Experimental Validation of Nonlinear Predictive Algorithms for Steering and Braking Coordination in Limit Handling Maneuvers

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1 AVEC 1 Experimental Validation of Nonlinear Predictive Algorithms for Steering and Braking Coordination in Limit Handling Maneuvers Paolo Falcone, a Francesco Borrelli, b H. Eric Tseng, Davor Hrovat c a Department of Signals and Systems, Chalmers University of Technology, Göteborg SE-1 96, Sweden. b Department of Mechanical Engineering, University of California, Berkeley, 97-17, USA. c Ford Research Laboratories, Dearborn, MI, USA. Corresponding author: Paolo Falcone Göteborg SE-1 96, Sweden Phone: Fax: falcone@chalmers.se Experimental results are presented for two low complexity Model Predictive Control (MPC approaches to autonomous path following via coordination of steering and braking in passenger vehicles. The two approaches, presented in [1 3], are validated in high speed double lane changes on snow. The results show that the proposed approaches are able to stabilize the vehicle up to 6 Kph and operate in wide regions of the tire characteristics. Topics/ Vehicle Dynamics Control, Real-Time Model Predictive Control 1. INTRODUCTION In classical vehicle stability control systems, an intervention is issued after the tire forces limits have been reached []. One cause of this is the lack of direct measurements of tire forces, for detecting the tire-road friction limits [5]. Alternatively, tire-road friction limits are commonly detected when yaw rate and velocities measurements deviate from the vehicle behavior predicted through linear vehicle reference models. As a drawback, tire maximum friction is detected only after it has been exceeded. Clearly, the availability of tire forces measurements and their achievable limits would enable early interventions, thus preventing the vehicle to enter possible unstable operating regions. Different sensing technologies have been recently investigated for measuring tire forces and moments [6]. In [7], the steering torque is used for estimating the tire forces. These approaches require the use of additional hardware. As alternative, in order shift from a reactive to a predictive nature of the vehicle stability systems, predictive control approaches can be used. In our previous works, we have focused on the design of predictive approaches to the control of yaw and lateral vehicle dynamics via the coordination of steering and independent braking at the four wheels. In [8] and [9] we have shown that the computational complexity of Model Predictive Control (MPC schemes is an obstacle for their real-time implementation even in the simplest case of front steering control (no brake intervention. The main reason resides in the real-time solution of a Non Linear Program (NLP problem whose complexity depends on the system nonlinearities and constraints. This fact led us to study in [1] and [1] lowcomplexity MPC schemes which could be real-time implementable for the vehicle dynamics control problems. In [1] we reduced the computational burden by online formulating and solving a convex Quadratic Program (QP instead of a non-convex NLP. The convex QP derives from an MPC scheme where linear approximations of the vehicle model and constraints are used. Since such linear approximations are updated every time step, we refer to this approach as Linear Time Varying MPC (LTV MPC. The resulting controller presented in [1] 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 and (iii the coupling between

2 AVEC 1 lateral and longitudinal tire forces in combined cornering and braking/driving manoeuvres are modeled. A different approach to complexity reduction was used in [1]. A Nonlinear MPC (NMPC with a reduced number of optimization variables is formulated and implemented on a sixth order nonlinear vehicle prediction model where (i the control inputs are the steering angle and the braking yaw moment, (ii the effects of braking on yaw dynamics only are modeled 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. the vehicle mathematical models are presented. In Section 3, the autonomous path following problem is formulated as a MPC problem. Experimental results are presented in Section, while Section5 closes the paper with final remarks.. MODELING For the sake of completeness we report next the two-tracks vehicle model presented in [1]. This is used, later in Sections. and.3, to derive simplified models 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.. 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. (a 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. (b Two actuators approach proposed in [1]. 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. In [], by using a similar approach as in [1], we have formulated a NMPC problem based on a sixth order nonlinear 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 and (ii the effects of braking on longitudinal and yaw dynamics are modeled and (iii the coupling between lateral and longitudinal tire forces are modeled. As shown in Figure 1(a, an outer logic splits the braking forces into braking torques at the four wheels. We refer to this approach as three-actuators approach. The underlying idea of the two- and three- actuators approaches is to formulate small online NLPs by using simpler vehicle prediction models without approximating the nonlinearities tire characteristics. The paper is structured as follows. In Section Fig.. The simplified vehicle dynamical model. 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 F l f,r sinδ f cosδ f F cr,l F cr,r mẍ=mẏ ψ (F l f,l F l f,r cosδ f sin δ f F lr,l F lr,r, I ψ=a [(F l f,l F l f,r sinδ f ] cosδ f (F cr,l F cr,r c b [( F l f,l F l f,r cosδ f F lr,l F lr,r ]. (1a (1b (1c F c f,r sinδ f

3 AVEC 1 The vehicle s equations of motion in the absolute inertial frame XY are I ψ=a b (F cr,l F cr,r c ( F ll F lr, (5c Ẏ = ẋsinψẏcosψ, 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. 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, whose right hand side is function of the braking torques. The model for tire longitudinal and cornering forces (3 used in this paper are described by a Pacejka model [11]. Further details can be found in [11,8,9]. 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 w (ξ(t,u(t, ( 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.. 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 sin δ 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 (5a (5b 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. By using the equations (-(5, the nonlinear vehicle dynamics can be described by the following compact differential equation: ξ(t= fs(t,µ(t w (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 The Simplified Bicyle Model In this section we derive a reduced order model from the four wheels model (1 that is called single track or bicycle model. It is based on the following set of simplifications Simplification 3 At front and rear axles, the left and right wheels are lumped in a single wheel. Simplification Braking application induces only yaw moment without longitudinal and/or lateral force changes. Simplification 5 In the lateral tire force calculation the tire slip ratio is assumed to be zero, i.e., F c = f c (α,,µ,f z. Simplification 6 The steering and braking effects on vehicle speed are negligible. Remark 3 Since braking is used for yaw stabilization only, minimum and single-sided usage of brakes is expected. The Simplifications 3-6 are therefore deemed reasonable. By the Simplifications 3-6 the simplified bicycle model can be re-written as follows mÿ= mẋ ψ F c f s= F cr s=, ẍ=, I ψ=a F c f s= b F cr s= M, (7a (7b (7c where M is the braking yaw moment that, in the four wheels model (1, is computed as follows M = c ( F x f,l F x f,r F xr,l F xr,r. (8 The forces in (7 can be computed through the equations (3, particularized for the front and rear axles. Remark We remark that the forces F y f and F yr in equations (7 represent the lateral components of the cornering tire forces F c generated by the contact of a single wheel with the ground.

4 AVEC 1 The nonlinear vehicle dynamics described by the equations (, (3, (7 can be rewritten in the following compact from: ξ(t= fµ(t w (ξ(t,u(t, (9 where µ(t=[µ f (t, µ r (t]. The state and input vectors are ξ=[ẏ, ẋ, ψ, ψ, Y, X] and u=[δ f, M], respectively. 3. MODEL PREDICTIVE CONTROL PROBLEM In this paper, experimental results of two MPC controllers, based on the vehicle models (6 and (9, respectively, are presented. The two approaches have been proposed in [1,] and [,3], respectively, where preliminary simulation and experimental results are shown. In this section, for the sake of completeness, we briefly recall the MPC problem formulation of the two controllers and refer the interested reader to [1 3] for further details. We consider the following discrete time vehicle dynamics: ξ(t 1= f(ξ(t,u(t, u(t=u(t 1 u(t, (1a (1b where the function f is derived from the vehicle models presented in Section, the vectors u(t and u(t are defined as in (6 and (9, and consider the following output map: 1 1 η(t=h(ξ(t= ξ(t. ( 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 (1 are known t. Moreover we consider the following cost function: J(ξ(t, U(t= H p i=1 η(t i ηre f (t i Q H c 1 u(t i S u(t i R, i= (1 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 (1-(11 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 (1-(11 and the cost function (1, is formulated and solved as in [8,9] to obtain the following state feedback control law u(t,ξ(t=u(t 1 u t,t(t,ξ(t. (13 Once the optimal value of the braking yaw moment and the braking forces at the two sides of the vehicle have been computed through the (13, for the two approaches, respectively, the braking torques at the four wheels are computed through the braking logics used in [1] and [], respectively. In particular, the algorithms in [1] and [] implement a single wheel braking logic and are 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 [1]. 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 [13].. 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 experimental results of the threeand two- actuators controllers presented in Section 3. We recall that in the three-actuators controller the control inputs are the front steering angle δ f and the braking forces at the two sides of the vehicle F ll and F lr, respectively. In the two-actuators controller, instead, the control inputs are the front steering angle δ f and a braking yaw moment M. In both controllers, the output tracking variables are the yaw angle, the yaw rate and the lateral position. In the three-actuators controller the longitudinal speed is an additional output tracking variable. The three- and the two- actuators 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 = 15; H c = 1. bounds: δ f,min =-1 deg, δ f,max =1 deg, δ f,min =-.85 deg, δ f,max =.85 deg.

5 AVEC 1 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 with Q 11 =, Q =, Q 33 = 1, Q = 3 and Q i j = for i j. R R 3 3 with R 11 = 1, R = 1, R 33 = 1 and R i j = for i j. S R 3 3 with S ii = 1 for i = j and S i j = for i j. Controller B. Controller presented in Section 3 with the following parameters: sampling time: T =.5s. horizons: H p = 15; H c = 1. 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 = 16 Nm, M max = 16 Nm. friction coefficient: µ =.3. weighting matrices: Q R 3 3 with Q 11 = 1, Q = 1, Q 33 = 3 and Q i j = for i j. R R with R 11 = 1 for i= j and R i j = for i j. S R with S 11 =.1, S = 1 and S i j = for i j. Figures 3-5 show the experimental results of the two controllers in a double lane change test at 6 Kph on snow. In particular, in Figure 3 the tracking variables are reported while the steering angle, the desired braking yaw moment and the braking forces at the two sides of the vehicle are showed in Figure, for the two controllers, respectively. Figure 5, shows the steering angle and the tire slip angles obtained with Controller A. The controllers have been tested with the experimental setup described in [9]. We point out that vehicle instability occurs, in the considered testing maneuver, when the same experiment is performed at entry speeds higher than 6 Kph. The output tracking variables in Figure 3 show that both controllers are able to stabilize the vehicle at the entry speed of 6 Kph on snow. Moreover, as shown in Figure, the steering and the braking commands are properly coordinated as a result of the MPC algorithm and the braking logics proposed in [1] and []. The results reported in Figure 5 show that Controller A is able to control the vehicle in a wide region of the nonlinear tire characteristics, while preserving the vehicle stability. Moreover, the steering angle reported in the upper plot of Figure 5 exhibits a hollow, between approximately 11.5 s and 1.5 s, when the rear tire slip angle reaches its maximum. A post-processing analysis of the predicted rear tire slip angle trajectories show that, by temporary decreasing the steering angle, the controller tries to bring the vehicle back to the linear region of the tire characteristics, i.e., to mantain the rear tire slip angle within the range [ 3,3] deg. Ψ [deg] dψ/dt [deg/s] Y [m] Ψ [deg] dψ/dt [deg/s] Y [m] (a Controller A (b Controller B Fig. 3. Experimental test at 6 Km/h entry speed. Output tracking variables. Reference (dashed and actual (solid signals. 5. CONCLUSIONS We have presented experimental results validating two low complexity MPC-based approaches to the problem of autonomous path following via combined steering and braking. The experimental results of the two MPC approaches, previously presented in [1 3], demonstrates that, despite of model simplifications, good path following performance can be achieved together with the capability of stabilizing the vehicle in a wide region of the nonlinear tire characteristics. REFERENCES [1] P. Falcone, F. Borrelli, H. E. Tseng, J. Asgari, and D. Hrovat. Mpc-based yaw and lateral stabilization via active front steering and braking. Vehicle System Dynamics, 6, Supplement:611 68, 8. [] P. Falcone, F. Borrelli, J. Asgari, H. E. Tseng, and D. Hrovat. Low complexity mpc schemes for integrated vehicle dynamics control problems. In Symposium on Advanced Vehicle Control (AVEC,

6 AVEC 1 δ f [deg] δ f [deg] F brl [N] α fl [deg] F brr [N] (a Controller A. Steering angle (upper plot and braking yaw moment (lower plot. δ f [deg] M [Nm] (b Controller B. In the upper plots, solid lines are the steering commands computed by the MPC algorithm, the dashed lines are the actual steering angles and the green dash-dotted lines are the steering commands from the driver. Fig.. Experimental test at 6 Km/h entry speed. Control input variables. Reference (dashed and actual (solid signals. Kobe, Japan, October 8. [3] P. Falcone, F. Borrelli, H. E. Tseng, and D. Hrovat. Automotive Model Predictive Control, chapter On Low Complexity Predictive Approaches to Control of Autonomous Vehicles, pages Lecture Notes in Control and Information Sciences. Springer Berlin / Heidelberg, 1. [] Y. H. Judy Hsu and J. Christian Gerdes. Envelop control: Keeping the vehicle within its handling limits using front steering. In 1 st International Symposium on Dynamics of Vehicles, on Roads and Trucks, 9. [5] A. T. van Zanten. Evolution of electronic control systems for improving the vehicle dynamic behavior. In 1 s t International Symposium on Advanced Vehicle Control (AVEC, pages 7 15, Hiroshima, Japan,. α rl [deg] Fig. 5. Experimental test at 6 Km/h entry speed. Controller A. Steering angle (upper plots, front (middle plot and rear (lower plot tire slip angles. [6] M. Gobbi, G. Mastinu, and F. Giorgetta. Sensors for measuring forces and moments with application to ground vehicle design and engineering. In ASME International Mechanical Engineering Congress and Exposition, Orlando, Florida, USA, December 5. [7] Y.-H. J. Hsu and J. C. Gerdes. A feel for the road: A method to estimate tire parameters using steering torque. In Symposium on Advanced Vehicle Control (AVEC, Taipei, Taiwan, 6. [8] 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/:65 91, 5. [9] 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. [1] 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. [11] E. Bakker, L. Nyborg, and H. B. Pacejka. Tyre modeling for use in vehicle dynamics studies. SAE paper # 871, [1] 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 7 51, Yokohama, Japan, 199. [13] E. Bedner, D. Fulk, and A. Hac. Exploring the trade-off of handling stability and responsiveness with advanced control systems. SAE, SAE , 7.