order driver model with parametric and unstructured uncertainty is proposed. We obtain and validate the uncertainty models using the data from real drivers.
Proceedings of the American Control Conference San Diego, California June 1999
DRIVER MODEL UNCERTAINTY Liang-kuang Chen and A. Galip Ulsoy Department of Mechanical Engineering and Applied Mechanics University of Michigan, Ann Arbor, MI 48109-2125 USA approximates (o,/s)exp(-Ts) around oc, where an effective time delay of the driver (T) is included. Many driver models can be regarded as different realizations of the crossover model, e.g., [1], and [2]. These models have similar characteristics around the crossover frequency and differ more at higher and lower frequency ranges. Regressive models of the driver steering control have also been reported in [ 6 ] . In general, higher order models can capture more characteristics of the real driver, and can be considered more “complete.” However, from the controller design perspective, it is preferable to use a linear, low order driver model to simplify the controller synthesis processes. Doing so suggests the consideration of model uncertainty. In general, model uncertainty can be divided into structured uncertainty (e.g.. parametric uncertainty) and unstructured uncertainty (e.g., additive uncertainty). Driver model uncertainty can also be addressed from three aspects: model order, parameter uncertainty, and nonlinearity, as suggested in [4]. The research work presented here will investigate the driver model with parametric and unstructured uncertainty.
Abstract This article presents a preliminary investigation of the modeling of driver uncertainty. A linear second order driver model with parametric and unstructured uncertainty is proposed. We obtain and validate the uncertainty models using the data from real drivers driving a fixed-base driving simulator.
1. Introduction Vehicle active safety systems are designed to help improve driving safety while the driver is still in control of the vehicle. For the design of such systems, the driver interaction can be significant and should not be neglected. Several different driver models have been developed in the literature. For vehicle lateral control, the steering wheel angle is the primary means for control actuation. Many driver models try to approximate the real driver’s road tracking performance, assuming certain driver inputs and outputs. Although these models approximate the driver behavior well, no driver model is expected to represent the real driver completely. Furthermore, for the purpose of controller design, it is common practice to use a low-order linear driver model with delay. Therefore, it is reasonable to expect that significant driver model uncertainty exists. This driver model uncertainty can have a significant effect on the performance of the designed control system. However, studies of driver model uncertainty have not been reported in the literature. This article aims to develop a model for the driver model uncertainty using the measured data from real drivers driving a fixed base driving simulator. Driver steering control model has been studied extensively; several driver models have been developed, e.g., [3], [7].A well-known result from the human factors research is the “crossover” model. The crossover model states that the open loop return ratio of the driver-vehicle combination approximates that of a transfer function o & s around the crossover frequency, where o,is the crossover frequency. More precisely, the open loop driver-vehicle combination
0-7803-4990-6/99 $10.00 0 1999 AACC
2. Uncertainty Modeling for Drivers The problem addressed in this paper is the uncertainty modeling of the driver steering control model. The main purpose of this section is to describe the procedure starting from the driving simulator data to the identification of the driver model and the uncertainty model. We will consider both structured and unstructured uncertainty of the driver model. The unmodeled nonlinearity is not considered here. We consider a black-box driver model where the lateral deviation from the centerline of the road (ydev)is treated as the input to the driver. The driver’s output is the steering wheel angle (6).The idea of this model is shown schematically in Figure 2.1. The objective of this research is to obtain a nominal driver model (GJ with parametric uncertainty and unstructured model uncertainty (A) from the driving simulator data. This idea is shown in Figure 2.2, where an additive model
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uncertainty is used. The data is collected from 12 drivers driving a fixed-base driving simulator [5].The data consists of two-hour runs for each driver and the measurements are sampled at 20 Hz.
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Figure 2.1: Black-box driver model
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Figure 2.2: Desired driver model In order to proceed with the driver model identification, we need to determine the structure and the order of the driver model. The ARX structure is one candidate that has a simple structure and can use a linear parameter estimation algorithm. However, the ARX structure Ay(t)=Bu(t)+e(t) is limiting, since e(?) must be white noise. The ARMAX model is chosen here to represent the driver model because its assumption of colored noise makes it useful for us to determine the uncertainty bound. The ARMAX driver model can be written as:
transfer function relating Ydev and n. This is based on the assumption that n approximates n’. Also assume that the signals Ydev and n have significant frequency contents for spectral analysis purpose. A spectral analysis with n and Ydev can give the estimated transfer function of IAI. The polynomial C is not used here because it is obtained based on the assumption that the uncertainty enters the system as a “filtered” version of the random signal, which does not result in the desired uncertainty model. In order to describe the time varying behavior of the driver, the two-hour data is divided into 60 twominute segments. The first segment of data is neglected because at the beginning of the experiments, some signals were not measured. As a result, 59 segments of data are available for each driver. For each segment of data, the ARMAX driver model parameters are estimated. We will use these 59 sets of data to identify and validate the driver model. The mean value, standard deviation, minimum, and maximum for the parameters of driver model can be calculated. This parametric variation represents the driver’s time-varying nature.
where A, B, and C are time-varying polynomials to be identified. The ARMAX model can be represented in the block diagram shown in Figure 2.3, where e is the white noise that goes into the driver model and Ltis the predicted driver’s output. We will use polynomials A and B to represent the nominal driver model.
- + Figure 2.3: Block diagram of ARMAX driver model Comparing Figure 2.2 and Figure 2.3, we propose that the 1A1 be approximated by the magnitude of the 715
3. Results and Discussion The driver model uncertainty developed is intended for controller design. Two different application scenarios are considered. The first scenario is to use the uncertainty model developed to design a steering assist system to be used for a specific driver. The advantage of this application is less uncertainty will be obtained, and this is a benefit for controller design. The cost for this advantage is that a “learning” system to estimate the particular driver preference of different drivers is needed. Another scenario is to generalize the driver model uncertainty to include the variation between different drivers, in addition to the variation within one driver. The driver model uncertainty developed for this purpose will be much more significant than the previous case, thus increasing the difficulty in the controller design phase. Therefore, we will consider the driver model uncertainty both within one driver and across different drivers. Although residual analysis shows that higher order ARMAX models yield better agreement with the white noise assumption, it is the purpose of this research to use a low order model. It is decided to keep two parameters in each polynomial, that is, the ARMAX model can be represented by (as described in equation (1))
A = 1+alq-'+a2q-2 B = (bl+bZq-l)q-"' c = c1+c2q-l Based on the method described in the previous section, the following two cases are considered: 1) Uncertainty within one driver: The system identification technique is used to estimate the ARMAX model using data from one driver. We use the odd numbered segments of the data for identification purposes. The even numbered segments of data will be used for validation purposes. There are cases where the identification algorithm yields significantly different results, that is, the outlier points. This may result from the fact that the particular segments of data are not suitable to be modeled by an ARMAX model, therefore those results are abandoned. Figure 3.1 shows the parameter variation of the driver model. The means and standard deviations of the parameters are summarized in Table 3.1. Their significance can be presented by the corresponding pole-zero locations, and DC gain, as shown in Figure 3.2. The results indicate that one of the system poles is very close to 1 and the location of the second pole lies between -1 and 1. Most of the time, both poles are real. This implies that one of the response modes is highly unpredictable. As a result, parametric uncertainty is significant. The only zero is found to be very close to 1. The DC gain can vary widely between -8 and 5.
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Figure 3.2: Poldzero and DC gain of driver model The estimate of unstructured uncertainty has been described in section 2. The resulting curve of the upper bound, obtained from the odd numbered segments of data, is shown in Figure 3.3. The upper bound can be represented as 400 * (0.25 - w2 + j w ) 400 - w2 + 40jw
(2)
AMPLITUDE PLOT, input # 1 output # 1
a1 a2 bl bz Odd # mean -1.1982 0.2261 2.9935 -2.9821 I lstd 10.4109 10.3844 13.8194 13.8018-pl IEven /mean 1-1.2469 10.2699 12.8599 1-2.8534 I # lstd 10.3606 10.3449 13.5509 13.5429 Table 3.1: Nominal driver model parameters
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Frequency (raasec) Figure 3 3 : Validating uncertainty bound: one driver -
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Figure 3.1: Driver model parameter variation
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Uncertainty across different drivers: The same analysis is applied to the data from six different drivers, and a more general driver model uncertainty is obtained. The resulting unstructured uncertainty bound is found to be larger than previous case, as shown in Figure 3.4. The parameters' statistics are listed in Table 3.2. The change in mean values of the parameters is expected, however, the decrease of the standard deviations is surprising. This 716
upper bound obtained from equation (2). Figure 3.3 shows that the unstructured uncertainty lies below IAI, thus validating the upper bound obtained in (2). For part 2) The unstructured uncertainty has been tested using the other six drivers' data. An example plot obtained from one driver's data is also shown in Figure 3.4. The results show that the upper bound obtained is appropriate to cover most of the unstructured uncertainty for this specific driver. The unstructured uncertainty exceeds the upper bound only at small portion of the frequency range of interest. The parametric uncertainty across many drivers is again affected by the outliers.
is due to the fact that the outliers contribute to much of the standard deviations. Since the number of outliers does not increase significantly, in effect, the standard deviations of the parameters become smaller because now we have much more samples. AMPLITUDE PLOT, input # 1 output # 1 1oa
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1o-2 1oo 10' . . Frequency (radsec) Figure 3.4: Validating uncertainty bound: many drivers 10.'
validation mean 1-1.4228 10.4376
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Figure 3.5: Sample residual analysis plots
3) Model validation: The ARMAX model is validated by the fact that the residual is very close to white noise. Figure 3.5 shows one example of the residual analysis plots. The upper plot is the auto-con-elation of the residuals and the lower plot is the crosscorrelation between the residuals and the input to the driver, i.e., the lateral deviation. Both plots lie within 99% confidence intervals. This shows that the residuals are very close to white noise, and therefore, the assumption for the ARMAX model holds. To validate the above uncertainty models developed, for part 1) The even numbered segments of data are employed using the same analysis algorithm. The resulting parameter distributions are also listed in Table 3.1. It is seen that although the means and standard deviations differ slightly from the odd numbered segment results, the two sets compare relatively well. The unstructured uncertainty is also computed for the even numbered segments of data. The spectral analysis results are also shown in Figure 3.3, in order to compare with the 71 7
4. Summary and Conclusions
1) Summary: System identification techniques are used to estimate the driver model with parametric and unstructured uncertainty, using data from real drivers driving a fixed-base driving simulator. The two different scenarios are considered: model uncertainty within one driver and across many different drivers. The models obtained are verified using different portions of driving simulator runs data. 2) Conclusions: The parameters of the ARMAX driver model vary significantly. The poles of the driver models show that the driver behavior is quite unpredictable. This feature implies that the parametric uncertainty is so significant that it may not be useful for controller design purpose. This also suggests that some kind of on-line driver parameters identification techniques may be necessary. The unstructured uncertainty
increases when we go from one driver to many drivers.
3) Future work: The validation of the uncertainty models needs to be investigated further. We are also considering other methods to determine the unstructured model uncertainty from the driving simulator data. The nonlinearity of the driver behavior is another issue to be addressed. The applications of the driver model uncertainty obtained also need to be explored. 5. Acknowledgements
The authors would like to thank the financial support from the Intelligent Transportation System Center (ITS), Research Center of Excellence (RCE). Also thanks to the experimental data provided by Tom Pilutti at Ford Research Laboratories.
References [ 11 Habib, M.S., ”Characterization of DriverVehicle Directional Control Using Three Models of Human Driver,” Proc. Intl. Symp. On Advanced Vehicle Control [AVEC], Tsubaki, Japan, Oct. 1994 [2] Hess, R. A. and Modjtahedzadeh, A., 1990, “A Control Theoretic Model of Driver Steering Behavior,” IEEE Control Systems Magazine, pp. 3-8. [ 3 ] MacAdam, C., “An Optimal Preview Control for Linear Systems,” ASME J. of Dynamic Systems, Measurement and Control. 1980, vol. 192, pp. 188190. [4] MacFarlane, D. C., Robust controller design usinx normalized covrime factor vlant descrivtions, Berlin ;New York : Springer-Verlag, 1990 [5] Pilutti, T., Lateral vehicle copilot to avoid unintended roadway departure. PhD. Thesis, University of Michigan, Ann Arbor, MI 1997. [6] Soma, H. and Hiramatsu, K., “Dynamic Identification of Driver-Vehicle System Using ARmethod,” Veh. Sys. Dyn. 24 (1993, pp. 263-282. [7] Weir, D. H. and McRuer, D. T., “Models for Steering Control of Motor Vehicles,” Proceedings, Fourth Annual NASA-University Conference on Manual Control (NASA SP-192) (pp. 135-169). Washington, DC:. 1968, U. S. Government Printing Office.
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