Algebraic Driver Steering Model Parameter Identification

Driver steering models describe the steering behavior of a human driver to follow a given path. They have been extensively studied since the 1970s [1–7] and are playing an increasingly important role in automotive dynamics and control applications. In particular, understanding individuals' steering characteristics enables advanced driver assistance systems (ADAS), e.g., lane-keeping assistance systems, to adapt to specific drivers, thereby serving each individual in a customizable manner [8,9]. Driver-adaptive ADAS can enhance driving safety, mitigate human-machine conflict, and improve user trust in the system [10,11].

Driver-adaptive ADAS requires identifying the parameters inside a driver steering model in real-time to account for driving characteristics variations caused by weather, lighting, road, or driver physiological conditions [12–14]. The general approaches for identifying a driver steering model's parameters online fall into the family of recursive least squares (RLS) and Kalman filter. For instance, authors in Refs. [14–16] utilize RLS to identify driver arm's viscoelastic properties, e.g., inertia, damping, and stiffness. Meanwhile, RLS is employed in Ref. [17] to identify the feedback steering gains of a driver model. In parallel, authors in Refs. [12] and [18] use extended Kalman filter and unscented Kalman filter to estimate the driver model's preview time, steering gains, and time lag. Albeit effective, the estimated parameters may demonstrate slow convergence due to their asymptotical nature [19,20].

Parameter identification can also be achieved from a differential algebra perspective. Unknown parameters can be identified if and only if they can be expressed as a function of system inputs, measurable outputs, and their finite order derivatives [21–23]. To illustrate this algebraic strategy, we adopt a driver steering model [24] with four parameters, i.e., feedback steering gain, feedforward steering gain, preview time, and first-order neuromuscular lag, as an example. We first perform a series of algebraic manipulations on the given model, which eventually express the model parameters in a linearly identifiable form. Then, we formulate a least-squares (LS) problem to solve the linear equation and analytically identify the parameters. Finally, we can periodically reset and solve the LS problem if time-varying parameters need to be updated online.

To verify the proposed algebraic approach, we identify the parameters with two sets of drivers' steering wheel angle data. In the first group, synthetic data, i.e., data generated from the driver steering model itself, are utilized to demonstrate the superior performance of the algebraic identifier over an RLS identifier in identifying constant parameters. In the second group, measurement data from human subject driving simulator experiments are used to illustrate the capability of the proposed algebraic approach in identifying the time-varying feedback and feedforward steering gains.

The rest of the paper is organized as follows: Sec. 2 illustrates the driver steering model under investigation. Section 3 describes the measurement data collected from human subject driving simulator experiments and then states how to create the synthetic data of driver steering wheel angle from the actual measurements. Section 4 illustrates how to algebraically identify all the parameters in the driver steering model. Using the synthetic data of driver steering wheel angle, we show the superior performance of the algebraic method over RLS. Afterward, Sec. 5 demonstrates how to algebraically estimate the time-varying feedback and feedforward steering gains online with the actual measurement data. Fitted steering wheel angles with offline-identified constant steering gains or the online-identified time-varying steering gains are contrasted. Finally, Sec. 6 concludes this paper.

The contributions of this paper are threefold. First, we explain how to algebraically identify the constant parameters inside a typical driver steering model. Second, we demonstrate the advantage of the proposed algebraic identifier over an RLS identifier in identifying constant parameters. Third, we exhibit the capability of the algebraic identifier in estimating time-varying steering gains.

The driver steering model under analysis comes from the authors' previous work [24]. As demonstrated in Fig. 1, it consists of a feedback term to compensate for the previewed path tracking error and a feedforward term to adapt driver steering maneuver per anticipated road curvature.

where $Y(t)$ is the current vehicle position.

where $Kff$ is the feedforward steering gain and $γd(t)$ represents the desired yaw rate at the previewed distance $L.$ Both $Yd(t)$ and $γd(t)$ are determined by the road geometry.

In Eq. (5), there are in total four parameters to identify, as: $Gh,$$Th,$$Tp,$ and $Kff.$

We adopt Eq. (5) as a benchmark to illustrate the algebraic approach for two reasons. First, the authors have validated its performance in fitting steering maneuvers of human subjects with different driving experiences. Second, vehicle dynamics states, e.g., $Y(t),$$vx(t),$$ψ(t)$ and road geometries $Yd(t),$$γd(t)$ from the human subject driving simulator experiments are internally accessible, which facilitates validation of the algebraic method.

Remark 1. The proposed algebraic parameter identification framework can be applied to a class of driver steering models whose parameters can be expressed in a linearly identifiable form. Several other noteworthy examples include [25–28], to name a few. Instead, highly nonlinear driver steering models based on machine-learning techniques, such as Refs. [29] and [30], are not suitable to analyze under the proposed algebraic framework.

In this section, we first describe the measurement data from the driving simulator experiments in Ref. [24]. Then, we explain how to derive the synthetic data of the driver's steering wheel angle based on the actual measurements.

Ten volunteers were recruited to operate a stationary driving simulator on a virtual urban track. The track is composed of two curves with different radii and angles of curvature. For smooth transitions, both curves were extended at the beginning and end with straight lines. Once the test kicked off, a driver was guided to follow the path centerline with a constant speed of 40 mph. The actual driver steering wheel angle $δswmes(t)$ vehicle coordinates $X(t),$$Y(t),$ yaw angle $ψ(t),$ longitudinal and lateral speeds $vx(t),$$vy(t)$ and path geometry information $Yd(t)$ and $γd(t)$ were recorded every 0.01 s. The recorded datasets were split for each driver per the two curves, summarized in Table 1.

With the real measurement data, the authors calibrated the model (5) by identifying the best-fitted constant parameters: $Gh,$$Th,$$Tp,$ and $Kff$ for each driver in both scenarios. Details of model parameter calibration, e.g., the measurement of performance, the goodness of fit, and the numerical optimization package, can be found in Ref. [24]. Finally, there existed four constant parameters per driver and scenario. Note that the calibration via numerical optimization is computationally expensive, which is not suitable for online execution. The synthetic driver steering data $δswsyn(t)$ that is, data generated from the driver steering model itself, can be deduced by substituting the best-fitted constant parameters and the recorded vehicle states and path geometries back into Eq. (5). For example, we present the measured and the synthetic driver steering wheel angle from the dataset Driver1_Curve I in Fig. 2. Note that because the synthetic steering wheel angle $δswsyn(t)$ comes from simulation, it is not contaminated by the measurement noise.

This section illustrates how to algebraically identify the four parameters: $Gh,$$Th,$$Tp,$ and $Kff$ inside Eq. (5), assuming they remain constant. The synthetic data of steering wheel angle from Sec. 3 are used to validate the algebraic method. Using synthetic data allows us to gauge the closeness between the identified parameters and their “true” values [31].

The workflow is illustrated in Fig. 3, where the blocks with back-slashes are executed in the time domain, whereas the blocks with vertical lines are executed in the frequency domain.

where we assign the nonlinear term $vx(t)sin ψ(t)$ as $θ(t).$

The terms $f(s)=L(f(t))$ in Eq. (7) indicate the Laplace transform of the corresponding terms $f(t)$ in Eq. (6). Note that the initial conditions $δsw(0)$ and $γd(0)$ are in practice, unknown.

Comparing Eqs. (9) and (8), we observe that all the derivate operators $s$ in Eq. (8) are removed. This step is crucial because derivative operators would amplify the measurement noises of the steering wheel angle in the time domain. Furthermore, we multiply $s−2,$ instead of $s−1,$ on both sides of Eq. (8). In this way, all the measured variables are integrated at least once after we return (9) from the frequency domain to the time domain. The (iterated) integrations serve as low-pass filters to wipe out high-frequency measurement noises [32,33]. Equation (9) corresponds to block 4 and block 5 in Fig. 3.

and $Θ=[Th,Gh,GhTp,Kff,KffTh]T.$ The initial conditions in Eqs. (12) and (13) are zero.

Note that in Eq. (13), we use $∫(n)f(t)$ to represent the iterated integrals $∫0t∫0σ1⋯∫0σn−1f(σn)dσn⋯dσ1.$ If $n=1,$ we simplify $∫(1)f(t)$ as $∫f(t).$

Once $Θ*$ in Eq. (18) becomes available, we can finally obtain $T̂h=Θ*(1),$$Ĝh=Θ*(2),$$T̂p=Θ*(3)/Θ*(2),$ and $K̂ff=Θ*(4).$ Note that Eq. (18) is derived from a series of algebraic operations performed on the driver steering model (5) itself. Therefore, it yields exact formulations of the model parameters. Unlike Kalman filter [34] or RLS [35], the proposed algebraic parameter identification method does not depend on the Lyapunov stability theory [36], and the identification process does not possess an asymptotic convergence stage. In fact, $Θ*$ can be directly determined if and only if the matrix $MPP$ in Eq. (19) is invertible. At $t=0,$ the matrix $MPP$ is singular as all its elements are zero. Consequently, the identified parameter vector $Θ*$ would demonstrate strong oscillation immediately after $t=0.$ However, the numerical conditions of the matrix $MPP$ will improve as time goes by because the matrix $MPP$ is positive semidefinite. Therefore, after a short transient period, the parameter vector $Θ*$ can be obtained [37]. The duration of the transient period after which $Θ*$ can be reliably identified depends on several factors, such as the machine precision, driver maneuver magnitude, and signal-to-noise ratio [20]. Section 5 introduces the convergence check criterion to determine if the identification results can be regarded as reliable.

Remark 2. In addition to $t=0,$ there also exist several trivial cases such that the matrix $MPP$ will remain singular. For instance

Case (a) $δsw(t)≡0,$ which implies that there is no steering maneuver from the driver.

Case (b) $vx(t)≡0,$ which would yield $θ(t)≡0$ and $p3(t)≡0.$

Case (c) The desired yaw rate $γd(t)≡0,$ implying the driver is driving on a straight road.

If $MPP$ remains singular for a long time, the estimated vector $Θ*$ would become highly fluctuating and unreliable. Therefore, it is suggested to disable the algebraic identifier if the driver is driving along a straight path with negligible steering angle input.

In the following, we demonstrate the effectiveness of the algebraic identifier and compare its performance with a standard RLS identifier [35]. We adopt the RLS module from matlab system identification toolbox. Both the algebraic (ALG) and the RLS identifiers were implemented on a real-time computing platform dspacescalexio, and the sampling period was fixed as $Ts=1 ms$⁠.

Before presenting the parameter identification result, we first show in Fig. 4 the task turnaround times (TATs) of the two identifiers. The red dashed line corresponds to the TAT entailed from the RLS identifier, and the blue solid line shows the counterpart of the algebraic identifier.

As shown in Fig. 4, both the ALG and the RLS identifiers can be executed in real-time, as their TATs are much lower than the sampling period. Interestingly, the average TAT of the proposed algebraic identifier seems smaller than RLS's. The small TAT of ALG is expected because the algebraic identifier only involves computationally lightweight algebraic operations, without iterative online optimization.

Then, as examples, Figs. 5–8 present the constant parameter identification results of driver 3, driver 6, diver 7, driver 9 on curves I and II, respectively. The synthetic datasets for generating Figs. 5–8 were randomly selected for demonstration purposes.

In Figs. 5–8, the dash-dotted lines termed “truth” represent the constant parameters identified from the offline model calibration. The dashed lines and the solid lines correspond to the identification results of RLS and the proposed algebraic method, respectively. We can observe that after a short period of fluctuation, the algebraic approach yields accurately identified parameters. Moreover, the algebraic identifier does not maintain an asymptotical convergence phase as RLS does.

To comprehensively reflect the enhanced identification speed of the algebraic approach over RLS, we introduce the concept of estimation period. The estimation period $τest(α),$$α=Th,$T_p$Gh,$ or $Kff$ is defined, similar to the “settling time” in control systems, as the time elapse from $t=0$ to a moment, after which the estimated value $α̂(t)$ always lays within $99%−101%$ of the true value of $α.$ We summarize the medians of $τest(α),$$α=Th,$$Tp,$$Gh,$$Kff$ of the two methods for all the twenty groups of synthetic datasets in Table 2. We calculate the median instead of the mean to avoid the extremum cases that the RLS can never yield accurate estimation within $99%−101%$ of the true values.

According to Table 2, we can conclude that the algebraic identifier substantially improves the estimation speed over a standard RLS identifier.

Section 4 illustrates how to algebraically identify constant parameters inside the driver steering model (5). Experimental results using the synthetic data demonstrate its superior performance over RLS. However, we cannot directly apply the identifier in Sec. 4 to treat the actual measurement data. First, a driver in reality only maintains a static steering pattern during a very short time interval [19]. Consequently, we cannot assume that the parameters always remain constant. Second, a driver steering model is only an approximation of the complex driver maneuver. In other words, modeling error always exists, as demonstrated in Fig. 2.

To estimate the time-varying parameters under the influence of modeling errors, we modify the algebraic identifier in Sec. 4 from two perspectives. First, we include a piecewise constant in Eq. (5) to represent the condensed modeling error [22]. Second, we periodically reset the algebraic identifier to update the identified parameters online.

In contrast to Sec. 4 where we identified all the driver steering model parameters in Eq. (5), we only identify two out of the four parameters in this section. As shown in Fig. 2, compared to the synthetic data, the measurement data were contaminated by noises, which reduce the signal-to-noise (SNR) ratio. With a decreased SNR ratio, the matrix $MPP$ in Eq. (19) would fall into the singularity from time to time. Simultaneously estimating fewer parameters allows using signals with a lower level of SNR [38]. We will identify the time-varying feedback and feedforward steering gains with the real measurement data. In the meantime, each driver's offline identified neuromuscular lag $Thoff$ and preview time $Tpoff$ will be substituted into the driver steering model (5) as known parameters. Choosing $Gh$ and $Kff$ to identify comes from the following two reasons. First, the steering gains are the principal adaptive elements of a driver [1]. Field test results in Ref. [18] also suggest that the steering gains are the most important parameters to identify online. Second, sensitivity analysis in Ref. [39] reveals that when driving on a curved road like our cases, the steering gains and preview time are among the most sensitive parameters for fitting the steering wheel angle. However, accurate estimation of the preview-time would require extra hardware, such as an eye-tracker [40].

Similar to Fig. 3, we illustrate in Fig. 9 the workflow to identify the time-varying steering gains. All the blocks in Fig. 9 are executed in the time domain.

where $Ts=1 ms$ is the sampling period.

Again, the integrals in Eqs. (25) and (26) play the role of a low-pass filter to wipe out high-frequency measurement noises in the time domain. The selection of $Tw$ trades-off between the filtering effect and the estimation bias. A larger $Tw$ can mitigate the measurement noise but also increase the estimation bias from the truncation error of the Taylor series. In practice, $Tw$ is typically selected as an integral multiple of the sampling period $Ts$ to facilitate the digital implementation [32].

and $Θgain=[Gh,Kff]T.$

Remark 3. Similar to Eq. (18), the matrix $∫0tPgain(τ)PgainT(τ)dτ$ will also remain singular if the driver is driving on a straight road with $γd(t)≡0,$ which makes $p2gain(t)≡0$ Again, it is suggested to disable the algebraic identifier in this case.

At each resetting moment $tk,$ all the elements in the matrix $∫tktk+1Pgain(τ)PgainT(τ)dτ$ and the vector $∫tktk+1Pgain(τ)qgain(τ)dτ$ are reinitialized from zero. Therefore, the matrix $∫tktk+1Pgain(τ)PgainT(τ)dτ$ becomes singular at each resetting moment. Consequently, we need to wait a short period of time $ε$ after each resetting moment, so as to avoid outputting fluctuating and unreliable parameter identifications (see Figs. 5–8). Several time-varying factors [20], like SNR and driver maneuver magnitude, can influence the convergence period $ε.$ The convergence check criterion in Refs. [20] and [42] (block 6 in Fig. 9) is applied to determine if the identification results after each resetting are reliable.

In Eq. (35), $E(α)$ is the moving-averaged identified parameter $α,$$α∈[Gh,Kff].$ It is calculated as $E(α)=∫t−Tavgtα(τ)dτ/Tavg$ with $Tavg$ being the window length. Then, $σ(α)$ is the standard deviation of the identified parameter $α,$ as: $σ(α)=E(α2)−E(α)2.$ Finally, $Δα$ is the convergence check threshold. The selection of $Δα$ trades off between the estimation accuracy and the estimation period. A smaller $Δα$ may yield more accurate estimation results but a longer estimation period, and vice versa. When negotiating a gentle curve, the feedback compensatory maneuver plays a major role [2]. Therefore, we assign a smaller threshold to $Gh$ and a larger one to $Kff$ as $ΔGh=0.01,$$ΔKff=0.1.$

After each resetting moment $tk,$ we monitor the fluctuation metric $|σ(α)/E(α)|.$ Once Eq. (35) is met, the moving-averaged $E(α)$ is outputted as $Θgain(k)$ at the current step k. On the contrary, if Eq. (35) remains unsatisfied even at the next resetting moment $tk+1,$ which implies that the estimation fails to converge within the current step k, then the estimation result when (35) was most recently met would be used as the identification output for the current step [38]. In other words, even though the “resetting” is periodically triggered, the update of the identified parameters is indeed event-triggered. Intuitively, the resetting period $Treset$ in Eq. (34) cannot be too short, as the estimation after each resetting requires a small amount of time to converge and make (35) satisfied. Meanwhile, $Treset$ cannot be too long either because this may lead to untimely parameter updates. In the following, we set $Treset=0.1 s$

In Fig. 10, we first compare the directly differentiated steering wheel angle and the algebraically estimated one from Eq. (25).

Clearly, directly differentiating the recorded steering wheel angle would amplify the measurement noise. In contrast, the integral inside Eq. (25) yields an accurate yet noise-attenuated result, which enhances the SNR of system input for the algebraic identifier.

The convergence check criterion (35) is then illustrated in Fig. 11. We use the measurement data from the dataset Driver2_ Curve II as an example.

From Fig. 11, we can observe that at each resetting moment, e.g., 3.6 s, 3.7 s, the fluctuation metrics of $Gh$ and $Kff$ become quite large. As we mentioned, this is caused by the singularity of the matrix $∫tktk+1Pgain(τ)PgainT(τ)dτ$ in Eq. (31). However, after a short period, the fluctuation metrics decrease below their respective thresholds: $ΔGh=0.01,$$ΔKff=0.1,$ which yields a timely update of the identified steering gains within each resetting step.

Following Fig. 11, we demonstrate the functionality of the convergence check criterion (35) in Fig. 12.

In Fig. 12, the solid lines represent the steering gains identified from Eq. (33). At each resetting moment, this raw identification produces a spike. This spike comes exactly from the singularity of the matrix (31). With the convergence check criterion, the estimation result of the last resetting step k-1 remains unchanged until (35) is triggered in the current resetting step k. Similarly, the steering gains identified at the current step k will remain unchanged until/if (35) is triggered again in the next resetting step k + 1. The dashed lines in Fig. 12 depict the finally identified piecewise constant steering gains without spike.

Afterward, in Fig. 13, we show the algebraically identified steering gains $Gh$ and $Kff$ of each driver tested on both curves in Table 1.

The overall trend of the two steering gains follows the same pattern: At the beginning, they increase from zero to certain values. Then, they fluctuate around. Finally, they decrease back toward zero. This trend is in line with the actual driving simulator experiments. At the beginning of the test, a human subject cruised along a straight entrance line with negligible steering maneuver. As the curve was on the horizon, the driver should gradually increase the steering gains to prepare for a smooth turning maneuver. When negotiating the curve at a constant cruise speed, the driver should somehow maintain steady steering gains. Finally, a straight exit line appeared at the end of the curve negotiation, which triggered the driver to gradually neutralize the steering wheel angle and the steering gains.

Several noteworthy phenomena in Fig. 13 are listed below. First, the feedback steering gain $Gh$ has a larger magnitude than the feedforward steering gain $Kff,$ which is consistent with the offline optimized results in Figs. 5–8. Second, for the short curve I with a smaller radius (150 m) and angle of curvature (45 deg), the identified steering gains exhibit strong oscillations after around 8 s. While for the long curve II with a larger radius (180 m) and angle of curvature (60 deg), the oscillations occur after approximately 12 s. As we explained in Sec. 2, both curves were connected with a straight exit line. Therefore, at the end of each test (approximately 8 s for the short curve and 12 s for the long curve), a human subject was either already driving on the straight exit line or preparing to stabilize the steering wheel to its neutral position. Under this circumstance, the actual steering wheel angle $δsw(t),$ the lateral position tracking error $Yd(t)−Yp(t),$ and the desired yaw rate $γd(t)$ should all be close to zero. As a result, the numerical condition of the matrix in Eq. (31) substantially degraded, which caused chattering and unreliable identification results. A possible remedy is to temporarily stop online identification when a driver's steering maneuver remains negligible. Thirdly, comparing the blue solid line with the red dashed line, we can observe that the feedback gain $Gh$ of at least five amid the ten drivers noticeably increased when they followed the curve with a more significant curvature (smaller radius). This finding agrees with Ref. [25] that the compensatory steering gain considering road geometry would increase if the curve becomes sharper.

To verify the identified feedback and feedforward steering gains, we contrast the fitted steering wheel angles with offline optimized constant steering gains or the online-identified time-varying steering gains. The results are compared in Fig. 14.

In Fig. 14, the blue solid line corresponds to the measured steering wheel angle $δswmes(t)$ The red dash-dotted line and the green dashed line indicate the fitted steering wheel angle $δ̂swfit(t)$ with either the constant steering gains from offline optimization or the algebraically identified time-varying steering gains. We can observe that the fitted steering wheel angle can better match the actual measurement with the time-varying steering gains identified online.

Finally, we compare the RMS of steering wheel angle's fitting errors in Fig. 15.

Therefore, using the online updated steering gains can reduce the fitting error of the steering wheel angle by more than 50%.

This paper describes an algebraic strategy to identify the parameters of a driver steering model. This algebraic method can yield fast parameter identification without an asymptotical convergence phase. Datasets from driving simulator experiments are used to validate the proposed strategy. We demonstrate its superior performance over an RLS identifier in identifying constant parameters and its ability to identify the time-varying feedback and feedforward steering gains. Applying the algebraic approach to identify driver steering model's parameters under various weather, road surface, and traffic scenarios will be studied in the next step.

• National Science Foundation (Award No. 1901632; Funder ID: 10.13039/100000001).