the main contribution of this paper. It is detailed by the flowchart in the right side of Fig. 2 and will be discussed in sections 3.1.1 and section 3.1.2. Sasaki in [START_REF] Sasaki | Cooperating grasping of a large object by multiple mobile robots[END_REF] treated a similar problem for optimal robots positioning taking into account two criterion: the payload stability and the energy consumption. It was considered that the positioning is optimal when the payload is stable and the robots consume the minimum of energy (according to the data received from the robots sensors). In the proposed strategy, the m-bots positioning is optimal when FCG and SSM are ensured. Finally, Step 4 corresponds to multi-robot transport the payload toward the assigned final pose, this part will be detailed in section 3.2. Step 1

Step 2

Step 3

Step 4

Step 3

Fig. 2 Flowchart for co-manipulation with C 3 Bots

Cooperative m-bots positioning and co-manipulation

Since the features of the payload are known (step 1 in Fig. 2) the minimum number of m-bots (m min ) is obtained while using equation 2 (step 2), the group of m-bots must be well positioned around the payload (step 3) to permit to safely lift it and to maintain a well stability of the payload in the top of the p-bot during the transportation phase (step 4). During this manipulation phase (sub-step 2 in step 4), FCG (cf. sub-section 3.1.1) as well as SSM (cf. sub-section 3.1.2) must be thus ensured to lift and transport safely the object (cf. details given for Step 3 in Figure 2).

Force Closure Grasping

Force closure grasping problem is extensively treated and studied for objects manipulation using multi fingered robotic hand [START_REF] Yoshikawa | Multifingered robot hands: Control for grasping and manipulation[END_REF][START_REF] Zheng | Limiting and minimizing the contact forces in multifingered grasping[END_REF]. This problem was adapted to mobile robot co-manipulation and transport in C 3 Bots project to ensure lifting and transport task. The co-manipulation problem (cf. section 2) is restricted to a 2D problem in plane (O, x, y) while robots are acting simultaneously and applying a tightening forces on the payload on the same plane (Fig. 3). The aim of this part is to ensure force closure grasping when choosing the m-bots positions which returns to fully constraint the payload motion with m min m-bots. In other words, the static equilibrium must be ensured while positioning the group of mobile robots. The problem of force closure grasping is studied under the following assumptions (cf. Fig. 3(c)):

• A contact force lies inside the friction cone centred about the normal direction to the contact surface with half angle α. • The tangent of α represent the friction coefficient.

• The friction cone of the m th contact is denoted C pm .

A necessary and sufficient condition to have force closure is that the intersection of three friction cones is not empty [START_REF] Liu | Qualitative test and force optimization of 3-D frictional form closure grasps using linear programming[END_REF]. This condition was extended to m min m-bots. In [START_REF] Liu | Qualitative test and force optimization of 3-D frictional form closure grasps using linear programming[END_REF], the treated problem concerns multi fingered hand grasping although the problem treated in this paper focuses on co-manipulation using a group of modular mobile robots. The proposed algorithm is based on ensuring force closure if forces and moments equilibrium satisfy (3) and when the payload center of mass is inside the friction cones intersection (4). The later condition allows to reduce the moments generated on the payload by the m-bots because the direction of the applied force on the plane is closer to the gravity center.

m min ∑ m=1 (P m G pl ⊗ f m,p,n ) = 0; m min ∑ m=1 f m,p,n = 0 (3) G pl ∈ Convexhull(∩C pm ) |m = 1..m min (4)
Where C pm presents the friction cone for the contact force on P m and f m,p,n is the applied normal on the payload (cf. Fig. 3(c)).

Static Stability Margin (SSM)

In this part, Static Stability Margin (SSM) is considered to ensure the payload stability during the transporting phase. Stability margins were extensively studied for walking mobile robots [START_REF] Queiroz | A Study on Static Gaits for a Four Legged Robot[END_REF][START_REF] Estremera | Continuous free-crab gaits for hexapod robots on a natural terrain with forbidden zones: An application to humanitarian demining[END_REF][START_REF] Wang | Mobility analysis of the typical gait of a radial symmetrical six-legged robot[END_REF]. In C 3 Bots project, to ensure a stable payload transport, the Static Stability Margin (SSM) is a crucial criterion for a successful task achievement. Before describing the proposed algorithm for m-bots positioning ensuring an optimal SSM during object transport using m-bots, let's detail the following assumptions (cf. Fig. 4):

• The payload shape from the top view is a closed curve (B) and defined by polar curve defined by P(θ ); θ ∈ [0, 2π]. • In function of the payload mass M p , m min is the minimum number of m-bots allowing to lift and transport the object. • The payload center of mass is denoted G pl .

Let R(G pl , x pl , y pl , z pl ) be the frame linked to the payload with respect to the reference frame R(O, x, y, z) (cf. Fig. 4). Cartesian coordinates will be used in the proposed algorithm. As given in section 2, P(θ ) be the parametric description of the payload closed boundary (B). P m|m=1..m min are the m-bots positions, H m,m+1 is the projection of the payload center of mass G on the edge linking two consecutive points P m and P m+1 and d m,m+1 is the stability margin on the same edge. P m and P m min +1 are confounded and as a consequence d m,m min +1 is equal to d m min ,1 .

The idea behind the algorithm is to run through (B) and to find the set of points P m ensuring a maximal SSM while maximizing the objective function [START_REF] Hichri | Design of Collaborative, Cross and Carry Mobile RoBots (C3Bots)[END_REF]. The constraint imposed by ( 6) must be satisfied for m min m-bots≥3 which gives a necessary condition to keep the center of mass G pl inside the polygon (P 1 ..P m )

f (θ m , ..θ m min ) = Min(d m,m+1 )|m = 1..m min (5) ⃗ f 1, p , n ⃗ f 2, p , n ⃗ f 4, p , n ⃗ f 3, p , n P 3 P 4 ⃗ x ⃗ y θ 1 ⃗ y pl ⃗ x
θ m+1 -θ m < π |m = {1...m min } (6)
In the case where we have only two m-bots to co-manipulate the object, the constraint expressed by ( 6) is not considered and the robots are positioned in opposed positions which means θ m+1θ m = π. For each configuration where n m-bots ≥ 3, the algorithm aims at determining the equation of the line P m P m+1 and at computing the shortest distance of G pl (x G pl , y G pl ) from it.

Then d m,m+1 is calculated by [START_REF] Wilcox | Athlete: A cargo handling and manipulation robot for the moon[END_REF] which represent the stability margin relative to each edge and the static stability margin SSM given by [START_REF] Hichri | Design of Collaborative, Cross and Carry Mobile RoBots (C3Bots)[END_REF]. P m coordinates are expressed in R(G pl , x pl , y pl , z pl ) (cf. Fig. 4).

Simulation results

The proposed algorithm allows to determine a sub-optimal configuration for a group of mobile robots in order to lift and transport a payload of any shape. Two criteria have been respected (FCG and SSM) which reduces the total configurations to be tested by the algorithm taking into consideration (3) and (4). The Algorithm was simulated by using an Intel Core i5 2400 CPU 3.1 GHz system. Fig. 5 presents the simulation results for the developed algorithm for robots positioning in order to guarantee an optimal static stability margin respecting the force closure condition. The blue bold polygon presents the polygon of support ensuring the optimal SSM (cf. subsection 3.1.2), the thin blue lines presents the friction cones sides and the intersection is presented by contrasted area resulted by the superposition of friction cones. It is shown how the algorithm keeps the payload center of mass G pl inside the intersection area and it allows to build a polygon of support ensuring the payload stability during the transport. The duration to find results depends on the chosen steps of θ m to run throw the payload curve. The payload stability during the lifting phase was simulated with respect to both criteria (SSM and FCG) using ADAMS multi-body dynamic software to validate the proposed algorithm (cf. Fig. 2) while testing the m-bots performances when they are positioned to co-manipulate the object. Fig. 6 shows that the robots ensure the payload lifting without loss of stability of the lift. Videos for simulation are visible under [33].

Multi-robot transport

After lifting the payload, which is positioned now on the top of the p-bot, the group of m-bots must transport the payload toward a final configuration. During this last phase (Step 4 in Fig. 2), and in order to guarantee the payload stability, the p-bot should navigate as rigid formation shape and for this, a virtual structure architecture was used [START_REF] Benzerrouk | Navigation of multi-robot formation in unstructured environment using dynamical virtual structures[END_REF]. After the end of Step 3, each m-bot receives its attributed position which insures the sub-optimal p-bot positioning that permits to ensure Force Clo-sure Grasping (FCG) and to maximize the Static Stability Margin (SSM) during the transport. For transport task, the m-bots have to reach their goals, computed using the algorithm presented in the previous section (cf.

Step 4 in Fig. 2). After reaching the desired positions, the transport task starts considering that the payload lays on robots bodies. To avoid payload slippage, the group of m-bots has to track a fixed position relative to the object when it follows a trajectory. In this section, a control law is proposed to solve the goal reaching problem (Pm in section. 2) and the navigation as Virtual Structure (VS) of the set of m-bots. In VS approach [START_REF] Van Den Broek | Formation control of unicycle mobile robots: a virtual structure approach[END_REF] [1], the entire formation is considered as a rigid body and the notion of hierarchy do not exist. The control law for each entity is derived by defining the VS dynamics and then translate the motion of the VS into the desired motion of each elementary robot.

The main advantages of this approach are its simplicity to prescribe the coordinate behavior of the group and the maintaining of the formation during manoeuvres. The result of the algorithm for a given object shape described by a parametric curve (B) is a set of n targets to be reached by the m-bots. Considering a unicycle mobile robot, the state vector X m = (x m , y m , θ m ) T denotes the position of the m th robot center of mass G m (x m , y m ) and its orientation θ m with respect to x axis of the global frame. The m-bots control inputs are the forward velocity V and the angular velocity ω. Let e be the error between the m-bot current pose and the desired pose defined by X dm = (x dm , y dm , θ dm ) T : e = X dm -X m .

After positioning the m-bots, they must keep their desired position (x dm , y dm ) with respect to the payload center of mass G pl and must respect the following conditions during the task achievement:

x dm = x G + l xm cos θ dml ym sin θ dm y dm = y G + l xm sin θ dm + l ym cos θ dm [START_REF] Grand | Stability and Traction Optimization of a Reconfigurable Wheel-Legged Robot[END_REF] where l xm and l ym are the relative distances G m G pl according the axis x and y respectively. These two distances define rigid links maintaining the robot position with respect to G pl . The used control law [START_REF] Benzerrouk | Navigation of multi-robot formation in unstructured environment using dynamical virtual structures[END_REF] is given by ( 9):

V m = V max -(V max -V d )e -(d 2 m /σ 2 ) ω m = ω Sm + kθ m (9)
• V m and ω m are the linear and angular velocities of the m-bot.

• V max is the maximum linear speed of the m-bot.

• V d is the desired velocity of the p-bot and considered to be constant.

• d m = e 2

x + e 2 y is the current distance between the m th robot and its desired target. • ω Sm is the angular velocity of set point angle θ Sm applied to the robot in order to reach the desired goal: ω Sm = θSm • σ , k are positive constants (control law gains).

The control law was simulated for a group of three m-bots transporting an object. Fig. 7(a) presents the goal reaching problem with k=22 and σ = 0.1. In order that the m-bots reach the desired positions, the desired speed when reaching the goal is set to zero and then the whole structure will navigate with a speed of 10 cm/s One can note that all m-bots keep a null position errors which means that the formation is properly maintained and that slippage avoidance and task performance are ensured. It is important to notice, that in this paper, we suppose a centralized control of the fleet of robots, thus, the movement of the virtual structure is already defined according to the configuration of the environment. Indeed, the focus of this paper is on the presentation of the virtual structure and the way how each elementary robot keeps the desired position relative to the payload.

CONCLUSIONS AND FUTURE WORK

This work takes place within the C3Bots project, that aims to design simple robot entities (m-bots) able to co-manipulate and transport payloads of any shape by ag- gregating in a modular way into a poly-robot (p-bot). This work has the ambition to combine two criteria in an original way:

• On one side, the Static Stability Margin (SSM), generally used for legged locomotion. • On the other side, Force Closure Grasping (FCG), used for stable multi-finger manipulation.

The m-bots used in this work include in their lower part a wheeled-axle, which is similar to a foot of a multi-leg mobile robot, and in their top part a manipulator acting like the finger of a robotic hand. The resulting p-bot ensures the stable payload grasping and transport. An algorithm was developed in order to search the optimal positions of n unicycle m-bots that ensure force closure grasping and maximize the static stability margin for the transport of a payload of any shape, defined by its closed curve boundary. Simulation results using a multi-body dynamic software validates our proposal and shows the ability of robots to maintain the payload stability during lifting process. A flexible control architecture was used to validate the target reaching problem while maintaining the chosen formation. This navigation was considered in a flat structured environment. Future works will consider the problem of payload manipulation and lifting in all terrain. Unreachable areas on the payload boundary will also have to be taken into consideration (as for example one side of a square object opposed to a wall).

  (a) Prototype for object lifting and transport [5] (b) Payload prehension by two m-bots (c) Payload lifted by two m-bots

Fig. 1

 1 Fig. 1 Co-manipulation of a box by a group of m-bots

  of the minimum number of m-bots to lift the payload (cf. Eq. 2) Generate the initial grasp (it=1) that ensures a SSM (cf. Eq. 4detection and estimation of Mpl and Gpl Determine the appropriate m-bots configuration Go forward to the obtained positions Lift the payload Transport the payload toward a final configuration while keeping a specific formation

Fig. 3

 3 Fig. 3 Applied tightening forces on the payload

Fig. 4

 4 Fig. 4 Support polygon formed by four robots positioned at P m|m=1..4

  d m,m+1 = d(G, (P m P m+1 )) = x G y P m+1 -y Pm x P m+1 -x Pmy G + y P mx P m y P m+1 -y Pm x P m+1 -x Pm ( y P m+1 -y Pmx P m+1 -x Pm ) 2 + 1[START_REF] Wilcox | Athlete: A cargo handling and manipulation robot for the moon[END_REF]

Fig. 5

 5 Fig. 5 Simulation results: a) 3 m-bots positioning; b) 4 m-bots positioning

Fig. 6

 6 Fig. 6 Multibody simulation results with ADMAS software: Top view (a and c), and 3D lifting phase (b and d).

Fig. 7 M

 7 Fig. 7 M-bots target reaching (TR): a) Trajectories of the m-bots reaching the desired positions; b) Position error for TR; c) Angle Error for TR.

Fig. 8

 8 Fig. 8 Virtual structure (VS) navigation: a) The p-bot is navigating as a rigid Virtual Structure (VS); b) The p-bot avoids the obstacle and keeps the same orientation; c) The p-bot avoids obstacle and changes the payload orientation.

Acknowledgements The C 3 Bots project acknowledges the following entities: LABEX IMobS3 Innovative Mobility: Smart and Sustainable Solutions, the French National Centre for Scientific Research (CNRS), Auvergne Regional Council and the European funds of regional development (FEDER).