12th World Congress on ITS, 6-10 November 2005, San Francisco
Paper 3416
Modeling of driver following behavior based on minimum-jerk theory Toshihiro HIRAOKA1∗ , Taketoshi KUNIMATSU2 , Osamu NISHIHARA1 , Hiromitsu KUMAMOTO1 1. Dept. of Systems Science, Graduate School of Informatics, Kyoto University, Yoshida Honmachi, Sakyo-ku, Kyoto 606-8501, JAPAN *TEL: +81-75-753-3369, FAX: +81-75-753-3293, E-mail:
[email protected] 2. Railway Technical Research Institute, 2-8-38 Hikari-cho, Kokubunji-shi, Tokyo 185-8540, JAPAN
ABSTRACT The acceleration and deceleration of ACC (Adaptive Cruise Control) system differ from driver’s intention to give them a feeling of wrongness, because the control laws are not designed based on the actual behavior. This paper assumes that the driver’s following behavior toward the forward vehicle matches a minimum-jerk model that represents a reaching movement of human arm and so on. In this paper, numerical simulations and driving simulator are performed to give the validity of the assumption. KEYWORDS Driver Behavior, Minimum-Jerk Model, Adaptive Cruise Control.
INTRODUCTION Recently in Japan, the number of traffic deaths decreases because passive safety technologies such as damage mitigating technologies have been developed. However, the number of traffic accidents itself increases year by year. Therefore, active safety technologies to prevent the traffic accidents are being researched and developed actively. An ACC (Adaptive Cruise Control) system is regarded as one of the active safety technology. It accelerates/decelerates the vehicle automatically to keep the target velocity and the headway distance set by driver. This system is expected to reduce not only rear-end collision accidents caused by the driver’s error in the processes of recognition, judgement and operation, but also driving workload because of the reduction of the frequent operations about acceleration and deceleration. Some vehicles are already equipped with the ACC system. However, there are some problems in the system: 1) driver’s overconfidence or distrust on the system[1], 2) a mode awareness error when the system consists of two types of ACCs; a high-speed range ACC and a low-speed range ACC[2], 3) a difference in timing of acceleration/deceleration between drivers and system[3], and 4) a feeling of wrongness caused by the difference between actual behavior and system behavior[4]. A main factor of these problems will be
Modeling of driver following behavior based on minimum-jerk theory
that the control algorithm of the ACC system does not match the driver’s actual behavior. Therefore, in this paper, the actual following behavior is assumed to have a resemblance to a reaching movement of human arm, and it is modeled by a minimum-jerk theory.
CONVENTIONAL DRIVER’S FOLLOWING BEHAVIOR MODEL A state of the vehicle running on the road is divided into two; 1) a free cruising state, and 2) a following state. The former is a state where a driver can run at his/her target velocity when there are no leading vehicles and no obstacles in a fixed distance. The latter is a state where the leading vehicles exist in a fixed distance and the velocity and the acceleration of the following vehicle are influenced by the leading vehicle. There have been existed many researches about the following behavior model for a long time in civil engineering such as traffic flow analysis. Some representative models are introduced in the following. GM model Gazis et al.[5] assumed that an acceleration of the following vehicle was in proportion to its own velocity and the relative velocity and was in inverse proportion to the headway distance. x¨(t + τd ) = cx˙ m (t) ·
˙ x˙ l (t) − x(t) [xl (t) − x(t)]l
(1)
where x(t), xl (t) represent positions of following and leading vehicles at time t respectively, τd is a driver’s response delay time. Parameters c, l, m define the driver’s characteristics. Equation (1) is commonly called as a “GM model of car-following”. However, estimates of parameters c, l, m, which are extracted from actual traffic flow data, differ in many researches. The main factors will be that the estimates are influenced by the initial values, and that collisions of vehicles may happen easily in the saturation flow or the low-speed flow. Helly model Helly[6] improved the GM model to propose the following equations. ⎧ ⎪ ⎨ ⎪ ⎩
x¨(t + τd ) = c1 (x˙ l (t) − x(t)) ˙ + c2 ((xl (t) − x(t)) − D(t + τd )) (2) D(t + τd ) = α + β x(t) ˙ + γ x¨(t)
where c1 , c2 , α, β, γ are parameters to specify the characteristic of the traffic flow. In this equation, the acceleration of the following vehicle is defined as the summation of a proportional term of relative velocity between a leading vehicle and a following vehicle and a proportional term of a deviation between a current headway distance and a target headway distance D(t). When using this model, the following vehicle can avoid the collision to the leading vehicle that is one of the problems in the GM model. Nevertheless, there remains a large individual variation in parameters of the model.
Modeling of driver following behavior based on minimum-jerk theory
ADAPTIVE CRUISE CONTROL SYSTEM Overview of ACC system The ACC system is an improved product of a conventional cruise control system, and it controls the throttle and brake to keep the target velocity and the target headway distance to the leading vehicle which are set by driver. Accordingly, the system automates the longitudinal vehicle control partially to reduce driver’s workload. Standard of ACC system ACC system is standardized in ISO (International Organization for Standardization) 15622 and JIS (Japan Industrial Standard) D 0801. Functional requirements for ACC system mentioned in JIS are summarized as follows. · Headway distance: τ ≥ 1.0[s] (steady state) · The lowest velocity that ACC system functions: vlow ≥ 5.0[m/s] · The lowest target velocity that a driver can set: vset ≥ 7.0[m/s] · The upper limit of acceleration: a ≤ 2.0[m/s2 ] · The upper limit of deceleration: a ≥ −3.0[m/s2 ] · The upper limit of jerk: |a| ˙ ≤ 2.5[m/s3 ] As mentioned above, the standard defines only limits of acceleration and jerk of ACC system and does not mention how it works. Therefore, the control rules of ACC system that are already implemented by many automobile manufacturers differ from each other.
MINIMUM-JERK THEORY Reaching movement A reaching movement is an action that human reaches out and grasps the target. Flash and Hogan[7] noted that a motion plan of the reaching movement is to maximize smoothness of the trajectory and proposed a model to minimize the sum of square of the jerk that is the first order derivative of acceleration. J=
tf 0
... ... (x)2 + (y)2 dt
(3)
Jerk is in proportion to the rate of change of force because acceleration is in proportion to force. Hence, the minimum-jerk model yields a trajectory to minimize the rate of change of force from origin to target.
Modeling of driver following behavior based on minimum-jerk theory
Formulation of minimum-jerk model A state vector is composed of position x, velocity v, and acceleration a. Input is jerk α. A state equation is written as follows. x˙ = Ax + Bα ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ x 0 1 0 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ x = ⎣ v ⎦, A = ⎣ 0 0 1 ⎦, B = ⎣ 0 ⎦ a 0 0 0 1
(4) (5)
An evaluation function J to minimize the jerk α of this reaching movement is defined as J=
tf 0
α2 dt
(6)
where tf denotes a time span. In this paper, the optimal control problem to minimize the evaluation function (6) is formulated by using the following Hamiltonian H. H = λT (Ax + Bα) + α2
(7)
Here, Euler’s equations become as follows. ∂H = λT B + 2α = 0, ∂α
∂H ˙ = x, ∂λ
−
∂H = λ˙ ∂x
(8)
These equations yield the following a differential equation.
x˙ λ˙
⎡
⎤ 1 T x BB A − ⎦ =⎣ 2 λ 0 −AT
(9)
Here, the variable λ is set to be as λ = [λ1 , λ2 , λ3 ]T . Substitution of Eq.(5) into Eq.(9) yields ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
x˙ v˙ a˙ ˙λ1 λ˙ 2 λ˙ 3
⎤
⎡
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
=
0 0 0 0 0 0
1 0 0 0 0 0
⎤⎡
⎤
0 0 0 0 x ⎥⎢ 1 0 0 0 ⎥⎢ v ⎥ ⎥ ⎢ ⎥ 1 ⎥ ⎢ ⎥ 0 0 0 −2 ⎥ ⎢ a ⎥ ⎥. ⎢ ⎥ 0 0 0 0 ⎥ ⎥ ⎢ λ1 ⎥ ⎥⎢ ⎥ 0 −1 0 0 ⎦ ⎣ λ2 ⎦ 0 0 −1 0 λ3
(10)
The solution of this optimal control problem is obtained as ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x = c0 t5 + c1 t4 + c2 t3 + c3 t2 + c4 t + c5 v = 5c0 t4 + 4c1 t3 + 3c2 t2 + 2c3 t + c4 a = 20c0 t3 + 12c1 t2 + 6c2 t + 2c3 α = 60c0 t2 + 24c1 t + 6c2
(11)
Modeling of driver following behavior based on minimum-jerk theory
where ci (i = 0, ..., 5) are constant values. The first equation of Eq.(11) represents that the position x of the reaching movement becomes the fifth order function of time t.
DRIVER FOLLOWING BEHAVIOR MODEL BASED ON MINIMUM-JERK THEORY Amano et al.[8] proposed that a target trajectory was derived from the minimum-jerk model when drivers steered to execute lane change or to avoid obstacles. The conventional models of driver following behavior have possibilities to accelerate abruptly, because the acceleration of the following vehicle is defined as a function that consists of the relative velocity and the relative distance to the leading vehicle. Generally, good drivers try to drive their vehicle smoothly to keep a comfortable ride. In other words, they regulate the vehicle velocity in consideration of the jerk. This is why the present paper models their following behavior to match the minimum-jerk model. T. C. Hwa et al.[9] designed a velocity pattern of an electromobile based on the minimumjerk theory. There are two differences between the research and this paper: 1) the velocity pattern is not influenced by the leading vehicle behavior, and 2) it is not designed based on the assumption that driver’s acceleration/deceleration behavior is similar to the reaching movement, but based on the controllability of the electronic motor. Target State A target state of the following vehicle is defined as follows: 1) the relative velocity and relative acceleration with the leading vehicle are zero, 2) the following vehicle keeps a proper headway distance that depends on its velocity. The proper headway distance S(t) is defined as v(t)2 vl (t)2 S(t) = P + τd v(t) + − (12) 2d 2dl where P is a headway distance when the both vehicles stop, τd is a reaction time, v, vl and d, dl are velocity and maximum deceleration of the following vehicle and the leading vehicle, respectively. Formulation Assume that an initial state of the following vehicle is assumed to be as x(0) = 0, v(0) = vf 0 , a(0) = af 0 , and the leading vehicle as xl (0) = S0 , vl (0) = vl0 , al (0) = al0 . After tf [s], the following vehicle go into the following state with the proper headway distance S(tf ). The time tf is defined as a reaching time in this paper. The distances where the following and leading vehicles run from time 0 to tf are vf 0 tf + af 0 t2f /2, vl0 tf + al0 t2f /2, respectively. The target headway distance at time tf is calculated as S(tf ). Here, the position x(tf ) of the following vehicle at time tf becomes x(tf ) = S0 + vl0 tf +
al0 t2f − S(tf ) 2
Modeling of driver following behavior based on minimum-jerk theory
= S0 − P + (tf − τd )vl0 +
(tf − 2τd ) al0 tf . 2
(13)
Substitution of the initial state and the target state into Eqs.(11) yields c0 , ..., c5 . c0 =
6L 3∆v + 6al0 τd ∆a − − 3, t5f t4f 2tf
c1 = −
15L 8∆v + 15al0 τd 3∆a + + 2 , t4f t3f 2tf
10L 6∆v + 10al0 τd 3∆a − , c2 = 3 − tf t2f 2tf af 0 , c4 = vf 0 , c5 = 0 c3 = 2
(14)
where L = S0 −P −τd vl0 , ∆v = vf 0 −vl0 , ∆a = af 0 −al0 . Reaching time For a comfortable ride, average drivers are assumed to accelerate/decelerate with as small acceleration/deceleration as possible, and not to repeat acceleration and deceleration before reaching the target following state. For convenience of description, assume that vf 0 > vl0 , af 0 = al0 = 0 and the leading vehicle velocity vl0 is constant. Then, x(t), v(t), a(t), α(t) of the following vehicle are obtained as ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x(t) = b0 t5 − b1 t4 + b2 t3 + v0 t v(t) = 5b0 t4 − 4b1 t3 + 3b2 t2 + v0 a(t) = 20b0 t3 − 12b1 t2 + 6b2 t
(15)
α(t) = 60b0 t2 − 24b1 t + 6b2
where b0 =
6L 3∆v 15L 8∆v 10L 6∆v − 4 , b1 = 4 − 3 , b2 = 3 − 2 . 5 tf tf tf tf tf tf
(16)
A reaching time tf of Eq.(16) still be unknown. In order to decrease the following vehicle velocity v(t) of Eq.(15) monotonously, the inequality 5 5 ≤p≤ 3 2
(17)
must be satisfied where tf = p(L/∆v). The following vehicle velocity decreases monotonously with minimum-jerk when the parameter p satisfies Eq.(17). The deceleration becomes maximum when the jerk α(t) of Eq.(15) is zero. Then, the maximum deceleration time tm is calculated as √ 8p − 15 − 19p2 − 75p + 75 tf tm = (18) 15(p − 2)
Modeling of driver following behavior based on minimum-jerk theory
when considering the constraint Eq.(17). Therefore, the maximum deceleration is minimized when the parameter p satisfies √ 2(6 − 6) p= 2.367. (19) 3 As a result, the reaching time tf to minimize the jerk, to decrease the velocity monotonously, and to minimize the maximum deceleration is described as √ 2(6 − 6)L . (20) tf = 3∆v
SIMULATIONS Numerical simulations The proposed minimum-jerk model yields a trajectory of the following vehicle when it approaches to the leading vehicle going at low-speed. The initial velocity of the following vehicle is v0 = 20[m/s], the initial velocity of the leading vehicle is vl = 10[m/s], the initial headway distance is S0 = 200[m], the headway distance at stopping is P = 6[m], the response delay is τd = 1.0[s], and the initial accelerations of the following and leading vehicles are a0 , al = 0[m/s2 ]. Assume that the maximum decelerations d, dl of Eq.(12) are equal, the target headway distance at the target state becomes 16[m]. Figures 1, 2 and 3 illustrate the simulation results when parameter p is set as 1.5, 1.667, 2.367 and 3.0, respectively. The vehicle velocities in the case of p = 1.667, 2.367, which satisfy Eq.(17), decrease monotonously. Moreover, the maximum acceleration in the case of an optimal value 2.367 expressed in Eq.(19) becomes smaller than the case of 1.667. The vehicle in the case of p = 1.5 decelerates after a small acceleration. Therefore, the time to reach the target state becomes shorter than others, but the maximum acceleration becomes the largest value. On the other hand, the vehicle of p = 3.0 accelerates a little just before reaching to the target state. Driving simulator experiments Driving simulator experiments were performed to verify the proposed following behavior model based on the minimum-jerk theory. Driving simulator As shown in Fig.4, a driving simulator has three screens in front to realize a visual field of 164[deg]. Vehicle dynamics is calculated by using CarSim which consists of 19 degrees of freedom vehicle model and a non-linear tire model to simulate a realistic vehicle behavior. Figure 5 illustrates a configuration of the driving simulator system. A general passenger vehicle model with an automatic transmission and 2.5 liter engine was used for this simulation. Experimental conditions
Modeling of driver following behavior based on minimum-jerk theory
Headway distance [m]
200 p=1.5 p=1.667 p=2.367 p=3.0 Target distance
150 100 50 0
0
10
20
30 Time [s]
40
50
60
Figure 1: Headway distance (closed-loop simulation) p=1.5 p=1.667 p=2.367 p=3.0 Target velocity
Velocity [m/s]
20
15
10 0
10
20
30 Time [s]
40
50
60
2
Longitudinal acceleration [m/s ]
Figure 2: Velocity (closed-loop simulation) 0 -0.2 -0.4
p=1.5 p=1.667 p=2.367 p=3.0 Target acceleration
-0.6 -0.8 -1
0
10
20
30 Time [s]
40
50
60
Figure 3: Longitudinal acceleration (closed-loop simulation)
The initial state of the experiments is that the leading vehicle runs at a low speed 7.5[m/s] and the following vehicle runs at 20[m/s] in the 150[m] rear of the leading vehicle. A driver decelerates to follow the leading vehicle. Five male subjects of twenties drove the following vehicle three times, and then, they chose one of the results, which was judged as the smoothest deceleration, as the data for analysis.
Modeling of driver following behavior based on minimum-jerk theory
Figure 4: Display system of driving simulator
Driving simulator
LAN(100BASE-T) Position data
Position data of other vehicles
BUS
PC Program
Position data of all vehicles
AD5410
Display system PC
Simulation SBC (CarSim RT-Linux) Control SBC
Calculation of other vehicle dynamics
Projector
DIO (AD7604) AD7121 A/D (AD7121-01) D/A (AD7121-02)
Calculation of vehicle dynamics
Meter Screen
Engine-speed, vehicle velocity
Sound system BUS
PC
A/D (AD16-16) LAN board sound board
7.1ch speaker
Reactive torque command Steering wheel angle Steering torque Steering wheel Brake pedal Brake pedal input Acceleration pedal input
Reactive DD motor Torque sensor, encorder Acceleration pedal
Interface
Figure 5: System configuration of diving simulator
Experimental results One subject did not decelerate smoothly because of pumping brake. The minimum-jerk model used in this paper represents a smooth movement. Therefore, this jerky data was
Modeling of driver following behavior based on minimum-jerk theory
Headway distance [m]
150 Average Minimum-jerk model GM model
100
50
0
0
5
10
15 Time [s]
20
25
30
Figure 6: Headway distance (driving simulator experiments)
Velocity [m/s]
20 Average Minimum-jerk model GM model
15
10
5
0
5
10
15 Time [s]
20
25
30
2
Longitudinal acceleration [m/s ]
Figure 7: Velocity (driving simulator experiments) 0.5 0 -0.5 -1 -1.5
Average Minimum-jerk model GM model
-2 -2.5
0
5
10
15 Time[s]
20
25
30
Figure 8: Longitudinal acceleration (driving simulator experiments)
excluded in this analysis. The rest of the experimental data are averaged and compared with the GM model and the minimum-jerk model. Parameters of the GM model were set as c = 0.9, l = 1.1 and m = 1, and parameters of the minimum-jerk model were p = 2.367, P = 6[m] and τd = 0.249[s] based on the experimental results. A time to begin deceleration was calculated as 9.225[s] and a headway distance at that time is 47.141[m]. This state was defined as an initial state to apply two models to analysis the vehicle behavior.
Modeling of driver following behavior based on minimum-jerk theory
Figures 6, 7 and 8 show the simulation results of headway distance, velocity and longitudinal acceleration, respectively. Small deceleration which continued from 0[s] to 9.225[s] means “engine brake and running resistance”. Deceleration of the experimental result decreases smoothly after gradual increase. The result of the minimum-jerk model is similar to the experimental result, especially in the first part of deceleration. On the other hand, the deceleration of the GM model becomes the largest value abruptly when it started to decelerate. The actual drivers decelerate a little harder than the minimum-jerk model, and accelerate a little in the end period of reaching the target state. It does not match the assumption that the deceleration decreases monotonously. It suggests that the actual drivers accelerate/decelerate their vehicle considering not only smoothness but also desire to hurry up. Accordingly, the reaching time of the actual data is suggested to be different from the optimal value p = 2.367. Headway distance of GM model at the following state is 12.899[m] that is about a vehicle length longer than the results of simulator experiment and minimum-jerk model. Hence, the minimum-jerk model can reproduce the actual following behavior better than the GM model as the conventional following model.
CONCLUSION This paper proposed the following behavior model based on the minimum-jerk theory. Numerical simulations and driving simulator experiments gave the results to suggest the validity of the proposed model. As future works, we have to verify the model using the actual vehicle and propose an ACC control rule based on the minimum-jerk model.
REFERENCES [1] H. Souma, T. Hoshi, S. Oota (2003). Drivers’ Trust in Low-Speed ACC Systems, JSAE Annual Congress Proceedings, No.51–03, pp.5–8 (in Japanese with English summary). [2] T. Inagaki, H. Furukawa, Y. Shiraishi, T. Watanabe (2003). Mode awareness of a dual-mode adaptive cruise control system, JSAE Annual Congress Proceedings, No.67-03, pp.25–27 (in Japanese with English summary). [3] S. Kojima, T. Hongo, Y. Uchiyama, N. Shiraki (2002). Comparison between the active speed reduction timing and the passive one to the slowdown of the preceding car, JSAE Annual Congress Proceedings, No.97-02, pp.5–8 (in Japanese with English summary).
Modeling of driver following behavior based on minimum-jerk theory
[4] Y. Kobayashi, Y. Yamamura, K. Akabori, S. Tange, H. Inoue (2000). A Study of a headway distance control system at low speeds, JSAE Annual Congress Proceedings, No.114-00, pp.1–4 (in Japanese with English summary). [5] D.C.Gazis, R.Herman, R.W.Rothery (1961). Nonlinear follow-the-leader models of traffic flow, Operation Research, Vol.9, No.4, pp.545–567. [6] W.Helly (1959). Simulation of bottlenecks in single lane traffic flow, Proceedings of the Symposium on Theory of Traffic Flow, pp.207–238. [7] T.Flash, N.Hogan (1985). The coordination of arm movements –an experimentally confirmed mathematical model, Journal of Neuroscience, Vol.5, No.7, pp.1688–1703. [8] Y. Amano, M. Hada, S. Doi (1998). A model of driver’s behavior in ordinary and emergent situations, Toyota Central R&D Review, Vol.33, No.1, pp.23–30 (in Japanese with English summary) [9] T. C. Hwa, S. Sakai, Y. Hori (2002). Proposal of a novel method of motion control of electric vehicles utilizing speed trajectory shaping, Proceedings of Japan Industry Application Society Conference, Vol.3, pp.1289–1292 (in Japanese with English summary)
