Modeling of Drivers Collision Avoidance Behavior based on Hybrid ...

To appear in paper Pre-version Proceedings ProceedingsofofIEEE IEEEInternational InternationalConference Conferenceon onSystems, Systems,Man Manand andCybernetics, Cybernetics,pp.3817-3822, pp.3817-3822,Oct. Oct.10-12, 10-12,2005, 2005,Waikoloa, Waikoloa,Hawai Hawai

Modeling of Drivers Collision Avoidance Behavior based on Hybrid System Model -An Approach with Data ClusteringTatsuya Suzuki, Susumu Yamada, Soichiro Hayakawa, Nuio Tsuchida, Taishi Tsuda, Hiroaki Fujinami

Abstract— This paper presents a development of the modeling of the human driving behavior based on the expression as Hybrid Dynamical System (HDS) focusing on the driver’s collision avoidance behavior. The driving data are collected by using the three-dimensional driving simulator based on CAVE, which provides stereoscopic immersive vision. In our modeling, the relationship between the measured information such as the sensory information of range between cars, range rate and lateral displacement between cars and the output of driver of the steering amount are expressed by the Piece-Wise Linear (PWL) model, which is a class of HDS. Then, we solve the identification problem for the PWL model by using the combination of Data Clustering and Support Vector Machine. By introducing the PWL model, it becomes possible to find not only coefficients in each submodel but also parameters in the logical (switching) conditions from the measured driving data. From the obtained results, it is found that the driver appropriately switches the ‘control law’ according to the sensory information. This enables us to capture not only the physical meaning of the driving skill, but also the decisionmaking aspect (switching conditions) in the driver’s collision avoidance behavior.

I. INTRODUCTION Recently, the modeling of driving behavior has attracted much attention by many researchers[1][2][3]. In order to model the driver’s driving maneuver, the nonlinear modelbased approaches such as the nonlinear regression models, the neural networks and fuzzy systems have been used so far. These techniques, however, have some problems as follows: (1) the obtained model often results in too complicated model, (2) this makes it impossible to understand the physical meaning of the driver’s maneuver. When we look at the driving behavior, it is often found that the driver appropriately switches the simple control laws instead of adopting the complex nonlinear control law. The switching mechanism can be regarded as a kind of driver’s decision-making in the driving behavior. Therefore, it can be a natural understanding that the model of the driving behavior involves both physical skill (operation) and the decision-making aspects (switching condition). This kind of T.Suzuki and S.Yamada are with the Department of Mechanical Science and Engineering, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Japan

t [email protected] S.Hayakawa and N.Tsuchida are with logical Institute, Hisakata, Tempaku-ku,

Toyota TechnoNagoya, Japan

s [email protected] T.Tsuda and H.Fujinami are with Toyota Motor Corporation, Toyota-cho 1, Toyota, Japan [email protected]

behavior expression can be achieved by introducing a model of Hybrid Dynamical Systems (HDSs). HDSs are systems, which consist of both continuous dynamics and logical conditions. The former are typically associated with the differential (or difference) equations, the latter with combinatorial logics, automata and so on [5][6][7][8]. Although many literatures have dealt with the description, stability analysis, control verification and parameter identification of the HDS, the application to the human behavior analysis has not been discussed as far as the authors know. In this paper, the Piece-Wise Linear (PWL) model, which is a class of the HDS, is adopted to understand the human driving behavior especially focusing on the driver’s collision avoidance maneuver. The driving data are collected by using the three-dimensional driving simulator based on CAVE, which provides stereoscopic immersive vision. In our modeling, the relationship between the measured information such as the sensory information of range between cars, range rate and lateral displacement between cars and the output of driver such as the steering amount are expressed by the PWL model. Then, we solve the identification problem for the PWL model by using the combination of Data Clustering and Support Vector Machine (SVM)[8]. By introducing the PWL model, it becomes possible to find not only coefficients in each submodel but also parameters in the logical (switching) conditions from the measured driving data. This implies that both physical operation and the decision-making aspects in the driving behavior can be identified simultaneously. II. C ONFIGURATION OF D RIVING S IMULATOR The configuration and appearance of the developed DS based on the CAVE are shown in Figs. 1(a) and (b). The display unit in the CAVE system provides the 3D virtual environment, and it is controlled by ONYX2. The display program was developed by making use of the CAVE library and the Performer. The cockpit is built by installing a handle, an accelerator and a brake in the CAVE system. The information on the driver’s output to the handle, accelerator and brake is transferred to the PC through the USB terminal, and the vehicle position and motion are calculated based on these inputs and vehicle dynamics implemented on the PC using the CarSim software. The results of the calculation are transferred to ONYX2 through the Internet (TCP/IP),

and the 3D visual image based on the position and motion of the vehicle is displayed. USB

Direct Input Speaker

PC Control Program Vehicle Dynamics TCP/IP

Magnetic Sensor Handle Accelerator Brake

ONYX2 (UNIX)

maneuver[4], we have adopted these variables as the sensory information. Also, one may argue that the braking operation must be included in the model. Actually, we have asked the examinees so as to avoid the preceding vehicle with emphasizing the steering operation rather than the braking operation. As the result, the examinees used braking just to secure the safety. In this case, the role of braking is considered to ‘assist’ the avoidance task with steering. Therefore, the steering operation seems to include higher-level decision making than the braking operation. In the following analysis, we focus only on the relationship between the sensory information and the steering operation.

Display Program CAVE Lib. Performer

CAVE

(a)

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Fig. 2.

(b) Fig. 1. The developed driving simulator. (a) Configuration of the DS. (b) CAVE system

III. DATA ACQUISITION IN COLLISION AVOIDANCE

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(

Road environment for experiments

The configuration of the experimental environment is shown in Fig.2. It has four intersections and two T-type junctions. The road has 1km length and 7m wide, and the pedestrian way is 1.5m wide. The friction coefficient of the road was set to be 0.8. The vehicle moves from the left side to the right side in Fig.2. The scene at 940[m] point is same as one at the start point. After 940[m] point, the same environment from the start point shows up again. Therefore, the driver feels that this virtual environment is straight road without any end point. There exist only two vehicles in this environment. One of them is a sedan-type car driven by the examinee. The other one is the big truck, which runs in front of the examinee and is controlled by the operator. The car used in the simulator has an engine with 3000cc displacement. Also, the car is supposed to have an ABS. The truck in front of the examinee runs at the constant speed of 50[km/h]. The maximum deceleration of the truck was supposed to be 7[m/s2 ].

A. Experimental environment and conditions

B. Procedure of experiment

In this paper, we focus on the driver’s collision avoidance behavior at the instance of the sudden stopping of the preceding vehicle. In order to model the driver’s collision avoidance behavior, the following sensory information is captured as the inputs: 1) Range between cars (x1,k ) 2) Range rate (x2,k ) 3) Lateral displacement between cars (x3,k ) The output of drivers is also specified as follows: 1) Steering amount (yk ) Since it is reported that the range between cars (x1,k ), range rate (x2,k ) and lateral displacement between cars (x4,k ) play important roles in the collision avoidance

The examinee drives the car with keeping constant distance to the preceding truck. The operator set the red or green parking vehicles on the right side at the intersection. The examinee are supposed to take a look at the right side at each intersection, then the examinee answers what colors the parking vehicle is and/or whether there are any parking vehicles. When the examinee looks right side at some intersection, the preceding truck is supposed to stop suddenly with maximum deceleration. Then, the collision avoidance behavior of the examinee is measured. By adopting this examination procedure, we can exclude the ‘effect of prediction’ of the examinee, which comes from the iterative experiments. The range between cars during the following mode was supposed to be 21[m]. Before the experiment, all examinees have practice to get used to the

driving simulator for 5 minutes. Each driver takes about 30 minutes to complete all of the trials.

y

feature vector i :Estimated Parameter mi :Mean Value

Transform to feature space

IV. M ODELING OF DRIVING BEHAVIOR A. Representation as PWL model

LDs i

In this section, the PWL model is introduced as a mathematical model of the driving behavior. We consider the following PWL model which consists of three modes:

Pi

x

(a)

(b) y

Clustering the feature space (K-means Method)

mode A (Avoidance) yk+1 = a1 x1,k + a2 x2,k + a3 x3,k if xk ∈ C1

Transform to sample space

(1)

mode B (Overtake) yk+1 = b1 x1,k + b2 x2,k + b3 x3,k if xk ∈ C2

(2)

x

(c)

mode C (Recovery) yk+1 = c1 x1,k + c2 x2,k + c3 x3,k if xk ∈ C3

(d) Fig. 3.

Outline of Clustering

(3)

where subscript k denotes the sampling index, xk = (x1,k , x2,k , x3,k )，and C1 , C2 and C3 represent subspaces in the input space. In this modeling, ai , bi and ci (i = 1, 2, 3) are unknown parameters. Also, the boundaries between subspaces C1 , C2 and C3 are unknown. This point makes the identification problem much more difficult than the standard parameter identification for the simple linear model. By introducing this PWL model, it becomes possible to capture not only the operation aspect (represented by coefficients ai , bi and ci (i = 1, 2, 3)）but also the decision making aspect（represented by boundaries between C1 , C2 and C3 ）of the human behavior． The time interval to be analyzed was set to be between the beginning of the steering operation and the ending of it. B. Mode classification by data clustering In order to identify the parameters in the PWL model, first of all, the input and output data are classified, and categorized into the corresponding modes. The outline of the clustering is described as follows (See [8] in detail): • Suppose that the sample data series {pi } (i = 1, 2 . . . , n) is given. For each sample data pi (i = 1, 2 . . . , n), collect the neighboring c data, generate the data set LDsi , and find the feature vector ξi (Fig.3(a)). Here, the feature vector ξi consists of the parameters calculated by applying least mean square method to the submodel in the PWL model, and the mean value of the data among the LDsi . • Transform the series of sample data {pi } to the series of feature vectors {ξi } (i = 1, 2 . . . , n) (Fig.3(b)). Apply the K-means algorithm to the {ξi }. (Fig.3(c)). • Transform the clustering results in the feature vector space to one in the original sample data space (Fig.3(d))．

The parameters ai , bi , ci (i = 1, 2, 3) are found by applying the standard least mean square method to the sample data set in each mode defined by the clustering procedure. C. Identification of switching plane by SVM The next step is to find the switching plane from one cluster to other, that is，the boundaries of C1 , C2 and C3 . In this work, the Support Vector Machine (SVM) is used for this purpose, where the plane is restricted to the linear hyper plane. The outline of SVM is briefly reviewed in the following: SVM is a kind of learning procedure which classifies data into two classes by finding linear hyper plane. Suppose that the training data xi and Label for class ti ∈ {−1, 1} are given. If the training data are distinguishable by the linear hyper plane, the following parameters {w, h} exist. wT xi − h ≥ 1 if ti = 1 wT xi − h ≤ −1 if ti = −1

(4)

SVM tries to find these parameters by maximizing the distance between two hyper planes wT xi − h = 1 and 1 (Maximization of wT xi − h = −1, which is given by w margin, see Fig.4). This problem can be formulated as the following quadratic programming. 1  w 2 2 subject to ti (wT xi − h) ≥ 1, min

L(w) =

i = 1, . . . , N (5)

If the training data are not distinguishable, the original quadratic programming is reformulated by introducing the nonnegative variable μi as follows:

Trial 1 Steering(degree)

wTx-h=1

1/||w||

0 −50 −100 30 −1 0 1 2 3

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1/||w||

Definition of margin

1 μi  w 2 +λ 2 i=1

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Subject A—Trial 1 Trial 2

(6)

The variable μi represents the penalty for exceeding the hyper plane, and λ denotes the weight for the penalty. This relaxation of the constraints is called a Soft Margin technique. In this work, we use this relaxed formulation to find the switching planes among C1 , C2 and C3 .

Steering(degree)

ti (wT xi − h) ≥ 1 − μi , μi ≥ 0, i = 1, . . . , N

0

mode A mode B mode C

50 0 −50 −100 30 −1 0 1 2 3

Range rate(m/s)

V. M ODELING AND ANALYSIS OF AVOIDANCE BEHAVIOR A. Information on examinees We have analyzed the driving data of two male examinees (20s years old). The driver A is unlikely to use braking, whereas the driver B tends to use braking. The driver A made four set of trials in which each set consists of 10 trials, i.e. totally 40 trials. The driver B made two set of trials since he has more experience of riding DS than the driver A.

30 20

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Fig. 6.

0 Range(m)

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Subject A—Trial 2 Trial 3

Steering(degree)

B. Parameter identification results in each mode The driving data used for the analysis are shown in Figs.5 to 10. These data show the profiles of three trials in each experiment set. The horizontal axis represents the range between cars, the vertical axes of the top, middle and bottom figures represent the steering, lateral displacement between cars and range rate, respectively. The right turn of the steering takes positive value. The lateral displacement between cars takes positive value when the driver’s vehicle moves to right side against the preceding vehicle. The range rate is the difference of the speed between the drivers vehicle and the preceding truck. All data are normalized before analysis. The interval of sampling was set to be 0.096[sec]. We can see that all data are appropriately decomposed into three modes, i.e. the ‘Avoidance’, ‘Overtake’ and ‘Recovery’ show up in a reasonable fashion.

100

Lateral displacement(m)

s.t.

10

Fig. 5.

100 mode A mode B mode C

50 0 −50 −100 30 −1 0 1 2 3

Lateral displacement(m)

min L(w) =

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Range rate(m/s)

Fig. 4.

mode A mode B mode C

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wTx-h=-1

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Fig. 7.

0 Range(m)

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Subject A—Trial 3

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mode A mode B mode C

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Lateral displacement(m)

Steering(degree)

Trial 4 100

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Mode B (Overtake) The magnitude of b1 (gain for the range) tends to become larger compared with mode A. This indicates the fact that the drivers pay more attention to the range in the mode B than in the mode A.

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Subject B—Trial 1 Trial 2 mode A mode B mode C

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TABLE I I DENTIFICATION RESULTS OF PARAMETERS

5

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Range rate(m/s)

Mode C (Recovery) The parameter set found in this mode shows large variances. This implies that no common control theoretic understanding exists. This is probably because the drivers cannot find any clear target during the mode C, i.e. in the recovery mode.

mode A mode B mode C

15

Fig. 9.

Steering(degree)

Mode A (Avoiding) The sign of a1 (gain for the range) is negative, the sign of a2 (gain for the range rate) is positive, and the sign of a3 (gain for the lateral displacement) is almost negative. This is the common characteristics between drivers. On the other hand, we can see some difference between drivers in the magnitude of the gains in the mode A. Although the magnitude of the gain a3 tends to be large in the driver A, the magnitude of the gain a2 tends to be large in the driver B. This implies that the driver A signifies the information on the lateral displacements, whereas the driver B signifies the range rate in the mode A. This difference may be caused by the difference in the use of braking.

−50 Lateral displacement(m)

Steering(degree)

Trial 1 100

The parameter identification results for each mode are shown in Table I. From Table I, we can figure out the following driving characteristics:

30 20

A-1 -0.550 1.604 0.743 3.012 -1.140 -0.367 -1.366 -0.840 0.100

A-2 -0.763 1.873 -1.000 1.077 -0.482 -0.805 -1.153 -0.720 0.028

A-3 -0.695 1.791 -3.302 -0.288 -0.790 0.344 0.024 0.523 -0.865

A-4 -0.918 1.767 -3.705 -1.088 0.689 -1.139 -0.078 0.562 -1.003

B-1 -0.534 1.294 0.905 2.289 -0.701 0.807 -1.679 -0.453 -1.360

B-2 -1.122 2.314 -0.335 2.143 -0.886 -0.780 -0.408 -0.349 -0.540

C. Identification results of switching plane 20

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a1 a2 a3 b1 b2 b3 c1 c2 c3

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Fig. 10.

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The identification results of the switching planes from one mode to the other are shown in Table II. The plane was supposed to have the form ax1 + bx2 + cx3 = h.

(7)

One of the obtained plane is shown in Fig.11. Note that x1 , x2 and x3 are normalized. The driver can be considered to switch the control law when he crosses this plane. Both drivers show the opposite sign in the coefficients a and b in the switching plane from the mode A (avoidance) to the mode B (overtake). This implies that the larger range rate leads to the earlier switching. Also, we can see the similar tendency in the relationship of coefficients a and

TABLE II PARAMETER ON H YPERPLANE mode A → B

a b c h mode B → C

a b c h

VI. C ONCLUSIONS

A-1 1.00 -1.28 -1.30 -0.50

A-2 1.00 -1.00 -0.83 -0.12

A-3 1.00 -1.42 -0.45 -0.72

A-4 1.00 -1.15 -0.71 -0.59

B-1 1.00 -1.24 -0.06 0.47

B-2 1.00 -0.93 -0.23 0.60

A-1 1.00 -0.80 -0.01 -0.59

A-2 1.00 -0.81 -0.08 -0.62

A-3 1.00 -0.07 -0.15 -0.41

A-4 1.00 -0.49 -0.13 -0.65

B-1 1.00 -0.58 0.17 -0.08

B-2 1.00 -1.22 -0.03 -0.69

In this paper, we have developed the modeling strategy of the human driving behavior based on the expression as PWL model especially focusing on the driver’s collision avoidance maneuver. The driving data was collected by using the driving simulator, which provides three-dimensional stereoscopic immersive virtual environment. Then, we have solved the identification problem for the PWL model by using the combination of Data Clustering and Support Vector Machine. By introducing the PWL model, it becomes possible to find not only coefficients in each submodel but also parameters in the logical conditions from the measured driving data. From the obtained results, it has been found that the driver appropriately switches the ‘control law’ according to the sensory information of range, range rate and lateral displacement between cars. This paper has enabled us to capture not only the operation aspect of the driving skill, but also the decision-making aspect (switching conditions) in the driver’s collision avoidance behavior. ACKNOWLEDGMENT This work was supported by the Space Robotic Center of the Toyota Technological Institute, where CAVE is installed. The authors would like to thank the researchers involved in the Space Robotic Center for their helpful suggestions. R EFERENCES

Fig. 11.

Switching hyperplane

c. However, we can see the difference in the magnitude of coefficient c. The driver B shows the smaller magnitude of c than that of the driver A. This indicates that the driver B feels less significance in the lateral displacement in the case of switching from the mode A to the mode B. In the switching plane from the mode B to the mode C, both driver show the large magnitude in the coefficients a and b, and the small magnitude in the coefficient c. This indicates that the switching from the mode B to the mode C is decided by referring the range and range rate. The switching planes projected on x1 − x3 are shown in Fig.12 together with the behavior of the vehicle.

Fig. 12.

Mode switching with vehicle behavior

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