A Robust Lateral Vehicle Dynamics Control

Transcription

1 AVEC 1 A Robust Lateral Vehicle Dynamics Control Giovanni Palmieri*, Miroslav Barić, Francesco Borrelli, Luigi Glielmo* *Engineering Department, Università degli Studi del Sannio, 821, Italy Department of Mechanical Engineering, University of California, Berkeley, , USA Benevento, 821, ITALY Phone: Fax: palmieri@unisannio.it The paper addresses the robust design of a lateral stability controller for ground vehicles. Nonlinear vehicle dynamics are modeled by an approximate hybrid piecewise affine model. Control of the vehicle yaw rate and lateral velocity of the center of gravity is achieved through steering and four wheels independent braking. Uncertainty in actuation is explicitly taken into account by introducing an additive, persistent disturbance taking values in a compact set which depends on the current control input. A robust control invariant(rci) set is computed for a nominal, linear dynamics of the vehicle. A time optimal robust control algorithm, that ensuresconvergenceofthevehicle sstatevectorintothercisetinafinitenumberofdiscrete time steps, is devised. The proposed control design is verified on simulations. Topics / Vehicle Control, Constrained robust control 1. Introduction Active safety is primarily used to avoid accidents and, at the same time facilitate better vehicle controllability and stability in emergency situations [12, 5, 4, 9, 14]. The available commercial active safety systems are mostly based on brake intervention. Antilock Braking Systems (ABS) and Electronic Stability Program (ESP) have been introduced to stabilize the longitudinal and yaw motion vehicle dynamics, respectively. The design and development of active safety systems tends to be traditional in nature [9], [4], [12]. Typically, local single input single output controllers are designed for each component or feature and, then, heuristic rules are used to coordinate their activations and the way they share computational resources. This requires extensive testing (both simulative and experimental) to deliver high performances and robustness at a significant cost by considering an enormous variety of drivers response and tire/road interaction. Furthermore, a formal verification of the safety of the resultant vehicle cyber-physical system (i.e., interconnection of vehicle, tire/road interaction and driver intervention) is computationally prohibitive with current methodologies. In this work, we present a preliminary study on systematic approach to yaw and lateral dynamics control. Nonlinear dynamics are approximated by a PWA discrete time system, whose state and control variables are subject to hard constraints. Uncertainty in actuation (braking and steering) deriving from neglected actuator dynamics, time delays and uncertain in control gains is taken into account by introducing and additive disturbance whose values belong to a compact set which depends on the currently applied control input. Finally, we design a controller which guarantees constraints satisfaction at all times for all possible disturbance realizations using set theoretic methods for control synthesis [2], in particular, the used technique is robust control invariance for discrete time systems affected by input dependent disturbances [1, 11]. To the best of our knowledge, this paper presents the first attempt to address hard constraints and uncertainties in the vehicle stability control system in a systematic and rigorous way through a robust constrained control design. We will refer to the proposed control framework as advanced ESP. The paper is structured as follows: in the section 2 we present the vehicle model, starting from the classical nonlinear bicycle model, we obtain a Piecewice Affine Hybrid realization. In the Section 3, we present the controller design while simulation results are illustrated in Section Model of the Lateral Vehicle Dynamics In this section, we derive a the PWA model of a

2 AVEC 1 lateral vehicle dynamics used for robust control synthesis. We start with the standard nonlinear bicycle model [9, Sec. 2.3] and approximate nonlinear Pacejka tire force characteristic [8] with a continuous PWA function. We use subscripts ( ) f and ( ) r to denote variables associated with the front and rear wheel, respectively. Also, the subscript () stands for both ( ) f and ( ) r. In literature, the bicycle model, based on small angle approximation, has been extensively used[4],[9], [3]: mÿ = mẋ ψ +2F cf +2F cr, I ψ = 2aF cf 2bF cr +M, (1a) (1b) where ẏ is the lateral speed, ẋ is the longitudinal velocity, ψ is the yaw rate, M is the braking moment, F c are the cornering tire forces, a is longitudinal distance from CoG to the front axle, b is longitudinal distance from CoG to the rear axle and I is yaw inertia moment of vehicle around the z-axis. The F lr F yr F cr F xr r y b Fig. 1: The simplified bicycle model of the vehicle. dependence of the cornering tire forces F c on the lateral speed ẏ and the yaw rate ψ needs to be specified. In general, this dependence is complex. In this paper, we opt for a simplified, semi empirical static model of the forces F c. In Pacejka s model [8] the cornering tire forces are defined by nonlinear mapping f c ( ): vc f x F l f a F c = f c (α,s,µ,f z ), (2) where s are slip ratios (the ratio between the longitudinal slip velocity and the forward speed of the wheel centre, [8, Sec. 2.2]), µ is the friction coefficient, α are tire slip angles (the angle between the longitudinal axis of the tire frame and the tire velocity, see Figure 1) and F z are normal tire forces. The mapping f c ( ) is given by the well-known Pacejka s formulas [8, Sec. 4.3]. A plot of the cornering (lateral) tire force F c for varying slip angle α and constant values of slip ratio, friction coefficient and normal force F z is shown in figure 2 (dashed line). In the same Figure one can see the piecewise-affine approximation of the cornering tire force used in the derivation of the PWA bicycle model of the vehicle, discussed next. F yf f v f F cf f v l f F xf F c [N] Original Pacejka Function PWA Approximation α [deg] Fig. 2: Lateral tire force. 2.1 PWA Hybrid Model The analytic form of the mappings f c ( ) as given by Pacejka s formulas is not suitable for synthesis of a robust, constrained control strategy using set theoretic methods. In order to facilitate the synthesis of a robust controller in the presence of constraints on state and input variables, we derive a PWA approximation of the nonlinear dynamical system (1). We work under the following assumptions: Assumption 1 The longitudinal velocity of the vehicle ẋ is constant and known. The friction coefficient µ is known, constant and the same for both wheels. Normal tire forces F z are assumed to be constant, known and the same for the front and the rear wheel. With Assumption 1 tire cornering forces F c, in pure cornering condition (s = ) depends on only the slip angles α. Then, in order to incorporate the nonlinear dependence of forces F c on the states ẏ and ψ, it is necessary to determine the dependence of the slip angles α on the lateral velocity ẏ and the yaw rate ψ. Side slip angle α (see Figure 1) is defined as α = tan 1 v c, /v l,. Assuming constant longitudinal vehicle velocity and the rear steering angle δ r =, we obtain the relations: ( ) v l,f = ẋ+ v c,f = ẋδ f + ẏ +a ψ ( ẏ +a ψ δ f, v l,r = ẋ, (3a) ) (, v c,r = ẏ b ψ ). (3b) Finally, substituting the equation (3) in tire side slip angle definition and considering the small angles assumption, we obtain the desired dependence of the lateral slip angels α f and α r on the yaw rate and the lateral velocity: α f = a ψ +ẏ ẋ δ f, α r = b ψ ẏ. (4) ẋ Theshapeofthemappingf c ( ), asshowninfigure2, is particulary amenable to piecewise affine approximation. In its simplest form such approximation

3 AVEC 1 comprises three affine parts: c s α +(c l +c s )α, if π 6 α α g (α ) = c l α, if α α α c s α (c l +c s )α, if α α π 6 (5) wheretheinterval[ α,α ]representsthelinearzone, the interval [ π/6, α i ] marks the negative saturation zone and the positive [α i,π/6] marks the positive saturation zone. We can now write the hybrid (PWA) bicycle model of the vehicle: [ÿ ] [ ] g f (α f )+g r(α r) mv = x g f(α f )a+g r(α r)b [ẏ ] mv x v x ψ g f(α f )a+g r(α r)b Iv x g f(α f )a 2 +g r(α r)b 2 ψ Iv [ ] x g f(α f )+g [δf r(α r) ] + m (6) M g f(α f )a+g r(α r)b I where v x = ẋ and expressions for g f (α f ) and g r (α r ) are defined by (5). Note that equations (5) (6) provide a hybrid dynamical system [13] with nine distinct dynamic behaviors described by linear (affine) differential equations. The actual dynamic behavior of the system is determined by the value of the side slip angles α according to (5). Using the relations (4) the hybrid bicycle model (5) (6) can be compactly written in the standard state space notation: 1 I ξ =A i ξ +B i u+f i, if (ξ,u) P i, (7) i {1,2,...,9}, where ξ = [ẏ ψ] T, u = [δ f M] T and {P i } 9 i=1 is a collection of polyhedral regions in R 4 defining the regions for each dynamic behavior and constraints on the state and control variables. The constraints imposed on states and control variables when the car is driving in the linear region are: a ψ +ẏ v x δ f.3, ẏ 1.67, b ψ ẏ.3, (8a) v x ψ.52, (8b) δ.41, M 88, (8c) where (8a) depends on the working mode of the vehicle, while (8b)-(8c) represent the physical limits of states and inputs (here we report the respective limits value for low (snow) friction coefficient). The projection of {P i } 9 i=1 onto (ξ,δ f) space is shown in Figure Uncertain discrete time hybrid model For the purpose of control synthesis we discretize the continuous time hybrid model (7) using the standard Euler method and obtain a discrete time hybrid system with the state transition equation: ξ + =A di ξ +B di u+f di, if (ξ,u) P i, (9) i {1,2,...,9}, Fig. 3: Mode regions of hybrid model in [ẏ, ψ,δ f ]-space.. where ξ + stands for the successor state, given the current state ξ and the control input u. For the sake of notational simplicity we adopted the same names for discrete time and continuos time variables. The final, discrete time model of the vehicle (9) is obtained under Assumption 1, which allows, for a simple model, and, at the same time, introduces the issue of discrepancy of the model and the actual vehicle. Such discrepancy can be taken systematically into account by introducing an uncertainty variable in the equations(9). Inthispaperweconsideranuncertainty in the actuation. In particular, the actual active steering systems generate a delay between the controller command and its physical realization in the actuator. Additionally, the joint braking moment M is distributed into four braking torques which are typically generated by an underlying hydraulic braking system, dynamics of which is neglected in the (9) for the sake of simplicity. The effect of all these uncertainties can be accounted for by extending the hybrid bicycle model with an uncertainty term w: ξ + =A di ξ +B di u+f di +Gw, if (ξ,u) P i, i {1,2,...,9}. (1) The uncertainty w is assumed to take values in a bounded set that depends on the applied control input according to: w W(u), (11) where W( ) is a set valued mapping used to represent the uncertainty bounds. At each time instant the value of the uncertainty w is not known to the controller. The controller, however, is aware of the bounds on w imposed by (11) and assumes that w can take an arbitrary value within the set W(u). For our purposes we will assume that each of the control inputs, the steering δ f and the braking moment M, is affected of ±1%uncertainty in its value. 3. ROBUST CONTROL DESIGN In this section we describe the robust control strategy for advanced ESP in the presence of uncertainties in the actuation, as specified in Section 2.2. Advanced ESP design aims primarily at inducing the nominal behavior of the vehicle, i.e., dynamic behavior obtained when neither front nor rear tires are

4 AVEC 1 saturated and, at the same time, tracking the given reference vector r as close as possible, for all possible realizations of the uncertainty w. We formalize these notions by set theoretic terms in particular robust control invariance and constrained controllability for the considered system class affected by input dependent uncertainties. Due to space limitations we provide only very limited discussion on these topics. For supplementary details the reader is referred to [2, 1, 11]. 3.1 Robust reachability for discrete time systems Consider a discrete time system: ξ + = f(ξ,u,w), (12) where ξ R n is the state vector, u R m is the control input and w R p is the uncertainty or disturbance vector. The variables (ξ, u, w) are subject to constraints: (ξ,u) C xu, w W(u), (13) where C xu R n R m are joint constraints on states and control inputs. The mapping W( ) is a set valued mapping that maps a control vector u R m into a non empty subset of the bounded set W and defines a bounded set of admissible disturbances w for a given control input u. The central role in robust control of the discrete time system (12) (13) isplayedbythe notionofone step controllable set. Definition 1 Given the dynamical system (12) subject to constraints (13), the one step controllable set with respect to target set X T is the set of states ξ for which there exists control u satisfying (ξ,u) C xu such that f(ξ,u,w) X T for all w W(u). We formalize the notion of one step controllable set for a family of target sets by introducing the controllability mapping: B(X) :={ξ: u such that (ξ,u) C xu and f(x,u,w) X for all w W(u)}. (14) The control mapping U X ( ) associated with B(X) is defined as: U X (ξ) :={u: (ξ,u) C xu and f(ξ,u,w) X for all w W(u)}. (15) Using the mapping B( ) the k step controllable set X k for a given target set X T can be constructed recursively: X = X T, X k = B(X k 1 ), k = 1,2,... (16) Algorithmic formulation of the mapping B( ) for general dynamical systems(12) subject to constraints(13) was initially given in [1] and subsequently published in [11]. We omit its formulation as the computational details are not relevant for the discussion and refer the reader to the original references. Using the constructs (14) (15) we now proceed to define the elements of our advanced ESP controller. 3.2 Robust control invariance and time optimal control strategy As stated above, advanced ESP is designed to preserve nominal behavior of the car for all possible uncertainties w that satisfy the uncertainty description given in Section 2.2. In other words, if at a time instant k the state control pair (ξ k,u k ) P 1, where P 1 is the polyhedral region corresponding to the nominal vehicle behavior ( linear mode) of the hybridsystem(1), the goalofthe advancedesp is to select the control input u k such that for the all possible successorstates ξ k+1 there exists the control u k+1 satisfying the constraint (ξ k+1,u k+1 ) P 1. We formalize this by introducing a mode 1 robust control invariant (RCI) set for system (1). Definition 2 A set R 1 R 2 is called mode 1 RCI set if for all ξ R 1 there exists a control u satisfying (ξ,u) P 1 such that A d1 ξ +B d1 u+f d1 +Gw R 1 for all w W(u). For our purposes it is desirable to characterize the maximal mode 1 RCI set R 1 which contains all other mode 1 RCI sets. If we define controllability mappings {B( ) i } 9 i=1 as B(X) i := {ξ: u such that (ξ,u) P i and A di ξ +B di u+f di +Gw X for all w W(u)}, (17) the maximal mode 1 RCI set R 1 can be obtained by the standard iterative algorithm [1]: Algorithm 1 Computation of the set R 1 R P 1 i = repeat i = i+1 R i B(R i 1 ) 1 until R i == R i 1 If the Algorithm 1 terminates in finitely many iterations i t, the set R 1 = R i t. The set R 1 can be actually empty, or for a general dynamical system the Algorithm 1 may not reach a fixed point in finitely many iterations. To the set R 1 we associate the control mapping U 1 ( ) defined on the domain R1 : U 1 (ξ) := {u: (ξ,u) P 1 and A d1 ξ +B d1 u+f d1 +Gw R 1, w W(u) }. (18) For a given state vector ξ, any control input selected forms the set U 1 (ξ) results in the successor state ξ+ being inside the set R 1. The maximal mode 1 RCI set is computed under certain assumption on uncertainies described by the vector w and advanced ESP controller can keep the state trajectorywithin the set R 1 using brakingand

5 AVEC 1 steering wheel angle for all admissible values of w. However, additional uncertainties that are not taken into account during the controller computation may drive the state vector outside the set R 1. This can be due, for instance, to short term variations of the friction coefficient or a sudden gust of wind. Also, the advanced ESP may be activated by the driver at the moment when the state ξ / R 1, e.g. when the vehicle is oversteering or understeering. In such situations the advanced ESP scheme should bring the state back to the set R 1. For that purpose we compute the k step controllable sets X k : X k+1 = 9 B(X k ) i, k =,1,2,...N. (19) i=1 with X = R 1. All the computations briefly outlined above are performed using Multi Parametric Toolbox for Matlab [6]. d y ref w 1 table of pre computed optimal control inputs. For details the reader is referred to the instructive manual of the Multi parametric Toolbox [6] and references therein. 4. Simulation The advanced ESP controller is tested for a scenario in which a driver performs a standard ATI9 manouver on an icy asphalt with corresponding friction coefficient µ =.3. In the ATI9 manoeuver, the driver changes steering angle from 9 to 9 deg and keeps the vehicle velocity constant (5 km/h in our case). We tested our strategy on an high fidelity model of a oversteering sport car. The tuning parameters are: (i) N = 3, (ii) sampling time T s = 2ms, (iii) the control weight matrix for the states variable ξ is diag[1, 1], (iv) the control weight matrix R is diag[.435,5e 1]. In Figure 5 we show the results of two ATI9 simulations: open loop (controller off) and closed loop (controller on). One can dψ/dt [deg/s] dy/dt [m/s] 1 1 Open Closed Loop7 2 Reference dψ/dt [deg/s] mode [ ]5 ref M b T b vˆ y,, y vw ij a, a 3.3 Notes on implementation The control scheme, standard in the literature[14] is depicted in Figure 4. The Vehicle, through its sensors and some Observers transmits the information on its dynamics to Reference Generator that computes state reference signals which the Advanced ESP needs to track by generating appropriate steering and brakingvalues, while keepingthe state in the set R 1 or driving it into R 1. In the actual implex Fig. 4: General control scheme of electronic stability controller. mentation for simulation, the control inputs of the Advanced ESP are computed as a selection from the corresponding control mappings. If the state ξ belongs to the set R 1 we solve the following optimization problem: min u U 1 (ξ)(ξ+ r) T Q(ξ r)+u T Ru, (2) where Q and R are suitable design matrices. In case when ξ belongs to some of the sets {X k } 3 k=1, one needs to solve a number of small scale convex quadratic programs with the same cost in order to select the appropriate control input. Currently, all computations are formulated and performed with the aid of the YALMIP Toolbox [7]. For a real time implementation one can use the Multi Parametric Toolbox to obtain the explicit solution using parametric programming, in which case the robust control evaluation would reduce to a search in a look-up c w 2 Fig. 5: Chassis behavior. From the top: (a) lateral velocity in OL simulation (dot-dashed line), lateral velocity (solid line) and its reference (dashed line)in CL simulation, (b) yaw rate in OL simulation (dot-dashed line), yaw rate (solid line) and its reference (dashed line)in CL simulation, (c) disturbance on yaw rate, (d) working mode. observe that the nominal behavior of the vehicle is preserved even if actuators are affected by uncertainties with a good tracking behavior. At the time instant, 3.4 s a passing over a patch of ice with the friction coefficient µ =.2 is simulated, causing the transition of the vehicle state outside the mode 1 and RCI set. The state variables are, now, in one of k- step controllable set described in (19) with N = 3 and, then, the advanced ESP controller brings the state variable back into the RCI set. Note that, during that time, the tracking performance is slightly degraded, as the primary goalis to restorethe nominal behavior of the vehicle. Figure 6 shows the time evolution of the control inputs, i.e. the steering angle and the braking moment both for the open loop case, where the steering angle is generated by the driver and the braking is zero, and the closed loop case, where the steering and the braking are generated by the advanced ESP controller. In the closed loop simulations the actuator uncertainty is simulated by introducing a delay in steering command of 1 [ms]. One can notice that, at the time instant when unmodelled disturbance happens, the advanced ESP controller forces

6 AVEC 1 M b [Nm] SWA [deg] 1 Driver Real Requested mode [ ] Fig. 6: Control Inputs: a) turn wheel angle of driver (dot-dashed line), turn wheel angle of controller (dashed line), delated turn wheel angle of controller (solid line), (b) braking moment, (c) working mode. the state variablew in R 1 imposes a slight countersteering, an action an experienced driver would typically do in such situation. Finally, in Figure 7 we compare the braking torque for each wheel requested by the controller and the applied braking torque for each wheel. The applied braking torque is affected by the disturbance within the nominal ±1% range. The braking torque on each wheel is obtained from braking moment M using the torque distribution scheme detailed in [3]. T bfl [NM] T brl [Nm] T bfr [Nm] T brr [Nm] Real Command Requested Command Fig. 7: Real Braking torques (solid line) vs Requested Braking torques (dashed line). Starting from top-left plot clockwise: a) Front-Left braking torque, b) Front-Right braking torque, c) Rear-Left braking torque, d) Rear- Right braking torque. 5. Conclusion In this paper, a novel vehicle lateral dynamic control approach has been presented utilizing differential braking and active front steering. The designed controller is able to guarantee constraint satisfaction for all admissible actuator disturbance realizations. To the best of authors knowledge, this paper presents the first attempt to address hard constraints and uncertainties in the vehicle stability control system in a systematic and rigorous way through a robust constrained control design. A future step will be to improve the control algorithm by introducing uncertainty on PWA approximation of cornering tire forces and dispensing the assumption of constant longitudinal velocity. REFERENCES [1] D. P. Bertsekas. Infinite-time reachability of state-space regions by using feedback control. IEEE Trans. Automat. Contr., 17:64 613, October [2] F. Blanchini and S. Miani. Set-Theoretic Methods in Control. Birkhäuser, 28. [3] P. Falcone, H. E. Tseng, F. Borrelli, J. Asgari, and D. Hrovat. MPC-based yaw and lateral stabilisation via active front steering and braking. Vehicle System Dynamics, 46, Supplement: , 28. [4] U. Kienke. Automotive Control Systems. Springer Verlag, 2. [5] P. Koleszar, B. Trencseni, and L. Palkovics. Development of an Electronic Stability Program Completed with Steering Intervention for Heavy Duty Vehicles. In Proc. of the IEEE Int. Symposium on Industrial Electronics, 25. ISIE 25, volume 1, pages , June 25. [6] M. Kvasnica, P. Grieder, M. Baotić, and M. Morari. Multi Parametric Toolbox (MPT). In Hybrid Systems: Computation and Control. Springer, 23. [7] J. Löfberg. YALMIP: A toolbox for modeling and optimization in MATLAB. In Proceedings of the CACSD Conference, Taipei, Taiwan, 24. Available from [8] H. B. Pacejka. Tire and Vehicle Dynamics. SAE International and Elsevier, 25. [9] R. Rajamani. Vehicle Dynamics and Control. Springer Verlag, 25. [1] S. V. Raković. Robust Control of Constrained Discrete Time Systems: Characterization and Implementation. PhD thesis, Imperial College, London, London, UK, 25. [11] S.V.Raković,E.C.Kerrigan,D.Q.Mayne,and J. Lygeros. Reachability Analysis of Discrete- Time Systems With Disturbances. IEEE Trans. Automat. Contr., 51(4): , April 26. [12] H. E. Tseng, B. Ashrafi, D. Madau, T. Allen Brown, and D. Recker. The development of vehicle stability control at Ford. IEEE/ASME T. Mech., 4(3): , sep [13] A. J. van der Schaft and J. M. Schumacher. An Introduction to Hybrid Dynamical Systems, volume 251 of Lect. Notes in Control and Information Sciences. Springer Verlag, 2. [14] A. T. van Zanten, R. Erthadt, and G. Pfaff. VDC, the vehicle dynamics control of Bosch. In Proc. of International Congress and Exposition, number in Proc. of SAE, pages 9 26, Detroit, MI, USA, February 1995.