2012 Intelligent Vehicles Symposium Alcalá de Henares, Spain, June 3-7, 2012
Backstepping based approach for the combined longitudinal-lateral vehicle control Lamri Nehaoua and Lydie Nouvelière of the dynamics model representing small motions in the neighborhood of the straight motion is considered [3]. In addition, the present project context considers the low speed automation where the system nonlinearities can be neglected. All variables are listed in table I.
Abstract— This paper presents an integrated control method of a light road vehicle driving on a known and high secured itinerary at low speed. A planning module sends a safe and low energy reference trajectory to a control module that permits to manage the trajectory tracking under 50 km/h. A backstepping procedure is presented and formulated to realize a coupled longitudinal and lateral control in lane change or collision avoidance maneuvers.
A. Lateral dynamics For the vehicle lateral dynamics, the following assumptions are made: • the vehicle is represented by one rigid body where the suspensions degrees of freedom (D O F) are neglected, • the equivalent wheel is considered at the longitudinal vehicle axis (this leads to the so-called bicycle model), • the roll D O F is neglected since it is mechanically limited and its impact is very limited within velocity range under 50[km/h], • only small disturbances around equilibrium are considered which leads to a linear model representation. Under this assumptions, the mathematical equations of the vehicle dynamics are expressed as follows:
I. INTRODUCTION In the last ten years, several research projects were centered on the development of Automated Driving Assistance Systems (ADAS) either for autonomous or cooperative light or heavy road vehicles (PREDIT ARCOS, FP6 EU PReVENT [1], FP6 EU HAVEit [2]). In the HAVEit project, a copilot was developed by integrating a scale from the manual driving to the automated driving. The ABV project (Automatisation Basses Vitesses1 ), a French National Research Agency project, aims at developing a copilot at low speed under 50 km/h, on an itinerary that can offers to the equipped vehicle a high degree of information (safety, environment, traffic, mobility). To enable this decision-making, the project covers the establishment of a high-level coupled controller for both longitudinal and lateral modes by taking into account the human-machine cooperation. This controller will ensure a global uniform stability of the vehicle and a robust tracking of the reference trajectories given by a planning module. The reference trajectories can take many forms depending on the considered maneuver: lane keeping, lane change, vehicle following and obstacle avoidance. The prototype vehicles involved in this project are equipped with by wire systems. This paper aims at developing a combined control algorithm for longitudinal and lateral dynamics by using the backstepping control technique. Section 2 will present the vehicle dynamics for a light road vehicle, section 3 will formulate the problem for a backstepping application while section 4 will show several simulations results under different conditions like a disturbed scenario by a wind force to demonstrate the robustness of the developed method.
˙ = Fyf + Fyr + Fwy m(v˙ y + vx ψ) Iz ψ¨ = lf Fyf − lr Fyr + lw Fwy
Fig. 1.
The use of a whole nonlinear model is hard upon to develop a suitable controller, even by using a nonlinear control method. For these reasons, a linearized version This work was supported by the ANR project ABV authors are with IBISC laboratory, EA 4526, CE 1455, 40 Rue du Pelvoux, 91020 Courcouronnes, University of Evry-Val d’Essonne, France,
{nehaoua,nouveliere}@ibisc.univ-evry.fr
x˙ = Ax + µBδ + DFwy
Speed Automation
978-1-4673-2118-1/$31.00 ©2012 IEEE
Equivalent bicycle model of the vehicle lateral dynamics
During motion, the main external forces acting on the vehicle are those of aerodynamics and the tire/road interaction). Here, the side wind force Fwy is regarded as an external perturbation while the tire’s lateral slip forces Fyi are modeled by linear stiffness with respect to the tires slip angles. Therefore, the state representation of the lateral vehicle’s dynamic is given by
II. V EHICLE DYNAMICS
1 Low
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˙ is the state vector, µ is the road friction where x = [vy , ψ] and δ is the tire steering angle.
Fig. 3.
Equivalent wheel model of the vehicle longitudinal dynamics
where Fx is the tire longitudinal slip force, τw is the driving/braking torque around each equivalent wheel and Fwx = −cx vx2 is the longitudinal aerodynamics resistive force. With the assumption of a null tire longitudinal slip ˙ equation (4) becomes: where vx = Rw ζ, 1 (5) τw + mvy ψ˙ + Fwx Rw in which, me is the equivalent mass and the longitudinal aerodynamics force Fwx is regarded as an external disturbance. me v˙ x =
Fig. 2.
Vehicle positioning with respect to the planned trajectory
III. BACKSTEPPING BASED VEHICLE C ONTROL From equations (1,5), one can see that the coupling between the longitudinal and lateral modes occurs at three levels: • Lateral and longitudinal dynamics are connected by both vx ψ˙ and vy ψ˙ terms, • The nominal charge transfer also creates a coupling between both modes via the road friction µ. Indeed, charge transfer is affected by the vehicle longitudinal acceleration. This issue will not be considered. • For some aggressive maneuvers and/or under limit driving conditions, the mobilized road friction is limited (friction ellipse). This limitation creates a strong coupling between the longitudinal and lateral tire forces. Nevertheless, in the context of low speed automation, this coupling level will not also be taken into account. From this assumptions, the longitudinal and lateral control can be done separately by considering a time varying longitudinal velocity vx .
In order to develop a controller for a driving assistance system, it is essential to know the vehicle positioning variables with respect to the road lanes. It had make-up a close tracking of the reference trajectories issued by a planning module and minimize the position and orientation errors relative to the planned trajectory. This means that at any time, the vehicle axis should be parallel to the planned trajectory tangent axis and the vehicle lateral deviation should be close to the nearest trajectory point, which implies that the heading error ψL and the lateral error ψL should converge in finite time to zero. In that case, the derivative of both variables is expressed by:
ψ˙ L = ψ˙ − vx ρ
(3)
y˙ L = vy + vx ψL + ls ψ˙
where ls is the preview sensor distance.
A. Longitudinal control The longitudinal control consists at generating a driving torque τw to track a reference velocity profile vx,r , defined by the trajectory planning module [5]. For this, let consider the error tracking variable ex = vx − vx,r and by using the concept of the control Lyapunov function [6], a suitable control τw can be deduced. Definition 3.1: [7] A smooth positive definite and radially unbounded function V is called a Control Lyapunov Function (CLF) if:
B. Longitudinal dynamics In this mode, the motion equations are described as follows [4]:
˙ = Fx + Fwx m(v˙ x − vy ψ) iy ζ¨ = τw − Rw Fx
(4)
∂V (x) V˙ = x˙ < 0 ∂x 396
∀x 6= 0
1) step 1: By using the definition (3.1) a CLF can be chosen as V1 = 1/2eT e. The derivative of this CLF is given by:
Definition 3.2: [7] Consider a system described by: x˙ = f (x, u), where u is the system control input and f (0, 0) = 0. The control feedback α(x) for the control variable u such that the equilibrium x = 0 of the closed-loop system x˙ = f (x, α(x)) is globally uniformly stable if and only if the CLF V (x) satisfies:
V˙ 1 = eT {f + gx + hρ} From the definition (3.2), the control x such that
∂V (x) V˙ = f (x, α(x)) < −W (x) ∂x where W (x) is a positive definite function. By using definition (3.1) a CLF can be chosen as V = me /2e2x . The derivative of this CLF is given by: V˙ = me ex {v˙ x − v˙ x,r }
gx = −kx e − f − hρ
(10)
reduces the error dynamics to the first order differentiable equation e˙ = −kx e. However, the vector x is not a real system input and its dynamics must be stabilized. 2) step 2: in this step, one defines the virtual input α ≡ x and looks for a control feedback such that the error z = x−α converges in finite time to zero under a globally uniformly bounded form. By introducing z, the CLF derivative V˙ 1 becomes V˙ 1 = −kx eT e + eT gz. Next, an augmented CLF V2 = V1 + 1/2z T z is defined and its derivative is expressed by:
(6)
From definition (3.2) and equation (5), the control τw with 1 τw = −(kx + γx )ex − mvy ψ˙ + me v˙ x,r Rw
(9)
(7)
in which kx and γx are positive constants, makes the CLF of equation (6) as follows
V˙ 2 = V˙ 1 + z T {Ax + µBδ + DFwy − α} ˙
(11)
A possible control δ such that:
1 V˙ = −kx e2x − γx ex − Fwx γx 1 2 F ≤ −kx e2x + 4γx wx
2
+
1 2 F 4γx wx
µBδ = −(kz I2 + γz DDT )z − Ax + α˙ − g T e
(12)
in which kz and γz are positive constants, makes the CLF of equation (11) to be:
From this last equation, V˙ is negative if :
1 2 V˙ 2 ≤ −kx eT e − kz z T z + kFwy k∞ 4γz
1 |ex | ≥ √ kFwx k∞ 2 γ x kx
which implies that the control law δ render the system (8) globally uniformly bounded.
Since Fwx is a bounded disturbance, then ex (t) is bounded since :
C. Control robustification
1 kFwx k∞ kex k∞ ≤ max |e(0)| , √ 2 γ x kx
In the previous subsection, the feedback control of equation (12) required an information of the road friction. If this one is not updated, the control δ may not render the system (8) globally uniformly bounded. One powerful advantages of the backstepping control is the possibility of choosing a CLF control which satisfies design requirement, performance and even estimation. It will be shown that this approach allows to design a control feedback robust against the external side wind force while estimating the road friction coefficient µ. Firstly, the matrix A in equation (2) can be split as A = µA1 + A2 where A1 and A2 are two free friction matrices. If the estimated friction is referred as µ ˆ, the control law of equation (11) is given by:
B. Lateral control In this mode, one look for ensuring a global uniform convergence of the vehicle heading error ψL and the lateral deviation yL to zero. For this, let e = [yL , ψL ]T be the error vector where its dynamics is derived from equations (2,3) as: e˙ = f + gx + hρ x˙ = Ax + µBδ + DFwy
(8)
where x is the vehicle lateral dynamics state vector. The same procedure, as in the previous section, is applied for each subsystem [7]. In the first step, the error dynamics e˙ is stabilized by considering the vector x as the system input and afterward, the vehicle state dynamics x˙ is stabilized by using the tire steering angle δ.
µ ˆBδ = −(kz I2 + γz DDT )z − Ax + α˙ − g T e
(13)
By reporting this equation in (12), the CLF derivative becomes: 397
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Fig. 4. (1) Versailles Satory test track (France) : (x,y) map. (2) road map curvature ρ. (3-4) longitudinal velocity tracking: without and with considering the wind force Fwx . (5-6) vehicle lateral deviation and heading error at preview point. (7) vehicle lateral deviation error yc at the vehicle center of gravity. (8) zoom on the trajectory tracking. (9) estimation of the road friction µ.
IV. S IMULATIONS R ESULTS µ ˜ V˙ 2 ≤ −ke eT e − kz z T z + z T (−kz z − A2 x + α˙ µ ˆ 1 2 µ ˜ T T T F −g e + −γz z DD z + µ ˆ 4γz wy
In order to assess the performances of the previous exposed approaches, a first simulation is carried-out by using a nonlinear vehicle dynamics model. This model allows to simulate 8D O F: longitudinal and lateral displacements, yaw and roll rotations and the four wheels rotation spin where the tires longitudinal slip is included. The road curvature (Fig.4.2) and the reference forward velocity (Fig.4.3) are obtained from a log file coming from a real experimentation carried-out on Versailles Satory test track (France) of Fig.4.1. Fig.4.3-4 represent respectively the tracking performance of the backstepping longitudinal controller. In the first case, the external disturbance is neglected and an exact convergence in finite-time is achieved while in the second case, the external disturbance which represents the resistive wind force is modeled as Fwx = −cx vx2 . By using the backstepping controller of equation (7), a time-finite asymptotic convergence is achieved and the effect of the wind force is damped by the constant γx . In figures 4.5-6, the lateral deviation error yL , at a forward distance ls from the vehicle center of gravity, and the heading errors ψL are represented. In that case the tire steering control of equation (13) is used where a damping coefficient γz is introduced to attenuate the effect of the side wind
where µ ˜ = µ−µ ˆ is the estimation error between the real friction µ and the estimated one µ ˆ. Afterwards, a new augmented CLF is defined as: V3 = V2 +
1 2 µ ˜ 2γµ
where γµ is a positive constant and the CLF derivatives verifies: 1 2 V˙ 3 ≤ −ke eT e − kz z T z + F 4γz wy T µ ˜˙ z −kz z − A2 x + α˙ − g T e − γz DDT z + +µ ˜ µ ˆ γµ
which implies that the control law δ renders the system (8) globally uniformly bounded if the last term is zero and, hence, an adaptive law can be deduced to update the friction variable. 398
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Fig. 5. Lane Keeping: (1) road line to track: (x,y) map. (2) road map curvature ρ. (2-3) vehicle lateral deviation and heading error at preview point. (4) tire steering angle (5) longitudinal velocity tracking. (6) reference acceleration force.
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Fig. 6. Lane changing: (1) road line to track: (x,y) map. (2) road map curvature ρ. (2-3) vehicle lateral deviation and heading error at preview point. (4) tire steering angle (5) longitudinal velocity tracking. (6) reference acceleration force.
force Fwy . In this simulation, two cases are considered: the nominal case and the disturbed case where an uncertainty of 30% is considered on the vehicle mass, inertia and lateral tires stiffness. The aim is to highlight the robustness of the
proposed control against external disturbances like the side wind force and the vehicle parametric variations. Fig.4.7 shows the vehicle lateral deviation error yc at the vehicle center of gravity computed by yc = yL − ls ψL . A zoom of 399
a lane keeping is given in Fig.4.8 for both cases. Finally, Fig.4.9 demonstrates the effectiveness of the adaptive law for the estimation of the real road friction. This estimation is used to robustify the lateral backstepping control, however, the achieved estimation is not exact. This effect results from the small road friction variation assumption where the friction estimation error dynamics is taken to be µ ˜˙ ≈ µ ˆ˙ .
VII. A PPENDIX TABLE I A BBREVIATIONS , NOTATION AND NUMERICAL VALUES vx , vy ψ, δ ψ L , yL µ, ρ Fx , Fy Fwx , Fwy m Iz Cf , Cr lf , lr iy , R w ls
longitudinal and lateral velocity vehicle yaw and steering angles heading and lateral error road friction and curvature longitudinal and lateral tire forces longitudinal and lateral wind gust vehicle mass [kg] (1500) vehicle Z inertia [kg.m2 ] (2454) tires stiffness [N/rad] (57500) wheelbase [m] (1.006, 1.462) tire spin inertia and radius (1.2, 0.3) preview distance [m] (5)
The different matrices and vectors in equation (8) used for the backstepping control of the lateral mode are defined as following:
Fig. 7.
Rtmaps communication co-system
The second simulation is carried-out on the home developed vehicle-infrasturcture-driver cosystem software S I VIC where the communication between S I VIC and the different modules of the co-system is done by RTM APS (Fig.7) [8], [9], [10]. In the first case, a lane keeping maneuver is considered and the combined controller presents a high tracking preformance. At each calculation step, two reference trajectories are generated by the planning module as shown in the last figure of Fig.5. The blue trajectory is the optimal recommended one since it allows to keep the vehicle in its lane whereas the white trajectory is also suggested to the driver in the case where he want to change the current lane. The case of the lane changing is described by figures Fig.6.
A=
"
B=
"
f=
Cr +Cf mvx Cr lr −Cf lf 2µ Iz vx # C 2 mf C l 2 Ifz f
−2µ
v x ψL 0
g=
Cr lr −Cf lf − vx mvx Cr lr2 +Cf lf2 −2µ Iz vx
2µ
1 ls 0 1
h=
#
0 −vx
R EFERENCES [1] European commitee, http://www.prevent-ip.org/, website of the FP6 EU IP PReVENT research project, ERTICO manager, 2004. [2] European commitee, http://www.haveit-eu.org, website of the FP7 EU IP HAVEit research project, VERDURE manager, 2011. [3] D. Mammar, “Two-Degree-of-Freedom H∞ optimization and scheduling, for Robust Vehicle Lateral Control", Vehicle Systems Dynamics, Vol. 34, pp. 401-422, 2000. [4] L. Nouvelière, S. Mammar and J. Sainte-Marie, “Longitudinal Control of Low Speed Automated Vehicles Using a Second Order Sliding Mode Control", Intelligent Vehicles Symposium, 2001. [5] S. Glaser, B. Vanholme, S. Mammar, D. Gruyer and, L. Nouvelière, “Maneuver-based trajectory planning for highly autonomous vehicles on real road with traffic and driver interaction", IEEE Transactions on Intelligent Transportation System, vol. 11, No. 3, 2010, pp. 589-606. [6] H.K. Khalil, “Nonlinear Systems", Prentice Hall, ISBN 0-13-0673897, 2002. [7] M. Krstic, I. Kanellakopoulos and P. Kokotovic, “Nonlinear and Adaptive Control Design", John Wiley and Sons, ISBN 0-471-12732-9, 1995. [8] D. Gruyer, C. Royere, N. du Lac, G. Michel, and J. M. Blosseville, “S I VIC and RTM APS, interconnected platforms for the conception and the evaluation of driving assistance systems"’, in Proceedings of ITS World Congress, London, U.K., 2006. [9] D. Gruyer, S. Glaser, B. Monnier, “S I VIC, a virtual platform for ADAS and PADAS prototyping, test and evaluation", in Proceeding of FISITA’10, Budapest, Hungary, 2010. [10] D. Gruyer, S. Glaser, R. Gallen, S. Pechberti, N. Hautiere, “Distributed Simulation Architecture for the Design of Cooperative ADAS", First International Symposium on Future Active Safety Technology, toward zero-traffic-accident, Tokyo, Japan, 2011.
V. C ONCLUSION AND FUTURE WORKS The ABV project aims at integrating several functions of automation under automated driving assistance systems (ADAS) in a human-machine cooperation mode at low speed (under 50km/h). A control law with a backstepping approach is presented to control the longitudinal and lateral modes in a coupling structure. Several simulation results are shown, as much in trajectory tracking as in lane changing.
VI. ACKNOWLEDGEMENT This work was supported by the Agence Nationale de la Recherche (ANR) in the framework of Automatisation Basse Vitesse project (ANR-09-VTT-01 ABV). A part of the presented results was obtained with the sensors and vehicle simulator S I VIC with the cooperation of D. GRUYER from IFSTTAR LIVIC - FRANCE. 400
