Optimal Control Based Modeling of Vehicle Driver Properties

Abstract

In this paper, we present a two-level driver model for the use in real-time vehicle dynamics applications. On the anticipation level of this model, nominal trajectories for the path and the speed profile of the vehicle along a given course are determined by reducing the driving task to a parametric optimal control problem and using an efficient direct collocation method for its solution. Typical optimality criteria and control-state constraints serve to depict driving properties of different driver types. On the stabilization level, a nonlinear position controller guides the full vehicle dynamics model along the prescribed trajectories in real-time. This synthetic driver model allows easy implementation of different driving strategies to simulate a wide range of driver types and vehicles. The expediency of the proposed model is shown by comparing simulation results with measured data from several drivers performing ISO double lane changes with a passenger car. INTRODUCTION Main tasks in the development of modern motor vehicles are the increase of driving safety and comfort as well as the relief of the driver by driver assistance systems. To guarantee reliability and robust design of the developed vehicle controllers, it is necessary to investigate the performance of the vehicle-controller system in major parts of the dynamic spectrum which is realized in practical driving situations. To meet the demands of all eligible drivers, handling studies must be carried out over a broad range of different driver types. Nowadays, software for the numerical simulation of the full car dynamics is indispensable for the development of vehicles and vehicle dynamics controllers. Besides virtual prototyping and conceptual design in the computer, vehicle dynamics simulation programs are employed for real-time applications in Hardware(HIL) und Software-in-the-Loop (SIL) environments. Realistic simulation results do not only require a comprehensive vehicle model and a detailed representation of the road conditions. For the investigation of the closed control loop of vehicle, driver and environment also a technical driver model is necessary which allows implementing driving maneuvers during normal operation and at the driving limits. In the following, we investigate an optimal control based model to depict the driver’s properties in vehicle guidance. Its application in real-time vehicle dynamics simulation is investigated and simulation results are compared to measurements from driving tests. FULL VEHICLE DYNAMICS SIMULATION For the computation of the full vehicle dynamics in real-time a simulation model is needed which realistically depicts the vehicle system behavior and requires little computational time. The vehicle model implemented in the vehicle dynamics program veDYNA [15] is a generic, fully parameterized multi-body system for the basic vehicle. The nonlinear elasto-kinematics of almost arbitrary axle types can be depicted either by kinematic look-up tables or by detailed geometric models, including the respective control arms, drag links, subframes and bushings. In addition, partial models are employed to account for intrinsic vehicle dynamics, such as of the drive train, the steering mechanism, and the tires (cf. Fig. 1). Figure 1: Multi-body model in veDYNA including steering system and geometrical axle model. Custom methods for treating multi-body systems use the descriptor form of the equations of motion and yield a system of differential-algebraic equations of index 3. The choice of suitable generalized coordinates in veDYNA, however, eliminates algebraic constraints and reduces the differential-algebraic system to ordinary differential equations. Due to their stiffness the numerical integration is performed with a semi-implicit Euler scheme which allows stable, real-time capable integration of the vehicle's equations of motion for step sizes up to several milliseconds [12]. For the representation of the vehicle environment the veDYNA Road model is used which allows depicting almost arbitrary road layouts with high accuracy [3, 16]. In this model, the horizontal course can be constructed synthetically according to a unit construction system or by specifying spatial road coordinates. The height profile allows segments with vanishing or constant slopes to be joined smoothly with arched pieces. For the road surface characteristic, geometric disturbances, such as potholes or rail tracks, may be depicted as well as different weather conditions and stochastic roughness. Figure 2: veDYNA Road layout for the Formula 1 course in Monza [3]. Besides openand closed-loop controls realistic driving maneuvers can be implemented with the veDYNA Driver [14, 16]. The versatile non-linear position controller which is described in the next section is able to guide the vehicle along arbitrary tracks and to handle the vehicle during demanding driving tasks. The driver controller shows to some extent natural human vehicle handling activities, but in a reproducible and adjustable way [8]. veDYNA allows the realistic simulation of the full vehicle dynamics in arbitrary driving situations. The program is employed among major car manufacturers and suppliers for rapid prototyping and parameter studies as well as for comfort and safety investigations on the PC. Applications in Hardwareand Software-in-the-Loop test benches include design, calibration and test of vehicle dynamics controllers, such as anti-lock braking and traction control systems, as well as reliability tests by endurance runs. OPTIMAL CONTROL BASED DRIVER MODEL According to Donges [4] human vehicular control can be separated into guidance and stabilization. Accordingly, a two-level driver model is suggested consisting of an anticipation level, where nominal trajectories for vehicle guidance are selected, and a stabilization level, where suitable control actions for adhering to the reference input variables are implemented. In the two-level driver model developed in the following, nominal trajectories for the path and the speed profile of the vehicle are determined by optimal control methods. This approach is motivated by the optimal control model of Baron, Kleinman and Levison [1] for the stabilization level. Accordingly, a “well-motivated, welltrained human operator behaves in a near-optimal manner subject to his inherent limitations and constraints, and his control task.” While common models for the driver anticipation level select the road center line and the maximum permissible vehicle speed as trivial reference variables, we formulate the driving task as a parametric optimal control problem. To depict the driver’s motivation, a combination of suitable optimality criteria, such as maximum traveled distance and minimum mean-square values of the vehicle’s deviation from the road center line and the lateral acceleration, is used. In this regard, the current work extends the driver model of Ehmann et al. [5, 17] for racing applications where time-optimal set variables are computed. The optimal control approach promises an easily parameterizable, synthetic driver model which is suitable to investigate the objective vehicle handling properties in the computer over a broad range of drivers, maneuvers, tracks and vehicles. On the stabilization level, a non-linear position control algorithm serves to guide the vehicle precisely along the prescribed trajectories for path and speed profile. Some controller parameters, such as preview, controller gain and steering delay, may be adjusted to represent human properties in vehicle guidance [8]. As compared to classic driver models from linear control theory, this approach provides vehicular control independent of the respective driving maneuver and offers a small number of meaningful driver parameters. For a detailed discussion of other driver controllers we refer to [2]. ANTICIPATION LEVEL The computation of nominal path and speed trajectories for vehicle guidance requires a vehicle dynamics model which can be treated with optimal control methods. Suitable for this application is the single track model [11] shown in Fig. 3. The latter is a planar vehicle model where front and rear wheels are condensed to one single wheel each and the center of gravity is situated on road height. Vertical dynamics, pitch and roll motions as well as displacements of the wheels relative to the vehicle body are neglected. Figure 3: Single track model [11]. y,r F x,r F β