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Vehicle Yaw Dynamics Control by Torque-Based Assist Systems Enforcing Driver’S Steering Feel Constraints

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Vehicle Yaw Dynamics Control by Torque-Based Assist Systems Enforcing Driver’S Steering Feel Constraints MITSUBISHI ELECTRIC RESEARCH LABORATORIES http://www.merl.com Vehicle Yaw Dynamics Control by Torque-based Assist Systems Enforcing Driver’s Steering Feel Constraints Zafeiropoulos, S.; Di Cairano, S. TR2013-040 June 2013 Abstract We investigate the control of torque-based steering assist systems for improving yaw rate track- ing and vehicle stabilization. As opposed to active front steering systems based on harmonic motors, torque-based steering assist systems are mechanically coupled with the driver. Thus, be- sides standard vehicle and actuators constraints, specific constraints related to the driver-actuator interaction need to be enforced. These constraints can be formulated to achieve a multiplicity of goals, including avoiding excessive strain in the driver’s arms, and preserving the driver’s ”feel for the road”. In order to achieve high control performance and constraints satisfaction, we implement controllers based on linear and switched model predictive control, where different types of driver’s steering feel constraints are enforced. The different controllers are evaluated in simulation maneuvers to analyze their capabilities and the impact of the constraints in terms of vehicle cornering, stabilization, and driver’s steering feel. American Control Conference (ACC) This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Research Laboratories, Inc. All rights reserved. Copyright c Mitsubishi Electric Research Laboratories, Inc., 2013 201 Broadway, Cambridge, Massachusetts 02139 MERLCoverPageSide2 Vehicle Yaw Dynamics Control by Torque-based Assist Systems Enforcing Driver’s Steering Feel Constraints Spyridon Zafeiropoulos, Stefano Di Cairano Abstract— We investigate the control of torque-based steering assist systems for improving yaw rate tracking and vehicle stabilization. As opposed to active front steering systems based on harmonic motors, torque-based steering assist systems are mechanically coupled with the driver. Thus, besides standard vehicle and actuators constraints, specific constraints related to the driver-actuator interaction need to be enforced. These constraints can be formulated to achieve a multiplicity of goals, including avoiding excessive strain in the driver’s arms, Fig. 1: Schematics of the subsystems and MPC controller. and preserving the driver’s “feel for the road”. In order to achieve high control performance and constraints satisfaction, we implement controllers based on linear and switched model predictive control, where different types of driver’s steering accounting for the effects of the steering assist system on the feel constraints are enforced. The different controllers are strain torque exerted on the driver. In order to achieve this, evaluated in simulation maneuvers to analyze their capabilities we develop model predictive control strategies (MPC) [12] and the impact of the constraints in terms of vehicle cornering, based on a model of the vehicle, the steering system, and stabilization, and driver’s steering feel. the driver, that aims at optimizing yaw rate tracking, while I. INTRODUCTION enforcing stability via slip angle constraints, and mechan- ical and driver’s steering torque feel constraints on the In the pursuit to achieve zero accidents on the road, torque actuator. These constraints limit the capabilities of the several active safety systems for vehicle stability control torque-assist system hence imposing a fundamental trade-off are being investigated. Electronic stability control (ESC) [1] between vehicle control capabilities and driver feels. exploits differential braking, and has been proved effective The paper is structured as follows. In Section II we in reducing single vehicle accidents [2]. Although not as formulate the steering and vehicle dynamics with respect effective as ESC in stabilization, steering-based driver assist to the tire sideslip angles, and in Section III we introduce systems are capable of improving both cornering perfor- a model for the driver and its interaction with the vehicle. mance and vehicle stability, and are less intrusive for the In Section IV we introduce vehicle stability constraints and driver. Among steering assist systems, active font steering constraints on the interaction between driver and steering- (AFS) [3], which modifies the relation between the steering assist system. In Section V we develop linear MPC and wheel angle (SWA) and the tire road wheel angles (RWA) at switched MPC (sMPC) strategies to trade-off between the the front tires, has been extensively investigated for vehicle vehicle dynamics performance and the driver torque feel. In control, also in combination with ESC [3]–[6]. Section VI we present simulation results for different types In this paper we consider vehicle cornering and stability of constraints in a specific maneuver. Conclusions and future control exploiting a torque-based steering assist system [7]. developments are summarized in Section VII. These systems are already present in several production Notation: for a discrete-time signal with sampling period vehicles via, for instance, Electric Power Steering (EPS), t , x(k) is the value of x at time kt , and a(h k) is the although not yet used for stability control. The torque applied s s | predicted value of a(k + h) basing on data at time k. by the assist system affects the torque felt by the driver Inequalities between vectors are intended componentwise. through the steering wheel, the so called driver steering We indicate a matrix of appropriate dimensions entirely (torque) feel. The driver steering feel is one of the primary composed of zeros by 0. ways drivers sense the road conditions and the current vehicle dynamics behavior [8], and hence it must be preserved. II. VEHICLE MODEL Preservation of the steering feel has been investigated The system we consider is composed of three subsystems, before, see e.g., [9]–[11]. Here, we combine vehicle cor- vehicle (chassis), steering system, and driver, that are con- nering and stability control with steering feel control by nected as shown in Figure 1. Spyridon Zafeiropoulos is a graduate student at the D. Guggenheim School of Aerospace Engineering, Georgia Inst. Technology, Atlanta, GA, A. Vehicle dynamics l [email protected] We focus on normal driving turning maneuvers, where the S. Di Cairano is with Mitsubishi Electric Research Laboratories, Cam- bridge, MA, [email protected]. vehicle dynamics can be conveniently approximated by the Spyridon Zafeiropoulos did this research while at MERL. bicycle model [13] (see Figure 2). The approximated model has reduced complexity with respect to a four-wheel vehicle with respect to the center of mass, ' [rad/s] is the steering model, while still capturing the relevant dynamics. angular rate, and Gg is the steering column reduction gear. We consider a reference frame that moves with the vehicle. Therefore, the continuous-time dynamics (1), (2), (3) are The frame origin is at the vehicle center of mass, with represented by the linear system the x-axis along the longitudinal vehicle direction pointing x˙ (t)=Acx (t)+Bcu (t) (4a) forward, the y-axis pointing to the left side of the vehicle, v v v c and the z-axis pointing upwards. The tire sideslip angles, or yv(t)=C xv(t). (4b) simply slip angles (the angles between tire directions and In (4) the state vector is xv =[↵f ↵r δ]0, the input vector velocity vectors at the tires), are denoted by ↵f [rad] and is uv = ', and the output vector is yv = r. ↵r [rad] for front and rear tires, respectively. Since we are interested only in normal driving, where the vehicle needs to B. Steering dynamics remain in the stable region, we approximate the tire forces The steering dynamics describe the rotational dynamics of as linear functions of the slip angles (see lower left corner the steering column due to the torques applied by the driver, of Figure 2) the steering assist system and the road-tire interaction. They Fj(↵j)=cj↵j, (1) are expressed by where j f,r , j = r for the rear tires, and j = f for the J'˙ =(T + T T ) ', (5) 2{ } drv mot − aln − front tires, and cj [N/rad] are identified from experimental where β [Nm s/rad] is the friction coefficient of the steering data or more detailed models [14]. The approximation in (1) 2 is considered valid for ↵ p , where p [rad] is the tire column, J [kgm ] is the column moment of inertia, and j j j saturation angle. Tdrv [Nm], Tmot [Nm], and Taln [Nm] are the torques generated by driver, steering assist system motor, and road- tire interaction (the so called alignment torque) reported at the steering wheel, respectively. Here, the alignment torque is modeled by a linear function of the front tire sideslip angle Taln = K↵↵f . (6) Even though (6) is an approximation, it can be seen in [16] and [8] to be appropriate in certain operating ranges and conditions. III. DRIVER-STEERING INTERACTION In this work we control the vehicle dynamics by a torque- based steering assist system while accounting for the driver’s feel through the steering wheel, where we define as driver’s Fig. 2: Schematics of the bicycle model of the vehicle, and qualitative approximation of the sideslip angle-tire force relation. steering torque feel (hereafter simply driver’s feel) the torque that is felt by the driver through the steering wheel. There- fore, we model the driver, the driver’s feel, and how the By following a procedure similar to [15], assuming the torque-based steering-assist system affects the driver’s feel.
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