Human Driver Modeling Based on Analytical Optimal Solutions: Stopping Behaviors at the Intersections

Han, Jihun; Karbowski, Dominik; Kim, Namdoo; Rousseau, Aymeric

doi:10.1115/1.4046575

Abstract

Safe and energy-efficient driving of connected and automated vehicles (CAVs) must be influenced by human-driven vehicles. Thus, to properly evaluate the energy impacts of CAVs in a simulation framework, a human driver model must capture a wide range of real-world driving behaviors corresponding to the surrounding environment. This paper formulates longitudinal human driving as an optimal control problem with a state constraint imposed by the vehicle in front. Deriving analytically optimal solutions by employing optimal control theory can capture longitudinal human driving behaviors with low computational burden, and adding the state constraint can assist with describing car-following features while anticipating behaviors of the vehicle in front. We also use on-road testing data collected by an instrumented vehicle to validate the proposed human driver model for stop scenarios at intersections. Results show that vehicle stopping trajectories of the proposed model are well matched with those of experimental data.

Introduction

Thanks to advanced vehicle technologies, vehicles can be connected with other vehicles and roadside infrastructure through communication, moreover they can be equipped with different levels of automation (e.g., levels 1–5). These connected and automated vehicles (CAVs) can be aware of the surrounding environment continuously and predict future situations accurately; thus, they can reduce collisions due to human errors through anticipative and cooperative car-following and save energy further through energy-efficient driving and powertrain operation. In recent years, this topic is rapidly growing, and many researchers have presented energy-efficient driving solutions for CAVs [1]. To evaluate the energy impacts of CAVs systematically, we have developed a new multi-vehicle tool, RoadRunner [2], which allows CAVs to interact with surrounding vehicles and infrastructure on real-world routes in a closed-loop fashion. Moreover, RoadRunner is based on autonomie [3], which is an established tool for examining vehicle energy consumption and performance, where powertrain models have been validated over 10 years using chassis dynamometer test data [4]. RoadRunner development is more timely now as it allows researchers to perform a large-scale analysis of the energy impacts of CAVs with high accuracy.

As a part of RoadRunner, human driver model development and validation are critical because simplified models may not capture detailed real-world vehicle state trajectories, moreover their unrealistic behaviors may exaggerate the energy-saving potential of CAVs. In the traffic flow research area, efforts have been devoted to developing microscopic and macroscopic traffic models that result from human driving behaviors since the 1950s. Generally, the microscopic models describe traffic flow from the point of individual drivers and vehicles, whereas macroscopic models describe the collective state in terms of spatiotemporal fields of the local density, speed, and flow [5]. A car-following component is a fundamental part of microscopic models, and this feature must be simple enough to compute quickly, while describing individual driving behaviors. Several papers provide a comprehensive and excellent survey on the car-following models [6–8]. According to these papers, the car-following models are divided into several types: stimulus-response models, desired measures models, safety-distance models, optimal velocity models, fuzzy logic models, and psycho-physical models, among others. Most of the models adjusting speed and/or distance to the vehicle in front work so well that they capture macroscopic aspects of traffic dynamics for a certain condition by aggregating individual trajectories. Furthermore, consideration of human factors resulting from imperfect control has led developers to improve their own models and present other types of car-following models [9,10].

In microscopic traffic simulation, there is no need to capture a wide range of individual trajectories if the macroscopic aspects are well captured within an acceptable level of accuracy. However, several papers pointed out that car-following models are not matched with experimental data even though they are calibrated [11,12]. The RoadRunner requires a simple but high-fidelity dynamic human driver model that can capture a wide range of different driving behaviors corresponding to individual driving style as well as the surrounding environment (e.g., traffic signal phase and timing), not limited to car-following behaviors. To this end, we assumed longitudinal human driving, and this assumption leads to the formulation of an optimal control problem minimizing jerk (the derivative of acceleration) energy. We derived analytical state-constrained optimal solutions as a function of driving-related parameters through Pontryagin’s minimum principle (PMP) [13], thereby satisfying the simplicity to be computationally efficient, while replicating real-world human-driven vehicles including car-following features if necessary. Unlike existing car-following models that update acceleration by using the relative distance and speed, the proposed model based on the optimal solutions updates jerk by using more driving-related parameters, which diversifies driving behaviors and guarantees the continuity in acceleration trajectory when road events occur.

The paper is organized as follows: the section on the human driving problem introduces assumptions and formulates human driving as an optimal control problem. In the section on analytical solutions, we address derivations of analytical optimal solutions and show several case studies. In the section on validation, the proposed model is validated through experimental data. Finally, in the last section, the conclusions and future work resulting from this paper are discussed.

Longitudinal Human Driving Problem

Assumptions.

We assume that drivers basically prioritize driving comfort, while avoiding any collisions with the vehicle in front and obeying traffic rules; maximizing driving comfort is considered as minimizing total jerk energy. As human drivers are able to anticipate behaviors of the vehicle in front of them, they plan and apply the control decision, such an anticipation is based on the assumption that the vehicle in front travels at the constant acceleration in a predictive time interval. Furthermore, vehicle longitudinal dynamics is simplified to the triple-integrator model by neglecting aerodynamic drag, road grade, etc. These assumptions facilitate derivation of analytical optimal solutions, which can be computed quickly. Anticipation, planning, and control decision are made at every time instant until arriving at the destination.

Optimal Control Problem Formulation.

Let us define a control variable u as jerk j. A cost function to minimize total jerk energy over the predictive time interval T is

J = min_{u} \int_{t_{k}}^{t_{k} + T} (l = \frac{1}{2} u^{2}) d t

(1)

where t_k is a current time. Furthermore, we consider a vehicle longitudinal dynamics model as the triple-integrator model

\dot{s} = v, \dot{v} = a, \dot{a} = j = u

(2)

where a system dynamics

f = [v, a, u]^{⊤}

⁠, and a state vector

x = [s, v, a]^{⊤}

⁠.

To avoid collisions with the vehicle in front, the position of the driver’s vehicle with a desired distance gap s_d must be smaller than the position of the vehicle in front s_p. Thus, a state variable inequality constraint (SVIC) h is defined as

h (t) = s + s_{d} - s_{p} = s + (v τ + s_{s}) - s_{p} \leq 0

(3)

where

s_{d} = s_{s} + v τ

⁠,

τ

is a desired time headway, which is the arriving time difference at the same point between successive vehicles, and s_s is the minimum safety distance gap at standstill conditions. Note that desired speed gap and acceleration gap are

v_{d} = s_{s} + a τ

and

a_{d} = u τ

⁠, respectively. Using assumptions made in the previous section, s_p can be expressed as

s_{p} = s_{p . k} + v_{p . k} t + \frac{1}{2} a_{p . k} t^{2}

⁠, where s_p.k = s_p(t_k), v_p.k = v_p(t_k), and a_p.k = a_p(t_k).

Boundary conditions are

\begin{aligned} s (t_{k}) = s_{k}, v (t_{k}) = v_{k}, a (t_{k}) = a_{k}, \\ s (t_{k} + T) = s_{f}, v (t_{k} + T) = v_{f}, a (t_{k} + T) = a_{f} \end{aligned}

(4)

where T is fixed. For simplicity, without loss of generality, t_k = 0 and k = 0 in the remainder of the paper.

Pontryagin’s Minimum Principle With SVIC.

To handle SVIC, we use a direct adjoining method [14] as it directly adjoins the SVIC in the Hamiltonian and provides the optimality conditions for the optimal solution through PMP; its optimality conditions are independent of the order of a pure SVIC form h(x, t) ≤ 0 in which u does not explicitly appear, where the order p is defined by $h_{u}^{(p)} = \partial / \partial u (d^{p} / d t^{p} h) \neq 0$ ⁠, $h_{u}^{(i)} = 0$ for i = 1, …, p − 1. When the SVIC is active, there may exist a sub-interval satisfying h(x, t) = 0 for t ∈ [t₁, t₂] with t₁ < t₂, called a boundary interval (t₁ and t₂ are entry time and exit time, respectively) or a point satisfying h(x, t₁) = 0, called a contact point (t₁ is contact time). Note that the entry, exit, and contact times are called junction times.

Hamiltonian H is defined first, and then a Lagrangian multiplier

η

is used to directly adjoin the SVIC to H in order to form Lagrangian L

H = l + λ^{⊤} f, L = H + η h

(5)

where

λ

are co-state variables and l has the same definition as in Eq. . The optimality condition is

L_{u} = 0, \dot{λ} = - L_{x}

(6)

where

η \geq 0

⁠,

η h = 0

⁠. Jump conditions at junction times t_j with a jump parameter

π

are

λ (t_{j}^{-}) = λ (t_{j}^{+}) + π h_{x} (x, t_{j}), H (t_{j}^{-}) = H (t_{j}^{+}) - π h_{t} (x, t_{j})

(7)

where π ≥ 0, πh = 0, and

t_{j}^{-}

and

t_{j}^{+}

indicate the left-hand side and the right-hand side of t_j, respectively.

Analytical Solutions

From the optimal control problem formulated by Eqs. , , , and , we define H and L as

\begin{aligned} H & = \frac{1}{2} u^{2} + λ_{s} v + λ_{v} a + λ_{a} u \\ L & = H + η (s + v τ + s_{s} - s_{p .0} - v_{p .0} t - \frac{1}{2} a_{p .0} t^{2}) \end{aligned}

(8)

Then, an optimal control policy is derived by the optimality condition as follows:

u * (t) = - λ_{a}

⁠. Co-state dynamics with jump conditions are derived as

\begin{matrix} {\dot{λ}}_{s} = - η & with λ_{s} (t_{j}^{-}) = λ_{s} (t_{j}^{+}) + π \\ {\dot{λ}}_{v} = - λ_{s} - η τ & with λ_{v} (t_{j}^{-}) = λ_{v} (t_{j}^{+}) + π τ \\ {\dot{λ}}_{a} = - λ_{v} & with λ_{a} (t_{j}^{-}) = λ_{a} (t_{j}^{+}) \end{matrix}

(9)

where h = 0 and π ≠ 0 if the SVIC is active, that is position and speed co-states must have a discontinuity at junction times, whereas the acceleration co-state is always continuous; otherwise,

η = 0

and π = 0.

From h = 0 and the last condition in Eq. , we add the constraints

\begin{aligned} s (t_{j}) + (s_{s} + v (t_{j}) τ) - s_{p .0} - v_{p .0} t_{j} - \frac{1}{2} a_{p .0} t_{j}^{2} = 0 \\ v (t_{j}) + a (t_{j}) τ - v_{p .0} - a_{p .0} t_{j} = 0 \end{aligned}

(10)

The SVIC here is of the second order; thus, when the SVIC becomes tighter, a contact point occurs first and then a boundary interval occurs (e.g., a decrease in s_p.0 gives the same conditions), which is theoretically proven in Ref. [15]. Therefore, to retain states on the boundary interval, the following constraint is also satisfied:

a (t_{j}) + u (t_{j}) τ - a_{p .0} = 0

(11)

Lastly, terminal state equality constraints (TSECs) from Eq. are

s (T) - s_{f} = 0, v (T) - v_{f} = 0, a (T) - a_{f} = 0

(12)

where v_f = a_f = 0 for stop scenarios.

In summary, boundary conditions determine whether and how SVIC is active, thus analytical solutions consist of three types: type 1 (inactive SVIC), type 2 (active SVIC at contact point), and type 3 (active SVIC on boundary interval). We heuristically determine an appropriate solution type. If type 1 does not violate the SVIC, then it is applied; otherwise, we check if there exists one feasible contact point for type 2. After this identification, if type 2 is feasible then it is applied; otherwise, type 3 is applied. The detailed description for all types of solutions is written as follows:

Type 1: Inactive SVIC.

When the SVIC is inactive,

η = 0

⁠. Thus, the optimal co-states are derived using Eq.

\begin{aligned} λ_{s} * (t) & = λ_{s .0} \\ λ_{v} * (t) & = - λ_{s .0} t + λ_{v .0} \\ λ_{a} * (t) & = \frac{1}{2} λ_{s .0} t^{2} - λ_{v .0} t + λ_{a .0} \end{aligned}

(13)

where

λ_{s .0} = λ_{s} * (0)

⁠,

λ_{v .0} = λ_{v} * (0)

⁠, and

λ_{a .0} = λ_{a} * (0)

⁠, while superscript * indicates optimal. Using the optimal control policy, it is possible to integrate the state dynamics in Eq. . Then, we obtain a system of three linear equations with three unknown variables (⁠

λ_{s .0}

⁠,

λ_{v .0}

⁠, and

λ_{a .0}

⁠). By enforcing the TSECs in Eq. , a solution to this system is

\begin{aligned} λ_{s .0} = \frac{720 (s_{0} - s_{f})}{T^{5}} + \frac{360 (v_{0} + v_{f})}{T^{4}} + \frac{60 (a_{0} - a_{f})}{T^{3}} \\ λ_{v .0} = \frac{360 (s_{0} - s_{f})}{T^{4}} + \frac{192 v_{0} + 168 v_{f}}{T^{3}} + \frac{36 a_{0} - 24 a_{f}}{T^{2}} \\ λ_{a .0} = \frac{60 (s_{0} - s_{f})}{T^{3}} + \frac{36 v_{0} + 24 v_{f}}{T^{2}} + \frac{9 a_{0} - 3 a_{f}}{T} \end{aligned}

(14)

Finally, we get

a * (t) = - \int λ_{a} + a_{0}

⁠,

v * (t) = \int a * + v_{0}

⁠, and

s * (t) = \int v * + s_{0}

⁠. Note that the above solution was also used in Ref. [16].

Type 2: Active SVIC at Contact Point.

When the SVIC is active at a contact time t₁, position and speed co-states jump to different values; thus, the resulting co-state trajectories with

η = 0

consist of two intervals as follows:

\begin{aligned} λ_{s} * (t) = {\begin{matrix} λ_{s .0} & t \in [0, t_{1}^{-}] \\ λ_{s .1}^{+} & t \in [t_{1}^{+}, T] \end{matrix}} \\ λ_{v} * (t) = {\begin{matrix} - λ_{s .0} t + λ_{v .0} & t \in [0, t_{1}^{-}] \\ - λ_{s .1}^{+} (t - t_{1}) + λ_{v .1}^{+} & t \in [t_{1}^{+}, T] \end{matrix}} \\ λ_{a} * (t) = {\begin{matrix} \frac{λ_{s .0}}{2} t^{2} - λ_{v .0} t + λ_{a .0} & t \in [0, t_{1}^{-}] \\ \frac{λ_{s .1}^{+}}{2} (t - t_{1})^{2} - λ_{v .1}^{+} (t - t_{1}) + λ_{a .1}^{+} & t \in [t_{1}^{+}, T] \end{matrix}} \end{aligned}

(15)

where

λ_{s .1}^{+} = λ_{s} * (t_{1}^{+}) = λ_{s} * (t_{1}^{-}) - π

⁠,

λ_{v .1}^{+} = λ_{v} * (t_{1}^{+}) = λ_{v} * (t_{1}^{-}) - π τ

⁠, and

λ_{a .1}^{+} = λ_{a} * (t_{1}^{+}) = λ_{a} * (t_{1}^{-})

from the jump conditions in Eq. . In the same way as in type 1, the state dynamics are integrated; but here, two interior-point constraints at t₁ from Eq. must be additionally satisfied. For this reason, we obtain a system of five nonlinear equations with five unknown variables (⁠

λ_{s .0}

⁠,

λ_{v .0}

⁠,

λ_{a .0}

⁠,

π

⁠, and t₁) as follows:

\begin{aligned} s * (t_{1}) + s_{s} + v * (t_{1}) τ - s_{p .0} - v_{p .0} t_{1} - \frac{1}{2} a_{p .0} t_{1}^{2} = 0 \\ v * (t_{1}) + a * (t_{1}) τ - v_{p .0} - a_{p .0} t_{1} = 0 \\ TSECs in Eq . (12) \end{aligned}

(16)

By solving the first four equations, we derive four variables (⁠

λ_{s .0}

⁠,

λ_{v .0}

⁠,

λ_{a .0}

⁠,

π

⁠) as a function of t₁, and then substitute them into the last equation to obtain one-ninth order of polynomial with one variable of t₁. From such a polynomial, we can compute a feasible t₁ satisfying a condition of 0 ≤ t₁ ≤ T.

Type 3: Active SVIC on Boundary Interval.

When the SVIC is active on the boundary interval [t₁, t₂], there are two jumps in position and speed co-states with jump parameters π₁ and π₂ at t₁ and t₂, respectively. The

η

is no longer zero and dominates co-state dynamics as the three constraints in Eqs. and must be held on [t₁, t₂]. After the third constraint is differentiated one time with respect to t, we substitute

λ_{a}

for u using the optimal control policy and obtain the following explicit solution:

λ_{a} * (t) = λ_{a .1}^{+} e^{- (t - t_{1}) / τ}

(17)

where

λ_{a .1}^{+} = λ_{a} * (t_{1}^{+}) = λ_{a} * (t_{1}^{-})

⁠. Using the above formula and the co-state dynamics in Eq. , we obtain the explicit solutions of other co-states as well:

\begin{aligned} λ_{s} * (t) & = λ_{s .1}^{+} e^{(t - t_{1}) / τ} + c λ_{a .1}^{+} (e^{- (t - t_{1}) / τ} - e^{(t - t_{1}) / τ}) \\ λ_{v} * (t) & = \frac{1}{τ} λ_{a .1}^{+} e^{- (t - t_{1}) / τ} \end{aligned}

(18)

with

η * (t) = - (1 / τ) λ_{s} * (t) + (1 / τ^{3}) λ_{a} * (t)

⁠, where

c = 1 / 2 τ^{2}

and

λ_{s .1}^{+} = λ_{s} * (t_{1}^{-}) - π_{1}

⁠. As

λ_{v} * (t_{1}^{+}) = (1 / τ) λ_{a .1}^{+}

⁠, the jump parameter π₁ from Eq. is obtained as

π_{1} = (1 / τ) λ_{v .1}^{-} - (1 / τ^{2}) λ_{a .1}^{+}

⁠. Based on the above boundary interval control, we summarize

\begin{aligned} λ_{s} * (t) & = {\begin{matrix} λ_{s .0} & t \in [0, t_{1}^{-}] \\ 1 st Eq . (18) & t \in [t_{1}^{+}, t_{2}^{-}] \\ λ_{s .2}^{+} & t \in [t_{2}^{+}, T] \end{matrix}} \\ λ_{v} * (t) & = {\begin{matrix} - λ_{s .0} t + λ_{v .0} & t \in [0, t_{1}^{-}] \\ 2 nd Eq . (18) & t \in [t_{1}^{+}, t_{2}^{-}] \\ - λ_{s .2}^{+} (t - t_{2}) + λ_{v .2}^{+} & t \in [t_{2}^{+}, T] \end{matrix}} \\ λ_{a} * (t) & = {\begin{matrix} \frac{1}{2} λ_{s .0} t^{2} - λ_{v .0} t + λ_{a .0} & t \in [0, t_{1}^{-}] \\ Eq . (17) & t \in [t_{1}^{+}, t_{2}^{-}] \\ \frac{1}{2} λ_{s .2}^{+} (t - t_{2})^{2} - λ_{v .2}^{+} (t - t_{2}) + λ_{a .2}^{+} & t \in [t_{2}^{+}, T] \end{matrix}} \end{aligned}

(19)

where

λ_{s .2}^{+} = λ_{s} * (t_{2}^{-}) - π_{2}

⁠,

λ_{v .2}^{+} = λ_{v} * (t_{2}^{-}) - π_{2} τ

⁠, and

λ_{a .2}^{+} = λ_{a} * (t_{2}^{-})

⁠. Similarly, we obtain a system of six nonlinear equations with six unknown variables (⁠

λ_{s .0}

⁠,

λ_{v .0}

⁠,

λ_{a .0}

⁠, π₂, t₁, and t₂):

\begin{aligned} s * (t_{1}) + s_{s} + v * (t_{1}) τ - s_{p .0} - v_{p .0} t_{1} - \frac{1}{2} a_{p .0} t_{1}^{2} = 0 \\ v * (t_{1}) + a * (t_{1}) τ - v_{p .0} - a_{p .0} t_{1} = 0 \\ a * (t_{1}) + u * (t_{1}) τ - a_{p .0} = 0 \\ TSECs in Eq . (12) \end{aligned}

(20)

After replacing four variables (⁠

λ_{s .0}

⁠,

λ_{v .0}

⁠,

λ_{a .0}

⁠, and π₂) in the last two equations, we can compute the remaining variables t₁ and t₂ satisfying a condition of 0 ≤ t₁ ≤ t₂ ≤ T.

Simulation Studies.

This section considers typical stop scenarios in the presence of the vehicle in front. We assume that the vehicle in front decelerates at the constant value and its initial setup is a_p.0 = −0.5 m/s², v_p.0 = 10 m/s, and $s_{p .0} = [60, 50, 40]^{⊤}$ m. For host vehicle, its initial setup is a₀ = −0.2 m/s², v₀ = 20 m/s, s₀ = 0 m, a_f = 0 m/s², v_f = 0 m/s, s_f = 100 m, T = 10 s, $τ = 1.2$ s, and s_s = 2 m.

In Fig. 1, the vehicle in front does not affect the driving behavior of the host vehicle. All co-state trajectories of type 1 are continuous, and the resulting optimal control policy enables the host vehicle to decelerate its speed for a stop in a way that minimizes total jerk energy. However, as initial distance gap decreases (i.e., s_p.0 decreases), the desired position trajectory (s + s_d) of the host vehicle starts surpassing the position trajectory of the vehicle in front (s_p) and then an actual rear-collision event occurs unless state-constrained solutions are used. As shown in Fig. 2, both position and speed co-states of type 2 are discontinuous at the contact time, which allows the desired position and speed of the host vehicle to match with the position and speed of the vehicle in front, respectively. On the other hand, the acceleration co-state is always continuous, which leads to a continuous optimal control policy. Lastly, in Fig. 3, type 3 builds up the boundary interval, jumps in the position, and speed co-states occur at both entry and exit times, whereas the acceleration co-state is continuous. Notably, co-state dynamics on the boundary interval are influenced by $η$ ⁠, which are different from that on other intervals, so that desired states of the host vehicle behave in exactly the same way as the vehicle in front.

Fig. 1

View large Download slide

State and co-state trajectories of type 1

Fig. 2

View large Download slide

State and co-state trajectories of type 2. Black dotted line represents the contact point.

Fig. 3

View large Download slide

State and co-state trajectories of type 3. Two black dotted lines represent entry and exit time, respectively.

Validation With Experimental Data

Experimental data were collected from an instrumented vehicle driven by a human driver and equipped with a dash video camera, global positioning system tracker, and a radar and then processed to make the data usable for validation. On-road testing was performed on specific sub-urban route near Argonne, and the conditions at that time are light traffic, mostly sunny day, dry road condition, normal driving style, etc.

In this paper, we considered a braking regime only for stop scenarios near the intersections, including situations with and without the vehicle in front. From the experimental data, the parameters required by the human driver model (i.e., boundary conditions) were obtained: braking distance (s_f − s₀), braking time (T), and vehicle states at the timing to start braking (a₀ and v₀). As a result of how drivers respond to surrounding environment (e.g., traffic signal phase and timing), we could have various stopping trajectories as shown in Fig. 4. For car-following parameters, s_s = 2 m and $τ = min (1, s_{p .0} - s_{s} - s_{0} / v_{0})$ s. Note that $τ$ could be optimally researched in a way to minimize the error between the model and data.

Fig. 4

View large Download slide

Various trajectories of the braking regime from experimental dataset

To validate the model, we used two measures: (1) normalized cross correlation power (NCCP) [17] and (2) normalized root-mean-squared error (NRMSE) [11]. The NCCP and NRMSE are defined as

\begin{aligned} NCCP = \frac{max [R_{x y} (τ)]}{max [R_{x x} (τ), R_{y y} (τ)]} \times 100 \\ NRMSE = \frac{\sqrt{\sum_{i = 0}^{T} (x_{i} - y_{i})^{2} / T}}{(y_{max} - y_{min})} \times 100 \end{aligned}

(21)

where

R_{x y} (τ) = lim_{T \to \infty} \int_{0}^{T} x (t) \circ y (t - τ) d t

⁠, and x and y represent speed signals of the model and data, respectively, while y_max and y_min are maximum and minimum values, respectively. Note that the NCCP value larger than 90% indicates that the two speed signals are highly correlated. Moreover, we also compared the proposed model with the intelligent driver model (IDM), well-known car-following model, with prescribed parameters, where the comfortable acceleration and deceleration are set to 1 and 1.5 m/s², the minimum spacing, the desired time headway, and desired speed are set to s_s,

τ

⁠, and v₀, and the exponent is set to 4.

In Fig. 5, for the proposed model and IDM, all NCCP values are larger than 90% and all NRMSE values are less than 10% (except for IDM in the case number 28). The proposed model outperforms IDM as the average value of both measures is 98.6% versus 95.4% in NCCP, and 2.33% versus 5.39% in NRMSE. Figure 6 shows two specific cases including car-following situations to see how the host vehicle adjusts its driving behaviors, where values of a pair (NCCP, NRMSE) are (97.8%, 4.68%) versus (91.5%, 9.94%) for case number 18 (left) and (99.9%, 1.58%) versus (98.6%, 2.41%) for case number 22 (right). In case number 18, the surrounding vehicle changes its lane at about 10 s and drives in front of the equipped vehicle, which causes a discontinuity in trajectories (red line). Note that the relative position is set to 250 m and relative speed is set to zero if the radar sensor detects nothing in the same lane. If there are no vehicles in front, IDM maintains constant speed from 0 s to 3 s, but it starts braking, which jumps the negative value from zero acceleration, because of considering a red traffic light as a standing object. At about 10 s, IDM is too close to the cut-in vehicle, thus it must have the maximum braking rate, which is set by 4 m/s², leading to the large discrepancy in trajectories compared to the testing data.

Fig. 5

View large Download slide

NCCP and NRMSE of the proposed model (blue bar) and IDM (magenta bar) for 54 braking-to-stop cases. Vertical cyan lines represent for car-following situations.

Fig. 6

View large Download slide

Comparison of state trajectories between models and data in case numbers 18 (left) and 22 (right). PV is the preceding vehicle, and the simple model is the model using only type 1 solution that cannot describe the car-following feature.

However, the proposed model can set different braking times (e.g., 37 s and 23 s in case numbers 18 and 22, respectively) and its trajectory is quite close to that of the experimental data, while describing the car-following feature without a discontinuity in acceleration trajectory when road event occurs (e.g., traffic signal state switches to red, the vehicle in front appears), as shown in Fig. 6. Note that the simple model uses only type 1 solution, thereby unrealistically overtaking the vehicle in front in the same lane. As shown in Fig. 7, the boundary conditions must be adjusted to ensure that they are feasible to compute analytical optimal solutions depending on driving behavior of the vehicle in front (e.g., a final position with the desired distance gap cannot be larger than the anticipated final position of the vehicle in front if there is no lane-change) [18]. Both cases adjust boundary conditions; however, case number 18 must use the state-constrained solutions, whereas case number 22 does not.

Fig. 7

View large Download slide

Solution type, final position, and final time trajectories in case number 18

Conclusions and Future Work

In this paper, we present a new approach for human driver modeling using an optimal control theory. The human driver is modeled based on analytical optimal solutions that maximize driving comfort for given boundary conditions, while considering the state constraint imposed by the vehicle in front. The proposed model is not only computationally efficient but also captures various stopping trajectories including car-following to keep the desired distance gap because its inputs (boundary conditions) directly represent one aspect of the driving behavior (e.g., braking time indicates braking level), respectively. Using experimental data, the proposed model is validated and also compared with IDM. Results show that the average values of NCCP and NRMSE for 54 braking-to-stop cases are about 98.6% and 2.33%, respectively, which indicates that the stopping behavior of the proposed model is highly correlated with that of the experimental data. The proposed model improves accuracy over IDM without a discontinuity which is seen in IDM when the road event occurs.

In future work, we would like to expand driving regimes (e.g., accelerating and cruising), rather than only focusing on braking, and we also consider road characteristics (such as curvature) to capture the speed reduction. We also would like to investigate if the proposed model can capture macroscopic aspects under high traffic conditions. Furthermore, another future research direction is to develop a model of perception and decision that can provide the timing and duration of each driving regime.

Acknowledgment

This report and the work described were sponsored by the U.S. Department of Energy (DOE) Vehicle Technologies Office (VTO) under the Systems and Modeling for Accelerated Research in Transportation (SMART) Mobility Laboratory Consortium, an initiative of the Energy Efficient Mobility Systems (EEMS) Program. DOE Office of Energy Efficiency and Renewable Energy (EERE) manager David Anderson played an important role in establishing the project concept, advancing implementation, and providing ongoing guidance.

Conflict of Interest

The submitted article has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory (“Argonne”). Argonne, a U.S. Department of Energy Office of Science laboratory, is operated under Contract No. DEAC02-06CH11357. The U.S. Government retains for itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly, and display publicly, by or on behalf of the Government.

Nomenclature

T =: predictive time interval
s_s =: minimum safety distance gap at standstill conditions
t_i =: contact time if i = 1 or entry and exit times if i = 1, 2
s, v, a, j =: position, speed, acceleration, and jerk
s_p, v_p, a_p =: position, speed, and acceleration of the vehicle in front
s_d, v_d, a_d =: desired distance, speed, and acceleration gaps
s_f, v_f, a_f =: final position, final speed, and final acceleration
u (:= j) =: control input variable (:= jerk)
H, L =: Hamiltonian and Lagrangian
$η$ =: Lagrange multiplier to directly adjoin the state constraint
$λ_{s}, λ_{v}, λ_{s}$ =: position, speed, and acceleration co-state variables
π =: jump parameter in co-state variables
$τ$ =: desired time headway

References

1.

Vahidi

,

A.

, and

Sciarretta

,

A.

,

2018

, “

Energy Saving Potentials of Connected and Automated Vehicles

,”

Transp. Res. Part C: Emerg. Technol.

,

95

, pp.

822

–

843

. 10.1016/j.trc.2018.09.001

Google Scholar

Crossref

2.

Kim

,

N.

,

Karbowski

,

D.

, and

Rousseau

,

A.

,

2018

, “

A Modeling Framework for Connectivity and Automation Co-Simulation

,”

SAE Technical Paper 2018-01-0607

.

3.

Argonne National Laboratory

,

2017

,

autonomie (computer software)

.

4.

Kim

,

N.

,

Duoba

,

M.

,

Kim

,

N.

, and

Rousseau

,

A.

,

2013

, “

Validating Volt PHEV Model With Dynamometer Test Data Using Autonomie

,”

SAE Int. J. Passenger Cars Mech. Syst.

,

6

(

2

), pp.

985

–

992

. 10.4271/2013-01-1458

Google Scholar

Crossref

5.

Treiber

,

M.

, and

Kesting

,

A.

,

2013

,

Traffic Flow Dynamics

,

Springer

,

Berlin

.

Google Scholar

Crossref

6.

Brackstone

,

M.

, and

McDonald

,

M.

,

1999

, “

Car-Following: A Historical Review

,”

Transp. Res. Part F: Traffic Psychol. Behav.

,

2

(

4

), pp.

181

–

196

. 10.1016/S1369-8478(00)00005-X

Google Scholar

Crossref

7.

Toledo

,

T.

,

2007

, “

Driving Behaviour: Models and Challenges

,”

Transp. Rev.

,

27

(

1

), pp.

65

–

84

. 10.1080/01441640600823940

Google Scholar

Crossref

8.

Saifuzzaman

,

M.

, and

Zheng

,

Z.

,

2014

, “

Incorporating Human-Factors in Car-Following Models: A Review of Recent Developments and Research Needs

,”

Transp. Res. Part C: Emerg. Technol.

,

48

, pp.

379

–

403

. 10.1016/j.trc.2014.09.008

Google Scholar

Crossref

9.

Ro

,

J. W.

,

Roop

,

P. S.

,

Malik

,

A.

, and

Ranjitkar

,

P.

,

2018

, “

A Formal Approach for Modeling and Simulation of Human Car-Following Behavior

,”

IEEE Trans. Intell. Transp. Syst.

,

19

(

2

), pp.

639

–

648

. 10.1109/TITS.2017.2759273

Google Scholar

Crossref

10.

Lindorfer

,

M.

,

Mecklenbrauker

,

C. F.

, and

Ostermayer

,

G.

,

2018

, “

Modeling the Imperfect Driver: Incorporating Human Factors in a Microscopic Traffic Model

,”

IEEE Trans. Intell. Transp. Syst.

,

19

(

9

), pp.

2856

–

2870

. 10.1109/TITS.2017.2765694

Google Scholar

Crossref

11.

Wagner

,

P.

,

2005

, “Empirical Description of Car-Following,”

Traffic and Granular Flow ’03

,

Hoogendoorn

,

S. P.

,

Luding

,

S.

,

Bovy

,

P. H. L.

,

Schreckenberg

,

M.

, and

Wolf

,

D. E.

, eds.,

Springer

,

Berlin

, pp.

15

–

27

.

Google Scholar

Crossref

12.

Sangster

,

J.

,

Rakha

,

H.

, and

Du

,

J.

,

2013

, “

Application of Naturalistic Driving Data to Modeling of Driver Car-Following Behavior

,”

Transp. Res. Record: J. Transp. Res. Board

,

2390

(

1

), pp.

20

–

33

. 10.3141/2390-03

Google Scholar

Crossref

13.

Pontryagin

,

L.

,

1987

,

Mathematical Theory of Optimal Processes

, 1st ed.,

Routledge

,

London

.

14.

Hartl

,

R. F.

,

Sethi

,

S. P.

, and

Vickson

,

R. G.

,

1995

, “

A Survey of the Maximum Principles for Optimal Control Problems With State Constraints

,”

SIAM Rev.

,

37

(

2

), pp.

181

–

218

. 10.1137/1037043

Google Scholar

Crossref

15.

Hamilton

,

W.

,

1972

, “

On Nonexistence of Boundary Arcs in Control Problems With Bounded State Variables

,”

IEEE Trans. Automat. Control

,

17

(

3

), pp.

338

–

343

. 10.1109/TAC.1972.1099982

Google Scholar

Crossref

16.

Da Lio

,

M.

,

Mazzalai

,

A.

,

Gurney

,

K.

, and

Saroldi

,

A.

,

2018

, “

Biologically Guided Driver Modeling: The Stop Behavior of Human Car Drivers

,”

IEEE Trans. Intell. Transp. Syst.

,

19

(

8

), pp.

2454

–

2469

. 10.1109/TITS.2017.2751526

Google Scholar

Crossref

17.

Meng

,

Y.

,

Jennings

,

M.

,

Tsou

,

P.

,

Brigham

,

D.

,

Bell

,

D.

, and

Soto

,

C.

,

2011

, “

Test Correlation Framework for Hybrid Electric Vehicle System Model

,”

SAE Int. J. Eng.

,

4

(

1

), pp.

1046

–

1057

. 10.4271/2011-01-0881

Google Scholar

Crossref

18.

Han

,

J.

,

Sciarretta

,

A.

,

Ojeda

,

L. L.

,

De Nunzio

,

G.

, and

Thibault

,

L.

,

2018

, “

Safe- and Eco-Driving Control for Connected and Automated Electric Vehicles Using Analytical State-Constrained Optimal Solution

,”

IEEE Trans. Intell. Veh.

,

3

(

2

), pp.

163

–

172

. 10.1109/TIV.2018.2804162

Google Scholar

Crossref

Human Driver Modeling Based on Analytical Optimal Solutions: Stopping Behaviors at the Intersections

Abstract

Introduction

Longitudinal Human Driving Problem

Assumptions.

Optimal Control Problem Formulation.

Pontryagin’s Minimum Principle With SVIC.

Analytical Solutions

Type 1: Inactive SVIC.

Type 2: Active SVIC at Contact Point.

Type 3: Active SVIC on Boundary Interval.

Simulation Studies.

Validation With Experimental Data

Conclusions and Future Work

Acknowledgment

Conflict of Interest

Nomenclature

References

Get Email Alerts

Cited By

ASME Journals

ASME Conference Proceedings

ASME eBooks

Resources

Opportunities

Human Driver Modeling Based on Analytical Optimal Solutions: Stopping Behaviors at the Intersections

Abstract

Introduction

Longitudinal Human Driving Problem

Assumptions.

Optimal Control Problem Formulation.

Pontryagin’s Minimum Principle With SVIC.

Analytical Solutions

Type 1: Inactive SVIC.

Type 2: Active SVIC at Contact Point.

Type 3: Active SVIC on Boundary Interval.

Simulation Studies.

Validation With Experimental Data

Conclusions and Future Work

Acknowledgment

Conflict of Interest

Nomenclature

References

Get Email Alerts

Cited By

Related Articles

Related Proceedings Papers

Related Chapters

ASME Journals

ASME Conference Proceedings

ASME eBooks

Resources

Opportunities

This Feature Is Available To Subscribers Only