G Blanc
email: gblanc@lasmea.univ-bpclermont.fr

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

P Martinet Lasmea

An image driven vision-based control framework for wheeled mobile robots

We deal in this paper with an original visual servoing method intended to control a nonholonomic wheeled mobile robot. These works are a part of a complete imagebased framework for navigation of a mobile robot, using only a monocular vision system embedded on the robot. The navigation system uses a set of relay images to enable the robot to reach a goal configuration expressed as a desired attitude of visual landmarks in the image plane. This paper particularly focuses on a suitable image-based control law of a nonholonomic robot for navigating in image database, with the assumption that the ground is obstacle free. We show how computing a suitable homographic transformation between the current camera image and a desired one allows to estimate an error vector on the state of the robot relative to a virtual straight line. This line is joined and followed thanks to a control law designed according to a well suited chained system of the state of the robot. Simulations illustrate this point, and experimental results for a two wheels driven robot confirm the validity of our approach.

I. INTRODUCTION

Navigation of a mobile robot includes many sub-tasks which are strongly dependent on each other. Given a global task of navigation, perception and control systems are used to make the robot succeed on it. From a perception module, with proprioceptive and/or exteroceptive sensors, a representation of the robot environment can be built, and it is then possible to locate the robot into this representation using the same perception system. This localisation allows to planify and generates adequated motions to achieve the specified task. Clearly, embedded sensors play a central role. Sonar and laser were the first sensors used for wheeled mobile robots navigation. More recently, the advent of fast and inexpensive digital imaging technology has allowed camera systems to be integrated as part of navigation systems. Given the advantages and shortcomings of each sensor, navigation approaches based on the fusion of observations provided by various sensors has been explored [START_REF] Borenstein | Mobile robot postioning -sensors and techniques[END_REF].

Computer vision can provide a powerful way of sensing the environment and controlling the robot motions. It is thus not surprising that more recent works have focused on the use of conventional or omnidirectional cameras as main sensors to navigate in partially or totally unknown environments. Indoor navigation tasks give a rise to a particular interest. Such workspaces are complex but well structured. Strong hypothesis such as parallelism of the floor and the ceiling or verticality of walls are generally respected. Such constraints are useful for 3D pose reconstruction methods, which allow the camera to be used as a global positioning system to locate the robot with respect to the observed environment. In order to obtain the 3D pose of the camera, several types of features can be used, such as straight lines [START_REF] Ait-Aider | Adaptation of Lowe's camera pose recovery algorithm to mobile robot self-localisation[END_REF] or rectangular patterns [START_REF] Hayet | A Visual Landmark Framework for Indoor Mobile Robot Navigation[END_REF].

From the control point of view, this kind of localization systems is useful because it allows to use directly state feedback based control laws designed for nonholomic robots [START_REF] De Luca | Robot motion planning and control[END_REF], which are proved asymptotically stable. A learn or planned path can be replayed with a control law adapted to path following, as it is done in [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single cp-dgps[END_REF] with a CP-DGPS.

In our approach, the whole problem of navigation of a mobile robot is formulated in the camera image space, as described in [START_REF] Blanc | Indoor navigation of mobile robot: An image based approach[END_REF] and represented by Fig. 1. Given an image database describing the robot environment, the robot motions are controlled from its current position to a desired one, specified through a target image selected in the image database. The image database is obtained and indexed off-line during a supervised learning step, and constitutes a visual memory of the robot. The contents of this memory is thus a set of images which provide a representative sample of all the points of view which could be reached during the displacement phase. When the robot is let staying at an unspecified place of its workspace, we determine which image of the memory is the closest from the observed one. This can be done, as Remazeilles and 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 of the visual memory. The task is specified as a target image to reach, chosen among the images of the database. A set of relay images describing a path from the current to the goal locations of the robot is then obtained thanks to the topological indexing of the database. The last step of this framework concerns the control scheme to bring an image of the given set to the next one and so on. This is the object of this paper.

The use of relay images to achieve large displacement positioning tasks is a promising approach. In [START_REF] Mezouar | Images interpolation for image-based control under large displacement[END_REF], the authors propose a multi-images framework to control the 6 dof of a robotic arm. The stabilization to a point is reformulated as a trajectory following in the image space through key images. Tsakiris et al [START_REF] Tsakiris | Extending visual servoing techniques to nonholonomic mobile robots[END_REF] have pointed out that image-based visual servoing techniques can be extended to nonholonomic mobile robots by adding degrees of freedom to the hand-eye system. The authors propose to embed the visual servoing control scheme in the task function formalism. Vision-based mobile robotic tasks such as wall following or self positioning with respect to landmarks is thus possible using this framework [START_REF] Kosecka | Visually guided navigation[END_REF]. Control laws for specific tasks have been developed, as for path following, if these extra degrees of freedom are not available. These methods relies on the uncoupling of the lateral and longitudinal control. Yi Ma et al present in [START_REF] Ma | Vision guided navigation for a nonholonomic mobile robot[END_REF] a theoretical framework to regulate the orientation of a wheeled mobile robot with respect to a ground curve using a single camera, by expressing in the image a state vector extracted from optical flow. More recently, motivated by the development of 2D 1/2 visual-servoing 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 estimation and epipolar geometry of the scene for visual-servoing of mobile robots. In [START_REF] Fang | 2.5d visual servoing of wheeled mobile robots[END_REF] and [START_REF] Chen | Homographybased visual servo tracking control of a wheeled mobile robot[END_REF], a partial estimation of the camera displacement between the current and desired views is used to design 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 asymptotic stabilisation on a desired image. In [START_REF] Chen | Homographybased 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 Lyapunovbased analysis. In our case, as a difference with a whole video sequence, we deal with a set of relay images which have been acquired from geometrically spaced out point of view. As a consequence, we design an homography-based visual-servoing control law appropriated to large displacements, i.e where the initial error is large. The authors of [START_REF] Mezouar | Images interpolation for image-based control under large displacement[END_REF] and [START_REF] Remazeilles | Robot motion control from a visual memory[END_REF] deal with this issue for a manipulator robot. Taking into account the nonholomic constraints of wheeled mobile robots, we propose a suitable formulation of our problem as a path following, with the assumption that the evolution plane of the robot is obstacle free. We show that computing a suitable planar homography between the current image and the next relay one allows to design a path connecting these images. This path can then be followed using a suitable control law. We propose to uncouple guidance and longitudinal control thanks to a chained form of the kinematics of the system. Leaving the longitudinal speed free allows more flexibility for the navigation task. In this paper, the evolved theory is illustrated in an indoor environment. However, other applications could be considered, as navigation of urban vehicles, which should move in a structured environment at low velocities. The remainder of this paper is organized as follows. In Section II, the problem we address and the models used through the paper are described. The Section III is devoted to the control law design. As an example of the generic proposed framework, the particular case of indoor navigation with a camera looking at the ceiling is treated in Section IV. In Section V, simulation and experimental results which confirm the validity of the proposed approach are presented.

II. PROBLEM FORMULATION AND MODELS

Throughout the paper, we consider a nonholonomic system with unicycle kinematics. Let F c and F m be the frames attached to the camera and to the mobile robot respectively and consider a plane Π on which is lying some natural or artificial landmarks. We note F * c the desired frame corresponding to the camera pose from which a relay image I * was shot. Our aim is to bring F c to F * c by moving the mobile robot along a nonholonomic path.

A. Geometric relations

The plane Π is supposed to contain some landmarks which can be tracked in the image I and are identified in I * . Let P be a set of points lying on one of these landmarks L in I, and P * the set of corresponding points in

I * . p i = u i v i 1 , and p * i = u * i v * i 1
denote the homogeneous coordinates in pixel of each point lying on P and P * respectively.

m i = x i y i 1 and m * i = x * i y * i 1
are the corresponding homogeneous coordinates in meter. A projective homography matrix G related to Π maps the points p i and p * i :

αp i = Gp * i ( 1
)
where α is a non zero scalar. If at least i ≥ 4, it is well known that G can be estimated by solving a linear system. In the remainder of this paper, the internal parameters matrix K of the camera is assumed to be known. Then the Euclidean homography can be computed up to a non zero scalar factor α:

αH = K -1 GK (2)
Let us define Π in F * c by n * d * where d is a constant distance from Π to the origin of F * c , and n * is the unitary normal vector to Π in F c . The Euclidean homography matrix H can be decomposed into a rotation and a rank 1 matrix:

H = R + t n * d * (3)
R and t are the rotation matrix and the translation vector that compose the homogeneous rigid transformation between F c and F * c . Note that when F c is confounded with F * c , H is proportional to the identity matrix. Finally, let us note that it is possible to compute the homography matrix in the more general case where at least the projection of eight 3D points are matched in I and I * .

B. Canonical camera configuration

First, consider that the camera frame F c is confounded with the control frame. In this configuration, which is named canonical in the sequel, the camera optical axis coincides with the rotation axis of the mobile robot and the camera optical center is the axle midpoint of the mobile robot. The camera frame and the mobile robot are thus subjected to the same kinematic nonholonomic constraints. Considering that the robot is moving on a perfect horizontal plane without any incidence on the image of the dynamics of the mechanical structure, what can be assumed if the robot moves enough slowly, the R matrix is reduced to a simple rotation matrix around the optical axis, and t has only two degrees of freedom on the ground plane. 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, it means the rotation R and the scalar translation vector t d * . The normal vector n * can also be determined, but the results are better if n * has been previously well estimated, as in the case of indoor navigation with a camera looking at the ceiling.

C. General camera configuration

Of course, in real cases the camera frame F c often presents a 6 degrees of freedom rigid transformation with respect to the robot frame F m . However we will show that it is possible to come down to the simple configuration previously described where F c and F m are coincident (refer to II-B). Consider a virtual camera in the previously canonical camera configuration and let us note F v = F m a frame attached on it, and I v its image plane. Let R m and t m be the rotation matrix and the translation vector between F c and F v = F m . They are mechanically fixed and represent the hand-eye camera parameters. As exposed in section II-A, an homography related to Π ties up I and I v . The points p i and p * i can thus be projected in the virtual camera image through the constant homography H m from the knowledge of R m , t m and n * . The problem of controlling the motions of the mobile robot using features extracted from the image of the real camera can thus be reformulated as the control of the virtual camera.

Therefore, in the remainder of this paper, we deal with the canonical configuration of the camera, where F c is in fact the frame F v above-mentioned.

III. CONTROL LAW DESIGN

Our aim is to generate robot motion in order to superimpose I and I * , what reverts to move F c to F * c with respect to the unicycle-like nonholomic constraints assigned to F c . The kinematic screw of F c is assumed to consist only of a longitudinal speed V along the X c axis and a rotational velocity ω c around Z c .

A. Design principle

In the previous section, we have seen that from the knowledge of matched points in a target image I * and the current image I, the relative displacement between F * c and F c can be extracted as a rotation matrix R and a scaled translation vector t d * . Since X c Y c -axis of F c and F * c are supposed to span a plane parallel to the ground plane, an estimation of the angular deviation θ between F * c and F c can be directly extracted from R. Consider a straight line Γ, parametrized according to the position F * c and the Y * c axis, as represented on Figure 3. We can get out from t 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 * c (see Figure 3). To achieve this control objective, chained systems 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]. In the considered cases, such a settling distance can be directly determined by the observed relative translations between F * c and F c at the initial configuration. This property is exploited to ensure that I and I * could be superimposed during the robot motion.

B. Control law design

s . F c w c X c Y c Y c X c F c * * * O y V θ M Γ

Fig. 3. Modelisation for control law design

In this part, a control law for straight line following is designed for cart-like vehicle moving on a perfect horizontal plane under the conditions of pure rolling and non-slipping. Let us note ω v the cart angular velocity around the center of its rear axle. We also define the point M as the orthogonal projection on Γ of the origin of F c . A curvilinear coordinate s is associated to M on Γ. The state of the cart in its plane of evolution is given by X v = s y θ (refer to Figure 3). The cart control vector is U = V ω c . These two vectors are related by the following state space model:

   ṡ = V cos θ ẏ = V sin θ θ = ω c (4)
Of course this state description is derivated from the more general case where the robot is following a geometric path (see [START_REF] Samson | Control of chained systems. application to path following and time-varying stabilization of mobile robots[END_REF], and [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single cp-dgps[END_REF] for an applicative example). Let us now convert the state space model (4) into a chained system of dimension 3 a 1 a 2 a 3 with a 2 dimensional control vector m 1 m 2

. The derivative of such a chained form with respect to time is:

   ȧ1 = m 1 ȧ2 = a 3 m 1 ȧ3 = m 2 (5)
The almost linearity of this system is then viewable when the chained system is derivated with respect to a 1 , with

m 3 = m2 m 1 :    a 1 = 1 a 2 = a 3 a 3 = m 3 (6)
Choosing a 1 = s, the linear system ( 6) is independent from V , since it is driven by a variable homogeneous to the distance covered by the robot. The input variable m 1 is given by (refer to equations ( 4) and ( 5)):

m 1 = ṡ = V cos θ (7)
Let us now define a 2 = y. Then it comes:

ȧ2 = V sin θ = a 3 m 1 ( 8
)
Therefore a 3 must be chosen as:

a 3 = tan θ (9)
Consequently, a 3 is not defined for θ = π 2 [π]. In the general case of geometric path following, this condition is avoided by planning an appropriate path to follow. In our application, this can be avoided by choosing judiciously the relay images during the off-line learning step. Finally, we deduce m 2 from m 2 = ȧ3 :

m 2 = ω c cos 2 θ (10)
Thanks to the linearity of the chained system (6), a natural and simple expression for the control law is: [START_REF] Ma | Vision guided navigation for a nonholonomic mobile robot[END_REF] This last relation combined to (6) leads to:

m 3 = -K d a 3 -K p a 2 (K p , K d ) ∈ R +2 (
a 2 + K d a 2 + K p a 2 = 0 (12)
Consequently, both a 2 and a 3 converge to zero, independently of the longitudinal velocity of the robot as long as V = 0. In view of a 2 = y and a 3 = tan θ, [START_REF] Ma | Vision guided navigation for a nonholonomic mobile robot[END_REF] ensures that both y and θ converge to zero. Taking into account that V can be negative, the expression of m 2 = m 3 m 1 can be obtained from [START_REF] Ma | Vision guided navigation for a nonholonomic mobile robot[END_REF] and it is given by:

m 2 = -|m 1 |K d a 3 -m 1 K p a 2 (13)
Therefore, ω c is extracted from [START_REF] De Luca | Robot motion planning and control[END_REF] in which ( 13) is inserted:

ω c (y, θ) = -V cos 3 θK p y -|V cos 3 θ|K d tan θ ( 14
)
The expression of the control vector U = V ω c can thus be computed. The guidance control law given by equation ( 14) has been designed independently of the longitudinal speed V . As highlighted with equation ( 11), only Kp and Kd theoretically set the performances of the control law. Then V can be set to a constant value during all the motion or be time-varying, seeing that V has to be non-zero. The chained form consists in a double integrator (refer to ( 6)) and then the asymptotic stability of the control law ( 14) is guaranteed. In practice, the success of this control law relies on the gains K p and K d . Indeed, K p and K d determine the dynamics of the second order system (12), i.e the impulse response of y with respect to the covered distance by the point M on Γ. An appropriate settling distance s r for the system [START_REF] Malis | 2 1/2 d visual servoing[END_REF] can be set to a typical one for the considered application. The convergence of y and θ to zero before F c reaches the desired attitude F * c should be ensured if the settling distance is chosen smaller than the distance between two shots of two consecutive images of the database.

IV. APPLICATION TO INDOOR NAVIGATION

Indoor environments abound in planes. Thus our framework should be well adapted to this kind of workspace. In this section, we address the particular application where Π is the ceiling, supposed to be parallel to the ground plane. In this case, the unitary normal vector n * to Π expressed in the canonical camera frame at a goal point of view F * c is known to be n * = [0 0 1]. The Euclidean homography matrix H has thus a very simple expression (refer to equation ( 3)). Indeed, as the displacement between F * c 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 ( 3)):

H =   cos θ -sin θ t Xc sin θ cos θ t Y c 0 0 1   (15)
Therefore, H has only three degrees of freedom. Only two points lying on Π and matched in I * and I are theoretically necessary to estimate H. From the computation of H, one can extract directly θ which is the angular deviation with respect to Γ introduced in section III. The lateral deviation y can also be estimated since it is the X * c coordinate of the origin of F c : 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 guidance control law designed in section III for a nonholonomic vehicle is then supplied. The next section illustrate and validate our approach with simulations and experimental results.

y = -t Xc cos θ -t Y c sin θ (16)

V. RESULTS

The proposed approach has been first tested in a simulation environment. In these simulations, both the vision system and the robotic plateform are assumed to be time-continuous, with a view to validating our approach from a theoretical point of view. The robotic task consists of moving the robot to a certain pose with respect to features in the desired image. The camera is embedded on the mobile robot and its optical center is positioned vertically with respect to the rear axle midpoint. A tilt angle of 0.1rad and a pan one of 0.01rad are introduced between the optical axis and the vertical axis of the frame attached to the mobile robot. The coordinates of the camera optical center with respect to F m are 0.01m, 0.01m and 0.1m along X m , Y m and Z m respectively. The camera is looking at three points. The control law presented in Section III is used to control the mobile robot. Figure 4 We first note that the paths described by the features converge to the desired positions. As can be shown in the Figure 5, the corresponding camera path confirms the validity of the control law. We can also notice that the mobile robot joins a straight line in the ground plane before passing through its desired attitude. As shown in Figure 6 (a) and (b), angular and lateral deviations of the robot are well regulated to zero. When the straight line is joined, the mobile robot stops rotating and follows straightly the path (Figure 6 (c)). However, we state that the mobile robot runs with a constant value of longitudinal velocity and it is not stopped-dead on the desired configuration. This is an inherent issue to the chained systems, which impose V = 0. That explains why the error in image points coordinates increases after having converged to zero (See Figure 6 (d)). Nevertheless, we do not mind this point when the reached configuration corresponds to a relay image. It is more cumbersome for the last step of stabilisation on the In view of these results, the proposed approach has been tested on a Pekee T M robot from Wany Robotics. A 1/3" CMOS embedded camera is turned to the ceiling and is approximatively placed on the canonical configuration exposed in subsection II-B. Only a motors speed regulation runs on the main electronic card of the robot. The whole application, from image acquisition to the control law computation, is processed at the rate of 25Hz on a external 1.8GHz PC. Thanks to a robust tracking based on particular filter (see [START_REF] Chateau | Robust real-time tracking of a vehicle by image processing[END_REF]), a set of four points lying on the ceiling is available at each image acquisition. The desired positions of these points are extracted from the relay image to reach. The two wheels driven robot is modeled as a unicycle. The length of the total displacement done on the ground plane is about 1.5m. The real initial 3D angular and lateral deviations, with respect to the desired attitude of the robot are respectively 0.5rad and 0.6m. The control gains Kp and Kd have been chosen for a settling distance of 0.7m (refer to equation ( 11)). The Figure 7(a) shows that the control law has prevailed the task on its objective. The tracked points reach their desired positions in the image plane, and then they follow a straight line. The Figure 7(b) confirms the good behavior of the visual features. As illustrated by Figures 8(a) and 8(b), lateral and angular deviations are regulated to zero, and the rotational velocity of the robot tends to be null (refer to Figure 8(c)). We observe very small oscillations around the desired state of the robot due to the robot internal control loop on velocities. However, they do not disturb the global task of path following, which objective is reached.

VI. CONCLUSION AND FUTURE WORKS

Our works deal with the topic of wheeled mobile robots navigation, using only one embedded camera. Given an ordered set of relay images, we aim to move the camera from a relay image to the next one, and so on until the desired image. In this paper, we have more especially addressed the problem of controlling the robot in order to bring an initial image to a desired one. The presented vision-based control law consists on the computation of an Euclidean homography between the current image and the desired one, from which we extract an error vector on the state of the mobile robot with respect to an adequately chosen straight line. The control law objective is thus formulated as a straight line following. However, the controller is designed so that the line is joined before reaching the desired relay image. We have validated our framework through simulations and experimental results on a Pekee T M robot. Future works will be devoted to the analysis of relay images switching, keeping the velocities continuous.

Fig. 1 .

 1 Fig. 1. Global strategy for indoor navigation

Fig. 2 .

 2 Fig. 2. Considered setup

Fig. 4 .

 4 Fig. 4. Visual features trajectories: (a) in the real camera image, (b) in the virtual camera image. Rounds are the initial features, squares are the desired ones.

  presents the visual features trajectories in the image plane: (a) curves are plotted in the real image plane, (b) curves are plotted in the virtual image plane.

Fig. 6 .

 6 Fig. 5. 3D trajectory of the camera

Fig. 7 .Fig. 8 .

 78 Fig. 7. (a) Tracked points trajectories in the image. Rounds are the initial features, squares are the desired ones.(b) behavior of the tracked points coordinates relatively to their desired position.


F Hao
email: hao@lasmea.univ-bpclermont.fr

R Lenain

B Thuilot

P Martinet

Sliding Mode Control of Automatic Guidance of Farm Vehicles in the Presence of Sliding

Satisfactory results of automatic guidance of farm vehicles have been achieved in previous works as long as vehicles move without sliding, but unfortunately in agricultural applications, sliding always occurs inevitably which causes loss of accuracy. In this paper the problem of path following control of autonomous farm vehicles subject to sliding is addressed. To take sliding effects into account, a vehicle-oriented kinematic model is built in which sliding effects are integrated in the form of additive disturbances to the ideal kinematic model. By transforming the vehicle-oriented kinematic model into a perturbed chained system, a sliding mode controller, which is robust not only to the sliding effects but also to the input noise, is designed with the help of the natural algebraic structure of chained systems. Simulation results show that the proposed control law can guarantee high path-following accuracy even in the presence of sliding.

I. INTRODUCTION

High-precision auto-farming and precise automatic guidance of agricultural vehicles have been the subject of research for a long time, since autonomous farm vehicles have some benefits

• Human factors such as the driver's ability to see the ground, driver comfort and operator safety do not need to consider when designing vehicles, so the manufacture cost may be reduced. • Remove human operators from a tired uncomfortable working environment. • Vehicles can run at a particular speed which guarantees the tracking accuracy without constraints of operator factors, increasing production efficiency. Recently with the development of GPS technology, more and more researchers apply GPS to automatic guidance systems of agricultural vehicles, since GPS can provide realtime absolute position with a centimeter accuracy and outdoor working environment of agricultural vehicles is suitable for using GPS. In our previous works we have solved the problem of curved path following with unique RTK GPS [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single CP-DGPS[END_REF], the vehicle kinematic model is built under the assumption of pure rolling, a nonlinear controller guaranteeing high lateral and orientation accuracy has been designed by converting the kinematic model into a chained system, Kalman filter is used to reconstruct system states only from GPS information, satisfactory results of path following have been obtained in [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single CP-DGPS[END_REF] providing the vehicle moves without sliding. However due to various effects such as slipping of tires, deformability or flexibility of wheels, the conditions of pure rolling without sliding are never strictly satisfied, especially when vehicles move on a slippery ground or on a slope following a curved path, sliding inevitably occurs which deteriorates performances of the automatic guidance and even system stability.

Abundance of satisfactory results of path following control have been achieved as long as there is no sliding between vehicles and ground, but there are very few papers dealing with sliding. [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 and assuming that the parameter characterizing the sliding effects is exactly known. In [START_REF] Leroquais | Vibrational control of wheeled mobile robots not satisfying ideal evlocity constraints: the unicycle case[END_REF], it is proven that the dynamic model of the unicycle WMR is not flat when slipping effects are considered, but the flatness of the robot can be recovered in average by using vibrational control, a controller is designed based on the averaged model allowing the tracing errors to converge to a limit cycle near the origin. In [START_REF] Zhang | Variable structrue control of a differentially steered wheeled mobile robot[END_REF] the effects of wheel slips and external loads are considered as the disturbances to the system, a variable structure control is used to eliminate the harmful sliding effects when the bounds of the disturbances 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, a simple discrete-time model of vehicles combined with sliding mode control leads to a controller which is robust against the sliding effects, guaranteeing the position errors and orientation error converge simultaneously. In [START_REF] D'andera-Novel | Control of wheeled mobile robots not satisfying ideal constraints: a singular perturbation approach[END_REF] a general singular perturbation formulation was developed which has led to robust results for linearizing feedback laws ensuring trajectory tracking in presence of sufficiently small slipping and skidding effects. In [START_REF] Lenain | Adaptive control for car like vehicles guidance relying on RTK GPS: rejection of sliding effects in agricultural applications[END_REF] an extended kinematic model integrating sliding effects is built, the sliding effects are rejected by re-scheming the desired path adaptively based on the steady control errors which is mainly caused by the modeled sliding effects.

In this paper we investigate the problem of path following of autonomous farm vehicles when subjected to sliding. The main idea of this paper is regarding sliding effects as additive disturbances to the ideal kinematic model, then sliding mode control theories are used to design an robust controller which has ability to reject sliding effects from the vehicle's path following performance. The structure of this paper is that, in section 2 a vehicle-oriented kinematic model considering sliding is built in the path frame. In section 3 chained system properties are reviewed by recalling our previous works. In section 4 by transforming the vehicle-oriented kinematic model into a chained form, a new sliding mode controller is designed and the stability of the closed-loop system is proven. In section 5, some comparative simulation results are presented to validate the proposed control law.

II. KINEMATIC MODEL A. Notation and problem description

For simplicity the vehicle is simplified with a bicycle model such that the two actual front wheels are equivalent to a unique virtual wheel located at the mid-distance between the actual wheels. The angle between the axis of the front wheels and the vehicle body is called the steering angle δ which is adjusted to allow the vehicle to follow the desired path. The direction of the rear wheels is fixed along the body axis.

In this paper 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 follow: (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, M exits and is uniquely defined if the path meets some conditions. • 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 is the orientation of the tangent to the path at point M .

• 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 new set of state vectors in the path frame is (y, s, θ), the path following problem consists of finding a feedback control law

δ = K(s, y, θ, v) (1)
such that

lim t→∞ y = 0 (2)
lim t→∞ θ = 0 (3) s = 0 θ d θ ṡ v η T η N M y δ 1 c(s) o R(s)

B. Kinematic model

Assuming the vehicle moves without sliding, the ideal kinematic model of the vehicle with the state (y, s, θ) 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 (4)
But when sliding effects are taken into account, the pure rolling constraint is violated, ( 4) is no longer valid. To describe the violation of the pure rolling constraint, a vehicle-oriented model is built in which the sliding effects are introduced as additive disturbances which may cause uncertain effects on the vehicle performances, the vehicleoriented kinematic model is

           ṡ = v cos θ 1 -c(s)y + ε 1 ẏ = v sin θ + ε 2 θ = v tan δ l - c(s) cos θ 1 -c(s)y + ε 3 ( 5
)
where ε is the vector depicting the violation of the ideal rolling without sliding constraints, ε is bounded by a known constant γ, that is

|ε i | ≤ γ i .
In actual applications, the longitudinal friction is much larger than the sliding force and the longitudinal velocity varies slowly enough, so in this paper the sliding effect ε 1 in the longitudinal direction is negligible, the vehicle-oriented kinematic model is rewritten as

           ṡ = v cos θ 1 -c(s)y ẏ = v sin θ + ε 2 θ = v tan δ l - c(s) cos θ 1 -c(s)y + ε 3 (6)

III. PREVIOUS WORKS

A. Chained system properties

As presented in [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single CP-DGPS[END_REF], in our previous work a path following controller has been designed by converting the model (4) into a chained system which allows using linear system theories to design nonlinear controllers without any approximation while still relying on the actual nonlinear system model (see [START_REF] Samson | Applicatoin to path following and time-varying point stabilization of mobile robot[END_REF]). For a 3-D nonlinear system with two control inputs, the general chained system is written as derivation w.r.t t

   ȧ1 = m 1 ȧ2 = a 3 m 1 ȧ3 = m 2 (7)
where A = [a 1 , a 2 , a 3 ] and M = [m 1 , m 2 ] are respectively the states and control inputs of the chained system. The general chained system can be converted into a singleinput linear system by replacing the time derivative with a derivation with respect to the state variable a 1 . Using the notation d da

1 a i = a ′ i and m 3 = m 2 m 1 (8) the general chained system is changed into derivation w.r.t a 1    a ′ 1 = 1 a ′ 2 = a 3 a ′ 3 = m 3 (9)
where m 3 is the virtual control input.

B. Automatic guidance based on chained form system

Considering the kinematic model ( 4), via state transformation as following

(a 1 , a 2 , a 3 ) = (s, y, (1 -c(s)y) tan θ) (10)
the ideal kinematic model is transformed into the general chained system [START_REF] Samson | Applicatoin to path following and time-varying point stabilization of mobile robot[END_REF] in which

                       m 1 = v cos θ 1 -yc(s) m 2 = d dt ((1 -yc(s)) tan θ) = -vc(s) sin θ tan θ -v dc(s) ds cos θ 1 -yc(s) tan θy + v 1 -yc(s) cos 2 θ tan δ l -c(s) cos θ 1 -yc(s) (11)
Form [START_REF] Lu | Sliding mode control for nonholonomic mobile robot[END_REF] [START_REF] Zp | A robust adaptive backstepping scheme for nonlinear systems with unmodeled dynamics[END_REF], the expression of the single-input linear system (9) can be obtained. In [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single CP-DGPS[END_REF] the virtual control input m 3 is designed to be a PD-type controller

m 3 = -K d a 3 -K p a 2 (K p , K d ) ∈ R +2 (12)
which leads to

a ′′ 2 + K d a ′ 2 + K p a 2 = 0 (13)
It is easy to prove that both the state a 2 and a 3 can converge to zero asymptotically by choosing K d , K p . Through inverse conversion, the physical control law is obtained as

δ(y, θ) = arctan l cos 3 θ (1 -yc(s)) 2 dc(s) ds y tan θ -K d (1 -yc(s)) tan θ -K p y +c(s)(1 -yc(s)) tan 2 θ + c(s) cos θ 1 -yc(s) (14)
Satisfactory path following results have been reported in [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single CP-DGPS[END_REF] provided the vehicles move without sliding. But in actual applications especially when vehicles move on a slippery ground or make a turn on a slope, the substantial sliding effects cannot be ignored which causes a significant lateral deviation.

IV. ROBUST CONTROL LAW DESIGN

In this section a sliding mode controller is designed based on the vehicle-oriented kinematic model [START_REF] D'andera-Novel | Control of wheeled mobile robots not satisfying ideal constraints: a singular perturbation approach[END_REF] in which sliding effects are considered as additive disturbances.

A. Sliding mode control for perturbed chained system [START_REF] Floquet | One-chained form and sliding mode stabilization for nonholonomic perturbed systems[END_REF] has designed a sliding mode controller to stabilize a nonholonomic perturbed systems, but all the disturbances have to satisfy a linear constraint. [START_REF] Lu | Sliding mode control for nonholonomic mobile robot[END_REF] has investigated the problem of designing robust controllers for general chained systems. A sliding mode controller was designed after the chained system was converted into a single-input and time-varying linear model by setting one input as a timevarying function, but unfortunately the system becomes no longer always controllable. To overcome this problem a new scheme is proposed in this paper to design a sliding mode controller with the help of the natural algebraic structure of chained systems.

Considering the kinematic model ( 6), a perturbed chained system (15) can be obtained when the same coordinates transformation (10) is used, derivation w.r.t t

             ȧ1 = v cos θ 1 -yc(s) = m 1 ȧ2 = v sin θ + ε 2 = a 3 m 1 + ε 2 ȧ3 = d dt ((1 -yc(s)) tan θ) = m 2 + η (15) where η = (1 -yc(s))ε 3 cos 2 θ -c(s)ε 2 tan θ (16)
Noting that in (15) a i and m i have the same expression as it in (10-11), ε 2 and η act as two additive disturbances to the ideal system [START_REF] Samson | Applicatoin to path following and time-varying point stabilization of mobile robot[END_REF]. So similarly (15) can be converted into a perturbed single-input linear system by computing the derivative with respect to the state variable a 1 .

derivation w.r.t a 1        a ′ 1 = 1 a ′ 2 = a 3 + ε 2 m 1 a ′ 3 = m 2 m 1 + η m 1 = u + η m 1 ( 17
)
where u is the virtual control input of the disturbed singleinput system (17). Remarking that u is velocity independent, ε2 m1 and η m1 are bounded. Because the single-input model (17) contains uncertain bounded disturbances, sliding mode control theories are applied to design a robust controller which may guarantee the system states converge to a neighborhood near the origin.

Concerning the states a 2 , a 3 for the path following problem, the sliding surface is defined as

z = Λa 2 + a 3 (18)
One condition that guarantees the system states reach the sliding manifold z = 0 in finite time and remain in this mode in future time is z ż < 0. Once the sliding manifold is encountered, the system stability is achieved with an exponential convergence. Theorem 2: Define a strictly increasing function

s(t) = t 0 v + (τ )dτ (19)
where v + is positive definite and use the notation that (⋄) ′ = d⋄ ds , if the sign of v + (τ ) is kept positive, then the condition zz ′ < 0 is equivalent to the reaching condition z ż < 0.

Prove:

zz ′ = z dz ds = z dz dt dt ds = z ż 1 v + (τ ) (20)
if zz ′ < 0 then it is easy to prove that the reaching condition

z ż < 0 is satisfied provided v + (τ ) is kept positive. 2
In our applications, s(t) is the curvilinear coordinates of point M , v + (τ ) is the linear velocity of point M along the desired path C. v + (τ ) = m 1 , since the orientation deviation θ of the vehicle with respect to the desired path C varies in the range of (-π 2 , π 2 ) and the vehicle remains closed to the desired path which means that 1yc > 0, from [START_REF] Zp | A robust adaptive backstepping scheme for nonlinear systems with unmodeled dynamics[END_REF] the condition of v + (τ ) > 0 is satisfied.

Theorem 3: Considering the system (15) where (a 1 , a 2 , a 3 ) = s, y, (1c(s)y) tan θ , define

z = Λa 2 + a 3 = Λy + (1 -yc(s)) tan θ (21)
the achievement of sliding motions on the sliding surface (21) can be guaranteed by the control law

u = -kz -Λa 3 -ρsign(z) (22)
where

ρ ≥ |ς| = Λε 2 + η m 1 (23)
Prove: For the states a 2 , a 3 , consider the reaching condition of sliding mode control:

zz ′ = z(Λa ′ 2 + a ′ 3 ) = z Λa 3 + u + Λε 2 + η m 1 = z Λa 3 + u + ς (24)
Applying ( 22) into (24), we get

zz ′ = z Λa 3 -kz -Λa 3 -ρsign(z) + ς < -kz 2 -(ρ -|ς|)|z| (25)
if we choose ρ following (23), then the reaching condition zz ′ < 0 is satisfied guaranteeing a sliding motion on the sliding surface (21). On the sliding surface (21), one has

z = Λa 2 + a 3 = 0 (26)
which leads to

a ′ 2 = -Λa 2 + ε 2 m 1 = -Λa 2 + ̟ (27)
The stability of system (27) has been analyzed in [START_REF] Zp | A robust adaptive backstepping scheme for nonlinear systems with unmodeled dynamics[END_REF] in detail, from (27) a 2 can be expressed as

a 2 = e -Λs a 2 (0) + ̟ Λ = a 2 (0)e -Λ t 0 v + (τ )dτ + ̟ Λ (28)
so the solutions of the resulting closed-loop system are globally uniformly ultimately bounded. 2 Due to the definition of a 2 , a 3 in (10), it is proven that the lateral deviation y and the orientation error θ are globally uniformly ultimately bounded in the presence of sliding effects.

Sliding mode control ( 22) is simple, robust and can guarantee transient performances, but the low level delay caused by hydraulic-driven steering systems always results in chattering responses which may wear down the actuator and excite unmodeled dynamics, possibly compromising performance and even stability. To mitigate the problem of chatter, the signum function is replaced by the hyperbolic tangent function tanh()

u = -kz -Λa 3 -ρtanh( 0.2785ρz σ ) σ > 0 (29)
Combining ( 11) with (29), the physical steering angle is obtained by inverse conversion of the virtual robust control law u.

δ(y, θ) = arctan l cos 3 θ

(1yc(x)) 2 dc(s) ds y tan θ

+ u + c(s)(1 -yc(s)) tan 2 θ + c(s) cos θ 1 -yc(s) (30)
Remarking that some constraints have to be added to the system to ensure the validity of the controller:

• Because of the definition of m 1 in (11) which requires 1yc(s) = 0, the vehicle is not allowed to pass the curvature center of the path c, which means that y < 1 c(s) holds and makes the definition of ρ in (23) reasonable.

• Because of the definition of m 2 in (11) which requires cos θ = 0, the vehicle body axis may not vertical to the path, the orientation error varies only in the range of (-π 2 , π 2 ). However the importance of the constraints is limited due to the small path curvatures and the restricted range of the steering angle in most practical cases.

V. SIMULATION RESULTS

In this section, some simulation results are presented to validate the proposed control law. In order to fully demonstrate the effectiveness of the controller, two reference paths consisting of straight lines and curves are followed (see figure 2 and5), the sliding effects are introduced to the system when the vehicle follows the desired path stepping into a curve. To simulate all the other external unconsidered disturbances, noises are always added to the system through the same channel as the control inputs. In the simulations, the gains used in the control law (22) are set as Λ = 0.3 ρ = 0.08, k = 0.3. To compare the control performances with the previous works, the control laws (14) are applied under the same condition except that we set the controller gains as k p = 0.09, k d = 0.6.

The simulation results of the lateral deviation for two path-following experiments are shown by figure 3 and6 respectively, the results of the orientation errors are shown in figure 4 and7. In the simulation results, the dashed line represents the results yielded by using the controller (14), the solid line depicts the results obtained by applying the sliding mode controller (30) proposed in this paper.

Because the control law (14) does not take sliding effects into account and from theoretical point of view, the PD-type virtual control law is not robust against disturbances, it is clear that it suffers from sliding greatly, when the sliding effects appear, its lateral deviation becomes more significant than the other. While the sliding mode control law proposed in this paper provides satisfactory simulation results, it has good transient responses and is robust against not only the sliding effects but also other input noises which inevitably occur in actual applications, the simulation results show that the sliding effects affect both the lateral deviation and the orientation error weakly.

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 disturbances is used. From this model, a particular perturbed chained system is evolved. The use of the attractive structure of chained systems together with the sliding mode control leads to a robust controller which is robust to both the sliding effects and external disturbances. The system states have been theoretically proved globally uniformly ultimately bounded. The advantage of this new scheme is that • The anti-sliding controller proposed in this paper is developed still relying on chained system properties, so abundant linear system theories are available to design more powerful controllers without loss of nonlinear features. • This scheme is a primary work of designing robust controllers for chained systems. Since chained systems have a perfect natural structure for the use of sliding mode control, some skillful high-dimensional sliding surfaces can be designed to fulfill more complicated tasks, for example longitudinal control or anti-sliding control of vehicles with four-wheel steering kinematic model. • This scheme does not require more sensors and costs less on-board computation which yields a easy actual implementation.

Fig. 1 .

 1 Fig. 1. Notation and path frame description

Fig. 2 .

 2 Fig. 2. Path to be followed

Fig. 5 .Fig. 7 .

 57 Fig. 5. Path to be followed