A control theoretic model of driver steering behavior - IEEE Control ...

A Control Theoretic Model of Driver Steering Behavior R. A. Hess and A. Modjtahedzadeh

Following well established feedback control design principles, a control theoretic model of driver steering behavior is presented. While accounting for the inherent manual control limitations of the human, the compensation dynamics of the driver are chosen to produce a stable, robust, closedloop driver/vehicle system with a bandwidth commensurate with the demands of the driving task being analyzed. A technique for selecting driver model parameters is a natural by-product of the control theoretic modeling approach. Experimental verification shows the a bility of th e m o d el t o p r o d u ce driver/vehicle responses similar to those obtained in a simulated lane-keeping driving task on a curving road. A technique for selecting driver model parameters is a natural byproduct of the control theoretic modeling approach. Experimental verification shows the ability of the model to produce driver/vehicle responses similar to those obtained in a simulated lane-keeping driving task on a curving road.

Control Technology for Automotive Engineering Active control technology, which is now routinely used in modern high-performance aircraft, is finding its way into the realm of automotive engineering [l]. The design and development of systems for four-wheel steering, active suspensions, active, indepenrdent braking and "drive-bywire" steering provide the engineer with considerably more freedom in altering vehicle handling qualities than existed in the past. Mathematical models of driver steering behavior can serve as useful tools in analytical investigations of proposed vehicle control An earlier version of this paper was presented at the 1989 IEEE Systems, Man, and Cybernetics Conference, Cambridge, MA, Nov. 17-19, 1989. A. A. Hess and A. Modjtahedzudeh are with the Dept. of Mechanical, Aeronautical and Materials Engineering, University of California, Davis, CA 95616.

I€€€ Control Systems Magazine

systems provided that they meet two criteria. First, they should be ab l e to predict driver/vehicle responses accurately enough to

Most, if not all, of the research described was directedatdevelopingdescriptivetoolsforthe researcher rather than predictive tools for the

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Fig. 1.Steering task geometry.

allow valid engineering decisions to be made concerning the acceptability of proposed vehicle control schemes, and second they should be relatively easy to use by an engineer who may not be a manual control specialist, i.e., they should require a minimum of "art"in their application. To be sure, the subject of mathematically modeling driver steering behavior is not new. A sampling of research in the area, dating from 1961 canbefoundin [2-12].Itisprobablysafe to say that none of the models presented in these references meet both criteria stated above. This is not intended as a criticism.

practicing engineer. The work described here places increased emphasis on the latter role, developing predictive tools for driver steering models.

Control Theoretic Framework C o n si d er Fi g . 1 , which shows a driver/vehicle system in a lane-keeping task on a curving roadway. Assume that the vehicle is under "cruise control" and is maintaining some desired speed uo.The driving task can be summarized as one of maintaining the lateral path error yE(t)near zero. This task suggests

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bandwidths of this magnitude to be obtained, it can easily be demonstrated that adequate stability margins will not be achieved. Simply adding a second lead term at frequencies above the crossover frequency would, of course, solve this problem, however the resulting Go@)would not be a good model for driver steering behavior, since the effects of higher frequency neuromuscular system modes have been neglected. If one hypothesizes that a two-loop feedback system is created by the driver, with an inner feedback loop controlling visually-sensed vehicle lateral velocity yv(t), and an outer feedback loop controlling visually-sensed vehicle lateral position yv(t), then the required bandwidth, itself, is unattainable. Thus, if our model is to meet the first of the two criteria mentioned in the preceding section, a new tack is required.

Proposed DrivedVehicle Model Vehicle Model

the feedback system of Fig. 2 which shows the driver operating on the perceived lateral error yE(t)and producing a corrective steering wheel input &. The characteristics of a "goodq drivedvehicle system can be succinctly described in control terminology as those which cause the output yv(t),to equal the input yR(t)over as broad a frequency range as possible while minimizing sensitivity to disturbances and variations in the characteristics of the vehicle or the driver. As is well-known, the closed-loop characteristicsjust mentioned translate into desirable frequency domain characteristics of the openloop return ratio or loop transmission Go(jw)Gv(jo) equals yv(jw)/eA(jw)as shown in Fig. 3 [13]. Perhaps not surprisingly, this desirable behavior of the open-loop return ratio around the crossover frequency oc.i.e., approximating that of a transfer function GoGv(s) = OC-/S, has been shown to be exhibited by almost all manually controlled systems. It has led to the formulation of what is called the "crossover" model of the human operator [ 141. Because of the inherent limitations of human sensing, processing, and actuation, an "effective" time delay is also included in the crossover model, and the open-loop return ratio of a single-loop manual control system around the crossover frequency can be accurately described by the transfer function (wc/s)exp{ -wI. The preceding discussion lays the groundwork for presenting a control theoretic model of driver steering behavior. The term "control theoretic" is not intended to suggest that new results in the theory of feedback control are forthcoming here, but rather to

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distinguish this approach from psychological models of human behavior or models which emphasize the visual information processing characteristics of the driver, e.g., [lo]. A simple, linear model for driver steering behavior corresponding to Fig. 2 might now be fo r mu l at ed as Go@) eq u al s [l/Gv(s)][(oJs)exp( - z ~ s ) ]Formostvehicles, . this Go@) would imply low-frequency lead generation on the part of the driver, i.e., G&) is approximately equal to (TLs+l)exp( -GS), with l/TLmuch less than wc. A problem arises with this approach in modeling aggressive driving tasks. In such tasks, lateral position bandwidths on the order of 2 rad/s are suggested by response data, e.g., [6]. While the f o r m of G&) j u s t g i v en may al l o w

A brief discussion of the vehicle model is in order before the driver model can be introduced. For purposes of exposition, the description of the vehicle model will be as simple as possible. Referring to Fig. 1, a linear, four-variable state space model was derived, with state variables v(t), r(t), y ( t ) and yy(t), defining the component of vehicle velocity in y B , the y body-axis, yaw rate, heading angle and lateral deviation, respectively. No roll or suspension dynamics were considered in the model although their inclusion would pose no difficulties except added vehicle model complexity. The characteristics of the particular vehicle being modeled were obtained from a study

Fig. 4 . The driverhehick model.

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summarized in [9] and represent those of a full-size car traveling at 50 km/h. For the purposes of this study, the vehicle dynamics are summarized by the following linear transfer function: 7.69 ( 52+2 (0.372) (5.96) s+ 5.962)

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-20

The Driver Model

d Fig. 4 is a block-diagram representation of the proposed drivedvehicle system, emphasizing the driver model elements. As the figure indicates, the driver model has been divided into high and low-frequency compensation elements, each of which are described in the following sections. As will be seen, thismodel will 1) allow the bandwidth and stability requirements of aggressive steering tasks to be met, 2) include a neuromuscular system mode, and 3) exhibit the desirable open-loop return ratio characteristics of Fig. 3.

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High Frequency Driver Compensation

We begin by adopting a model which has been successfully employed in modeling human pilots in well-defined flight control tasks: the so-called "structural model" of the human operator [ 151. This model constitutes the high-frequency driver compensation, where high-frequency refers to frequencies within an approximate one decade range around the crossover frequency of the overall driver/vehicle open-loop return ratio. Space does not permit a detailed discussion of the genesis of the structural model, and the interested reader is referred to [15]. The block labeled GNMis a simple second-order representation of the neuromuscular system of the driver's arms. Block Gpl has, as its input, the output of the neuromuscular system 6swwhich is the driver's steering input to the vehicle. Block GP2receives its input from the output of GPI.Both of these dynamic elements represent feedback of variables derived from the motion of human limbs and muscle tissue, and are referred to as "proprioceptive" feedback elements. The block GLis a time delay representing the inherent human signal processing delays. This delay is not equivalent to the zEmentioned previously, in that the latter delay includes the phase lag effects attributable to the remainder of the dynamics in the structural model, namely those emanating from the feedback loop involving GNMand GPI. Although the high-frequency driver compensation involves an eight parameter model, experience in modeling the human operator in IEEE Control Sntems Magazine

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Fig. 6. A visual guidance cue for the driving task.

Table I Nominal Parameter Values for StructuralModel compensation

(integral) (proportional) (derivative)

k

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*Selected to achieve desirable crossover characteristics. a variety of manual control tasks, e.g., [16], has shown that four of these parameters can be considered invariant, with the remaining four dependent only upon the human operator compensation (proportional, derivative, or integral) which will cause they& transfer func-

tion in Fig. 4 to follow the dictates of the crossover model in the high-frequency region just defined. Table I summarizes these parameters. For the vehicle dynamics of (l), high frequency derivative compensation is required tocancel thevehicle pole at s = -4.44.

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Fig. 7 .Curving roadway from the simulation of [9].

Note that the vehicle pole at s = -5.33 is essentially cancelled by the complex zero. This means that, in Table I, k = 2, and T = V4.44 s, and the remaining model parameters are listed in the last row of the table. Note that no feedback loop involving visually sensed lateral vehicle velocity has been employed in the model, despite the fact that we have formed the structural model based upon yv/&w vehicle dynamics. The reason for avoiding a driver model with two feedback loops using visually (as opposed to proprioceptively) sensed variables has been pointed out previously. This is an important step in the modeling procedure. The high-frequency compensation in the driver model which is based on the yV/hw transfer function, simply serves to tailor the high-frequency part of the open-loop return ratio. Note further that the feedback variables in the structural model are internal to the human and are assumed to

involve proprioceptive organs such as muscle spindles, joint angle receptors, etc. [ 171.

of 1 r ad s has been chosen. The amplitude peaking evident around 101 1 rad/s is a result of the "proprioceptive"feedback loops around the neuromuscular system G N Min the structural model in Fig. 4 . As will be seen, the maximum bandwidth achievable with the system of Fig. 5 is considerably greater than that possible with either of the two feedback possibilities discussed earlier. With the linkage between Ky,UT3, and a-,we see that only one parameter in the driver/vehicle model is used to account for driver adaptation to different steering tasks with any set of vehicle dynamics. That parameter is the open-loop return ratio crossover frequency %. Visual Guidance Cue

Low-Frequency Driver Compensation

An examination of the transfer function yv/u in Fig. 4 at this point indicates that the desired open-loop return ratio characteristics of Fig. 3 can be obtained with the addition of the block Gc in Fi g . 4, where G c = Ky[s+( 1/T3)].For modeling purposes, the zero at - l/T3will be maintained a decade below the crossover frequency of the open-loop return ratio yv/eA.This decade separation represents areasonable choice as it provides the desirable large, low-frequency magnitude shown in Fig. 3, without adversely affecting the gain and phase margins. Fig. 5 shows the resulting amplitude and phase characteristics of the open-loop return ratio when, for the sake of comparison with Fig. 3, a crossover frequency

A simple visual guidance cue, first proposed in [ 1 8 ] ,could be used by the driver in closing the outer feedback loop of Fig. 4. By realizing that &t) = U O v R ( t ) and, for typically configured vehicles &(t) = uovv(t) the variable U in the model of Fig. 4 is equivalent to a weighted sum of heading and lateral displacement errors between the desired and actual vehicle paths. Thus, in the time domain, and referring to Fig. 6,

Th us, the variable U is proportional to vu, defined as the angle between the vehicle body axis and an "aim point" on the tangent to the lane centerline, a distance MOT3 = 10udoc ahead of the vehicle. The simple guidance cue defined in (2) is valid regardless of the driving task. Of course, a combination of high speed and low crossover frequency can lead to an aim point location far ahead of the vehicle. This simply means that v , ( t )would be quite small as compared to vE(t), and the latter would become the primary visual cue in the driving task.

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Driving Simulator Data Comparison

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Driver/vehicle response data from a driving task summarized in [9] can now be used to evaluate the accuracy of the driver/vehicle model just described. The task used the vehicle dynamics summarized here by the lateral path to steering input transfer function of ( 1 ) with a constant speed uo= 50 km/h. The task consisted of lane-keeping on the curving roadway shown in Fig. 7 . All the driver model

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parameters associated with the high-frequency compensation have been chosen in the preceding section independently of the driving task. As just mentioned, 0~ (and consequently, Kyand 1/T3)is the only driver/vehicle model parameter which will be varied to tune the model responses to those from the simulation experiment of [9]. This tuning procedure was accomplished by employing a simulation of the drivedvehicle system of Fig. 4 (including nonlinear roadway kinematics) and varying 0~ from run to run until the standard deviations of lateral deviation and heading errors of the driver/vehicle model were close to those obtained in the experiments of [9]. The resulting low-frequency compensation, Gc(s),is given by:

G ~ ( s=) 1.75(~+ 0.325).

0 two-wheel steering 0

four-wheel s t e e r i n g

(3)

The crossover frequency oc for the driver vehicle system for this Gc(s)is 3.25 rad/s, with gain and phase margins of 5.4 dB and 24", respectively. The corresponding closed-loop bandwidth defined as the lowest frequency for which

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0

20

40

60

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was found to be 5.35 rad/s. The error standard deviation comparisons are:

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heading lateral displacement

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120

140

160

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ig. 9. Driverfvehicle crossover frequency as a function of velocityfor direrent steering systems.

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The statistics from the experiment were the result of eight simulation runs by each of six, well-trained test subjects. A comparison of low-frequency steering wheel inputs, i.e., those which correspond to the roadway curvature, indicated very close agreement between model and experiment. However, the experimental hW contained a low-amplitude high-frequency component which can be attributed to driver "remnant" [ 141.The effect of such remnant injection is not considered significant in the evaluation of overall vehicle handling qualities, and hence, remnant was not included in the modeling effort here. Fig. 8 shows the steering input time histories for one of the drivers from the experiment and that from the model, with the former exhibiting the high-frequency remnant component. It is interesting to note that the crossover frequency, gain and phase margins of the "tuned" driver/vehicle model are also representative of values which have been measured in single-loop manual tracking tasks in which the dynamics of the plant require high-frequency lead compensation on the part of the

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human similar to that required here [ 191. This result suggests that, in the absence of driver simulation data with which to choose wc, the wealth of theoretical and experimental information in the manual control literature may provide adequate guidance.

Exercising the Model As an example of the utility of the model of driver steering behavior, we can analyze the same vehicle used in the previous section, but now including a four-wheel steering system. One "control law" which has been proposed for such a steering system, e.g., [20], automatically determines the steering angle of the rear wheels so that the component of the vehicle velocity in the ye body axis remains zero (no vehicle "sideslip") when the driver turns the front wheels in normal maneuvering. Fig. 9 compares predicted driver/vehicle crossover frequencies wc at a number of velocities for the vehicle with conventional two-wheel steering and the same vehicle with the fourwheel steering system just described. In both cases, the control theoretic model for driver steering behavior was formulated with gain

and phase margins fory,/e, of at least 25O, and 6 dB, respectively.The superiority of the fourwheel system is evident in the results. As can be seen from the figure, crossover frequency, and hence, maneuverability,for the two-wheel system decreases with increasing velocity, while that for the four-wheel system, does not. Of course, care must be exercised in extrapolating the results of such an analysis to maneuvers of a severity that would invalidate the simplifying assumptions of the linear vehicle model used here.

Conclusions The control theoretic model of driver steering behavior has been developed consisting of low- and high-frequency compensation elements, with the latter obtained from application of a structural model of the human. A single parameter, the crossover frequency of the open-loop return ratio, was used to tune model response statistics to those from a simulation experiment involving lane keeping on a curving roadway. The remaining parameters in the driver model were selected on the basis of well-established feedback con-

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trol design principles dependent only upon the vehicle dynamics. A simple visual guidance cue can account for the manner in which the driver closes the outer feedback loop in steering tasks. Research is currently underway in employing the model to study the effects of different vehicle steering systems on handling qualities.

[9] L. D. Reid, E. N. Solowka, and A. M. Billing, "A systematic study of driver steering behavior, Ergonomics, Vol. 24, No. 1, pp. 447-462, 1981. [ 101 U. Kramer and G. Rohr, "A model of driver behavior," Ergonomics, Vol. 25, No. 10, pp. 891907, 1982. [ll] J. Godthelp, "Precognitivecontrol: Open and closed-loop steering in a lane change maneuver," Ergonomics, Vol. 28, No. 10, pp. 1419-1438,1985.

References [l] M. Iguchi, "Applicationof active control technology to motor vehicle control," Int. J. Vehicle Des., Vol. 9, No. 3, pp. 287-294, 1988. 121 J. G. Wohl, "Man-machinesteering dynamics," Human Factors, Vol. 3, No. 4, pp. 222-228, 1961.

[12] T. Legouis, A. Laneville, P. Bourassa, and G. Payre, "Characterization of dynamic vehicle stability using two models of the human pilot behavior," Vehicle Syst. Dynamics, Vol. 15, No. l, pp. 1-18, 1986. [ 131Maciejowski, Multivariable Feedback Design. New York: Addison Wesley, 1989.

[3] W. W. Wienville, G. A. Gagne, and J. A. Knight, "An experimental study of human operator models and closed-loop analysis methods for highspeed automobile driving," IEEE Trans. Human Factors Elecrron., Vol. HFE-8, pp. 187-201, 1967.

[ 141D. T. McRuer and E. Krendel, "Mathematical models of human pilot behavior," AGARDograph No. 188. Jan. 1974.

[4]

D. H. Weir and D. T. McRuer, "Models for steering control of motor vehicles," in Proc. Fourth Ann. NASA-Univ. Conf. on Manual Control, Mar. 1968,pp. 135-169.

[I51 R. A. Hess, "A model-based theory for analyzing human control behavior," in Advances in ManMachine Systems Research, W. B. Rouse, Ed., Vol. 2. London: JAIPress, 1985, pp. 129-175.

[5] E. R. F. W. Crossman and H. Szostak, Manmachine models for car steering," in Proc. Fourth Ann. NASA-llniv. Con$ on Manual Control, Mar. 1968,pp. 171-195.

[16] R. A. Hess, "Theory for aircraft handling qualities based upon a structural pilot model," J. Guidance, Control Dynamics, Vol. 12, No. 6, pp. 792-797, 1989.

[6] D. T. McRuer, R. W. Allen. D. Weir, and R. H. Klein, "New results in driver steering control models," Human Factors, Vol. 19, No. 4, pp. 381397, 1977.

[17] D. T. McRuer, "Human dynamics in manmachine systems,'' Automatica, Vol. 16, No. 3, pp. 237-253,1980.

[7] G. A. Bekey, G. 0. Burnham, and J. Seo, "Control theoretic models of human drivers in carfollowing," Human Factors, Vol. 19, No. 4, pp. 393-413, 1977.

[18] R. W. Allen, "Stability analysis of automobile driver steeringcontrol,"in Proc. Seventeenth NASAUniv. Conf. Manual Control, pp. 597-609, Oct. 1981.

[8] E. Donges, "A two-level model of driver steering behavior." Human Factors, Vol. 20, No. 6, pp. 691-707.1978,

[19] D. E. Johnston and D. T. McRuer, "Investigation of limb-sidestick dynamic interaction with roll control,'' J. Guidance, Control, and Dynamics, Vol.

IO, NO.2, pp. 178-186, 1987

[20] E. C. Yeh and R. H. Wu, "Open-loop design for decoupling control of a four-wheel steering vehicle,"Int. J. of VehicleDes., Vol. 10,No. 5,1989. A . Hess received the B.S., M.S., and Ph.D. degrees in aerospace engineering from the University of Cincinnati, in 1965, 1967, and 1970, respectively. In 1982he joined the faculty of the Department of Mechanical, Aeronautical, and Materials Engineering at the University of California, Davis, where is currently a Professor in the Division of AeronauticalScience and Engineering. His current research interests lie in the areas of automatic and manual control and in madmachine systems. He is an Associate Fellow of the AIAA, a member of IEEE, SigmaXi, and Tau Beta Pi and is an Associate Editor of the Journal of Aircraf, and the IEEE Transactions on Systems,Man, and Cybernetics. He is a Vice-presidentof the IEEE Systems, Man, and Cybernetics Society and chairman of the Society's Manual Control Technical Committee. He is a member of the AIAA Technical Committee on Atmospheric Flight Mechanics.

Ronald

Ali Modjtahedzadeh

was born in 1960in Iran. He received the B.S. degree in mechanical engineering from the Universityof California, Irvine, in 1983, the M.S. degree in mechanicalengineering from San Jose State University in 1985, and is presently working towards the Ph.D. degree in the Department of Mechanical,Aeronautical,and Materials Engineering at the University of California, Davis. His research interests are focused in system dynamics and control, with emphasis upon mathematical modeling and control design. His current research area is manual control theory, applied to vehicular control.

1990 SMC Conference The 1990 IEEE International Conference on Systems, Man, and cybernetics will be held November 4-7, 1990 at the Sheraton Universal Hotel in Los Angeles, CA. Conveniently located on the Universal Studios lot atop the Hollywood Hills, the Sheraton provides easy access to cultural and entertainment centers of Los Angeles. The conference theme, Models and Media in Human-Machine Systems, h as been selected to reflect the growing importance of these aspects in decision support, training and information systems. Presentations dealing with theoretical perspectives, innovative design and simulation methods, and emerging

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modeling paradigms are encouraged. Plenary and technical sessions will emphasize this year's theme. Papers will be presented relevant to the following areas: enterprise modeling and simulation; concurrent engineering; manufacturing systems; human-machine systems; telerobotics and autonomous systems; behavioral decision making; artificial neural systems, knowledge systems; computer aided software/systems engineering; visual programming and graphical interfaces; networked simulators and distributed simulation; collaborative design; decision support systems; automation in manned systems; aviation

safetylautomation; information and decision systems; risk management and human engineering; workload analysis and modeling; image interpretation and computer vision; team decision making and training; intelligent tutoring and training systems; human computer interaction in complex systems; automated performance measurement. For further information, contact: Amos Freedy, Conference Chairman, or Azad M. Madni, Program Chair. Perceptronics, Inc. 21135 Erwin Street, Box 4198 Woodland Hills, CA 91365-4198 (818) 884-3485.

August 1990