Low Complexity MPC Schemes for Integrated Vehicle Dynamics Control Problems

Transcription

1 AVEC 8 Low Complexity MPC Schemes for Integrated Vehicle Dynamics Control Problems Paolo Falcone, a Francesco Borrelli, b H. Eric Tseng, Jahan Asgari, Davor Hrovat c a Department of Signals and Systems, Chalmers University of Technology, Göteborg SE-41 96, Sweden. b Department of Mechanical Engineering, University of California, Berkeley, , USA. c Ford Research Laboratories, Dearborn, MI, USA. Corresponding author: Paolo Falcone Göteborg SE-41 96, Sweden Phone: Fax: falcone@chalmers.se A low complexity Model Predictive Control (MPC approach to the problem of autonomous path following via combined steering and independent braking is presented in this paper. We start from the simpler approach in [5 and significantly improve the performance by better modeling the longitudinal dynamics and slightly increasing the number of optimization variables, i.e., the computational complexity. In order to assess the performance improvement, simulations are presented and compared against the results of the simpler approach in [5. Moreover, experimental results are shown and discussed. Topics/ Vehicle Dynamics Control, Real-Time Model Predictive Control 1. INTRODUCTION In this work we focus on the problem of controlling yaw and lateral vehicle dynamics by means of lowcomplexity Model Predictive Control (MPC schemes coordinating steering and independent braking at the four wheels. In our previous works [1 and [ we have shown that the real-time implementation of MPC approaches is challenging for such problem even in the simplest case of front steering control (no brake intervention. The main challenge resides in the real-time solution of a Nonlinear Programming (NLP problem at each sampling times (typically between and 5 ms by using a hardware with limited computational resources. Challenged by this issue, in our works [3, [4 and [5 we have proposed an analyzed alternative lowcomplexity MPC schemes which could be real-time implementable for the type of control problems addressed in this work. In particular, in [3, [4 we reduced the computational burden by formulating and solving a convex Quadratic Program (QP at each time step. The problem convexity derives from the linear approximations of the vehicle model and constraints. Since such linear approximations are updated at each time step, we refer to this approach as Linear Time Varying MPC (LTV MPC. The resulting controller (Figure 1(a has been experimentally tested and and it is based on a tenth order vehicle prediction model where (i the control inputs are the front steering angle and the braking torques at the four wheels, (ii the effects of braking on lateral, longitudinal and yaw dynamics are modeled and (iii the coupling between lateral and longitudinal tire forces in combined cornering and braking/driving manoeuvres are modeled. A different approach to complexity reduction was used in [5. A nonlinear MPC with a reduced number of optimization variables is formulated and implemented based on a sixth order nonlinear vehicle prediction model where (i the control inputs are the steering angle and the braking yaw moment, (ii braking is assumed to affect only yaw dynamics and (iii the coupling between lateral and longitudinal tire forces are neglected. The computed braking yaw moment is split into braking torques at the four wheels through an outer logic as shown in Figure 1(b. We refer to this approach as to the two-actuators approach. Objective of this work is to continue the study of the approach presented in [5. In particular, we formulate an NMPC problem based on a sixth order nonlinear

2 AVEC 8 (a Five actuators approach proposed in [3,4. δ f is the steering angle and T b,, with { f,r} and {l,r}, are the braking torques at the four wheels. (b Two actuators approach proposed in [5. δ f is the steering angle and M is the braking yaw moment. Fig. 1. Two different approaches to the integrated vehicle dynamics control problem. vehicle prediction model where (i the control inputs are the front steering angle and the braking forces at the left and right sides of the vehicle, (ii the effects of braking on longitudinal and yaw dynamics are modeled and (iii the coupling between lateral and longitudinal tire forces are neglected. As shown in Figure, an outer logic splits the braking forces into braking torques at the four wheels. We refer to this approach as threeactuators approach. The underlying idea of the two- and threeactuators approach is to formulate small online NLPs by using a vehicle prediction model which is very simple but at the same time captures the nonlinear tire characteristics. In the two-actuators approach, presented in [5 the contact lateral forces are computed through a Pacejka tire model [6 and nonlinearities arising from lateral tire force saturation are properly modeled. Nevertheless the effects of braking on longitudinal dynamics, as well as the coupling between lateral and longitudinal tire forces in combined steering and braking are neglected. In the three-actuators approach presented in this paper the model is improved. In particular, we model the effects of the braking forces at the two sides of the vehicle on longitudinal dynamics. We point out that reducing the problem size through model simplification, as proposed in our study, is a challenging trading off between problem complexity and modeling accuracy. In particular, reducing the model complexity while modeling relevant vehicle nonlinearities (e.g., the coupling between longitudinal Fig.. Three actuators approach. δ f is the steering angle and F b, with {l,r}, are the braking forces at the two sides of the vehicle. and lateral tire forces might require the use of additional non convex constraints (e.g., the Kamm s circle limiting the maximum total tire force. Furthermore, we observe that the inclusion of the braking logic in the optimization algorithm would require the use of a mixed-integer optimization framework. In this paper the performance of the three-actuators approach is compared against the two-actuators approach. Performance is measured in terms of maximum longitudinal entry speed at which the vehicle can autonomously drive along a predefined path on a slippery road without spinning. Moreover, we present experimental results obtained with a controller based on the two-actuators approach presented in [5 showing that the vehicle can be stabilized at 6 Kph on a snow covered road while performing a double lane change. The paper is organized as follows: in Section we present the vehicle models used in the MPC approach formulated in Section 3. In Section 4 we present simulation results. Finally Section 5 closes the paper with remarks and ideas for future works.. MODELING For the sake of completeness we report next the two-tracks vehicle model presented in [4,5. Later in Section.1 we derive a simpler model based on a set of simplifying assumptions..1 The Two-Tracks Vehicle Model The nomenclature used in the following refers to the model depicted in Fig. 3. Moreover, two subscript symbols are used throughout the rest of the paper to denote variables related to the four wheels. In particular, the first subscript { f,r} denote the front and rear axles, while the second {l,r} denotes the left and right sides of the vehicle. As example, the variable ( f,l is referred to the front left wheel. The longitudinal, lateral and yaw dynamics of the vehicle are described through the following set of differential equations: mÿ = mẋ ψ + (F l f,l sinδ f + cosδ f + F cr,l + F cr,r (1a

3 AVEC 8 The model for tire longitudinal and cornering forces (3 used in this paper are described by a Pacejka model [6. Further details can be found in [6,1,. Using equations (1-(3 and the additional wheel dynamics mentioned in Remark 1, the nonlinear vehicle dynamics can be described by the following compact differential equation, assuming a certain road friction coefficient µ = [ µ f,l, µ f,r,µ r,l, µ r,r vector: ξ(t = fµ(t 4w (ξ(t,u(t, (4 where ξ = [ẏ, ẋ, ψ, ψ, Y, X, ω f,l, ω f,r, ω r,l, ω r,r, u = [δ f, T b f,l, T b f,r, T br,l, T br,r and T b, are the braking torques at the four wheels. Fig. 3. The simplified vehicle dynamical model. mẍ = mẏ ψ + (F l f,l cosδ f (1b + sinδ f + F lr,l + F lr,r, I ψ = a [(F l f,l sinδ f (1c + cosδ f b (F cr,l + F cr,r + c [( F l f,l cosδ f + F c f,r sinδ f F lr,l + F lr,r. The vehicle s equations of motion in the absolute inertial frame XY are Ẏ = ẋsinψ + ẏcosψ, Ẋ = ẋcosψ ẏsinψ. (a (b The cornering F c, and longitudinal F l, tire forces in (3 are given by F c, = f c (α,,s,,µ,,f z,, F l, = f l (α,,s,,µ,,f z,, (3a (3b where α, are the tire slip angles, s, are the slip ratios, µ, are the road friction coefficients and F z, are the tires normal forces. In the following we assume constant normal tire load, i.e., F z, = constant. The equations of α, and s, are not relevant for the rest of the paper and are omitted here. The interested reader is referred to [4 for further details. Remark 1 We point out that the tire slip ratios s, at the four wheels are nonlinear functions of the wheel angular speeds. The latter can be computed as the solution of a nonlinear differential equations system [4, whose right hand side is function of the braking torques.. The Simplified Two-Tracks Vehicle Model The two-tracks vehicle model presented next is based on the following set of simplifications Simplification 1 Small angle approximation is used, i.e., cosδ f = 1 and sinδ f =. Simplification Single wheel braking is considered on each side of the vehicle, i.e., F l f, F lr, =. Remark By the Simplifications 1 and, the effects on the longitudinal and yaw dynamics of the longitudinal tire forces F l, can be described though the forces F l = F l f, + F lr,, with {l,r}. By the Simplifications 1-, the equations (1 can be rewritten as follows: mÿ = mẋ ψ + F c f,l + F cr,l + F cr,r mẍ = mẏ ψ + F ll + F lr I ψ = a b (F cr,l + F cr,r + c ( F ll + F lr, (5a (5b (5c where F ll and F lr are the longitudinal forces induced by braking at the left and right sides, respectively, of the vehicle (see Remark. Using the equations (-(5, the nonlinear vehicle dynamics can be described by the following compact differential equation: ξ(t = f s(t,µ(t (ξ(t,u(t, (6 where ξ = [ẏ, ẋ, ψ, ψ, Y, X and u = [δ f, F ll, F lr, respectively and where s(t = [ s f,l, s f,r,s r,l, s r,r (t is the vector of slip ratios at the four wheels at time t and µ(t = [ µ f,l, µ f,r,µ r,l, µ r,r (t is the vector of road friction coefficients at the four wheels at time t. 3. MODEL PREDICTIVE CONTROL PROBLEM In order to formulate a finite dimensional optimal control problem we discretize the system dynamics (6 with a fixed sampling time T s : ξ(t + 1 = f s(t,µ(t (ξ(t,u(t, u(t = u(t 1 + u(t, (7a (7b

4 AVEC 8 with u(t = [ δ f (t, F ll (t, F lr (t [ and u(t = δ f (t, F ll (t, F lr (t, and consider the following output map: 1 1 η(t = h(ξ(t = ξ(t. (8 1 1 In the following we will make use of the following assumption Assumption 1 Measurement of the tire slip ratios and road friction coefficient are assumed to be available for each wheel, i.e., the vectors s(t and µ(t in (7 are known t. Moreover we consider the following cost function: H p J(ξ(t, U(t = η(t + i ηre f (t + i Q i=1 (9 H c 1 + u(t + i S + u(t + i R, i= where U(t = [ u(t,..., u(t + H c 1 is the optimization vector at time t, η(t + i denotes the output vector predicted at time t + i obtained by starting from the state ξ(t and applying to system (7-(8 the input sequence u(t,..., u(t + i. η re f is the output reference signal and H p and H c denote the output prediction horizon and the control horizon, respectively. A model predictive control problem, based on the discrete time vehicle model (7-(8 and the cost function (9, is formulated and solved as in [1, to obtain the following state feedback control law u(t,ξ(t = u(t 1 + u t,t(t,ξ(t. (1 Once the optimal value of the braking forces at the two sides of the vehicle has been computed through the (1, the braking torques where T b, at the four wheels are computed through the Algorithm 1 [5 where α f and α r are the tire slip angles at the front and rear axles, respectively, and r is the wheel radius. We observe that, in order to induce the desired braking yaw moment with minimum longitudinal dynamics effect, the Algorithm 1 implements a single wheel braking logic. The braking logic in Algorithm 1 is based on the following well known results: Outside wheel braking induces understeer while inside wheel braking induces oversteer. Left/right brake distribution is more effective in steering the vehicle than front/rear distribution [7. Braking at the rear inside corner is most effective in inducing an oversteer yaw moment, and braking at the front outside corner is most effective to induce an understeer yaw moment [8, [9, [1. Algorithm 1: Braking torques calculation Input: F ll, F lr, α f, α r. Output: T b f,l, T b f,r, T br,l, T br,r. begin if F ll = and F lr = then T b f,l =, T b f,r =, T br,l =, T br,r = ; else if α f α r > then /* understeering */ T b f,l =, T b f,r =, T br,l = rf ll, T br,r = rf lr ; else /* oversteering */ T b f,l rf ll, T br,l = rf lr, T b f,r =, T br,r = ; end end end 4. RESULTS We considered a scenario where the objective is to follow a desired path as close as possible on a snow covered road (µ =.3 at a given desired speed. The control inputs are the front tire steering angle and the brake torques at the four wheels and the goal is to follow the trajectory as close as possible by minimizing the vehicle deviation from the target path. The experiment is repeated with increasing entry speeds until the vehicle loses control. Next we show simulation results of the three- and two- actuators controllers presented in Section 3 of this paper and in [5, respectively. We recall that in the twoactuators controller in [5 the control inputs are the front steering angle δ f and a braking yaw moment M and the output tracking variables are the yaw angle, the yaw rate and the lateral position. The three- and the twoactuators approach are next referred to as Controller A and Controller B and are defined as follows: Controller A. Controller presented in Section 3 with the following parameters: sampling time: T =.5s. horizons: H p = ; H c =. bounds: δ f,min =-1 deg, δ f,max =1 deg, δ f,min =-.85 deg, δ f,max =.85 deg. F l min = 15 N, F l max = N, F l min = 18 N, F l max = 18 N, with {l,r}. friction coefficient: µ =.3. weighting matrices: Q R 4 4 with Q 11 =, Q = 1, Q 33 =.1, Q 44 = 3 and Q i j = R R 3 3 for R ii = 1 for i = j and R i j = S R 3 3 for S ii = 1 for i = j and S i j = Controller B. Controller presented in [5 with the following parameters: sampling time: T =.5s.

5 AVEC 8 horizons: H p = ; H c =. bounds: δ f,min =-1 deg, δ f,max =1 deg, δ f,min =-.85 deg, δ f,max =.85 deg. M min = 1 3 Nm, M max = 1 3 Nm, M min = 146 Nm, M max = 146 Nm. friction coefficient: µ =.3. weighting matrices: Q R 3 3 with Q 11 = 1, Q = 1, Q 33 = 1 and Q i j = R R for R 11 = 1, R = 1 and R i j = S R with S 11 = 1, S = 1 and S i j = We recall that, in Controller A, brake intervention, yaw and lateral stabilization and vehicle slowing down (i.e., deviation from a desired vehicle velocity can be traded off by varying the weights on the main diagonals of the matrices Q and S. In Controller B, instead, the deviation from a desired velocity cannot be included in the cost function since the longitudinal dynamics are not modeled. This is an attracting features of the approach used in Controller A. We point out that, for the sake of fair comparison between the two approaches, the element Q 11 in Controller A (the weight on the deviation from the desired velocity has been set to. The remaining tuning parameters of the two controllers are the same. In Figures 4-5 the simulation results of the two controllers are reported when the vehicle performs a double lane change on a snow covered road at 7 Kph. In particular, in Figure 4 the output tracking variables are shown while in Figure 5 the braking torques at the four wheels are presented. Figure 4 shows that the behaviors of the two controller are very similar. Nevertheless, we observe that the braking torques computed by the two controllers are different. In particular, the Controller A brakes on both left and right sides (instead of single side braking and slows down the vehicle. We finally point out that the Controller B becomes unstable for entry speeds higher than 7 Kph, while the Controller A stabilizes the vehicle up to 9 Kph. Figure 6 shows the experimental results of the Controller B in a double lane change test at 6 Kph on snow. In particular, in Figure 6(a the tracking variables are reported while the steering angle and the desired braking yaw moment are showed in the upper and lower plots, respectively, of Figure 6(b. The controller has been designed according the design procedure in [5 and tested with the experimental setup described in [. We point out that vehicle instability occurs, in the considered testing manoeuvre, when the same experiment is performed at entry speeds higher than 6 Kph. This result is close to the maximum entry speed of 55 Kph obtained in the simulations in [5. In Figure 6(a we observe good tracking performance as well as an intuitive coordination of the steering angle and the braking yaw moment in Figure 6(b. Ψ [deg dψ/dt [deg/s Y [m Ψ [deg dψ/dt [deg/s Y [m (a Two actuators approach (b Three actuators approach. Fig. 4. Simulation results at 7 Km/h entry speed. Tracking variables. Reference (dashed and actual (solid signals. 5. CONCLUSIONS We have presented a low complexity MPC-based approach to the problem of autonomous path following via combined steering and braking. The approach presented in this paper is an improvement of the simpler two-actuators approach presented in [5. The main difference between the two approaches is the improvement in the modeling of the longitudinal dynamics. In particular, by better modeling the longitudinal dynamics, the saturation of the lateral tire forces can be predicted since the tire slip angles are functions of the longitudinal speed (see equations (3. We showed that, by slightly increasing the computational complexity (i.e., adding one more control input, significant performance improvement are achieved in terms of maximum entry speed. REFERENCES [1 F. Borrelli, P. Falcone, T. Keviczky, J. Asgari, and D. Hrovat. MPC-based approach to active steering for autonomous vehicle systems. Int. J. Vehicle Autonomous Systems, 3(/3/4:65 91, 5.

6 AVEC Tb fl [Nm Tb fr [Nm Ψ [deg dψ/dt [deg/s Tb rl [Nm 3 1 Tb rr [Nm 3 1 Y [m (a Two actuators approach (a Tracking variables. Reference (dashed and actual (solid signals Tb fl [Nm Tb fr [Nm δ f [deg Tb rl [Nm Tb rr [Nm (b Three actuators approach Fig. 5. Simulation results at 7 Km/h entry speed. Braking torques. [ P. Falcone, F. Borrelli, J. Asgari, H. E. Tseng, and D. Hrovat. Predictive active steering control for autonomous vehicle systems. IEEE Trans. on Control System Technology, 15(3, 7. [3 P. Falcone, F. Borrelli, H. E. Tseng, J. Asgari, and D. Hrovat. Integrated braking and steering model predictive control approach in autonomous vehicles. Fift IFAC Symposium on Advances of Automotive Control, 7. [4 P. Falcone. Nonlinear Model Predictive Control for Autonomous Vehicles. PhD thesis, Universitá del Sannio, Dipartimento di Ingegneria, Piazza Roma 1, 81, Benevento, Italy, June 7. [5 P. Falcone, F. Borrelli, J. Asgari, H. E. Tseng, and D. Hrovat. Mpc-based yaw and lateral stabilization via active front steering and braking. Vehicle System Dynamics In press, 8. [6 E. Bakker, L. Nyborg, and H. B. Pacejka. Tyre modeling for use in vehicle dynamics studies. SAE paper # 8741, M [Nm (b Control signals. Steering angle (upper plot and braking yaw moment (lower plot. Fig. 6. Two actuators approach. Experimental test at 6 Km/h entry speed. [7 S. Motoyama, H. Uki, K. Isoda, and H. Yuasa. Effect of traction force distribution control on vehicle dynamics. In Proc. 199 Int. Symp. Advanced Vehicle Control (AVEC9, pages , Yokohama, Japan, 199. [8 H. E. Tseng, B. Ashrafi, D. Madau, T. A. Brown, and D. Recker. The development of vehicle stability control at ford. IEEE-ASME Trans. Mechatronics, 4(3:3 34, [9 G. Bahouth. Real world crash evaluation of vehicle stability control (vsc technology. In 49th Annual Proceedings, Association for the Advancement of Automotive Medicine, pages 19 34, 5. [1 E. Bedner, D. Fulk, and A. Hac. Exploring the trade-off of handling stability and responsiveness with advanced control systems. SAE, SAE , 7.