PARAMETER DETERMINATION FOR A CROSSOVER DRIVER MODEL by D. L. Wilson R. A. Scott Technical Report Department of Mechanical Engineering and Applied Mechanics UM-MEAM- 8 3 -17

1 INTRODUCTION Manual control of automobiles has attracted much attention in recent years. One of the more useful developments has been the development of "describing-function" models of driver behavior, notably the "crossover" model initially proposed by McRueri (1973). These models, which describe the behavior of the driver as a linear element in a control system, are attractive because of their simplicity and because they provide a useful framework for evaluating experimental results. Various describing function models have been proposed by McRuer (1977), Donges (1978) and Allen (1982) and others. This paper will largely focus on the work of Mc Ruer. One problem which has faced potential users of these models has been that of determining appropriate parameters to characterize the behavior of a typical driver in a particular vehicle. In a previous paper, (Garrott, Wilson and Scott, 1982a) the authors presented a mathematical development of a "crossover"-type driver model and general expressions for deriving driver parameters from physical parameters of a vehicle. This paper presents a systematic procedure for determining driver parameters from measured or simulated frequency response data, based in part on experimental findings reported by Allen (1982). STEERING CONTROL LAW In its most general form, feedback control involves multiple inputs and loops with the entire state vector. Here, as illustrated in Figure 1,

2 a relatively simple form is adopted in which it is assumed that: (1) State-vector disturbances are caused by control variable disturbances 6d. (2) The desired state X(s) is known by the driver and is therefore not an independent input. In the figure, V(s) and D(s) represent the vehicle and driver transfer functions, respectively, where s denotes the Laplace transform parameter. It has been shown experimentally (McRuer, 1973; McRuer and Klein, 1976) that for disturbances at frequencies near the "crossover frequency", wc, the open-loop transfer function of the driver-vehicle system may be approximated by the "crossover law", i.e., V(s)D(s) = Te /s (1) where T is a pure time delay. A simple driver model for regulation of a vehicle about a straight line path was advanced by McRuer and Klein (1976). It is based on the following assumptions: (1) The kinematic quantifies of significance for control are lateral position error Y, heading angle error i, and time derivitives of these quantities. (2) The vehicle dynamics can be adequately described by a constant speed, linear, two-degree-of-freedom model. The transfer functions for this equivalent model can be reduced to the following "first-order" form for neutral-steer automobiles. V<(s) = (s)/6sw(s) = K /[S(l+T s)] (2) V6(s) = Y(s)/6 () = ' T eq VY(s) Y(s)/6 (s) -= K u'[s2(l+Ts) (3) 6 S6W( c eq

3 where u is the forward speed, K is the steady turning yaw rate gain, 6Sw is the steering wheel angle, T is the equivalent yaw time constant, and eq the sub- and superscripts on V refer to inputs and outputs, respectively. It has been proposed by McRuer, Allen, Weir and Klein (1977), that the principal feedback variables for regulation ("inner loop closure") ijr disturbances at frequencies near w are X and its time derivatives. C Letting D denote the transfer function which relates.steering wheel angle to heading angle for the-driver, it is therefore assumed that V' 6 and D satisfy Equation (1) so that 6 - w Ts/e-S/S V6D = wce /s. (4) C Substituting Equation (2) into Equation (4) yields 6 Sw/ = [(l+TeqS/K]ce. (5) It is further assumed that for straight line regulation the driver's response to a lateral position error Y is to define a desired path conerr sisting of a line between the current position and an "aim point" on the road centerline at a distance Xap as diagrammed on Figure 2. The lateral position error is interpreted as an equivalent heading angle error eq given by 1 = Y /X. (6) eq err ap The effective heading angle error AL is given by

4 =- 4 + +Y /X (7) AL Kerr eq err err ap and it is this angle that is operated upon by Do in Equation (5), Hence s(S) err(s)+KyYerr (s)[+T s]e TS(W/Kc) (8) where K = 1/X. y ap This simple crossover model is further extended, following a proposal of Allen (1982). An "extended crossover model" is obtained by multiplying Equation (8) by an "integral trim" term (l+K'/s), where K' is the "trim" gain. The control law in the time domain is then 6Sw(t) = [er (t-) + T rr(t-T) err eq err + KY (t-T) + KT eY (t-T)]w /K y err y eq err c c rt-T + K' l 6S ()dS (9) where the dot denotes a time derivative. The effect of the integral term is to reduce errors for turning paths and in situations in which the vehicle does not travel in a straight path even though 6Sw = 0 (such as would occur in a crosswind). The control law in Equation (9) is still inadequate for a curved path. If the vehicle is perfectly positioned, i.e. r = = Y e err err = Y =0, and the integral term is zero, then 6Sw 0, which is not err SW correct. Following Garrott, Wilson and Scott (1982a) it is assumed that the driver is able to perceive path curvature

5 K and vehicle speed u and is able to determine and implement the steering angle necessary to cause a steady turn of that curvature. This is achieved by the addition of the following term 6Sw = (GR)[l+(KD)u2] L K (10) where (GR) is the steering gear box ratio, (KD) is the vehicle understeer factor and L is the vehicle wheel base. It is also assumed that a pure time delay, equal to T, occurs in implementing this curvature correction. The final extended crossover model for curved paths used here is given by 6Sw(t) = (GR) [l+(KD)u2]LK(t-T) + Sw(corr) t + K' TSw(corr)( (11) where + C ) (t) + KyT /K12) KY err eq err c c Details on the computation of the curvature and error terms in Equations (11) and (12) are given in Garrott et al (1982a, b). DETERMINATION OF DRIVER PARAMETERS There are seven parameters in the driver control law given by Equation (11), namely, c, T, K, Teq Ky K' and KD. All of these

6 parameters except KD are potentially related to the behavior of the "equivalent first-order vehicle" (KD is strictly a property of the actual vehicle being driven). The first-order vehicle is defined by three parameters, namely, Kc, T and u. Experimental results on closed-loop response to steering diseq turbances have been analyzed by McRuer et al (1977) and Allen (1982), using the extended crossover model, for a wide range of drivers and vehicles. It has been found that: (1) T, wC and K all depend on T, c y eq but are not strongly related to K. (2) For a given driver and vehicle C K is nearly independent of vehicle speed for speeds below 22 m/s (50 mph). (3) K' is approximately constant. Straight-line fits to data presented by Allen (1982) produce the following relationships K' = 0.5 rad/sec (13) K u = 0.5 rad/sec + 0.068/T - (14) eq 4.0 rad/sec, 1/T < 5.0 rad/sec C = 4.0 rad/sec + (/Teq - 5.0 rad/sec), 1/Teq > 5.0 rad/sec. (15) It is felt that for speeds above 22 m/s, the "look ahead" distance X will increase and consequently K will decrease. The following exap pression is proposed to reflect the above considerations

7 Cy/22, u < 22 m/s KY ~ > (16) Cy/u, u > 22 m/s where Cy = 0.5 rad/sec + 0.068/Teq. (17) The lag is given by T - 0.30 sec - 0.023 sec2/T. (18) eq K and T will now be addressed. The problem of determining typical c eq driver parameters for a particular vehicle becomes one of defining the parameters for the equivalent first-order model. For a neutral-steer vehicle, these can be computed simply from physical parameters such as mass, wheel base, etc., (Weir and McRuer 1973, Garrott et al 1982a). However most vehicles are not neutral and an alternative procedure for obtaining driver parameters is desirable. A procedure will be described here for reflecting parameters for the equivalent first order vehicle from measured or simulated frequency response curves. Recalling that multiplication by s in the Laplace transform domain corresponds to differentiation in the time domain, it can be seen from Equations (2) and (3) that [Vr (s)] s[V W(s)]eq = K/(l+T s). (19) 6 eq eq c eq where r is the yaw rate.

8 Therefore, [Gr(W)] = K I +Te 2 (20) G6 eq c eq [ r()] = -tan- (T ). (21) [ % 6 eq eq Yaw rate frequency response curves are presented in Figure 3 in the form of "Bode plots," namely 20 log G versus log w and ( versus log w, where G and w denote the dimensionless numerical values of the corresponding (dimensional) gain and frequency. Corresponding frequency response plots for an understeering vehicle are presented in Figure 4. These plots were obtained by computer simulation with a three-degree-of=freedom vehicle model, using a sinusoidal steering input.l It may be seen that the first-order form can never fit the actual curves, since they lack the "hump" in the yaw rate gain curve, and the rapid changes in 6 near log w = 0.8. A procedure is now suggested for selecting the parameters K and T to obtain a "best fit" of the first-order form to the data. The eq yaw-rate curves are fit by emphasizing the region w = w for the following reasons: (1) The "crossover law" is most accurate near w = w. (2) C w and 1/T are of the same order of magnitude for most drivers and c eq vehicles, so both can be emphasized by fitting in this region. (3) The driver-vehicle system given by Equation (1) will oscillate at neutral stability at w, suggesting that this frequency range is vital for stability considerations. The vehicle parameters used represented a 1971 Ford Mustang. See Wilson and Scott (1982).

9 The following steps are performed to determine the parameters: (1) Locate an wT such that 6(WT) = -/4. The value of -r/4 was chosen since this central value tends to minimize the global error in the approximation. Then Te = l/w. (22) (2) Compute w using Equation (15). (3) Determine Gi ). Then, from Equation (20) r 2 r )~+..... 2 22 Kc = G(w) l+T. (23) c 6 c eq c (4) Recalling that for steady turns K r/u, Equation (10) may be rewritten u 2 ( s/r)L(GR) 1 + (KD)u. (24) The steady turning value of r/6Sw is Gr(o), which is known. Hence, using this value, KD can be determined from Equation (24). The process is illustrated in Figure 5 for the measured data of Figure 4. Shown also are curves obtained from the equivalent first- order model and their straight line asymptotic representations. These asymptotes are not accurate at low or high frequencies.- If, however, as has been done in the past, K and T were selected to fit the measured data c eq asymptotically, the curves would differ considerably near wa (in this case by nearly 5dB, or a factor of 1.8). The resulting parameter values are T = 0.158 s eq

10 xc = 5,0 rad/s c K = 0.285 s c -1 Ky = 0.0416 m z = 0.15 s -3 2 2 KD = 3.4 x 103 s2/2 EXAMPLES As tests, several simulation runs were made using the three-degreeof-freedom vehicle model (Wilson and Scott 1982) and the extended crossover driver model employing the set of parameters above. Example 1 is a straight line path following task. The forward speed was held constant by adjusting drive torque with a simple speed control routine. The desired path and speed are Y = 0 and u = 22.4 m/s. A heading angle perturbation of ) = 0.025 rad (1.4~) is specified as an initial condition. Other initial conditions are X = Y = v = r = 0 u = 22.4 m/s. o Figure 6 shows the lateral position and steering wheel angle time histories. A very stable system is seen, with the driver completing the assigned task in about 2 seconds.

11 Example 2 is identical to the first except that now the driver time delay is 0.30s instead of 0.15s simulating an intoxicated or non-alert driver. Clearly, as Figure 7 shows, this system borders on instability, with the driver unable to complete the task in a reasonable time. Example 3 demonstrates the ability of the driver model to follow, while maintaining constant speed, a curved path, involving an 80~ turn with a minimum radius of about 100 m. It was generated by running the simulation under open-loop control with the input 1..5t/2.0, t<2.0 sec 6S (rad) = 1.5, 2.0<t<6.0 sec 1.5(8-t)/2.0, 6.0<t<8.0 sec. Figure 9 shows the steering wheel angle versus time for the open and closed loop cases. Note that between 8 and 10 seconds the vehicle returned to, and slightly overshot,the desired path. These examples indicate that the specified systematic procedure does in fact yield reasonable values for driver control parameters. Some question of predicting stability of the driver-vehicle system with this driver control law will be addressed in a later paper.

12 REFERENCES Allen,R.W. Stability and performance analysis of automobile driver steering control. SAE Paper No. 820303, 1982. Donges, E. A two-level model of driver steering behavior. Human Factors, 1978, 20, 691-707. Garrott, W.R., Wilson, D.L., and Scott, R.A. Closed loop automobile maneuvers using describing function models. SAE Paper No. 820306, 1982a. Garrott, W.R., Wilson, D.L., and Scott, R.A. Closed loop automobile maneuvers using preview-predictor models. SAE Paper no. 820305, 1982b. McRuer, D.T., Allen, R.W., Weir, D.H., and Klein, R. New results in driver steering control models. Human Factors, 1977, 19, 381-397. McRuer, D.T. and Klein, R. Effects of automobile steering characteristics on driver/vehicle performance for regulation tasks. SAE Paper No. 760778, 1976. Weir, D.H. and McRuer, D.T. Measurement and interpretation of driver steering behavior and performance. SAE Paper No. 730098, 1973. Wilson, D.L. and Scott, R.A. Further developments in the simulation of automobile handling. Dept. of Mechanical Engineering and Applied Mechanics, University of Michigan, Technical Report No. UM-MEAM-82-2, 1982.

13 I U a) rn LI Cl Lo r (rad) 0 (rad) -10 -15 -20 -25 0 -7T/4 logw (w in radians) logw (w in radians) FIGURE 5 SELECTION OF PARAMETERS TO EQUIVALENT SIMPLE 2-DOF MODEL FOR SIMULATED VEHICLE RESPONSE

14 6 -— E dURE 1 FIGURE I / I. r I V sI I S OSW ^/..-/ -.^, ~Sw SIMPLIFIED SINGLE-VARIABLE DRIVER-VEHICLE SYSTEM BLOCK DIAGRAM

15 L 1 _ _ e q '1 centerline FIGURE 2 SCHEMATIC DIAGRAM OF EQUIVALENT YAW ANGLE AND AIM POINT DISTANCE

16 p Co c t,o I X 0 (%4 w = 1/Tq ii ii r/2-I logw FIGURE 3 BODE PLOT OF YAW RATE RESPONSE FOR FIRST-ORDER MODEL

17 -10 T r-4 54co I o ul C0 U cn rC 0V ** - 90 + + + + + + + + + + + I 0 - - + + + + + + + + + r (ra) (rad) -Tr/4 - + + + -r/2. -0.5 I 0 0.5 1.0 1.5 logw (w: rad/sec) FIGURE 4 BODE PLOT OF YAW RATE RESPONSE FOR 3-DOF MODEL

18 I z H F-4 -e E-4 Dq m TIME (SEC) e.oO 6.00 2.00 3.00 TIME (SEC) FIGURE i LATERAL POSITION AND STEERING JWHEEL ANGLE TIME HISTORY FOR YAW ANGLE PERTURBATION.

19 z H 2; a ff A. _ _ _ __ __ 1.U 2.00 3.00 4.,O 5.0 6.00 TIME (SEC) 1.00 200 3,00 4.00 5o0 6e,C TIME (SEC) LATERAL POSITION AND STEERING WHEEL ANGLE TI1E HISTORIES FOR YAW ANGLE PERTURBATION, INCREASED DRIVER TIME DELAY (T=0.30 s) 'IGURE 7

20 X (IN) 3nOO SFCCHCO 6SECC&CO.CO trajectory desired path C) C) C~ FIGURE 8 VEHICLE TRAJECTORY FOR CLOSED-LOOP CURVED PATH FOLLOWING TASK.

21 I0 2.00 4.00 6.00 8.00 10.0 T (SEC) STEERING WHEEL TIME HISTORY FOR CLOSED-LOOP CURVED PATH FOLLOWING TASK, STANDARD DRIVER PARAMETERS FIGURE ~