A New Microscopic Traffic Model Using a Spring-Mass-Damper-Clutch System

TL;DR

This paper introduces a novel microscopic traffic model based on a spring-mass-damper-clutch system that captures vehicle interactions and can be scaled for macroscopic traffic analysis, with real-time parameter identification for vehicle control.

Contribution

The paper proposes a physically interpretable microscopic traffic model using a spring-mass-damper-clutch system and develops an efficient online parameter estimation algorithm, enhancing traffic flow understanding and vehicle control.

Findings

Model captures ego vehicle resistance and impact on lead vehicle.

The proposed model is scalable for macroscopic traffic analysis.

The PRLS-IQR algorithm enables real-time, stable parameter estimation.

Abstract

Microscopic traffic models describe how cars interact with their neighbors in an uninterrupted traffic flow and are frequently used for reference in advanced vehicle control design. In this paper, we propose a novel mechanical system inspired microscopic traffic model using a mass-spring-damper-clutch system. This model naturally captures the ego vehicle's resistance to large relative speed and deviation from a (driver and speed dependent) desired relative distance when following the lead vehicle. Comparing to existing car following (CF) models, this model offers physically interpretable insights on the underlying CF dynamics, and is able to characterize the impact of the ego vehicle on the lead vehicle, which is neglected in existing CF models. Thanks to the nonlinear wave propagation analysis techniques for mechanical systems, the proposed model therefore has great scalability so that multiple mass-spring-damper-clutch system can be chained to study the macroscopic traffic flow. We investigate the stability of the proposed model on the system parameters and the time delay using spectral element method. We also develop a parallel recursive least square with inverse QR decomposition (PRLS-IQR) algorithm to identify the model parameters online. These real-time estimated parameters can be used to predict the driving trajectory that can be incorporated in advanced vehicle longitudinal control systems for improved safety and fuel efficiency. The PRLS-IQR is computationally efficient and numerically stable so it is suitable for online implementation. The traffic model and the parameter identification algorithm are validated on both simulations and naturalistic driving data from multiple drivers. Promising performance is demonstrated.

Tables

Table 1. TABLE I : Parameters for vehicle following simulation.

$M_n\left[kg\right]$	$k_n\left[N/m\right]$	$c_n\left[N\cdot s/m\right]$	$s_n\left[s\right]$	${\tau }_n\left[s\right]$	$\Delta t\left[s\right]$
1000	100	500	5	0.5	0.1

Table 2. TABLE II : Prediction performance summary

Driver ID	# of trips	avg. RMSE	worst RMSE
1	9	0.28	0.33
2	3	0.32	0.42
3	5	0.26	0.35
4	3	0.46	0.52
5	2	0.42	0.48

Equations

$$a_n(t)=f(\Delta x(t-{\tau }_n),\Delta v_n(t-{\tau }_n),v_n(t-{\tau }_n)),$$

$$X_0(v_n)=\{\begin{array}{ll}{X}_{0,\min },& \text{mbox}if{v}_n<v_{n,1}\\ s_n\cdot v_n,& \text{mbox}if{v}_{n,1}\le v_n\le v_{n,2}\\ X_{0,\max },& \text{mbox}otherwise,\end{array}$$

$$\Delta {\dot{x}}_n(t)$$

$$M_n{\dot{v}}_n(t)$$

$$+c_n\Delta v_n(t-\tau ).$$

$${\dot{v}}_n(t)=k_n/M_n(\Delta x_n(t-\tau )-s{v}_n(t-\tau ))+c_n/M_n\Delta v_n(t-\tau ).$$

$${\dot{x}}_1(t)$$

$${\dot{x}}_2(t)$$

$$+c_n/M_n(u(t-\tau )-x_2(t-\tau )).$$

$$\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right]=\left[\begin{array}{cc}0& -1\\ 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1(t)\\ x_2(t)\end{array}\right]+\left[\begin{array}{cc}0& 0\\ \alpha & -(s\alpha +\beta )\end{array}\right]\left[\begin{array}{c}{x}_1(t-\tau )\\ x_2(t-\tau )\end{array}\right]$$

$$+\left[\begin{array}{c}u\\ \beta u\end{array}\right],\mathrm{or}\dot{\mathbf{x}}=\mathbf{Ax}(\mathrm{t})+\mathbf{Bx}(\mathrm{t}-\tau )+\mathbf{f}(\mathrm{t}),$$

$$\mathbf{x}(t)=\mathbf{\Phi }{\mathbf{c}}_m,\text{ and }\mathbf{x}(t-\tau )=\mathbf{\Phi }{\mathbf{c}}_{m-1},$$

$$(\dot{\mathbf{\Phi }}-\mathbf{A}\mathbf{\Phi }){\mathbf{c}}_m=\mathbf{B}\mathbf{\Phi }{\mathbf{c}}_{m-1}+\mathbf{ϵ},$$

$$\left(\underset{0}{\overset{\tau }{\int }}\mathbf{\Psi }\left(\dot{\mathbf{\Phi }}-\mathbf{A}\mathbf{\Phi }\right)dt\right){\mathbf{c}}_m=\left(\underset{0}{\overset{\tau }{\int }}\mathbf{\Psi }\mathbf{B}\mathbf{\Phi }dt\right){\mathbf{c}}_{m-1},$$

$${\mathbf{c}}_m$$

$$=\mathbf{\Gamma }{\mathbf{c}}_{m-1}.$$

$${\mathbf{x}}_{\mathrm{stst}}={\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]}_{\mathrm{stst}}=\frac{u}{\alpha }\left[\begin{array}{c}-1+\alpha s+\beta \\ \alpha \end{array}\right].$$

$$\frac{v_n(k)-v_n(k-1)}{\Delta t}$$

$$+c_n/M_n\Delta v_n(k-d),$$

$$[\Delta x_n(k-d)v_n(k-d)\Delta v_n(k-d)]\left[\begin{array}{c}{\alpha }_n(d)\\ {\beta }_n(d)\\ {\gamma }_n(d)\end{array}\right]$$

$$=\frac{v_n(k)-v_n(k-1)}{\Delta t}.$$

$$\underset{p_n(d)}{\min }{\Vert A_n(d)p_n(d)-B_n(d)\Vert }^2,$$

$$p_n^{\ast }(d)=(A_n^{\text{T}}(d)A_n(d){)}^{-1}{A}_n^{\text{T}}(d)B_n(d).$$

$$J(d,k+1)=(1-\alpha )J(d,k)+\alpha |y(k+1)-x(k+1)p_n^{\ast }(d)|,$$

$$d^{\ast }(k)=\arg \underset{d}{\min }J(d,k).$$

$$SNR=10{\log }_{10}\frac{{\sum }_{k=1}^N(y(k)-e(k){)}^2}{{\sum }_{k=1}^N{e}^2(k)}.$$

Full text

I Introduction

Traffic congestion has been one of the most prevalent and stubborn challenges in urban areas for decades, causing a spectrum of issues including wasted time and economic loss [1], elevated driver stress and frustration [2], and increased air pollution [3]. It is estimated that in 2017, traffic congestion costs U.S. more than $300 billion and drivers in big cities spent more than 100 hours in congestion [1]. To alleviate traffic congestion, various traffic control technologies have been proposed, including variable speed limits [4, 5], dynamic traffic light control [6, 7], and ramp metering [8, 9]. It is worth noting that these technologies all require accurate real-time traffic estimation and prediction. It is therefore of critical importance to have good understanding of the traffic flow to enable those traffic control systems.

As such, numerous traffic models have been proposed to investigate traffic characteristics and flow evolution. The traffic models are generally grouped into two categories, macroscopic and microscopic. Macroscopic models are concerned with the macroscopic traffic flow characteristics such as traffic density, average speed, and traffic volume [10]. These traffic models are inspired by continuum fluid flow theories and under different assumptions, they are further classified as kinematic models [11, 12, 13], dynamic models [14, 15], and lattice hydrodynamic models [16, 17]. On the other hand, microscopic traffic models are concerned with individual vehicles and study the local vehicle interactions in terms of speed, relative distance, and acceleration. Microscopic models can be further categorized as cellular automata (CA) models and car-following (CF) models where CA models are based on stochastic discrete event system with the ability to characterize the lane change behaviors [18, 19] while CF models study the ego vehicle’s interaction with its preceding vehicle in a single lane [20, 21]. CF models have great implications to the design of driving assistant systems such as adaptive cruise control [22] and is the focus of this paper.

The development of CF models can date back to the 1950s [20]. Among the many CF models, the arguably most well-known model is the Gazis-Herman-Rothery (GHR) model, which was developed by the General Motors research lab in the late 1950s [21]. The model is based on the hypothesis that the acceleration of the ego vehicle is proportional to the relative speed and inversely proportional to the relative distance, assessed at time $\tau$ earlier with $\tau$ being the delay due to reaction time. Parameters including the orders of the speed term and relative distance term, as well as a gain, were calibrated using data from wire-linked vehicles. Since then, many variants of GHR models have been developed, proposing different combinations of “optimal” parameters on various sets of experimental data [23, 24, 25]. Another class of widely-used CF models are the optimal velocity models (also referred to as Helly model), which considers a speed and/or acceleration dependent desired spacing and explicitly incorporates an error term [26]. Several variants have also been proposed and calibrated on different experimental datasets [27, 28]. Besides the above models, some other types of models are also proposed, including collision avoidance models [18, 19], psychophysical models [29, 30], and fuzzy logic-based models [31]. A comprehensive review of the CF models can be referred to [22]. Despite the many afore-mentioned CF models, the available relationships are still not rigorously understood and proven [22].

In this paper, we propose a new microscopic CF model, inspired by the mechanical mass-spring-damper-clutch system. There are natural similarities between the CF dynamics and the mass-spring-damper-clutch system: 1) the ego vehicle tends to accelerate when the relative distance to the lead vehicle is too large and tends to decelerate when the relative distance is too small, which resembles a mechanical spring between two masses; 2) the ego vehicle tends to follow a similar speed as the lead vehicle, resisting large speed difference. This phenomenon resembles a mechanical damper between two masses; 3) drivers tend to have delayed actions due to reaction time, which resembles a mechanical clutch whose engagement induces delays. Therefore, we propose a mass-spring-damper-clutch system to model the CF dynamics. In [32], a mass-spring system is proposed, which is oversimplified and neglects the delayed reaction and resistance to relative speed. With the mass-spring-damper-clutch model, we further conduct stability analysis on the time delays and the related parameters using spectral element method [33].

Real-time driving prediction has shown to be critical to improve fuel efficiency and road safety in advanced driving assistant systems [34]. In this study, we develop a parallel recursive least squares with inverse QR decomposition to identify the model parameters in real-time. The algorithm is very computationally efficient and numerically stable that is suitable for the use of real-time prediction [35]. We validate the parameter identification framework in both simulations and naturalistic driving data of three drivers. Promising performance is demonstrated.

The contributions of this paper include the following. First of all, we develop a novel mechanical system inspired mass-spring-damper-clutch system to model the CF dynamics. The new model incorporates the impacts of the ego vehicle on the lead vehicle and can be extended to estimate and predict macroscopic traffic flow with wave propagation techniques on chained mass-spring-damper systems. Secondly, we perform stability analysis on the proposed model using the spectral element method to determine the system parameter set that retains stability under different time delays. Last but not least, we develop a parallel recursive least square with inverse QR decomposition to estimate the model parameters in real time with great computational efficiency and numerical stability. Promising results are demonstrated both in simulations and on naturalistic driving data.

The remainder of this paper is organized as follows. In Section II, we present our mechanical system inspired CF model, followed by the stability analysis of the model in Section III. In Section IV, we present an online parameter identification algorithm using parallel recursive least squares with inverse QR decomposition. The validation of the parameter identification framework is presented in Section V, both in simulation and on experimental data. Finally, conclusion remarks are drawn in Section VII.

II Mechanical System Inspired CF Model

The car-following dynamics is illustrated in Figure 1(a), where vehicle $n$ follows vehicle $n-1$ in a single lane. By convention, we name vehicle $n-1$ the lead vehicle and vehicle $n$ the follow/ego vehicle. The speeds of the ego vehicle and the lead vehicle are $v_{n}$ and $v_{n-1}$, respectively. The relative distance (or range) between the two vehicles is denoted as $\Delta x_{n}$. A car following model is characterized in terms of the ego vehicle’s acceleration as a function of relative distance, vehicle speed, and relative vehicle speed [21]:

[TABLE]

where $a_{n}$ is the acceleration of the ego vehicle (vehicle $n$), $\tau_{n}$ is the delay due to driver reaction time and vehicle response time of the ego vehicle , and $\Delta v_{n}=v_{n-1}-v_{n}$ is the relative speed between the lead vehicle and the ego vehicle. Various mathematical models have been proposed to characterize the relationship [21, 23, 26, 28]. However, these models are mainly based on data regression and lack insights on the system dynamics [22].

In this paper, we propose a new mechanical system inspired CF model as shown in Figure 1(b), where vehicle $n$ and vehicle $n-1$ are, respectively, represented as rigid bodies with masses $M_{n}$ and $M_{n-1}$. The two masses are connected with a spring with stiffness $k_{n}$, a damper with damping coefficient $c_{n}$, and a clutch that induces time delay $\tau_{n}$. Note from the observation that drivers have different desired relative distances at different vehicle speeds [26, 27], the spring in the model has the following speed-dependent relaxation length $X_{0}(v_{n})$, which is illustrated in Figure 2.

[TABLE]

where $v_{n,1}$ and $v_{n,2}$ represent the lower and upper threshold points, respectively; and the slope is denoted as $s_{n}$.

The mass-spring-damper-clutch system naturally characterizes human driving when following a vehicle. First, drivers tend to resist large relative speed (positive or negative), which is captured by the damper that exerts forces responding to relative speed between the two vehicles. Second, drivers tend to follow a desired speed-dependent distance from the lead vehicle, which is captured by the spring with a speed-dependent relaxation length that exerts forces responding to deviations from the relaxation equilibrium. Third, the delay due to driver reaction time and vehicle response time is captured by the clutch which induces a time delay for engage and disengage.

Remark II.1**.**

Compared to existing CF models [21, 23, 24, 26, 27, 28], the proposed model offers several advantages. First, unlike existing models that are mainly derived from data regression, this mechanical system inspired model provides interpretable physical insights on the CF dynamics. Second, the proposed model can characterize the impact of the ego vehicle on the lead vehicle, i.e., the lead vehicle tends to accelerate (if not changing lane) if the ego vehicle stays too close. This phenomenon is neglected in existing models and therefore it will cause issues when chaining CF models to represent macroscopic traffic. Third, thanks to the wave propagation techniques in mechanical mass-spring systems, the proposed model has good scalability when the mass-spring-damper-clutch systems are chained together to model macroscopic traffic flow, e.g., study the impact of shock waves.

Based on the mass-spring-damper-clutch model and the Newton’s law, the equations of motion of the system can be written as:

[TABLE]

In a normal highway car-following case, i.e., $v_{n,1}\leq v_{n}\leq v_{n,2}$, the second equation in (3) becomes

[TABLE]

Defining $x_{1}=\Delta x_{n}$, $x_{2}=\Delta v_{n}$, and $u=v_{n-1}$, (3) can be written as

[TABLE]

In the following section, we investigate the stability of the model on the system parameters and time delay.

III Stability Analysis

In this section, we perform stability analysis of the proposed CF model in Section II. From (5) and introducing the new variables $\alpha=k_{n}/M_{n}$ and $\beta=c_{n}/M_{n}$, and setting the lead vehicle’s speed to be constant according to $u(t)=u(t-\tau)=u$, equation (5) can then be written as

[TABLE]

where $A$, $B$, and $f$ are corresponding matrices. Equation (6) is a Delay Differential Equation (DDE) with a constant point delay $\tau$. The state-space for these equations is typically taken as the space of continuous functions. Consequently, due to the infinite dimensional nature of this state-space the stability analysis of Eq. (6) is more difficult than its delay-free counterpart. Nonetheless, there are several methods available for the stability analysis of this problem including the semi-discretization method [36], Chebyshev polynomials [37], and the spectral element method (SEM) [33]. In this paper we will use the SEM approach due to its flexibility and efficiency [38].

The main idea of the SEM is to discretize the state space of Eq. (6) and then construct a dynamic map over one period where the length of this period for autonomous systems is typically taken to be the length of the time delay $\tau$. The eigenvalues of the resulting matrix that describes this dynamic map must be within the unit disc of the complex plan in order for the corresponding DDE to be stable. While convergence in the SEM can be obtained either by using multiple temporal elements ($h$-refinement) or by increasing the order of the interpolating polynomial ($p$-refinement), in this study we use one temporal element and only increase the order of the polynomials to achieve convergence.

Since the vector $\mathbf{f}$ in Eq. (6) only affects the steady state solution but does not affect the stability analysis, we drop it from the subsequent discussion. Let $\mathbf{T}=\{t_{i}\}_{i=1}^{n+1}$ be a set of $n+1$ distinct temporal mesh points on $[0,\tau]$, and let $\mathbf{c}_{m}$ and $\mathbf{c}_{m-1}$ be $2\times(n+1)$ vectors containing the values of the states $\mathbf{x}(t)$ and the delayed states $\mathbf{x}(t-\tau)$, respectively, evaluated on $\mathbf{T}$. We choose a barycentric Lagrange interpolation [37] to represent the states according to

[TABLE]

where $\boldsymbol{\Phi}=\boldsymbol{\Phi}(t)=(\boldsymbol{\phi}(t)\otimes\mathbf{I})$, and $\boldsymbol{\phi}(t)$ is the vector of barycentric Lagrange interpolating polynomials $\boldsymbol{\phi}(t)=[L_{1}(t),L_{2}(t),\ldots,L_{n+1}(t)]$, $\mathbf{I}$ is the $2\times 2$ identity matrix, while $\otimes$ is the Kronecker product. We now substitute the state approximations into the DDE to obtain

[TABLE]

where $\boldsymbol{\epsilon}$ is the vector of approximation errors. Let $\boldsymbol{\psi}=[\psi_{1}(t),\psi_{2}(t),\ldots,\psi_{n+1}(t)]$ be a vector of linearly independent test functions. This set of functions is then used in a Galerkin approach where the errors are required to be perpendicular to the space spanned by the set $\boldsymbol{\psi}$. The result is the $2(n+1)\times 2(n+1)$ system of equations

[TABLE]

where $\boldsymbol{\Psi}=(\boldsymbol{\psi}\otimes\mathbf{I})^{T}$. Note that the integrals in Eq. (9) are often difficult to evaluate analytically which necessitates using numerical integration as described in [33]. Equation (9) can then be used to construct a dynamic map $\boldsymbol{\Gamma}$ according to

[TABLE]

In order to ascertain the stability of Eq. (6), we examine the eigenvalues of $\boldsymbol{\Gamma}$: if all the eigenvalues are within a modulus of less than one in the complex plane then the system is asymptotically stable.

In this paper we used the SEM with a $100\times 100$ grid in the $(\alpha,\beta)$ plane where $\alpha\in[0.01,2]$, $\beta\in[0.01,8]$, while 6 equally spaced values of $\tau$ were considered in the range $\tau\in[0.2,2]$. The temporal mesh used consisted of $21$ Legendre-Gauss-Lobatto points which correspond to an interpolating polynomial of order $20$, while the trial functions were the shifted Legendre polynomials. Increasing the order of the interpolating polynomial beyond 20 did not change the results, thus indicating the convergence of the solution.

Figure 3 shows the stability diagram in the $(\alpha,\beta)$ plane for 6 different values of $\tau$. The shaded region under each curve is stable, while the area above the curves are unstable. It can be seen that the triangular stability regions shrink as the value of the delay parameter increases from $0.2$ to $2$. In order to show the resulting stable and unstable system responses two points were simulated in Figs. 3b and c using Matlab’s dde23 function and a value of $u=20$ in Eq. (6). The chosen stable point is $(\alpha=1,\beta=2,\tau=0.2)$, while the selected unstable point is $(\alpha=1.6,\beta=2,\tau=0.2)$. The history function used in the simulation was the perturbed steady state solution $\tilde{\mathbf{x}}=\mathbf{x}_{\rm stst}(1+0.10)$ where $\mathbf{x}_{\rm stst}$ is the constant steady state solution given by

[TABLE]

Figure 3b shows that the perturbed system goes back to the steady state solution as the system evolves. In contrast, the perturbed, unstable system of Fig. 3c deviates from the steady state solution and grows exponentially.

IV Online Parameter Identification

Vehicle speed prediction is essential in automated longitudinal control for improved fuel efficiency and safety [39]. Existing studies on cooperative cruise control generally assume that the dynamics of the preceding vehicles are available [40, 41]. While this assumption is valid for fully autonomous vehicle platoon, it does not hold if human drivers are involved. Therefore, it is important to accurately predict human driver maneuvers to provide a “comprehensive preview”. In this paper, we develop a framework to identify the parameters and predict vehicle speed changes online.

We first discretize (4) using the explicit Euler method with sampling time $\Delta t$, which gives

[TABLE]

where $d=\text{round}(\tau_{n}/\Delta t)$ is the corresponding delayed steps.

For a specific delay $d$, define $\alpha_{n}(d)=k_{n}/M_{n}$, $\beta_{n}(d)=-(k_{n}s_{n})/M_{n}$, and $\gamma_{n}(d)=c_{n}/M_{n}$, then (12) can be written as the following linear equation:

[TABLE]

Given a time series data of $K$ steps, $K>d$, and define the parameter vector $p_{n}(d)=[\alpha_{n}(d);\beta_{n}(d);\gamma_{n}(d)]$, then the parameters can be identified by solving the following least-square problem:

[TABLE]

where $A_{n}(d)=\begin{bmatrix}\Delta x_{n}(0)&v_{n}(0)&\Delta v_{n}(0)\\ \Delta x_{n}(1)&v_{n}(1)&\Delta v_{n}(1)\\ \vdots&\vdots&\vdots\\ \Delta x_{n}(N-d)&v_{n}(N-d)&\Delta v_{n}(N-d)\end{bmatrix}$ and $B_{n}(d)=\frac{1}{\Delta t}\cdot\begin{bmatrix}v_{n}(d)-v_{n}(d-1)\\ v_{n}(d+1)-v_{n}(d)\\ \cdots\\ v_{n}(N)-v_{n}(N-1)\end{bmatrix}$ are the data matrices of vehicle $n$. Note that for different delay $d$s, the data matrices are different, which leads to different identified parameters. In this paper, we consider the possible range of the delay $\tau_{n}\in\left[\tau_{\min},\tau_{\max}\right]$. With the sampling time $\Delta t$, the range of the discrete time delay is $d\in\{d_{\min},d_{\min}+1,\cdots,d_{\max}\}$, where $d_{\min}=round(\tau_{\min}/\Delta t)$ and $d_{\max}=round(\tau_{\max}/\Delta t)$.

It is straightforward to show that the optimal solution to (14) is

[TABLE]

Note that the sizes of matrices $A_{n}$ and $B_{n}$ increase as the data length grows, causing computational issues if implemented online. Therefore, recursive computation is needed. In this paper, we exploit a recursive least squares with inverse QR decomposition algorithm (RLS-IQR) for online identification, which has great numerical stability and computational efficiency [35]. Specifically, at each time step $k$, $k>d$, the algorithm takes in the input vector $x(k)=[\Delta x_{n}(k-d)\;v_{n}(k-d)\;\Delta v_{n}(k-d)]$ and output $y(k)=\frac{v_{n}(k)-v_{n}(k-1)}{\Delta t}$, and then update the parameters $p_{n}(d)$. The details of the update is shown in Algorithm 1. Since there are multiple possible delays, we run the RLS-IQR in parallel for each $d\in\{d_{\min},d_{\min}+1,\cdots,d_{\max}\}$. To determine the best delay $d$ and the corresponding parameter $p_{n}^{*}(d)$ for prediction, we accumulate the prediction error as:

[TABLE]

with $J(d,0)=0$ for all $d\in\{d_{\min},d_{\min}+1,\cdots,d_{\max}\}$ and $\alpha\in(0,1]$ is the learning rate. To determine the best delay parameter $d$, we use the delay parameter corresponding to the minimum accumulated error:

[TABLE]

Then the parameter for prediction is chosen as $p^{*}(k)=p_{n}(d^{*})$. The process is summarized in Algorithm 1.

V Simulation Validation

In this section, we perform simulation to validate the developed online parameter identification algorithm. Towards that end, we use the mass-spring-damper-clutch system with the parameters listed in Table I. The vehicle following is simulated over a 50 seconds horizon with initial ego vehicle speed 5 $m/s$ and initial distance headway 20 $m$. The lead vehicle speed profile is set as $v_{l}=15-5\exp(-0.05t)$. Following the dynamics (5), the ego vehicle speed and the relative distance is obtained. The speeds of the lead vehicle and ego vehicle are shown in Figure 4.

We apply the Parallel RLS-IQR algorithm in Section IV on the simulated data. We consider the reaction time range as $\tau_{\min}=0.2[s]$ and $\tau_{\max}=1[s]$. With the sampling time $\Delta t=0.1$, the corresponding delays are $d_{\min}=2$ and $d_{\max}=10$. As a result, we run a parallel of 9 RLS-IQR for each of the possible delays. The forgetting factor $\lambda$ is chosen as 0.95 and the inverse matrix initialization parameter is chosen as 10. The accumulated prediction error with learning rate $\alpha=0.05$ for all the delays are shown in Figure 5. It can be seen that $d=4$ gives the lowest prediction error, which matches the specified time delay in the simulation.

The online estimated parameters are shown in Figure 6. It can be seen that in the ideal simulation case, i.e., no model uncertainty and perfect measurement, it only takes 10 steps (1 second) to identify the parameters. This justifies the use of online prediction. However, in reality sensor noises exist in the measurement channel. We next investigate the sensitivity of algorithms to the measurement noises.

Towards this end, we inject measurement noises in the measurement channels. We consider 4 levels of noises with respect to signal-to-noise ration (SNR). The SNR of a discrete signal $y$ of length $N$ with noise $e$ is defined as

[TABLE]

In the simulation, we inject Gaussian noises to ego vehicle speed, relative distance, relative speed, and acceleration of the ego vehicle. We performed the online system identification on noise levels: no noise, SNR 30 dB, SNR 15 dB, and SNR 5 dB. The parameter estimation performance is shown in Figures 7-9. It can be seen that the algorithm is robust to noises and have fast convergence rate even under large noise levels.

VI Model validation on Naturalistic Driving Data

In this section, we validate the proposed mass-spring-damper-clutch CF model and the online parameter estimation algorithm on a naturalistic driving dataset from the Integrated Vehicle based Safety System (IVBSS) program [42]. The main objective of the IVBSS program was to investigate the effectiveness of driving assistant systems such as Lane Departure Warning, Curve Speed Warning, and Forward Crash Warning. A diverse group of 108 drivers participated in the program with balanced age and gender. The participants drove the experimental vehicles for their personal use for about six weeks. The experimental vehicles are equipped with data collection instruments to record driving data including vehicle speed and acceleration, as well as the relative distance and relative speed to the leading vehicle that are estimated using Mobileye [43]. Data including vehicle speed, acceleration, relative distance and relative distance are recorded every 0.1 second (10 Hz). The vehicle fleet and some instrumentations are illustrated in Figure 10.

To validate the proposed model, we use the naturalistic driving data from five randomly selected participants. For each driver, we extract around five car-following episodes where the cruise control is disengaged and there is no relative distance jumps due to lead vehicle lane change or other vehicle cut-ins. We apply the parallel RLS-IQR algorithm for online acceleration prediction for delays varying from 2 steps to 10 steps. The relative distance, vehicle speed, and relative speed are scaled by 1/40, 1/30, and 1/4, respectively to make the three inputs at similar level. A sample trajectory and the prediction with best estimated delays from each driver is shown in Figure 11. It can be seen that the online prediction offers promising prediction performance. The prediction error statistics for the five drivers are listed in Table II. The online identified parameters for driver 1 are shown in Figure 12.

VII Conclusions

In this paper, we developed a novel mechanical-system inspired microscopic traffic model using a mass-spring-damper-clutch system. This model naturally captures general CF behaviors and offers physical interpretations of the CF dynamics. It also considers the impact of the following vehicle on the lead vehicle, which is neglected in existing microscopic CF models and causes issues when chaining vehicles for macroscopic traffic modeling. We develop a parallel recursive least square with inverse QR decomposition algorithm for online parameter identification. The parameter identification has been validated in both simulations and on naturalistic driving data with promising performance.

Future work will include the consideration of nonlinear spring/damper to further improve the model. We will also chain the vehicles together and study the wave propagation on the vehicle platoons.