Robust Adaptive Control of Automatic Guidance of Farm

Vehicles in the Presence of Sliding H. Fang, R. Lenain, B. Thuilot and P. Martinet LASMEA 24, av. des Landais 63177 Aubiere Cedex France hao@lasmea.univ-bpclermont.fr Abstract-High-precision autofarming is rapidly becoming a reality with the requirements of agricultural applications. Lots of research works have been focused on the automatic guidance control of farm vehicles, satisfactory results have been reported under the assumption that vehicles move without sliding. But unfortunately the pure rolling constraints are not always satisfied especially in agriculture applications where the working conditions are rough and not expectable. In this paper the problem of path following control of autonomous farm vehicles in presence of sliding is addressed. To take sliding effects into account, a vehicle-oriented kinematic model is constructed in which sliding effects are introduced as additive unknown parameters of the ideal kinematic model. Based on backstepping method a stepwise procedure is proposed to design an adaptive controller in which timeinvariant sliding effects are learned and compensated by parameter adaptations. It is theoretically proven that for the farm vehicles subject to sliding, the lateral deviation can be stabilized near zero and the orientation errors converge into a neighborhood near the origin. To be more robust to disturbances including external noises and unmodeled time-varying sliding components, the adaptive controller is refined by integrating Variable Structure Controllers (VSC) or projection mappings. Simulation results show that the proposed robust adaptive controllers can reject sliding effects and guarantee high path-following accuracy.

Index Terms-Robust Adaptive Control, Nonholonomic Systems, Backstepping Control

I. INTRODUCTION

High-precision auto-farming and precise automatic guidance of agricultural vehicles have been the subject of research for a long time. Lots of satisfactory results of path following control have been achieved provided vehicles satisfy the pure rolling constraints [START_REF] Stombaugh | Automatic Guidance of Agricultural Vehicles at Higher Speeds[END_REF]-[3]. In our previous works we have solved the problem of curved path following of farm vehicles with unique RTK GPS [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single CP-DGPS[END_REF], the ideal vehicle kinematic model is constructed, a nonlinear controller guaranteeing high lateral and orientation accuracy has been designed by converting vehicle kinematic model into a chained system.

However due to various effects such as slipping of tires, deformability or flexibility of wheels, the pure rolling constraints are never strictly satisfied. Especially in agriculture applications when farm vehicles are required to move on all-terrain grounds including slippery slopes, sloppy grass grounds, sandy and stony grounds, where contact conditions between wheels and grounds are unexpectable and change greatly, sliding inevitably occurs which deteriorates performances of automatic guidance and even system stability.

Until now there are very few papers dealing with sliding. [START_REF] Ackermann | Robust control prevents car skidding[END_REF] prevents cars from skidding by robust decoupling of car steering dynamics which is achieved by feedback of the integrated yaw rate into front wheel steering, but acceleration measurements are necessary and the steering angle is assumed small. [START_REF] Motte | Control of sliding mobile robots : a slow manifold approach[END_REF] copes with the control of WMR (Wheeled Mobile Robot) not satisfying the ideal kinematic constraints by using slow manifold methods, but the parameter characterizing the sliding effects is assumed to be exactly known. Therefore [8][9] are not realistic for agriculture applications. In [START_REF] Leroquais | Vibrational control of wheeled mobile robots not satisfying ideal velocity constraints: the unicycle case[END_REF] a controller is designed based on the averaged model allowing the tracking errors to converge to a limit cycle near the origin. In [START_REF] D'andrea-Novel | Control of wheeled mobile robots not satisfying ideal constraints: a singular perturbation approach[END_REF] a general singular perturbation formulation is developed which lead to robust results for linearizing feedback laws ensuring trajectory tracking. But above two schemes only take into account sufficiently small skidding effects and they are too complicated for real-time practical implementation. In [START_REF] Zhang | Variable structure control of a differentially steered wheeled mobile robot[END_REF] [START_REF] Fang | Sliding Mode Control of Automatic Guidance of Farm Vehicles in the Presence of Sliding[END_REF] variable structure control is used to eliminate the harmful sliding effects when the bounds of the sliding effects have been known. The trajectory tracking problem of mobile robots in the presence of sliding is solved in [START_REF] Corradini | Experimental testing of a discretetime sliding mode controller for trajectory tracking of a wheeled mobile robot in the presence of skidding effects[END_REF] using discrete-time sliding mode control which is robust against the sliding effects. But the controllers [START_REF] Zhang | Variable structure control of a differentially steered wheeled mobile robot[END_REF]- [START_REF] Corradini | Experimental testing of a discretetime sliding mode controller for trajectory tracking of a wheeled mobile robot in the presence of skidding effects[END_REF] counteract sliding effects only relying on high-gain controllers which is not realistic because of low level delay introduced by steering systems of farm vehicles. In [START_REF] Lenain | Adaptive control for car like vehicles guidance relying on RTK GPS: rejection of sliding effects in agricultural applications[END_REF] the sliding effects are rejected by re-scheming the desired path adaptively based on the steady control errors which are mainly caused by the modeled sliding effects, the actual implementation has validated its effectiveness but the proof of convergence is still open.

In above references most research works treated sliding as disturbance, but alternatively sliding can be also regarded specifically as time-varying parameters. On the other hand backstepping methods which have been used widely in controller design are proven powerful in controlling nonholonomic systems with uncertain parameters [START_REF] Jiang | A recursive technique for tracking control of nonholonomic systems in chained form[END_REF] [START_REF] Zhang | Backstepping control of linear time-varying systems with known and unknown parameters[END_REF], so the purpose of this paper is to use backstepping methods to design a practical path following controller in presence of sliding with small control gains. It has been well known that vehicle positions and velocities cannot be measured by relative localization systems for example incremental encoders when sliding occurs, so absolute localization is necessary for anti-sliding control. The absolute localization technologies include active beacon, GPS, laser range scans and multi-sensor data fusion etc. In addition, with the development of GPS technology more and more researchers apply GPS to automatic guidance systems of agricultural vehicles [START_REF] Conner | Automatic Steering of Farm Vehicles Using GPS[END_REF]- [START_REF] Cordesses | GPSbased Control of a Land Vehicle[END_REF], since GPS can provide realtime absolute positions with centimeter accuracy and outdoor working environments of agricultural vehicles are suitable for using GPS. Due to those two points and our previous works [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single CP-DGPS[END_REF], in this paper the problem of anti-sliding controller design is addressed for the farm vehicles with GPS localization systems.

The main idea of this paper is to regard sliding effects as unknown parameters added to the ideal kinematic model. Based on backstepping method an adaptive controller is designed to estimate and compensate modeled time-invariant sliding components. Furthermore two approaches are proposed to refine the adaptive controller, making it more robust to the time-varying sliding effects and disturbances. This paper is organized as follows, in section II a vehicleoriented kinematic model considering sliding is constructed in the path frame. In section III by assuming sliding is timeinvariant, an adaptive controller is designed using backstepping methods. In section IV robust adaptive controllers are obtained by integrating VSC or applying projection mapping. In section V some comparative simulation results are presented to validate the proposed control law.

II. KINEMATIC MODEL

A. Notation and Problem Description

In this paper the vehicle is simplified into a bicycle model. The kinematic model is expressed with respect to the path in frame (M, η t , η n ), variables necessary in the kinematic model are denoted as follows: (see figure 1)

• C is the path to be followed.

• O is the center of the vehicle virtual rear wheel.

• M is the orthogonal projection of O on path C. • η t is the tangent vector to the path at M . • η n is the normal vector at M . • y is the lateral deviation between O and M • s is the curvilinear coordinates (arc-length) of point M along the path from an initial position. • c(s) is the curvature of the path at point M .

• θ d (s) is the orientation of the tangent to the path at point M with respect to the inertia frame. • θ is the orientation of the vehicle centerline with respect to the inertia frame. • θ = θθ d is the orientation error.

• l is the vehicle wheelbase.

• v is the vehicle linear velocity.

• δ is the steering angle of the virtual front wheel So the vehicle movement can be described by (y, s, θ). In this paper path following control law (2

δ = K(s, y, θ, v) (1) s = 0 θ d θ ṡ v η t η n M y δ θ C 1 c(s) o R(s)
)
and θ is bounded in presence of sliding.

B. Kinematic Model

When the vehicles move without sliding, the ideal kinematic model of the vehicles is (see [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single CP-DGPS[END_REF]).

           ṡ = v cos θ 1 -c(s)y ẏ = v sin θ θ = v tan δ l - c(s) cos θ 1 -c(s)y (3)
But when the vehicles move on a steep slope or the ground is slippery, sliding occurs inevitably, (3) is no longer valid. The violation of the pure rolling constraints is described by introducing the lateral sliding velocity v y and steering angle bias δ b (figure 2). Similar developing methods lead to

           ṡ = v cos θ 1 -c(s)y - v y sin θ 1 -c(s)y ẏ = v sin θ + v y cos θ θ = v tan(δ + δ b ) l - c(s) cos θ 1 -c(s)y - v y l + c(s)v y sin θ 1 -c(s)y (4) Since δ b is quite small, a linearized kinematic model can be obtained    ẏ = v sin θ + ψ(t) cos θ θ = v tan δ l - c(s) cos θ 1 -c(s)y + v l η(t) + c(s) sin θ 1 -c(s)y - 1 l ψ(t) (5)
where ψ(t) = v y , η(t) is linked to δ b including inaccuracy due to linearization approximation.

III. ADAPTIVE CONTROL LAW DESIGN

In actual agriculture applications, farm vehicles always move with smooth velocities and most paths to be followed are straight lines and circles. The accelerations and the path curvature vary quite slowly with time, hence ψ(t), η(t) can be approximated as where the sliding components ψ s , η s are almost timeinvariant, ψ(t), η(t) are time-varying variables. In this section by assuming ψs = ηs = 0 and ψ(t) = η(t) = 0, an adaptive controller will be designed. The inaccuracy of this approximation mainly consisting of trivial timevarying sliding effects is not important, it will be treated as disturbances and rejected by robust controller design in forthcoming section IV.

ψ(t) = ψ s + ψ(t) η(t) = η s + η(t) (6)

A. Backstepping-based Control Design Scheme

In the model ( 5), the lateral deviation y is not directly controlled. To overcome this problem the idea of backstepping is used; see [START_REF] Kanellakopoulos | Nonlinear and Adaptive Control Design[END_REF] for details. Using backstepping we propose a stepwise control design procedure for this 2-order nonholonomic system with unknown parameters.

step 1: Consider the lateral subsystem of ( 5) and assumption of time-invariant sliding, the Lyapunov function candidate is chosen as

V 1 = 1 2 y 2 + 1 2 ( ψs -ψ s ) T Γ -1 ( ψs -ψ s ) (7)
where Γ is positive definite, ψs indicates the estimation of ψ s . The time derivative of V 1 along the kinematic model is

V1 = y(v sin θ + ψs cos θ) + ( ψs -ψs) T Γ -1 ( ψs -Γy cos θ) (8)
Regard u 1 = sin θ as the virtual control input of the first step. If choose u 1 as

u 1d = -k 1 y -ψs cos θ v ( 9
)
and ψs = Γy cos θ

then we have

V1 = -k 1 y 2 ( 11
)
u 1d is the desired value of the virtual control input u 1 for the first step. If u 1 tracks (9) precisely, then the lateral subsystem of (5) can be stabilized asymptotically.

In the closed-loop system u 1 is not the actual control input, tracing u 1d with some errors indeed, so an error variable ũ1 is defined as

ũ1 = u 1 -u 1d (12)
The time derivative of ũ1 can be computed as

u1 = m 1 v tan δ l - c(s) cos θ 1 -c(s)y + m 3 η s + m 2 ψ s + ψs cos θ + k 1 (v sin θ + ψ s cos θ) v ( 13
) where

m 1 = cos θ - ψs sin θ v m 2 = c(s) sin θ 1 -c(s)y - 1 l m 3 = v l ( 14
)
Note that

m 1 = 1 cos φ cos( θ + φ) ( 15
)
where φ is the estimation of the slip angle defined in the figure 2. By assuming | θ + φ| < π 2 , one get m 1 > 0. Remark: For simplicity it is assumed that v is constant, in case v is time-varying, only variation is adding

(-k1y-ψs cos θ) v v 2
in (13). step 2: Considering the Lyapunov function as

V 2 = V 1 + 1 2 ũ2 1 + 1 2 (η s -η s ) T γ -1 (η s -η s ) ( 16
)
where γ is positive definite, ηs indicates the estimation of η s . Regard u 2 = tan δ as the virtual control input of the second step, then the time derivative of V 2 along (8)( 13) is

V2 = y(vu1 + ψs cos θ) + ( ψs -ψs) T Γ -1 ( ψs -Γy cos θ) +ũ 1m1 v( u2 l - c(s) cos θ 1 -c(s)y ) + m 3ηs + m2ψs +ũ1 k1(v sin θ + ψs cos θ) + ψs cos θ v + (η s -ηs) T γ -1 ηs (17)
Substituting (9)(12) into (17), we have the following equation (see [START_REF] Fang | Automatic Guidance of Farm Vehicles in the Presence of Sliding[END_REF] for detail)

V2 = -k1y 2 + ũ1m1 v( u2 l - c(s) cos θ 1 -c(s)y ) + m 3 ηs + m2 ψs +yvũ1 + ũ1 k1(v sin θ + ψs cos θ) + ψs cos θ v +(η s -ηs) T γ -1 ( ηs -ũ1γm1m3) +( ψs -ψs) T Γ -1 ( ψs -Γy cos θ -Γk1 ũ1 cos θ v -Γũ 1m1m2) (18)
In ( 18) let

ψs = Γy cos θ + Γk 1 cos θ v ũ1 + Γũ 1 m 1 m 2 ηs = γm 1 m 3 ũ1 (19)
and choose u 2 as

u 2 = l(- y m 1 - k 2 ũ1 m 1 v - m 3 ηs + m 2 ψs v + α + β) (20) where α = - k 1 (v sin θ + ψs cos θ) + ψs cos θ v 2 m 1 (21) β = c(s) cos θ 1 -yc(s) (22)
then we get

V2 = -k 1 y 2 -k 2 ũ2 1 (23)
Finally the vehicle steering angle δ is determined by δ = arctan u 2 (24)

B. Stability Analysis

From [START_REF] Fang | Automatic Guidance of Farm Vehicles in the Presence of Sliding[END_REF] it is known that the solutions y, ũ1 and estimation errors ψs , ηs are bounded on [0, +∞). The direct application of LaSalle invariance principle yields that all the solutions converge to the set Ω with Ω = {(y, ũ1 , ψs , ηs ) : y = 0, ũ1 = 0}

(25)

From ( 25), one gets that the lateral deviation will converge to zero asymptotically, simultaneously the steady orientation error θ is bounded by

tan θ = - ψs v ( 26
)
It is obvious that when vehicles move without sliding ( ψs = 0), the orientation error will converge to zero.

IV. ROBUST ADAPTIVE CONTROLLER DESIGN

A. Kinematic Model with Time-varying Sliding

We have designed an adaptive controller when assuming that the sliding effects are time-invariant, but from (5)( 6) the precise kinematic model with time-varying sliding is

   ẏ = v sin θ + ψ s cos θ + ε 1 θ = v tan δ l - c(s) cos θ 1 -c(s)y + m 3 η s + m 2 ψ s + ε 2 (27) where ε 1 = ψ(t) cos θ + ς 1 ε 2 = m 3 η(t) + m 2 ψ(t) + ς 2 (28)
ς 1 , ς 2 are unknown external disturbances. It is still an open problem to design controllers using backstepping algorithm for a system with time-varying parameters in presence of bounded disturbances and noise. In the latest literature [START_REF] Zhang | Backstepping control of linear time-varying systems with known and unknown parameters[END_REF] a new parametrization and filter structure that take into account parameter variation lead to a new backstepping controller when the time variation of the parameters is known. It is indicated that the uncertainty in the parameters can be counteracted by increasing the values of the design parameters. But high-gain controllers are not realistic for farm vehicles because of constrained control inputs and limited bandwidth of the steering systems.

In agriculture implementations the time-varying sliding components ψ(t), η(t) have small amplitudes and we also assume that external disturbances are bounded, so it is reasonable to assume that ε 1 , ε 2 are bounded

|ε i | < ρ i (29)
Using the same backstepping procedures as section III-A, we can prove that in the first step

V1 = y(v sin θ + ψs cos θ) + ( ψs -ψs) T Γ -1 ( ˙ψ s -Γy cos θ) + yε 1 (30)
Moreover applying the resulting adaptive laws [START_REF] Koshkouei | Adaptive Backstepping Control of Nonlinear Systems with Unmatched Uncertainty[END_REF] and the controller [START_REF] Yao | Smooth robust adaptive sliding mode control of robot manipulators with guaranteed transient performance[END_REF], we can prove that

V2 = -k 1 y 2 -k 2 ũ2 1 + yε 1 + ũ1 (m 1 ε 2 + k 1 ε 1 v ) (31)
which implies that the closed-loop system is globally uniformly bounded.

B. Robust Adaptive Controller with Variable Structure Controller

Here we are in the place to design a controller which is robust to ε i . Combining classical robust controllers with adaptive controllers is straightforward [START_REF] Koshkouei | Adaptive Backstepping Control of Nonlinear Systems with Unmatched Uncertainty[END_REF]. So in the first step we design u 1d as a variable structure controller from (30). Furthermore sign(•) is replaced by tanh(•) in this paper which is continuously differentiable in the stepwise procedures.

u 1d = -k 1 y -ψs cos θ -ρ 1 tanh( y σ3 ) v (32)
Then following the similar stepwise procedures of section III-A, we get a robust adaptive controller

u2 = l(- y m1 - k2 ũ1 m1v - m3 ηs + m2 ψs v + α + β + s 1 + s2) (33)
where β, ηs are defined as before and

α = - (v sin θ + ψs cos θ) + ψs cos θ v 2 m 1 s 1 = - ρ 1 v 2 m 1 tanh( ũ1 σ 1
)

s 2 = - ρ 2 v tanh( ũ1 σ 2 ) ˙ψ s = Γy cos θ + Γ cos θ v ũ1 + Γũ 1 m 1 m 2 = k 1 + 1 -tanh 2 ( y σ 3 ) ρ 1 σ 3 (34)
where σ i > 0, is related to the derivative of tanh(), s 1 is developed to reject ε 1 appearing in the derivative of ũ1 , s 2 is to reject ε 2 in θ. The controller (33) can guarantee the closed-loop system converge to Ω. (see [START_REF] Fang | Automatic Guidance of Farm Vehicles in the Presence of Sliding[END_REF] for detail)

C. Projection Mapping for Parameter Adaptation

However the controller (33) has some drawbacks. It needs a large amount of on-board computation and highorder derivative of sign-like functions may lead to oscillating control inputs.

Another approach to design robust adaptive controllers is utilizing projection mapping for the parameter adaptation procedures. The projection mapping P roj ξ (•) is defined by [START_REF] Yao | Smooth robust adaptive sliding mode control of robot manipulators with guaranteed transient performance[END_REF], [START_REF] Sastry | Adaptive Control: Stability, Convergence, and Robustness[END_REF]

P roj ξ (•) =    0 if ξ = ξ max and • > 0 0 if ξ = ξ min and • < 0 • otherwise (35)
By using projection mapping P roj ξ (•), the robust adaptive laws become

ψs = P roj ψs Γ[cos θ k 1 cos θ v + m 1 m 2 ][y ũ1 ] T (36) ηs = P roj ηs γm 1 m 3 ũ1 (37)
The system will converge into a neighborhood of zero.

(36)(37) can guarantee the estimation of sliding effects stay within a defined range regardless of external disturbances. The prior information on the bounds of the sliding effects ψ s , η s can be obtained off-line after performing large number of absolute coordinates measurements under different typical working conditions.

V. SIMULATION RESULTS

First a classical "U" path with a perfect circular arc (path #1) is followed to test the adaptive controller. In the simulations, the gains used in [START_REF] Koshkouei | Adaptive Backstepping Control of Nonlinear Systems with Unmatched Uncertainty[END_REF], [START_REF] Yao | Smooth robust adaptive sliding mode control of robot manipulators with guaranteed transient performance[END_REF][START_REF] Sastry | Adaptive Control: Stability, Convergence, and Robustness[END_REF] are set as k 1 = 0.15, k 2 = 1.14, Γ = 0.15, γ = 0.02. In actual implementations, the gains should be tuned gradually to make an optimal compromise between transient characteristic and limited bandwidth of the steering system. The constant sliding is introduced with v y = -0.1, δ b = -0.048. The control law in our previous work [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single CP-DGPS[END_REF] is applied also with the controller gains k p = 0.09, k d = 0.6. Because this control law does not take sliding effects into account, in figure 3 when the sliding appears the lateral deviation (dashed line) becomes significant. While the controller (20) can compensate sliding effects through estimating them on line, the lateral deviation (solid line) converges to zero with small offsets (due to linearization approximation in ( 5)) after deviating from the desired path. Since small control gains are used, the vehicle movements are kept smooth. The remarkable overshoots at the beginning and end of the curve are caused by "jump change" of the sliding effects and low level delay. The robust adaptive controller with VSC (33) is simulated also. Because the modeling inaccuracy is counteracted by VSC, the lateral deviation can converge to zero with a good transient response (dotted line in figure 3). The bounded orientation errors are shown by figure 4. As analyzed by section III-B the orientation errors do not converge to zero, indeed they are bounded by (26), it is normal when sliding occurs known as "crab sliding". The evolution of the sliding parameters ψ (solid line), η (dashed line) is shown by figure 5. It is clear that at the beginning and end of the circle, ψ varies greatly which explains the overshoots of the lateral deviation, but as the vehicle follows the circle, ψ,η evolve smoothly close to the real values. To simulate the actual working condition, a set of real measurement data is used in the simulation to reconstruct the actual v y and δ b . In figure 6 the (robust) adaptive controllers yield small lateral deviation with zero mean value, while the lateral deviation of the controller [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single CP-DGPS[END_REF] is significant and has obvious bias.

In order to compare the performances between the robust controllers (33) and projection mappings (36-37), another reference path #2 consisting of straight lines and curves is followed (see figure 7), the lateral deviation is shown by figure 8. The experimental data indicates that the robust controller with VSC yields better transient performances at the expense of non-smooth movements (dotted line) especially when low level delay is considered. While the controller with projection mapping yields a movement with less oscillation following the reference path (solid line). So for the vehicles with good low-level characters, the controller with VSC is favorable, but for the vehicles whose bandwidth is limited, the robust controller with projection mapping is preferred.

VI. CONCLUSION

The path following problem of autonomous agricultural vehicles in the presence of sliding is investigated in this paper. A vehicle-oriented kinematic model which integrates the sliding effects as additive unknown parameters is constructed. From this model, a practical adaptive controller is designed based on the backstepping method which can stabilize the lateral derivation near zero and guarantees the orientation error converge into a neighborhood near the origin. In addition two approaches are proposed to refine the adaptive controller, allowing it to be robust to the time-varying sliding components and external disturbances. Experimental comparative results show the effectiveness of the proposed control laws. The advantages of this scheme are that

• When no sliding occurs, it provides a path following controller which can guarantee lateral deviation and orientation error converge to zero. • Instead of using high gains or powerful functions to counteract sliding effects, small controller gains are used to stabilize the system and the sliding effects are • Backstepping procedures are used to design a pathfollowing controller for a 2-order nonholonomic system, it can be extended easily to high order nonholonomic systems. The prospective works include extending backstepping methods to lateral-longitudinal control and using predictive control to decrease overshoots of lateral deviation [START_REF] Lenain | Model Predictive Control of vehicle in presence of sliding Application to farm vehicles path tracking[END_REF].

Fig. 1 .

 1 Fig. 1. Notation and path frame description

Fig. 2 .

 2 Fig. 2. Notations of sliding effects

Fig. 4 .Fig. 6 .

 46 Fig. 3. Lateral deviation of path #1

Fig. 8 .

 8 Fig. 7. Path #2 to be followed


Guillaume Blanc

Youcef Mezouar

Philippe Martinet

Indoor Navigation of a Wheeled Mobile Robot along Visual Routes

Keywords: Visual Navigation, Visual Memory, Visual Servoing for Mobile Robots Ψ 3

When navigating in an unknown environment for the first time, a natural behavior consists in memorizing some key views along the performed path, in order to use these references as checkpoints for a future navigation mission taking a similar path. This assumption is used in this paper as the basis of a navigation framework for wheeled mobile robots in indoor environments. During a human-guided teleoperated learning step, the robot performs paths which are sampled and stored as a set of ordered key images, acquired by a standard embedded camera. The set of these obtained visual paths is topologically organized and provides a visual memory of the environment. Given an image of one of the visual paths as a target, the robot navigation mission is defined as a concatenation of visual path subsets, called visual route. When running autonomously, the robot is controlled by a visual servoing law adapted to its nonholonomic constraint. Based on the regulation of successive homographies, this control guides the robot along the reference visual route without explicitly planning any trajectory. Real experiment results illustrate the validity of the presented framework.

I. INTRODUCTION

Vision is a central clue of most of recent mobile robots navigation frameworks. The authors of [START_REF] Desouza | Vision for mobile robot navigation: A survey[END_REF] accounts of twenty years of works at the meeting point of mobile robotics and computer vision communities. Often used among more "traditional" embedded sensors -proprioceptive sensors like odometers as exteroceptive ones like sonars -it provides accurate localization methods. In many works, and especially those dealing with indoor navigation as in [START_REF] Hayet | A Visual Landmark Framework for Indoor Mobile Robot Navigation[END_REF], computer vision techniques are used in a landmark-based framework. Identifying extracted landmarks to known references allows to update the results of the localization algorithm. These methods are based on some knowledges about the environment, such as a given 3D model or a map built online. They generally rely on a complete or partial 3D reconstruction of the observed environment through the analysis of data collected from disparate sensors. The mobile robot can thus be localized in an absolute reference frame. Both motion planning and robot control can then be designed in this space. The results obtained by the authors of [START_REF] Royer | Towards an alternative gps sensor in dense urban environment from visual memory[END_REF] leave to be forcasted that such a framework will be reachable using a single camera. However, although an accurate global localization is unquestionably useful, our aim is to build a complete vision-based framework without recovering a the position of the mobile robot with respect to a reference frame. The authors of [START_REF] Desouza | Vision for mobile robot navigation: A survey[END_REF] call this kind of framework a qualitative approach of navigation. The principle of this approach is to represent the robot environment with a bounded quantity of images gathered in a set called visual memory. In [START_REF] Remazeilles | Robot motion control from a visual memory[END_REF], this concept is exploited to control the 6 dof of a robotic arm under large displacements. A set of images is extracted from a previously learnt database which describes successive targets for a global visual servoing task. The authors do not consider the kinematic constraints of a mobile robot. In the context of mobile robotics, [START_REF] Matsumoto | Visual navigation using view-sequenced route representation[END_REF] also proposes to use a sequence of images, but recorded during a human teleoperated motion, and called View-Sequenced Route Reference. This concept underlines the close link between a human-guided learning and the performed paths during an autonomous run. However, the automatic control of the robot in [START_REF] Matsumoto | Visual navigation using view-sequenced route representation[END_REF] is not formulated as a visual servoing task. In this paper, we propose a complete image-based framework (i.e from environment learning to control) for mobile robots navigation. A sequence of images, acquired during a human-guided learning, allows to derive paths driving the robot from its initial to its goal locations. In order to reduce the complexity of the image sequences, only key views are stored and indexed on a visual path. The set of visual paths can be interpreted as a visual memory of the environment. The visual memory is structured as a graph which takes into account the environment topology. A navigation task consists then in performing autonomously a visual route which is a concatenation of visual paths. The visual route connects thus in the sensor space the initial and goal configurations. Section II details more precisely this point. The Section III deals with the vision-based control scheme designed to control the robot motions along a visual route. The nonholonomic constraints of most current wheeled mobile robots makes the classical visual servoing methods unexploitable since the camera is fixed on the robot [START_REF] Tsakiris | Extending visual servoing techniques to nonholonomic mobile robots[END_REF]. However, motivated by the development of 2D 1/2 visualservoing method proposed by Malis et al (see [START_REF] Malis | 2 1/2 d visual servoing[END_REF]), some authors have investigated the use of homography and epipolar geometry to stabilize mobile robots [START_REF] Fang | 2.5d visual servoing of wheeled mobile robots[END_REF], [START_REF] Chen | Homography-based visual servo tracking control of a wheeled mobile robot[END_REF]. In this paper, because the notions of visual route and path are very closed, we turn the nonholonomic visual-servoing issue into a path following one. The designed control law In Section IV, experiments on a small indoor wheeled mobile robot illustrate the implementation of the proposed framework.

1 Ψ 1 Ψ1 2 1 Ψ 1 1 (a) ( b) Ψ 1 2 Ψ 2 2 (c) Ψ 2 1 Ψ 1 1 Ψ 1 2 Ψ 1 2 Ψ 1 3 Ψ 3 1 (d)

II. VISUAL MEMORY AND ROUTES

In [START_REF] Desouza | Vision for mobile robot navigation: A survey[END_REF], approaches using a "memorization" of images of the environment taken from an embedded camera are ranked among mapless navigation systems. Indeed, as proposed in [START_REF] Matsumoto | Visual navigation using view-sequenced route representation[END_REF] or in [START_REF] Jones | Appearance based processes for visual navigation[END_REF], any notion of map nor topology of the environment appears, neither to build the reference set of images, nor for the automatic guidance of the robot. The first step of our framework consists on a learning stage to build the visual memory. The visual memory is structured according to the environment topology to reduce the computational cost. We assume that the features observed by the camera are quasi-static. However, the remainder of the environment is not restricted to staticity, since both perception and control algorithms are robust to dynamic constraints such as occlusions.

A. Structure of the visual memory

The learning stage relies on the human experience. The user guides the mobile robot into its workspace. In the case of an urban vehicle for instance, as proposed in [START_REF] Royer | Towards an alternative gps sensor in dense urban environment from visual memory[END_REF], the user manually drives the robot along all the streets where it will go down in an autonomous way. To each street is associated a visual path, which is a set of key images ordered from the beginning to the end of the street. We use a similar learning approach in an indoor environment. The user guides the mobile robot along one or several paths into each room where the robot is authorized to go (see Figure 1 (a),(b) and (c)). A visual path r Ψ p is then stored and indexed as the p th learnt path in the r th room.

1) Visual paths: A visual path r Ψ p is composed of n key images: r Ψ p = { r I p i |i = {1, 2, . . . , n}} For control purpose (refer to Section III), the authorized motions during the learning stage are assumed to be limited to those of a car-like vehicle, which only goes forward. The following Hypothesis 2.1 formalizes these constraints.

Hypothesis 2.1: Given two frames R F i and R F i+1 , respectively associated to the mobile robot when two successive key images I i and I i+1 of a visual path Ψ were acquired, there exists an admissible path ψ from R F i to R F i+1 for a car-like vehicle whose turn radius is bounded, and which only moves forward.

Moreover, because the controller is vision-based, the robot is controllable from r I p i to r I p i+1 only if the hereunder Hypothesis 2.2 is respected.

Hypothesis 2.2: Two successive key images I i and I i+1 contain a set P i of matched visual features, which can be tracked along a path performed between R F i and R F i+1 .

During the acquisition of a visual path, the Hypothesis 2.2 constrains the choice of the key images. Two main approaches have been implemented in our laboratory. The first one, described in [START_REF] Aider | Indoor autonomous navigation using visual memory and pattern tracking[END_REF], uses planar patterns tracking. A new key image I i+1 is stored when a pattern, which has been tracked since I i was acquired, is likely to leave the image. In the second approach, proposed in [START_REF] Royer | Towards an alternative gps sensor in dense urban environment from visual memory[END_REF], I i+1 is stored if the camera motion between I i and I i+1 is sufficient to ensure a good 3D reconstruction of matched interest points, and if the number of interest points is upper than a fixed threshold. In this case, P i is the set of all correctly matched points between I i and I i+1 . As a consequence of Hypothesis 2.1 and 2.2, each visual path r Ψ p corresponds to an oriented edge which connects two configurations of the robot's workspace. Moreover, the number of key images of a visual path is directly linked to the human-guided path complexity. According to this parameter, we define the value of a visual path as its cardinal.

2) Visual memory vertices: In order to connect two visual paths, the terminal extremity of one of them and the initial extremity of the other one must be constrained as two consecutive key images of a visual path. The paths are then connected by a vertex, and two adjacent vertices of the visual memory are connected by a visual path (see Figure 1 (d)).

Proposition 2.1:

Given two visual paths Ψ p1 = {I p1 i |i = {1, 2, . . . , n 1 }} and Ψ p2 = {I p2 i |i = {1, 2, . . . , n 2 }}, if the two key images I p1 n1 Ψ 3 1 Ψ 2 1 Ψ 1 3 Ψ 1 1 Ψ 1 2 Ψ 1 2 Ψ 3 1 ' Ψ 1 2 '
Starting point of the visual route

Ξ

Last key image Fig. 2. A visual route:

Ξ = 1 Ψ 3 ⊕ 2 Ψ 1 ⊕ 3 Ψ 1 ⊕ 2 Ψ 1 . 1 Ψ 3 and 2 Ψ 1
are subsets of respectively 1 Ψ 3 and 2 Ψ 1 . 1 Ψ 3 is splitted at the closest key image to the initial Ic, while the last key image of 2 Ψ 1 is a desired image to reach by navigating onto the visual memory.

and I p2 1 abide by both Hypothesis 2.1 and 2.2, then a vertex connects Ψ p1 to Ψ p2 .

We also assume this Proposition 2.1 in the particular case where the terminal extremity of a visual path Ψ p1 is the same key image as the initial extremity of another visual path Ψ p2 . This is useful in practice, when building the visual memory.

3) A strongly connected digraph: According to Sections II-A.1 and II-A.2, the visual memory structure is defined as a digraph, whose arcs are the visual paths. It is yet necessary that this digraph is strongly connected. Indeed, this condition warrants that any vertex of the visual memory is attainable from every others, through a set of visual path.

B. Visual route

A visual route describes the robot's mission in the sensor space. Given two key images of the visual memory I c and I g , corresponding respectively to the current and goal locations of the robot, a visual route is a set of key images which describes a path from I c to I g , as presented in Figure 2. In practice, I c is not exactly a key image. It is then necessary to determine which stored key image is the closest one to I c . This can be done in an off-line stage, as Remazeilles et al propose in [START_REF] Remazeilles | Robot motion control from a visual memory[END_REF], by comparing the photometric invariants of the request image with those of the images store onto the visual memory. Assuming that the user is able to point out which is the room r where the robot is staying, the search of the closest image is bounded to the learnt visual paths in r. The visual route is the minimum length path of the visual memory connecting two vertices associated to I c and I g . According to the definition of the value of a visual path, the length of a path is the sum the values of its arcs. The minimum length path is obtained in a very simple way, using Dijkstra's algorithm. Consequently, the visual route results from the concatenation of indexed visual paths. Given two visual paths Ψ p1 and Ψ p2 , respectively containing n 1 and n 2 indexed key images, the concatenation operation of Ψ p1 and Ψ p2 is defined as follows:

Ψ p1 ⊕ Ψ p2 = I p1,2 j |j = {1, . . . , n 1 , n 1 + 1, . . . , n 1 + n 2 } I p1,2 j = I p1 j if j n 1 I p2 j-n1 if n 1 j n 1 + n 2
The visual route describes a set of consecutive states that the image has to reach in order that the robot joins the goal configuration from the initial one. The robot motions are controlled along the visual route using the data provided by the embedded camera. The next section deals with this issue.

III. VISUAL ROUTE FOLLOWING

Visual-servoing is often considered as a way to achieve positioning tasks. Classical methods, based on the task function formalism, are based on the existence of a diffeomorphism between the sensor space and the robot's configuration space. Due to the nonholomic constraints of most of wheeled mobile robots, under the condition of rolling without slipping, such a diffeomorphism does not exist if the camera is rigidly fixed to the robot. In [START_REF] Tsakiris | Extending visual servoing techniques to nonholonomic mobile robots[END_REF], the authors add extra degrees of freedom to the camera. The camera pose can then be regulated in a closed loop. In the case of an embedded and fixed camera, the control of the camera is generally based on wheeled mobile robots control theory [START_REF] Samson | Control of chained systems. application to path following and time-varying stabilization of mobile robots[END_REF]. In [START_REF] Ma | Vision guided navigation for a nonholonomic mobile robot[END_REF], a car-like robot is controlled with respect to the projection of a ground curve in the image plane. The control law is formalized as a path following problem. More recently, in [START_REF] Fang | 2.5d visual servoing of wheeled mobile robots[END_REF] and [START_REF] Chen | Homography-based visual servo tracking control of a wheeled mobile robot[END_REF], a partial estimation of the camera displacement between the current and desired views has been exploited to design visionbased control laws. The camera displacement is estimated by uncoupling translation and rotation components of an homography matrix. In [START_REF] Fang | 2.5d visual servoing of wheeled mobile robots[END_REF], a time-varying control allows an asymptotical stabilization on a desired image. In [START_REF] Chen | Homography-based visual servo tracking control of a wheeled mobile robot[END_REF], a trajectory following task is achieved. The trajectory to follow is defined by a prerecorded video and the control law is proved stable using Lyapunov-based analysis. In our case, unlike a whole video sequence, we deal with a set of relay images which have been acquired from geometrically spaced out points of view. A visual route following can be considered as a sequence of visual-servoing tasks. A stabilization approach could thus be used to control the camera motions from a key image to the next one. However, a visual route is fundamentally a path. In [START_REF] Mezouar | Images interpolation for image-based control under large displacement[END_REF], the authors propose to plan the trajectories of image features directly in the sensor space from the first image to the last one. These trajectories are then used as references to control a robotic arm. To design the controller, described in the sequel, the key images of the reference visual route are considered as consecutive checkpoints to reach in the sensor space. The control problem is formulated as a path following to guide the nonholonomic mobile robot along the visual route.

A. Assumptions and models

Let I i , I i+1 be two consecutive key images of a given visual route to follow and I c be the current image. Let us note

F i = (O i , X i , Y i , Z i ) and F i+1 = (O i+1 , X i+1 , Y i+1 , Z i+1
) the frames attached to the robot when I i and I i+1 were stored and

F c = (O c , X c , Y c , Z c )
a frame attached to the robot in its current location. Figure 3 illustrates this setup. The origin O c of F c is on the axle midpoint of a cart-like robot, which evolutes on a perfect ground plane.

The control vector of the considered cart-like robot is u = [V, ω] T where V is the longitudinal velocity along the axle Y c of F c , and ω is the rotational velocity around Z c . The hand-eye parameters (i. e. the rigid transformation between F c and the frame attached to the camera) are supposed to be known.

According to Hypothesis 2.2, the state of a set of visual features P i is known in the images I i and I i+1 . Moreover P i has been tracked during the learning step along the path ψ between F i and F i+1 . The state of P i is also assumed available in I c (i.e P i is in the camera field of view). The task to achieve is to drive the state of P i from its current value to its value in I i+1 .

B. Principle

Consider the straight line Γ = (O i+1 , Y i+1 ) (see Figure 4). The control strategy consists in guiding I c to I i+1 by regulating asymptotically the axle Y c on Γ. The control objective is achieved if Y c is regulated to Γ before the origin of F c reaches the origin of F i+1 . This can be done using chained systems. Indeed in this case chained system properties are very interesting. A chained system results from a conversion of a mobile robot non linear model into an almost linear one [START_REF] Samson | Control of chained systems. application to path following and time-varying stabilization of mobile robots[END_REF]. As long as the robot longitudinal velocity V is non zero, the performances of path following can be determined in terms of settling distance [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single cp-dgps[END_REF]. The settling distance has to be chosen with respect to robot and perception algorithm performances. The lateral and angular deviations of F c with respect to Γ to regulate can be obtained through partial Euclidean reconstructions as described in the next section.

C. Evaluating Euclidean state

In the sequel, we suppose that

P i = {p ik , k = 1 • • • n} is a
set of image points of P i+1 . These points are matched with the set of image points

P c = {p ck , k = 1 • • • n} of I c .
Let Π be a 3D reference plane defined by three 3D points whose projections onto the image plane belong to P i (and P c ). The plane Π is given by the vector π T = [n * d * ] in the frame F i+1 , where n * is the unitary normal of Π in F i+1 and d * is the distance from Π to the origin of F i+1 . It is well known that there is a projective homography matrix G, relating the image points of P i and P c [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF]:

α k p ik = Gp ck + β k e
where α k is a positive scaling factor and β k is a scaling factor, null if the target point belongs to Π. Given at least four matched points belonging to Π, G can be estimated by solving a linear system. If the plane Π is defined by 3 points, at least five supplementary points are necessary to estimate the homography matrix [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF]. Assuming that the camera calibration K is known, the Euclidean homography of plane Π is estimated as H = K -1 GK and it can be decomposed into a rotation matrix and a rank 1 matrix:

H = R c i+1 + t c i+1 n * d * (1)
As exposed in [START_REF] Faugeras | Motion and structure from motion in a piecewise planar environment[END_REF], it is possible from H to determine the camera motion parameters, that is R c i+1 and

t c i+1 d * .
The normal vector n * can also be determined, but the results are better if n * has been previously well estimated (note that it is the case in indoor navigation with a camera looking at the ceiling for instance). In our case, the mobile robot is supposed to move on a perfect ground plane. Then an estimation of the angular deviation θ between F c and F i+1 can be directly extracted from R c i+1 . Furthermore, we can get out from t c i+1 d * the lateral deviation y up to a scale factor between the origin of F c and a straight line Γ. As a consequence, the control problem can be formulated as following Γ in regulating to zero y and θ before the origin of F c reaches the origin of F i+1

D. Control law

Exact linearization of nonlinear models of wheeled mobile robot under the assumption of rolling without slipping is a well known theory, which has already been applied in many vehicle guidance applications, as in [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single cp-dgps[END_REF] for a carlike vehicle, and in our previous works (see [START_REF] Blanc | An image driven visionbased control framework for wheeled mobile robots[END_REF]). The used state vector of the robot is X = s y θ , where s is the curvilinear coordinate of a point M, which is the orthogonal projection of the origin of F c on Γ. The derivative of this state give the following state space model:

   ṡ = V cos θ ẏ = V sin θ θ = ω c (2)
The state space model ( 2) is converted into a chained system of dimension 3 a 1 a 2 a 3 . Deriving this system with respect to a 1 gives an almost linear system. By choosing a 1 = s and a 2 = y, and thanks to classical linear automatics, it is then possible to design an asymptotically stable guidance control law, which performances are theoretically independent to the longitudinal velocity V :

ω(y, θ) = -V cos 3 θK p y -|V cos 3 θ|K d tan θ (3)
K p and K d are gains which set the performances of the control law. They must be positive for the control law stability. Their choice determine a settling distance for the control, i. e. the impulse response of y with respect to the covered distance by the point M on Γ. However, as y is estimated up to a scale factor 1 d * , this modifies the performances of the control law [START_REF] Chen | Homography-based visual servo tracking control of a wheeled mobile robot[END_REF]. In practice, to alleviate this difficulty, we choose K p and K d for a given d * that we though maximum. As said before, the control performances are independent to V . Then V can be fixed or tuned by a supervisor. However, V has to be non-null to allow the regulation. Then, in practice, V is controlled in an open loop at the first and the last image of the visual route. The proposed framework is implemented on a Pekee TM robot which is controlled from an external PC. A small 1/3" CMOS camera is embedded on the robot and looks at the ceiling. Ceiling images are used for two main purposes. Firstly, extracted features from an image of the ceiling or from the top of the walls during a visual path learning ought to be recovered in an autonomous run under similar conditions. Secondly, the ceiling is generally a plane parallel to the ground plane. Then, it is quite easy to give a good approximation of the normal vector n * to the reference plane Π in order to evaluate an homography. If the camera frame is confounded with the robot frame F c , we can assume that n * = 0 0 1 . The Euclidean homography matrix H has thus a very simple expression. Indeed, as the displacement between F i+1 and F c only consists of one rotation θZ c and two translations t Xc X c and t Y c Y c , H is given by (refer to equation ( 1)):

IV. EXPERIMENTS

H =   cos θ -sin θ t Xc sin θ cos θ t Y c 0 0 1   (4)
Therefore, H has only three degrees of freedom. Only two points lying on Π and matched in I i+1 and I c are theoretically necessary to estimate H. The angular deviation θ with respect to Γ can be estimated directly from the computation of H. The lateral deviation y can also be estimated since it is the X i+1 coordinate of the origin of F c :

y = -t Xc cos θ -t Y c sin θ (5)
Thus, from the computation of the Euclidean homography H, the state of F c with respect to the straight line Γ can easily be determined. The Figure 6 illustrates the evolution of planar patterns

(1) (2) (3) (4) [...] (5)
(6) (7) (8)
Fig. 6. Evolution of the image space when the robot is regulated between two consecutive key image: in each image, the yellow square is the current state of the tracker, the red one is the state to reach. At image [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF], a new reference state is given for the tracker. The image ( 6) is thus considered close to the previous reference key image: the control has succeeded.

tracked during the robot motion along a given visual route. These tracked planar patterns have been extracted while the user was creating a visual path which is included into the visual route to be followed. They result from the extraction of a set of interest points, regarding oriented gradient maxima into a rectangular area selected by the user in the current image. An algorithm, based on particle filtering, provides a robust tracking of planar patterns with respect to partial occlusion and to illumination changes. Details on both pattern extraction and tracking can be found in [START_REF] Aider | Indoor autonomous navigation using visual memory and pattern tracking[END_REF].

To create a visual path, the user specifies the beginning of the learning. Then he teleoperates the robot and selects, into the current image, areas which contain possible interesting features. The remainder of the key image selection is then automatic (see [START_REF] Aider | Indoor autonomous navigation using visual memory and pattern tracking[END_REF]). The user indicates when the path to learn has been performed. Then, the new visual path is stored.

At the first step of an autonomous run, the current camera image has to be located into the visual memory. The tracking of learnt planar pattern in this image is then automatic. As a consequence, the user must have chosen at least one reference attitude of the robot which has to be associated with one key image. If the robot has ever achieved a mission since it has been started up, the current image is already supposed to be closed to a key image. At each frame, the tracker provides the coordinates of a current tracked planar pattern. H is then computed thanks to the knowledge of this pattern in the key image to reach I i+1 .

A key image is assumed to be reached when a distance between the current points coordinates and the desired one goes under a fixed threshold. The reference path, which is represented on the Figure 5 by the white squares which are lying on the ground, has been acquired as a visual route of fifteen key images. The corresponding path length is about 10m. The longitudinal velocity V was fixed to 0.2m.s -1 . When the robot stops at the end of the visual route, the final errors in the image corresponds to a positioning error around 5cm and an angular error about 3 • . Nevertheless, note that the robot has been stopped roughly, by setting V to zero since the last key image of the visual route has been detected. Moreover, both camera intrinsic and hand-eye parameters has been roughly determined. The positioning accuracy depends above all on the threshold which determines if a key image is reached. Our future works will improve that point.

V. CONCLUSION This paper presents an original image-based navigation framework dedicated to nonholonomic mobile robots. The approach is illustrated in the context of indoor navigation. We propose to learn the environment as a graph of visual paths, called visual memory. A navigation mission into this visual memory is a visual route. A visual route is made of a sequence of key images of the environment which describes, in the sensor space, an admissible path for the robot. This visual route can be performed thanks to a visual-servoing control law, which is adapted to the robot nonholonomy and does not require any absolute geometrical localization of the robot. Future works will be devoted to relax the staticity constraint of the environment. We will try to analyse and to take into account environment modifications, which may occur between learning steps and autonomous runs, in both visual route building and following.

Fig. 1 .

 1 Fig. 1. Building a visual memory: Into the rooms (a) and (b) and the corridor (c), the paths r Ψp have been learnt by teleoperating the robot. As a result, the graph (d) represents the topological organization of the visual memory. The blue circles show the vertices

Fig. 3 .

 3 Fig. 3. Frames and images: I i and I i+1 are two consecutive key images, acquired along a teleoperated path γ

Fig. 4 .

 4 Fig. 4. Control strategy

Fig. 5 .

 5 Fig. 5. Following a visual route: the previously learnt visual path, about 10m long, is materialized on the ground. The pictures were taken during an autonomous run

ACKNOWLEDGMENT

These works are supported by a grand of the WACIF RNTL (2002)(2003)(2004)) project of the French government.


J Bom
email: bom@lasmea.univ-bpclermont.fr

B Thuilot

F Marmoiton

P Martinet

Nonlinear Control for Urban Vehicles Platooning, Relying upon a Unique Kinematic GPS

Keywords: mobile robots, nonlinear control, platooning, Automatic Guided Vehicles, RTK GPS

In order to solve problems of traffic saturation in cities, new alternative "Urban Transportation Systems" are based on electric vehicles in free-access. One necessary functionality of such systems is their ability to move in a platoon fashion. Platooning of these automatic guided vehicles, relying on RTK-GPS sensors and inter-vehicles communication, is addressed in this paper. More precisely, vehicles platoon is expected to follow a curved reference path. Relying on nonlinear control theory, lateral and longitudinal control are fully decoupled, and therefore addressed independently. To ensure passengers comfort, additional monitoring functions supervise our control system. Then, simulations followed by experiments carried out with urban vehicles, are presented.

I. INTRODUCTION

Nowadays, congestion of vehicles traffic in cities, pollution that is linked, and people security problems are current and important concerns. As a consequence, new alternative transport solutions must be found. Numerous researches are addressed in large projects so-called "Urban Transportation Systems". Urban electric vehicles, available for example in a car-sharing concept, seem to be an interesting solution. On one hand, they reduce considerably the cited nuisances; and on the other hand, they propose a great flexibility, which could satisfy users. Then, their potential applications appear wide: they appear welcomed in urban structure, where they can be used in the transports of goods, in public transport, to visit attraction resorts, etc. Such systems have been developed since the mid-90's, e.g. Praxitèle in France [START_REF] Laugier | Towards autonomous vehicles for future intelligent transportation systems[END_REF]- [START_REF] Daviet | Platooning for vehicles and automatic parking by scheduling robotic actions[END_REF], CarLink in the USA [START_REF] Shaheen | Carlink II: a commuter carsharing pilot program final report[END_REF], Crayon in Japan [START_REF] Evans | New energy sources for the car : how toyota sees the future[END_REF]... In order to transport many passengers, to drive vehicles along main urban roads, to bring back vehicles to their base station, etc., urban mobile robots with fully automated driving capabilities must be able to move in a platoon fashion. Our research interest is platooning of vehicles composed of a leader followed by vehicles in a single file. To confer autonomy to a platoon, it is necessary to localize accurately the automatic guided vehicles in their environment. Some applications, requiring equipped infrastructures, are in developing or developed: automatic vans in suspension over a guideway, thanks to Electro Magnetic forces, are described in [START_REF]The M 3 urban transportation system[END_REF], fleet of urban shuttles detecting magnetic track integrated in the road pavement is described in [START_REF] Gillet | BURST : Bright urban system for transportation[END_REF]. Unluckily, the cost of implementing the infrastructure and the fact that these systems can not be used outside of the equipped area appear restricted. RTK GPS (Real Time Kinematic Global Positioning System) sensors do not present these drawbacks, and since they can provide in realtime vehicle localization with a centimeter accuracy, they appear very attractive, at least for guiding the leader vehicle. Nowadays RTK GPS sensors seem to be expensive and risks of canyoning effect in urban environment is real. However, the evolution of car equipment market and a wider satellites cover thanks to EGNOS and Galileo constellations, are going to reduce the costs and rise up the performances of that sensor. In platooning applications, relative distance and speed with respect to the preceding vehicle are also needed. Such information could be provided by direct sensors (as cameras [START_REF] Daviet | Platooning for small public urban vehicles[END_REF], radar [START_REF] Kuroda | An adaptive cruise control system using a milimeter wave radar[END_REF], . . .), which also do not require an equipped environment. However drawbacks of such sensors are their field of vision too small when platoons describe reference paths with high curvature, and the imperious necessity that no obstacle is present between the vehicles of the platoon. To overcome such problems, a conceivable solution is that vehicles share absolute localization measurements provided by their own RTK-GPS sensor via inter-vehicles communication. Relevancy of this approach is demonstrated in this paper. Urban electric vehicles called Cycabs (Fig1), which serve as development products in several French laboratories, have been used, and intervehicles communication relies on WiFi technology. In the forthcoming experimental results, the leader vehicle is driven automatically, and each vehicle in the platoon follows the same reference path. Nevertheless, since intervehicles communication is available, it could also be possible that the leader vehicle was driven manually and that the other vehicles objective was to follow leader vehicle evolution.

Some results in vehicles platoon control relying on RTK GPS can be found in the literature. In [START_REF] Cannon | Precise GPS sensor subsystem for vehicle platoon control[END_REF], each vehicle in the platoon is equipped with two GPS sensors in order to determine precisely vehicle location and heading. The reference path for this three-vehicles platoon is defined by the evolution of the first vehicle. The platoon is operating at a constant speed fixed between 20km/h and 90km/h and the control objective is to maintain a safe driving distance of approximatively 2s. Reported experiments present joining, leaving and passing the platoon. In [START_REF] Ohtomo | Automatic following system utilizing vehicle-to-vehicle communication[END_REF], an automatic following device, making use of GPS and vehicleto-vehicle communication, enables an unmanned vehicle to follow a driven leading one. These works are more developed upon the technological sensor aspects (GPS and communication respectively) rather than upon control design ones, contrarily to the work that is presented here.

Since urban applications are addressed, vehicles velocities between 10km/h and 20km/h are considered (this is exactly the velocity range of Cycab vehicles). The objective in platooning is to control vehicles velocity in order to respect some constraints: generally a constant direct distance or time between the cars is regulated (see e.g. [START_REF] Girault | Stability analysis of a longitudinal control law for autonomous vehicles[END_REF]). In this paper, the platooning objective is to keep a constant curvilinear distance within vehicles. The main advantage of curvilinear distance is that it agrees with the distance travelled (monotonous behavior) and is perfectly consistent when following reference paths with high curvature (which is not the case with direct distance). Platooning control design relies on nonlinear techniques, as in [START_REF] De Luca | Feedback control of a nonholonomic car-like robot[END_REF], instead of control approaches based on linear approximations (e.g. [12] [13]). Since no approximation is achieved, performances provided by the nonlinear control law are more satisfactory and more robust than those offered by linear control. Moreover, this control approach allows to fully decouple longitudinal and lateral controls. In [START_REF] Thuilot | Accurate automatic guidance of an urban electric vehicle relying on a kinematic GPS sensor[END_REF], automatic lateral guidance has been presented for one vehicle. Thanks to this decoupling feature, lateral guidance of each vehicle in the platoon can indeed be achieved independently, by relying on that control law. Therefore, the present paper addresses only longitudinal control. Finally, in order to ensure passengers security and comfort, a monitoring approach is set to manage saturations of the nonlinear platooning control law.

The paper is organized as follows: first our experimental platform is described . Then, vehicle modelling is addressed. After that, platoon control law design is presented. Simulation, and finally full-scale experiments are then reported.

II. EXPERIMENTAL VEHICLE

In this section, RTK GPS receiver and experimental vehicles constituting the platoon are successively presented.

The RTK GPS receiver is a "Sagitta" unit from Thales Navigation. It provides position measurements at a 10Hz sampling frequency, with a 2cm accuracy. The GPS reference station has been located on a 15m-height building. Each Cycab is equipped with a unique GPS sensor located on the roof of the vehicle straight up the mid-distance between the rear wheels. The radio receiver is in front of the vehicle.

Experimental vehicles, named Cycab, are depicted on fig1. Two passengers can be transported simultaneously. They have been designed specifically for car-free cities.

Indeed, the small dimensions (length : 1.90m, width : 1.20m) are advantages in urban traffic. These vehicles are entirely under control computer. They can be driven manually with a joystick, or automatically. Cycabs have two hours autonomy, and the used kinematic configuration is the same as a car-like vehicle. Its maximum speed is 18km/h. In our experiments, a personal computer is added in each Cycab. This high-level computer collects information received from GPS, mounted by serial link, and from front vehicles, via WiFi. The steering values and the vehicle velocities computed by control laws are transmitted to the Cycab computer by an Ethernet network and reach the two MPC555 micro-controllers (one for front wheels, the other one for rear wheels) by CAN. • C is the common reference path, defined in an absolute frame

[A, X A , Y A ). • O i is the center of the i th vehicle rear axle. • M i is the closest point on C to O i . • s i is the curvilinear coordinate of point M i along C, c(s i )
denotes the curvature of path C at this point, and θ c (s i ) stands for the orientation of the tangent to

C at M i , with respect to frame [A, X A , Y A ). • θ i is the heading of i th Cycab at point O i , with respect to frame [A, X A , Y A ). • θi = θ i -θ c (s i )
denotes the angular deviation of the i th vehicle with respect to C. • y i is the lateral deviation of the i th vehicle with respect to C. • δ i is the orientation of the i th vehicle front wheel with respect to its centerline.

• l is Cycab wheelbase.

• v i is the i th vehicle linear velocity at point O i .

• n is the number of vehicles in the platoon.

B. State Space Model Derivation

The vector (s i , y i , θi ) describes the state of the i th vehicle. The celebrated tricycle model is (see [START_REF] Samson | Control of chained systems: application to path following and time-varying point stabilization of mobile robots[END_REF], [START_REF] Daviet | Platooning for small public urban vehicles[END_REF], [START_REF] Thuilot | Accurate automatic guidance of an urban electric vehicle relying on a kinematic GPS sensor[END_REF]):

ṡi = v i cos θi 1 -y i c(s i ) (1) ẏi = v i sin θi (2) θi = v i tan δ i l - c(s i ) cos θi 1 -y i c(s i ) (3)
Control objectives are to bring and maintain y i and θi to 0, thanks to δ i , and (s is i+1 ) to a fixed value, thanks to v i . State variables s i and y i are directly provided by GPS sensor. θi could a priori be inferred from vehicle velocities, also provided by GPS sensor, but velocities measurements are too noisy, so that θi has to resort to a Kalman reconstructor, see [START_REF] Thuilot | Accurate automatic guidance of an urban electric vehicle relying on a kinematic GPS sensor[END_REF] for more details.

IV. CONTROL LAW DESIGN

First, it is shown that longitudinal and lateral controls can be decoupled. Then longitudinal control law is designed. Moreover, it is considered that : y i = 1 c(si) (vehicle is not on the reference path curvature center) and θi = π 2 [π]. In practical situations, if the n vehicles are well initialized, such difficulties never arise.

A. Decoupling feature

Via invertible state and control transformations, nonlinear Cycab model ( 1)-( 2)-( 3) can be converted, in an exact way, into the following so-called chained form, see [START_REF] Samson | Control of chained systems: application to path following and time-varying point stabilization of mobile robots[END_REF]:

ȧ1i = m 1i ȧ2i = a 3i m 1i ȧ3i = m 2i
where A = (a 1i , a 2i , a 3i ) T = Θ(s i , y i , θi ) is the chained state vector and M = (m 1i , m 2i ) T = Υ(v i , δ i ) is the chained control vector. From this chained form, a large part of linear systems theory can be used (but, since the transformations are exact, it is not required that vehicle configuration is in a specific configuration, contrarily to tangent linearization techniques). More precisely, it can be noticed that lateral control (i.e. control of a 2i and a 3i ) can be achieved by designing only m 2i . Since it can be shown that m 2i is related in an invertible way to δ i (provided that v i = 0), lateral control is fully decoupled from longitudinal one: in lateral control, v i appears as a free parameter, that can now be used to achieve longitudinal control. Details and performances of lateral control are presented in [START_REF] Thuilot | Accurate automatic guidance of an urban electric vehicle relying on a kinematic GPS sensor[END_REF].

B. Longitudinal Control

1) Standard Mode: Longitudinal (or platooning) control aims at preserving a constant curvilinear distance d, between the i th and the (i + 1) th vehicles. So, the control objective is to keep e i equal to 0, with

e i = (s i -s i+1 ) -d (4)
When differentiating e i , it can be obtained that:

ėi = v i cos θi 1 -y i c(s i ) - v i+1 cos θi+1 1 -y i+1 c(s i+1 ) (5)
Just as in lateral control design, exact linearization techniques can also be used: the actual control variable is the follower velocity v i+1 . Let us however introduce auxiliary control u i+1 , related to v i+1 via:

v i+1 = 1 -y i+1 c(s i+1 ) cos θi+1 v i cos θi 1 -y i c(s i ) -u i+1 ( 6
)
Error evolution is then simply given by ėi = u i+1 . Therefore, it is intuitive to design the auxiliary control law as:

u i+1 = -k e i ( 7
)
with (k > 0). Reporting [START_REF] Girault | Stability analysis of a longitudinal control law for autonomous vehicles[END_REF] in [START_REF] Gillet | BURST : Bright urban system for transportation[END_REF] provides us with the actual platoon control law:

v i+1 = 1 -y i+1 c(s i+1 ) cos θi+1 v i cos θi 1 -y i c(s i ) + ke i ( 8
)
Longitudinal control performances can then be adjusted at will by tuning gain k. The desired gain value, denoted k max , has however to be altered when longitudinal error e i is large, in order to guarantee that control variable v i+1 always meets the application constraints :

0 ≤ v i+1 ≤ v max ( 9
)
(i.e. the follower vehicle is not supposed to go backward even if e i is largely negative, nor to climb beyond a prespecified maximum velocity v max even if e i is largely positive). Therefore, control gain k max is here decreased according to the smooth adaptive gain tuning depicted on fig 4. Straightforward computations can display that constraints (9) are then satisfied, at least in standard situations where lateral errors are small (i.e. when 1-yi+1 c(si+1) cos θi+1

≈ 1

and cos θi 1-yi c(si) ≈ 1). 9). However, vehicle acceleration is not upper-bounded, which may be unpleasant to passengers. Therefore, a monitoring scheme, shown on fig 5, has been introduced to take care of passengers comfort, as far as their security is guaranteed.

More precisely, the notation a comf on Figure 5 stands for the maximum acceleration/deceleration comfortable to a passenger aboard. Then, if the acceleration/deceleration of vehicle i + 1, denoted a i+1 , is superior to that value, a security test is performed, in order to investigate if a i+1 can or cannot be safely limited to ±a comf :

• the case a i+1 > a comf > 0 may occur if vehicle i + 1 is far behind vehicle i. In such a situation, a i+1 can be limited to a comf without any collision risk. • the case a i+1 < -a comf < 0 may occur if vehicle i slows down abruptly. The final distance di between vehicle i + 1, if it was stopped with a deceleration -a comf , and vehicle i, if it was remaining immobile, is then computed. → if di is superior to a security distance d secur , then a i+1 can also be safely limited to -a comf . → if di < d secur , the urgency deceleration -a urg < -a comf leading to a final distance equal to d secur is computed, and a i+1 is limited to that value. Passengers security is therefore guaranteed. When computing di and a urg , delays introduced by actuator features and transmission latencies are considered. In simulations, gains of the lateral control law are tuned to impose that lateral deviation converges to 0 within 15m. Gains of the longitudinal control law are set to k max = 0.6, v max = 4m/s and a comf = 1m/s 2 . This last value is ensued from studies reported in [START_REF] Inc | Simulated on-the-road emergencies used to test stopping sight distance assumptions[END_REF]. The desired curvilinear distance is chosen equal to 8m and d secur is set equal to 3m. Finally RTK-GPS sensor features are introduced in simulation via a white noise with a standard deviation of 2cm added to position measurements.

Simulation results are reported on fig 6 to 9. Vehicles are initially parked besides the reference path. The objective is: when the leader arrives close to the second vehicle, it hooks it. Then, the second vehicle hooks the third one and so on up to the 6 th one. The leader velocity is specified by the user : it starts from 0m/s and reaches progressively 1m/s. During platoon formation, the influence of adaptive gain can be observed: the more the error is important, the smaller the gain is. By this way, vehicles velocities remain also inferior to v max , as desired. Once the platoon has been constituted, (fig7 and fig9), it can be noticed that each longitudinal error tends to zero, and consequently, gain k reaches its maximal value. At time t = 75s, leader velocity is increased up to 2.5m/s. Velocities of the follower, as expected, stay close to the leader one (fig9). However, limitations of near-tonear approach (only information relative to the preceding vehicle are used to compute v i+1 ) can be visualized: prior to t = 75s, velocity evolutions of the last followers are much more noisy than the evolutions of the first ones, due to errors accumulation from vehicle to vehicle. Therefore, it appears utopian to control a platoon composed of numerous vehicles with a control law based on a near-to-near approach, which can only be seen as a first step in platoon control design. Finally, it can be observed that reference path curvature has no influence on vehicles velocities, and inter-distances. This was expected from lateral/longitudinal decoupling features exhibited by chained form model.

VI. EXPERIMENTAL RESULTS

Several experiments have been carried out at "Campus des Cézeaux", in Clermont-Ferrand area. The experiments presented here involve only two Cycabs. However, experiments with more vehicles are planned at short term, since our lab will soon be equipped with additional vehicles.

The reference path is shown on fig10. The desired vehicle inter-distance and the values of the control gains are chosen identical to those described in the simulation section. The first experiment presents a hooking, and then the control law (in standard mode described by ( 8)) maintains the desired gap. At initial time, the first Cycab is close to the reference path and the second is waiting for the passage of the leader, see fig10. The leader velocity is 1m/s. Accuracy of the lateral control law is very satisfactory: the maximum lateral deviation is close to 3cm during straight lines and 10cm during curves. This is in accordance with lateral performances presented in [START_REF] Thuilot | Accurate automatic guidance of an urban electric vehicle relying on a kinematic GPS sensor[END_REF] and these small errors are due to some delays at the actuators level. Longitudinal behavior is also the one expected: the second vehicle is evaluating its curvilinear distance to the leader one, thanks to GPS and WiFi communication. Once the desired curvilinear distance is close to be reached, it starts progressively, hooks the leader and joins it on the reference path. The longitudinal deviation can be observed on fig11. The convergence to the desired gap is progressive, thanks to adaptive gain, see fig12. Once the second vehicle is hooked, the gap between the two vehicles presents a satisfactory mean value inferior to 1mm and a standard deviation of 4.8cm. We can also notice that the follower trajectory converges progressively to the reference one, see fig10. There is no violent turning of the vehicle towards the closest point on the reference path. This was expected since the follower has to converge to the reference path within 15m, according to the lateral control gains.

In the second experiment, monitoring mode is investigated. Control parameters are still the same as previously. At the beginning, the longitudinal control law is in standard mode. Then, the leader vehicle stops abruptly. The security test shown on fig 5 ensures that the follower vehicle can be stopped safely with deceleration -a comf . Therefore, follower decelerates at -1m/s 2 until it stops, see fig15. Due to the constant deceleration, velocity decreases linearly and the curvilinear distance parabolically. During this abrupt stop, monitoring mode privileges passengers comfort instead of longitudinal performances. As a consequence, the braking distance is quite large, and the curvilinear distance is under the desired one. Finally, a delay between the instants when the leader stops, and when the follower begins to decelerate can be observed. It follows from actuation delays.

In the third experiment, security distance is raised up to 6.50m instead of 3m (others parameters are unchanged). In practical situations, we could imagine that the leader is pulling a trailer, and that collision occurs if the interdistance is inferior to 6.50m. Initially, the two Cycabs move at the same speed, with a constant gap close to 8m. Then, once more the leader stops abruptly. Security test depicted on fig 5 concludes that the follower vehicle cannot match d secur if it is stopped with deceleration -a comf . Therefore, an urgency deceleration equal to -1.2m/s 2 is computed and applied, see fig16 and fig17.

When the two vehicles are stopped, it can be noticed that the curvilinear distance is 40cm over d secur . This just follows from the fact that actuator and transmission delays considered in -a urg computation are over-estimated. In this paper, platooning control of automatic guided vehicles has been addressed. The case of urban electric vehicles, equipped with a RTK-GPS sensor, and exchanging data via WiFi communication, has been considered. Nonlinear control techniques have been used to fully decouple lateral and longitudinal control. Monitoring has been sketched in order to guarantee passengers security and comfort. Finally, performances of this control scheme are investigated via full-scale experiments.

Nevertheless, possible losses of signal satellites, due to canyoning effect in urban environment, are a major concern. To deal with this difficulty, data fusion combining GPS information (when available) to dead reckoning from inertial sensors is investigated to maintain continuously localization measurements. An other investigated solution consists to reconstruct the vehicles state by vision, see [START_REF] Royer | Towards an alternative GPS sensor in dense urban environment from visual memory[END_REF]. Finally, as seen in simulations, platoon stability with numerous vehicles can not be guaranteed in a satisfactory way as long as vehicles are controlled from near-to-near. We are presently considering the case where the follower receives information, not only from the preceding vehicle, but also from every vehicle that is ranked before it in the platoon. Control design is further investigated in order to benefit from these new information.

ACKNOWLEDGMENT

This work was funded by CPER Auvergne 2002-04 program.

Fig. 1 .

 1 Fig. 1. Our experimental vehicles : Cycabs

Fig. 2 .

 2 Fig. 2. Architecture

Fig. 3 .

 3 Fig. 3. Model tricycle description

Fig. 4 .

 4 Fig. 4. Adaptive gain tuning

Fig. 5 .

 5 Fig. 5. Monitoring scheme

Fig. 6 .Fig. 7 .Fig. 9 .

 679 Fig. 6. Path following achieved by the platoon

Fig. 10 .Fig. 11 .Fig. 12 .Fig. 13 .

 10111213 Fig. 10. Reference trajectory and initial states -First Experiment

Fig. 16 .

 16 Fig. 16. Curvilinear distance between cars -third experiment

Fig. 17 .

 17 Fig. 17. Velocities -third experiment


Gabriel Olive

Wouter Gilles Olive
email: gilles.olive@eicvn.be

Gabriel Olive

Gilles Olive

Wouter Dekonninck

come

 2016), here an inventory to complete these ant lists is reported. A sampling site in the east part of the city was studied because this area contains different ecological areas mainly suburban and Garigue. Two methods and various sites were selected to have the most species observed.

Aphaenogaster splendida (Roger, 1859) Messor capitatus (Latreille, 1798)

Results

Hand sampling

Pitfall 1 to 4 with sweet white wine as attractive Passive pitfall traps "swimming pool"

muellerianus Inventory of ant species in the Marseilles suburbs

  Gabriel OLIVE, Gilles OLIVE and Wouter DEKONINCKIntroductionMarseilles is a big city and in its surroundings you can find several ecological areas, like urban, suburban and Garigue areas (type of low, deciduous Mediterranean forests). Besides a list on AntArea and a list composed by BERVILLE et al. (

								The numbers in the cells indicate
								the month of observation for
	Aphaenogaster dulciniae Emery, 1924		X			12		Hand sampling
	Aphaenogaster senilis Mayr, 1853			7				Pitfall 1
	Aphaenogaster splendida (Roger, 1859)							4, 8	8, 10 Pitfall 2
	Aphaenogaster subterranea (Latreille, 1798)					8, 11		Pitfall 3
	Camponotus aethiops (Latreille, 1798)	7	X					Pitfall 4
	Camponotus cruentatus (Latreille, 1802)			5				4, 8, 8	10	Pitfall "swimming pool"
	Camponotus foreli Emery, 1881					7
	Camponotus lateralis (Olivier, 1792)	7	X			7		8
	Camponotus piceus (Leach, 1825)	7	X
	Camponotus sylvaticus (Olivier, 1792)		X
	Cataglyphis cursor (Fonscolombe, 1846)			5				8
	Chalepoxenus muellerianus a (Finzi, 1922)					12
	Crematogaster auberti Emery, 1869		X
	Crematogaster scutellaris (Olivier, 1792)	7	X			12	4	3, 4, 8, 11	1
	Crematogaster sordidula (Nylander, 1849)		X	5
	Formica cunicularia Latreille, 1798					7
	Formica rufibarbis Fabricius, 1793		X		5
	Goniomma blanci (André, 1881)					9
	Hypoponera eduardi (Forel, 1894)		X		5	10, 12
	Lasius gr. alienus (Foerster, 1850)							8
	Lasius cinereus Seifert, 1992			5		12
	Lasius grandis Forel, 1909		X
	Lasius lasioides (Emery, 1869)					8
	Lasius niger (Linnaeus, 1758)				5
	Linepithema humile (Mayr, 1868)		X		5
	Messor barbarus (Linnaeus, 1767)	8	X
	Messor bouvieri Bondroit, 1918	7	X			7
	Messor capitatus (Latreille, 1798)							8
	Messor structor (Latreille, 1798)		X					8, 8	8, 8
	Myrmecina graminicola (Latreille, 1802)					10
	Myrmica sp. Latreille, 1804		X
	Myrmica ruginodis Nylander, 1846		X
	Pheidole pallidula (Nylander, 1849)		X	5				8, 11	4, 8	8
	Plagiolepis pygmaea (Latreille, 1798)		X	5	5	7		8
	Plagiolepis xene Stärcke, 1936			5		7
	Solenopsis sp. Westwood, 1840		X
	Solenopsis fugax (Latreille, 1798)				5	10
	Strumigenys tenuipilis Emery, 1915					11
	Tapinoma madeirense Forel, 1895					11
	Tapinoma magnum Mayr, 1861					7
	Tapinoma nigerrimum (Nylander, 1856)		X			12
	Temnothorax aveli (Bondroit, 1918)					7, 12
	Temnothorax kraussei (Emery, 1915)		X
	Temnothorax lichtensteini (Bondroit, 1918)		X		5	7, 9, 12
	Temnothorax luteus (Forel, 1874)			5		7
	Temnothorax niger (Forel, 1894)		X			9
	Temnothorax parvulus (Schenck, 1852)					7, 9, 10
	Temnothorax racovitzai (Bondroit, 1918)		X
	Temnothorax recedens (Nylander, 1856)					9
	Temnothorax unifasciatus (Latreille, 1798)					7, 12
	Tetramorium immigrans Santschi, 1927					9
	Tetramorium semilaeve André, 1883					7, 8
	Tetramorium gr. caespitum (Linnaeus, 1758)		X					8
	References	AntArea AntArea	BERVILLE	AntArea AntArea AntArea AntArea AntArea This Work This Work This Work This Work This Work

a Current Valid Name: Temnothorax