Nonlinear Identification of Lateral Dynamics of an Autonomous Car Vehicle ^†

Abstract

In this paper, the nonlinear identification of the lateral dynamics of a road vehicle and the velocity dependence of the dynamics are presented. One of the most useful methods to define the mathematical model is system identification based on measured data. A test vehicle for autonomous driving was constrained to move in a straight line while the vehicle’s steering servo was artificially excited. The input of the system is therefore the sum of the artificial excitation and the control signal of the autonomous function, and the output is the lateral acceleration of the vehicle. The measurements are used to identify Wiener and Hammerstein models of the lateral dynamics at different speeds using nonlinear methods. The aim is to investigate the velocity dependence of the dynamics.

1. Introduction

Knowledge of vehicle dynamics is essential for the design of driving dynamics, ride comfort and vehicle control. The use of model-based control systems is necessary for the design of driver assistance systems and autonomous vehicle functions. The development of such systems requires both the use of models developed and verified from vehicle mechanics and the use of models developed from system identification (ID) based on measurement results.

There are many studies on lateral vehicle dynamics, ranging from the simplest kinematic and bicycle-type models to high-dimensional multi-body descriptions [1,2,3].

A rich toolbox is available for identifying and managing a linear, dynamic system, as the field of linear modeling has been extensively explored over the past decades [4]. Most problems can be solved efficiently using commonly known methods, which are usually traced back to the determination of the impulse response. However, the majority of real dynamical systems are nonlinear. These nonlinearities can be neglected in many cases, but there are cases in which the degree of precision or nonlinearity and its unique properties require consideration [5].

If the input to a dynamical system is $u\left(t\right)$ and the output is $y\left(t\right)$, then in the mathematical model of the system, the past inputs are related to the outputs. If the system is subject to stochastic disturbance, this mathematical model can only predict an approximation $\widehat{y}\left(t\right)$, as shown in Formula (1):

$$\widehat{y}\left(t\right)=\tilde{g}\left(y\left(t-1\right),u\left(t-1\right),y\left(t-2\right),u\left(t-2\right),\dots \right).$$

(1)

This model is nonlinear if the function $\tilde{g}$ is nonlinear. Various different methods are available for modeling such systems [6].

2. The Nonlinear ID Methods

For nonlinear ID, several different methods are available, such as Nonlinear AutoRegressive Moving Average with eXogenous inputs (NARMAX) models, nonlinear state-space models and Wiener and Hammerstein models.

NARMAX is a mathematical framework and modeling approach used to describe and analyze dynamic systems, particularly nonlinear systems, by representing the relationships between input, output, and exogenous variables [7].

The approximation of nonlinear systems by linear dynamics and static nonlinearities has long been, and still is, an area of research. Wiener and Hammerstein models are block-oriented nonlinear methods [8]. The following is an early result following Wiener’s lead [9].

Suppose $N$ is a nonlinear (so-called fading memory) operator $y=Nu$. Then there exists an I/O operator $\widehat{N}$ which is realizable in the form

$$\dot{x}=Ax+bu, y=p\left(x\right),$$

(2)

where $A$ is an exponentially stable matrix of dimension $M$ and $p\left(x\right)$ is a $R^M\to R$ polynom. The above theorem does not say anything about the dimension, the definition of which is still under research.

We can identify the linear model $\widehat{N}$ from the data as the model that approximates the system. One of the most common forms (especially in practice) is the bilinear model

$$\dot{z}=Ez+Fzu+Gu, y=Hz$$

(3)

and the so-called block-oriented models. Bilinear time-series models have been studied and parameter estimation have been evaluated e.g., [10]. The Wiener–Hammerstein models (see Figure 1) are the most common, if only because of their availability in the MATLAB ID Toolbox routines.

The general form of these (discrete-time models) is

$$y_t=N_W\left(z_t,\beta \right)+{\mu }_t,$$

(4)

$$z_t=G\left(q,\zeta \right)w_t+{\nu }_t,$$

(5)

$$w_t=N_H\left(u_t,\alpha \right),$$

(6)

where $q$ is the shift operator, $\mathsf{\theta }=\left[\mathsf{\beta },\mathsf{\zeta },\mathsf{\alpha }\right]$ are the model parameters, $N_W,N_H$ are static nonlinearities, and ${\mathsf{\mu }}_t,{\mathsf{\nu }}_t$ are noise processes with zero expected value and constant variance. The ${\displaystyle \frac{\partial }{\partial \mathsf{\alpha }}}{N}_H\left(\cdot ,\mathsf{\alpha }\right)$, ${\displaystyle \frac{d}{\partial \mathsf{\beta }}}{N}_W\left(\cdot ,\mathsf{\beta }\right)$ derivatives are assumed to exist, but need not necessarily be invertible.

Estimation of $\mathsf{\theta }$ parameters for linear models can be done using Prediction Error (PE) and Maximum Likelihood (ML) methods. The most general description can be found in [11,12].

Subspace ID is a system ID method that can also be used to estimate the parameters of Linear Time-Invariant (LTI) and Wiener–Hammerstein models from input and output data. It is a powerful technique for modeling and understanding dynamic systems, particularly in control theory, signal processing, and structural dynamics. Subspace ID methods are often used to extract a model directly from observed data [13,14].

3. The Test Vehicle and the Experiment Conditions

The experimental test vehicle is a Nissan Leaf with full autonomous driving capability. The adaptation of the test vehicle and its sensor system involved collaborative work by professionals from the Vehicle Industry Research Center at Széchenyi István University. As part of the vehicle modification process, they installed a safety system, both upper and lower-level control systems, a data acquisition system for measuring data, and a sensor system.

In the experiments, the vehicle was forced to drive in a straight line at various constant vehicle speeds using the autonomous function. To ensure proper excitation of the dynamic system, the vehicle’s steering control was excited by an artificial disturbance signal. The input to the dynamics system was therefore the sum of the control signal sent by the autonomous function to the steering servo and the artificial disturbance signal, and the output was the lateral acceleration of the vehicle. We chose lateral acceleration as the output of the system because it has the biggest impact on ride comfort. The experiments were conducted at a uniform sampling frequency of 0.02 Hz.

The vehicle’s control system comprises both a lower-level and an upper-level control system. The upper-level control ensures the path’s tracking defined by the autonomous function, and the reference signal of lower-level control is the vehicle’s steering angle. A visual representation of the vehicle control system is shown in Figure 2.

In Figure 2. HLC is the high-level control, LLC is the low-level control, NL is the nonlinearity of the steering servo, Servo is the steering servo, VD is the vehicle dynamic, Path is the vehicle’s route, PRBS is the artificial excitation, ${\mathsf{\delta }}_{ref}$ is the steering angle reference, $v_{ref}$ is the voltage reference, $v_{in}$ is the input voltage of steering servo, $\mathsf{\delta }$ is the steering angle, $x, y$ is the vehicle position, $\mathsf{\psi }$ is the vehicle heading, $\ddot{y}$ is the lateral acceleration, and $\stackrel{.}{\mathsf{\psi }}$ is the yaw rate.

The data collection necessary for characterizing the lateral behavior of the test vehicle was conducted at the ZalaZone testing facility [15]. The vehicle was directed along predetermined straight paths at different, yet constant velocities. A pre-existing two-level control system guarantees autonomous path following and speed regulation.

To perform system ID, the system undergoes excitation using a filtered Pseudo Random Binary Sequence (PRBS). The Auto Power Spectral Density (APSD) of the PRBS remains consistent within a specific frequency range, ensuring a signal with maximum energy while adhering to amplitude constraints. This external excitation signal is essential because without it, the vehicle’s dynamics would not be properly excited, making system ID infeasible. Figure 3 shows a visual representation of the PRBS in both the time and frequency domains.

The system’s dominant frequency range extends up to approximately 0.16 Hz. It is important for trajectory control to perform effectively within this range while also maintaining robustness for higher-frequency model uncertainties. Therefore, a PRBS bandwidth of 1.6 Hz was sufficient for identifying the necessary models. The steering servo control operates with a maximum voltage signal of 0.5 V amplitude. Based on preliminary measurements, an artificial excitation of 0.3 V amplitude was found to be necessary for proper system excitation, hence the PRBS amplitude was set to 0.3 V.

4. Nonlinear ID Results

This section focuses on the outcomes of nonlinear model ID with Wiener and Hammerstein structures. The full procedure for estimating the model structure is not detailed; this section focuses on the characteristics of the best performing models. These models are characterized as statistical model validation and are examined in relation to the vehicle’s speed. It is known that lateral vehicle models depend on the longitudinal velocity. This would make the application of LPV models necessary. In our experiment, the gridding approach was applied and several LTI Wiener and Hammerstein models were identified. The velocity dependence was analyzed by computing the static gains and the pole location maps. The Matlab System Identification Toolbox was used for the calculations [16].

The models were evaluated in the time domain (correlation function of residual series), AIC (Akaike Information Criterion), BIC (Bayesian Information Criterion), FPE (Final Prediction Error), parameter space (T-tests on the estimated parameters for hypothesis $E\widehat{{\mathsf{\theta }}_i}=0$, using the parameter’s covariance matrix), pole-zero maps, and the frequency domain (by comparing the APSD estimates calculated from the data and the Bode diagrams of the frequency functions calculated from the models). The FPE and BIC values are given by the following equations:

$$FPE=MSE{\displaystyle \frac{\left(1+\raisebox{1ex}{$n_p$}\!\left/ \!\raisebox{-1ex}{$n$}\right.\right)}{\left(1-\raisebox{1ex}{$n_p$}\!\left/ \!\raisebox{-1ex}{$n$}\right.\right)}},$$

(7)

$$BIC=n·\ln \left(MSE\right)+n\left(n_y\ln \left(2\mathsf{\pi }\right)+1\right)+n_p·\ln \left(n\right),$$

(8)

where $MSE=\raisebox{1ex}{${\mathsf{\Sigma }}_{i=1}^n{\left(y_i-\widehat{y_i}\right)}^2$}\!\left/ \!\raisebox{-1ex}{$n$}\right.$, $n$ is the number of samples, $y_i$ is the measured output, $\widehat{y_i}$ is the estimated output, $n_p$ is the number of estimated model parameters, and $n_y$ is the number of output vectors ($n_y=1$). Since the one-sample T-test of the hypothesis $E\widehat{{\mathsf{\theta }}_i}=0$ and the element number of the sample is 1, the T-value is the estimated parameter divided by its standard deviation. Based on literature recommendations, parameters are considered appropriate if they are at least 3–4 times as large as their deviation.

The Wiener and Hammerstein models were examined separately, and in each case polynomial nonlinearities were assumed. This type of model assumes nonlinearity at the input. In the calculations, second- and third-order polynomial nonlinearities were assumed. The linear part of the Hammerstein IDs were OE (Output Error) models. Again, the reference signal for the ID was the excited voltage signal applied to the steering servo and the output was the lateral acceleration. The IDs were performed using a base sampling frequency of 50 Hz and resampled data at 10 Hz, and with second- and third-order assumed polynomial nonlinearities. In the parameter estimations, the best T-test results were obtained for the second-order linear models with third-order polynomial nonlinearization. The pole-zero maps, the static gains and the BIC and FPE values of the Hammerstein models are shown in the Figure 4.

The Wiener models are very similar to the Hammerstein model, with the difference that the static nonlinearization is applied to the output of the linear core. The poles, zeros and other statistical measures obtained with this model are similar. The T-test results are worse in the high-speed tests, so the T-test results for the identification of the highest-speed measurements (28.8 km/h) are presented in

Table 1. T-values based on the identification of measurements at 28.8 km/h.

Parameter	Hl = 2, nl = 2	Hl = 2, nl = 3	Hl = 3, nl = 2	Hl = 3, nl = 3	Wl = 2, nl = 2	Wl = 2, nl = 3	Wl = 3, nl = 2	Wl = 3, nl = 3
Nominator parameters	–56.1457	–7.1443	–0.8480	–3.0824	–87.6743	–29.4370	–2.9308	–0.7883
			0.1229	1.5652			–0.1977	0.4430
Denominator parameters	–177.4374	–346.7861	–0.4372	–90.8882	–76.8269	–241.9313	–1.2948	–3.6107
	97.6223	184.1067	–0.1123	–79.7580	40.6095	125.5478	–0.2668	–2.4565
			0.3071	113.1682			0.8325	4.6002
Polynomial parameters	11.7585	9.9925	0.8462	1.9901	5.5570	4.8526	1.0156	0.1965
	10.7016	–7.3557	0.8485	–1.9397	10.1343	–4.7426	2.0357	–0.2958
	–11.8026	–8.9740	–0.8447	–1.7691	–38.9045	1.8464	–38.1026	0.5763
		–9.1832		–1.9803		–32.4591		–33.4737

In the table header, l is the degree of the linear model, and nl is the degree of the polynomial nonlinearity.

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5. Conclusions

Based on the results presented, it can be concluded that there is no significant speed dependence of the identified dynamics in the speed range below 30 km/h. The only speed-dependent component of ID models is that of the gains with increasing speed. Of course, the gains presented here are only the gains of the internal linear OE model. At higher speeds, the heeling angle of the vehicle body depends on lateral acceleration. For this reason, the accelerometer sensor data at higher speeds showed a higher proportion of the effect of the G force due to the smaller than 90° angle between the sensor and the ground. The observation that the lateral dynamics of the vehicle are not speed-dependent at low speeds will help to develop autonomous vehicle functions that require low-speed driving, e.g., parking, obstacle avoidance in urban environments, etc.