Dynamic car following data collection and noise cancellation based on the Kalman smoothing Xiaoliang Ma∗ and Ingmar Andréasson
†
∗ Center
for Traffic Simulation Research Royal Institute of Technology, Stockholm, Sweden 10044 Telephone: 0046-8-7908426, Fax: 0046-8-7908461 Email:
[email protected] † Center for Traffic Simulation Research Royal Institute of Technology, Stockholm, Sweden 10044 Email:
[email protected] Abstract— This paper will introduce a data collection method that we used in a project on modeling driver behavior in microscopic traffic simulation. A modern instrumented vehicle was employed to study a crucial element of driver behavior, that of car following, on Swedish roads. The collected car following data shows noisy patterns. To eliminate the measurement noise, Kalman smoothing algorithm is applied to the state-space formulation of the physical states (acceleration, speed and position) of tracked vehicles. The smoothed data shows clear car following patterns and has been further applied in our car following model calibration and validation study.
I. I NTRODUCTION Driver behavior is important for the performance evaluation of transportation systems. With the advances of new technologies in form of Intelligent Transportation System (ITS), it becomes even more important to understand the normative behavior response of drivers. Traffic simulation has become a cost-effective option for the evaluation of infrastructure improvements, on-road traffic management systems and invehicle driver support systems. Over the past decade, a rapid evolution of the sophisticated microscopic simulation models leads to applications of micro-simulation in transportation and traffic engineering. However, the credibility of the models, especially the behavior model of drivers, is of vital importance to successful application of them. Therefore, calibration of those models based on real measurements attracts increasing attention in simulation applications in transportation and traffic planning.
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A standard car following situation.
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A. Car following behavior Car following is one key element of driver behavior and describes the longitudinal action of driver when they are following another car and trying to maintain a safe distance to the leading car. Figure 1 shows the situation when a following car n is behind a leading car n − 1. Under normal circumstances, drivers in the car following stage are assumed to forward without intentions of changing lane or overtaking. Research of driving behavior in traffic science can be retrospected to early 1950s when the basic traffic flow theory was initially developed and car following model was extensively studied. The well-known mathematical car following model introduced by Gazis et al. in 1961 [4] was both an extension of the early model by Chanlder et al. [3] and a summary of the early idea of stimulus-response type car following models. The model takes the form of a delay type differential equation an (t + τn ) = α
vn (t + τn )β [vn−1 (t) − vn (t)] [sn−1 (t) − sn (t)]γ
(1)
where s(t), v(t) and a(t) are the position, speed and acceleration of the cars. τn is the term called reaction time of the driver and is always assumed to be a fixed value for a certain driver n; α, β and γ are constant parameters. This model is often called the nonlinear General Motor (GM) model and has the intuitive hypothesis that the follower’s acceleration is proportional to the speed difference term, dv(t) = vn−1 (t) − vn (t), and the exponent of its own speed but being deviated from that of distance headway, D(t) = sn−1 (t) − sn (t). The speed difference part at the right hand side is always translated as ‘stimulus term’ while the fraction between exponent of following speed and that of distance headway is called ‘sensitivity term’. Although more factors [10] might be involved in the follower’s decision in the real traffic environment, the variables above show strong correlation to the driver’s decision and are relatively easier to capture by modern equipments. Previous time series data on car following were mainly collected using static laboratory simulators or vehicles on test tracks due to the lack of availability of cheaper technology for
Fig. 2.
The interface of Volvo ERS software designed for the instrumented vehicle.
measurement in real traffic. Nowadays, instrumented vehicles capable of collecting data in real traffic environment become available and had been reported promising in some research applications [1]. B. Objective In this study, car following data is collected in form of a multivariate time series. The experiment was conducted on roads of Stockholm using a modern instrumented vehicle. This article will introduce the experiment method in details. Moreover, the noise cancellation method based on Kalman smoothing algorithm is applied to the measured physical states of vehicles (position, speed and acceleration) in order to filter measurement noise. The smoothed data has been recently applied in our car following model calibration and validation work [12]. This article is organized as follow: in the second section, the car following data collection method using an instrumented vehicle is introduced in detail; the third section demonstrates the principle and practice of noise cancellation process by Kalman smoothing; the fourth section will shortly illustrate a simulation environment in which the car-following model is going to calibrated based on the collected data; at last, the results of the experiments will be summarized with conclusions. II. DATA COLLECTION METHOD An instrumented car developed by Volvo Technical Development (VTD) was used in the car following experiment. The method is similar to our early study in EVITA project [13]
[14], but both the hardware and software were updated and the experiment was more carefully designed for the purpose of car following study. The instrumented vehicle was equipped with a GPS-based navigation system and an advanced on-board trip computer for recording travel time, distance, speed, fuel economy etc. Two lidar sensors and video cameras were installed to observe, at most, eight objects in both front and rear sides. A Volvo ERS software installed on a portable laptop computer can log all information infused from those equipments at a maximum frequency of 50 times per second (Hz). Figure 2 shows the interface of Volvo ERS software where runtime data can be analyzed together with the synchronal video recording. For car following study, a multivariate time series can be obtained from the output file of the Volvo ERS software. The time series can be written as x(t) = [si (t) vi (t) ai (t) D(t) dv(t) da(t)]T where si (t), vi (t) and ai (t) are travel distance, speed and acceleration of the instrumented vehicle i, and D(t), dv(t) and da(t) are relative distance, speed and acceleration between the instrumented vehicle and the observed one. Therefore, the physical state of the observed vehicle j can be derived by si (t) − D(t) sj (t) vj (t) = vi (t) − dv(t) . (2) ai (t) − da(t) aj (t) In the testing experiments, we are more interested in measuring the random followers’ behavior by the lidar hiding at the rear, which can continuously measure the sampled distance
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The configuration of Swedish ’2+1’ type road.
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Fig. 4. The plot of raw data measured by the instrumented vehicle: the speed profiles (upper), the acceleration profiles (middle) and the distance between the two cars (lower).
headway, relative speed and acceleration to the following vehicles driven by unknowns. The reasons that we were interested in random vehicles behind were that we wanted to measure subjects who did not know that their behaviors were observed and that we intended to observe a random population of drivers instead of fixing the observed subjects in advance. However, to collect the data being able to use in car following study in platoons, another lidar sensor and a video camera are installed in the front of the car to track objects in front. The test runs have been conducted on one part of the motorway E18 near Stockholm. On the test road section, there are mainly two types of speed limits: 70 kmph and 90 kmph. To measure ‘pure’ car following behavior, the typical Swedish ‘2+1’ type motorway sections have been adopted and its configuration is illustrated in figure 3. It is apparent that the vehicle running on a single lane and being in platoon mode shows ‘pure’ car following characteristic. Meanwhile, the car following data on right lane of the double-lane motorway have also been recorded in order to analyze the behavioral difference when there are opportunities to do lane changing. The measuring frequency adopted in our experiment is either 25 or 50 times per second (Hz), which means that the time interval is either h = 0.04 or h = 0.02 sec. This high measuring frequency should give enough resolution to our study on car following. In our previous car following data collection experiment [14], a less advanced laser equipment with a maximum fre-
quency of 5 Hz was applied and there was no software for the comparison between the logged data and real time video records. Hence the estimated acceleration and speed of the observed car, which are approximated from the velocity of the instrumented car and the relative distance, are less reliable. The improvements of this experiment to the previous one offer us a chance to obtain car following data of higher quality than previous studies. Figure 4 shows the results of the measurement: the speed and acceleration of the two cars and the distance between them. From the pictures, we can see, however, that the measured states of both cars, especially the observed car, show noisy patterns and are therefore not appropriate to be directly applied in the model calibration and validation. In fact, the affordable lidar sensor can only measure distance by laser beam scanning. Due to the high measuring frequency, the relative speed dv(t) and relative acceleration da(t) can be derived by the numerical approximation of the first and second order derivatives of distance headway. Meanwhile, the Volvo ERS software uses an online adaptive filtering algorithm to give smoother pictures, though it results in the delayed time series, not smooth enough for our car following studies. In the next section, we will introduce an offline smoothing algorithm to cancel the noise in the measurement. III. N OISE CANCELLATION OF CAR FOLLOWING DATA Whenever the state of a system must be estimated from noisy sensor information, some kind of state estimator is neeeded to extract the data from different sensors to produce an accurate state estimate of the true system. As far as is known, there are different methods of estimating the real signal or time series from the noise-corrupted signal or time series. Taking the moving average values or using low pass filter to deal with high frequency noise are often used in data preprocessing in traffic engineering [16] and other fields. In fact, these methods can be generalized as filtering data series with a linear time invariant (LTI) filter [6] y(t) =
q X
φ(l)x(t − l).
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However, LTI filter assumes that the data process is stationary and the filter parameters are predetermined without optimization. These bring risks of losing information of real signal. Wiener filtering, on the other hand, applies the linear filtering approach and also minimizes the mean square error, but it has the limitations in its applicability to the non-stationary and multivariate signals. A. Kalman filter and state space model Kalman filter [9] is an optimal linear filtering approach and can treat with non-stationary data process in an iterative way, and it is specially appropriate for the state-space model form X(t + 1) Y(t)
= F (t) · X(t) + G(t) · V(t) = H(t) · X(t) + W(t).
(3)
where X(t) is the state vector and Y(t) is the measurement vector at time t. F (t) is state transition matrix and H(t) is
autocovariance function of the acceleration of the car n
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an (t)]T , V(t) = [0 0 v(t)]T , H = I and h h2 /2 1 h . 0 φ
It is worth mentioning that we present a general case, full order physical states (position, speed and acceleration), in the observation equation.
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Fig. 5. Autocovariance and partial autocorrelation function of the acceleration of car n.
relation matrix between measurement and state vector. Here we have not considered any control input in the state equation. Futhermore, the Kalman filtering approach has been extended to treat with nonlinear systems using EKF method [15], in which the model is linearized in order that the traditional linear equations can be applied. In this study however, we intend to apply the Kalman smoothing algorithm, an extension from classical linear Kalman filter, to estimate the states of vehicles being tracked by our equipped car. Before introducing the smoothing algorithm, the state-space model needs to be formulated at first. For a traveling car being tracked, it is natural to formulate a state space model due to the physical state relation as follow 1 sn (t + 1) = sn (t) + vn (t)h + an (t)h2 2 vn (t + 1) = vn (t) + an (t)h. Nevertheless, we need an equation on the acceleration to complete the state-space model. Referring to the autocovariance function (ACF) and partial autocovariance function (PACF) of the noise corrupted time series of the acceleration in figure 5, we then find an evidently strong correlation between an (t) and an (t−1). According to the order selection approach of AR models [2], it is suggested to represent the acceleration time series as the first order autoregressive (AR) model an (t + 1) = φan (t) + v(t)
(4)
where v(t) is a white noise WN(0, σ 2 ). The time series an (t) will be stationary AR(1) process as far as |φ| < 1. If the unit root exists e.g. φ = 1, the time series will be an first order ARI process. This model represents a random walk process in which acceleration at next time interval is acceleration at current time interval with an addition of a random noise term. In summary, the state space equation for our problem can be
B. Kalman smoothing algorithm Besides Kalman’s famous paper, there are many other texts (such as [2], [6] and [7]) that describe Kalman filtering algorithm and its extensions. Different notations have been adopted in their formulations. Here we only summarize the algorithm based on those texts and notations that we are familiar with. The basic Kalman filter for the prediction in the state space model of equation (3) can be represented by follow update equations ˆ ˆ t+1 = F X ˆ t + Θt ∆−1 X t (Yt − H Xt ) T Ωt+1 = F Ωt F T + GRV GT − Θt ∆−1 (7) t Θt ˆ t := Pt−1 (Xt ) defines the projection of Xt on the linwhere X ear space of Yt−1 := {Y(k)|k = 0...t − 1} i.e. Pt−1 (Xi ) := ˆ t )(Xt − X ˆ t )T ] represents the P (Xi |Yt−1 ); Ωt = E[(Xt − X posterior error covariance matrix; ∆t = HΩt H T + RW and Θt = F Ωt H T ; it is noticeable that both the process noise V(t) and the measurement noise W(t) are assumed to be white and hence RV and RW are the covariance matrix of signals V(t) and W(t) respectively. For the off-line smoothing problem, the complete observation in the interval Y0 , ..., YN is commonly known and therefore non-causal information can be used for the noise cancellation to get maximum likelihood solution by Kalman fixed interval smoothing algorithm, also called Rauch-TungStriebel smoother [5]. It is an extension of the Kalman prediction algorithm. The smoothing estimator is defined by the following recursive equations ˆ t−1|N = X ˆ t|t−1 + Qt|N F T Ω−1 ˆ ˆ X t (Xt|N − Xt|t−1 )
Qt−1|N
−1 = Qt−1 − Qt−1 F T Ω−1 t (Qt|N − Ωt )Ωt F Qt−1 (8)
ˆ i|j := P (Xi |Yj ) is similar to the definition of where X ˆ t|t )(Xt − Kalman prediction algorithm above; Qt = E[(Xt −X T ˆ Xt|t ) ] is the error covariance matrix while Qt|N = E[(Xt − ˆ t|N )(Xt − X ˆ t|N )T ] is the error covariance matrix when the X information of the whole data series is used; Ωt is defined as same as in the Kalman prediction recursion. In essence, the smoothing algorithm uses both forward prediction result of conventional Kalman filter and the backward estimation above.
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Fig. 6. The speed (upper) and acceleration profiles (middle) estimated online and after noise cancellation by Kalman smoothing algorithm and the smoothed space headway (lower).
C. Practical results In practice, we have applied the Kalman smoothing algorithm to the logged data series of both the instrumented and observed vehicle. Due to the short time interval h or high measuring frequency, random walk model is applied for the acceleration time series of both vehicles, i.e. φ = 1 for equation (4). To cancel the noise by Kalman smoother, the covariance matrices, RV and RW , for the noise processes need to be known. Noises are assumed to be white ones and uncorrelated to each other, and two covariance matrices are therefore diagonal. Hence we only need to know the power levels, or variances, of the noises in both equations. The data of the instrumented car are smoothed at first. The noises for the position and speed are zeros and so only the noise term in the random walk equation (4) needs to be determined. The equipment in the instrumented car can directly estimate the position and speed of the car but the acceleration is derived from the speed data. Therefore, it is not necessary to include acceleration measurement in the observation equation, that is, Y(t) = [ˆ sn (t) vˆn (t)]T and 1 0 0 . H= 0 1 0 Since we do not know the noises in the observation data of traveling distance and speed, it becomes a filter design problem in which we have to choose appropriate noise covariance
matrices to realize a trade-off between tracking ability and noise rejection capability of the filter. According to literature [7], the performance of the filter is mainly determined by the ratio of the covariance matrices. By adjusting the ratio, we can obtain more reasonable estimation of state profiles. In fact, the measurement noises of the speed and position for the instrumented car are rather small in comparison with those for the observed objects, which make estimated results vary within a small range. Next, we apply the smoothing algorithm to cancel noises in the states of the observed car. As we have discussed in advance, the relative speed and acceleration in the measurement output of the following car are in fact derived from the distance measurement from laser beam scanning and an online adaptive filter is applied to smooth the data. Therefore, only the space headway measurement shall be adopted in our Kalman smoother since the online estimation of relative speed and acceleration can not improve our estimation. Then the measurement equation becomes a first order form of the general one Y(t) = H · X(t) + W(t) (9) where Y(t) = sˆn (t) and H = [1 0 0]. As far as is known, noises for velocity and position in the state update equation are zero but we give them a very small value to avoid the singular matrix. The key becomes determining the ratio between the power of the measurement noise for distance and that of the
Fig. 7.
Illustration of the TPMA simulation environment
noise term in the random walk model of equation (4). Nevertheless, the power level of the noise term for the acceleration in state equation can be estimated by computing the variance of the process v(t) = an−1 (t) − an−1 (t − 1) where an−1 (t) is the acceleration process of the instrumented vehicle. The noise in the distance measurement is approximately known from the producer and so the parameters of the Kalman smoother can be tested out with small adjustments. If there are remarkable changes of the noise level in the measurement which means the non-stationary characteristic of the system, smoothing different parts of the time series separately should give better result than smoothing the whole time series at one stroke. Figure 6 shows one of the results of the speed and acceleration profiles of the instrumented car and the observed car before and after applying the Kalman smoothing algorithm. IV. A SIMULATION ENVIRONMENT In this section, we will shortly introduce TPMA, a microscopic traffic simulation environment, that has been developing in our research center. TPMA is a simulation tool which has been developed in the previous project of Traffic Performance on Major Arteries (TPMA) at Stockholm [8]. Recently, the software has been re-developed using Delphi on Windows platform and intelligent agent design approach has been applied in the new development [11]. The data collected in this study will be applied to compare different car following models, including both behavior based and blackbox models such as neural network models, in this simulation environment. The
calibrated models can not only be used as standard models of the simulation tool but also be applied as part of the general Adaptive Cruise Control algorithms. Figure 7 illustrates this microsimulation tool by simulating a large network. V. S UMMARY AND CONCLUSION Studying driver behavior based on data from real traffic has recently become a promising direction for understanding behavioral response of drivers, for modeling driver behavior in traffic simulation and for designing new ITS system such as Adaptive Cruise Control. In this paper, we presented a dynamic car following data collection method based on an advanced instrumented vehicle in detail. To obtain car following data of high quality, the experiment was conducted in the ’2+1’ type road section of a Swedish motorway where ‘pure’ car following information could be obtained. To eliminate the measurement noise, the Kalman smoothing algorithm, an extension of the conventional Kalman prediction algorithm, was applied to the measured data. The smoothed result shows clear car following patterns and has been applied in our model calibration and validation study. In this paper, only one phase of our study on driver behavior is presented and we will continue to report our model calibration and validation work based on the car following data collected in this study in the near future. ACKNOWLEDGMENTS The author would like to thank Professor Karl-Lennart Bång at the Department of Transportation and Logistics for
his support on data collection using the advanced instrumented vehicle and his helpful advice on the experiment design and research methodology. Special thanks go to the researchers at Volvo Technical Development for sharing their expert knowledge on the equipments. This study is mainly financed by Swedish Road Administration and Center for Traffic Simulation Research at KTH. R EFERENCES [1] M. Brackstone and M. McDonald. Dynamic behavioural data collection using an instrumented vehicle. Transportation Research Record, 1689:9– 17, 1999. [2] P.J. Brockwell and R.A. Davis. Introduction to Time Series and Forecasting. Springer-Verlag, New York, 1996. [3] R.E. Chandler, R. Herman, and E.W. Montroll. Traffic dynamics: studies in car following. Operational Research, 6:165–184, 1958. [4] D.C. Gazis, R. Herman, and R.W. Rothery. Nonlinear follow-the-leader models of traffic flow. Operational Research, 9, 1961. [5] S. Hakin. Kalman Filtering and Neural Networks. John Wiley and Sons, New York, 2001. [6] M.H. Hayes. Statistical Digital Signal Processing and Modeling. John Wiley Sons, Toronto CA, 1996. [7] H. Hjalmarsson and B. Ottersten. Lecture Notes in Adaptive Signal Processing. Royal Institute of Technology, Stockholm, 2002. [8] F. Davidsson T. Sauerwein P. Blad I. Kosonen, A. Gutowski and D. Aalto. Traffic performance on major arterials. Final report part (b-f), Royal Institute of Technology, Sweden, Center for Traffic Simulation Research, 2001. [9] R.E. Kalman. A new approach to linear filtering and prediction problems. ASME Transaction: Journal of Basic Engineering, D:35–45, 1960. [10] T. Kim, D.J. Lovell, and Y. Park. Limitation of previous models on carfollowing behaviours and research needs. In Proceedings of the 82th TRB annual meeting, Washington D.C., 2003. [11] X. Ma. Report on the development of TPMA 1.0: a manual for the developers. internal report (draft), Royal Institute of Technology, Sweden, Center for Traffic Simulation Research, March 2005. [12] X. Ma. Driver reaction time estimation from the real car following data and its application in GM-type model evaluation. In Submitted to the 85th Transportation Research Board (TRB) annual meeting, Washington D.C., 2006. [13] X. Ma and I. Andréasson. Predicting the effects of ISA penetration grades on pedestrian safety by simulation. Accident Analysis and Prevention, article in press, 2005. [14] X. Ma, L. Engelsson, I. Andréasson, and G. Lind. Evaluation of safety effects of various ISA vehicle penetration grades by microscopic simulations. Technical report, Royal Institute of Technology, Sweden, Center for Traffic Simulation Research, February 2004. [15] H.W. Sorenson. Kalman filtering: theory and applications. IEEE Press, 1985. [16] M.A.P. Taylor and W. Young. Traffic Analysis. Hargreen Publishing, Melbourne, 1988.
