Data-Driven-Based Eco Approach for Connected and Automated Articulated Trucks in the Space Domain

Abstract

Since conventional eco approach systems can only achieve longitudinal automation, they may be disabled due to the impedance of the slow-moving vehicle. In addition, they could sacrifice a lot in travel time and it is hard to control the articulated truck with a complex dynamic model. This research presents an enhanced eco approach system for connected and automated articulated trucks, and is able to: (i) overtake slow-moving vehicles for sustainability and mobility; (ii) efficiently optimize the travel duration approaching a signalized intersection; (iii) achieve the trade-off between fuel saving and vehicle mobility; and (iv) improve computational efficiency and optimality for the articulated truck control. To achieve these features, the problem was formulated as an optimal control problem. A longitudinal and lateral coupled truck dynamic model was utilized for enabling the truck to own the automatic overtaking capability. The data-driven-based Koopman operator theory was adopted to globally linearize the truck dynamic model for reducing the computational burden while ensuring optimality. The optimal control problem is transformed from the time domain to the space domain in order for optimizing travel duration and considering the signal timing constraint. A quantitative evaluation was conducted to validate the performance of the Koopman system dynamics. In addition, the simulation experiment was designed to compare the proposed controller against human drivers and the conventional eco approach, which only has longitudinal automation. The results demonstrate that the proposed controller improves the fuel efficiency by 5.12–67.15%, and outperforms the two baseline controllers by 9.09–32.65% in terms of fuel saving. This range is caused by the different arrival times of the ego articulated truck.

1. Introduction

Worldwide, greenhouse gas (GHG) emissions and air pollution are continuously receiving considerable attention. As one of the main sources of emissions and pollution, the transportation sector accounts for the largest share of fuel consumption and GHG emissions [1]. According to the statistics of the International Energy Agency (IEA), all over the world, transportation had the highest reliance on fossil fuels and accounted for 37% of CO₂ emissions among all economic sectors in 2021 [2]. More notably, IEA forecasts that emissions from transportation will continue to rise in the future. Furthermore, the largest source of transportation-related emissions is road transport, including passenger transport and freight transport. This source accounts for over one-half of the emissions and fuel consumption of the entire transportation sector [1]. Therefore, there is a great need to find out a countermeasure to reduce fuel consumption and emissions in road transportation.

The U.S. Environmental Protection Agency (EPA) has proposed some guidelines on emission reduction opportunities in the transportation sector, such as fuel switching, operating practice improvement, and travel demand reduction [1]. In recent years, with the rapid development of Connected and Automated Vehicles (CAV), the improvement of operating practice for ecological purposes has gained considerable attention and has been gradually applied in road transport. The most representative measure for the improvement of operating practice in the field of road transport is eco driving. A CAV application maintains an optimum vehicle driving state to minimize fuel consumption in response to specific situations, such as changing slopes and incoming traffic. The measures of eco driving include avoiding rapid acceleration and braking, reducing engine idling, accelerating moderately, and maintaining driving space [3]. Existing studies have demonstrated that, in the urban driving cycle, eco driving could generate savings of up to 58% in fossil fuel consumption [4,5]. Accordingly, GHG emissions and air pollution could also obtain significant improvements.

Eco driving can be classified into three categories: eco cruise, eco approach, and eco departure. Among them, eco cruise is a type of cruise control that enables the CAV to be driven in a fuel-efficient manner in uninterrupted highways [6,7,8,9,10,11,12], while eco approach and eco departure are upgrades of eco cruise, and are oriented to interrupted signalized arterials [13,14,15]. They are able to coordinate the driving behavior of CAV with signal timing in order to achieve fuel saving and emission reductions. Since the signalized intersection on arterials, such as a bottleneck interrupting traffic flow, has a more negative impact on driving behavior and sustainability than highways, the research and application of eco approach or eco departure are of greater importance than those of eco cruise. However, differently from eco cruise, eco approach and eco departure highly rely on SPaT (Signal Phase and Timing) message, and thus the support of connected infrastructure is indispensable. Fortunately, in 2017, a SPaT challenge [16] was activated by USDOT (U.S. Department of Transport) to encourage the deployment of DSRC infrastructure with SPaT broadcasts. Within a year, a total of 26 states committed 216 operating signals in over thirty corridors. By 2020, the number of signals with SPaT broadcasts increased to approximately 2100. With the deployment of such a large number of DSRC infrastructure, the eco approach will be in a position to be applied practically on a large scale in the future.

A majority of the existing studies on eco approach are on the basis of Model Predictive Control (MPC), which is prevailing in automotive control [17,18,19]. MPC is an advanced process control based on the iterative, receding finite horizon optimization of a known system dynamic model. The advantage of MPC is that it has the ability to integrate the forward information predicted by a system model to generate a series of optimal control commands, best achieving certain objectives while satisfying a set of constraints on control inputs and system states. For automotive control, particularly for the eco approach, its process is limited to multiple constraints, such as speed limit, signal timing, and the dynamic environment. Hence, considering the advantage of MPC, it gradually becomes the preferred control method in eco approach studies.

However, the existing MPC-based eco approach systems achieve automation only in the longitudinal direction [20,21,22,23,24,25]. They tend to focus on the optimization of vehicle speed or gearbox in an ideal environment where no any other general vehicle impedes the normal driving operation of the CAV equipped with an eco approach controller. Factually, in the practical traffic environment, the operation of other general vehicles has an adverse influence on the eco approach controller with only longitudinal automation. For example, if a slow-moving vehicle is driven directly in front of the ego CAV (the host CAV being controlled), as the eco approach could just automate the ego CAV in the longitudinal direction, the ego CAV has to decelerate and follow the impeding vehicle. It means that the eco approach function is almost disabled. On the contrary, if the ego CAV is able to automatically adopt a lane-changing maneuver and overtake the slow-moving vehicle, the eco approach function could be maintained and the fuel efficiency could be improved. Therefore, it is of utmost importance to develop an eco approach controller that is able to achieve both longitudinal and lateral automation.

In fact, commercial vehicles, especially heavy-duty trucks, have a higher demand for eco approach functions with bi-directional automation than passenger vehicles. The reason for this is based on the fact that heavy-duty trucks account for a large share of road-transport-related fuel consumption and emissions [26]. In addition, heavy-duty trucks are more likely to be cut in and impeded by other light-duty vehicles. Hence, there is a greater need for heavy trucks to be equipped with an advanced eco approach controller. Furthermore, to save on logistics costs and reduce emissions, many countries around the world actively keep promoting freight transport automation and have been inclined to apply CAV technology to heavy trucks first [27,28]. This motivation offers an opportunity for heavy-duty trucks to receive the advanced eco approach function earlier.

Nevertheless, the complexity of the truck dynamic model brings a great challenge to heavy-duty truck control, particularly to articulated truck control. Differing from the model of the passenger vehicle that is easier to be linearized based on the small-angle assumption [29,30], as a kind of typical and presentative heavy-duty truck, the articulated truck has a bi-directional coupled and highly-nonlinear dynamic model that provides a considerable burden on the computation efficiency and optimality. For solving this control problem in the articulated truck, some researchers proposed more assumptions to linearize the dynamic model [31,32], but this results in a serious loss of model accuracy. Some researchers considered local linearization and designed an iterative algorithm to solve nonlinear MPC problems [33,34], but the computation efficiency is unable to support online operation. Koopman operator theory was proposed by Bernard Koopman in 1931 [35,36]. It is a global linear transformation method that lifts the dynamics of a finite-dimensional nonlinear system to an infinite-dimensional function space where the evolution of the original system is in linearity. In recent years, with the rise of data science, Koopman operator theory has obtained wide attention again. Researchers could utilize rich data to construct a finite-dimensional approximation for the Koopman operator so as to achieve the global linearization of the complex dynamic system [37,38,39,40,41,42]. Therefore, in terms of the eco approach controller design for the articulated truck, a data-driven-based Koopman operator would offer great potential to solve the control challenge presented by the nonlinear dynamic system.

Furthermore, there is another shortcoming in the majority of eco approach studies. It is that, for most MPC-based eco approaches, the optimization time horizon is provided rather than being optimized [5,43]. In other words, the travel duration of the ego CAV is fixed so that the vehicle can make it through the signal light. The pre-setting travel duration restricts the flexibility of the CAV’s maneuver, thus resulting in fuel saving not being able to achieve optimality. Even though some researchers assumed the duration as an optimization variable [6,44,45,46], these studies do not take into account the signal timing constraints and are not concerned with eco approaches. More importantly, the efficiency of the variable horizon optimal control problem does not support real-time computation in receding horizon. Consequently, it is critical, for the eco approach controller, to design a high-efficient optimal control algorithm that considers the signal timing constraints and optimizes the travel duration.

Considering the shortcomings of past studies on the eco approach, this research proposes to enhance eco approach technology in terms of the consideration of vehicle impedance, dynamic model of articulate trucks, and travel duration optimization. The proposed eco approach controller enables the articulated truck to pass the green light ecologically and has the following features, which existing eco approach controllers do not possess:

• Is able to overtake slow-moving vehicles in order to achieve sustainability and mobility.
• Is Able to efficiently optimize the travel duration approaching a signalized intersection.
• Achieves the trade-off between fuel saving and vehicle mobility.
• Improved computational efficiency and optimality for the articulated truck control.

The remainder of the paper is organized as follows: The Section ‘Research scope’ proposes the research scope and the highlights; the Section ‘Problem formulation’ provides a detailed formulation of the proposed eco approach controller; the Section ‘Solution Method’ offers an algorithm to numerically solve the optimal control problem; the Section ‘Verification’ identifies all the specifics of the evaluation and presents the results and findings; the Section ‘Conclusion and future research’ concludes this research and discusses future studies.

2. Research Scope

The target of the proposed eco approach controller was to improve the fuel efficiency of a CAV articulated truck approaching an isolated signalized intersection. There are three main highlights of this proposed controller:

• No more waiting for the slow-moving vehicle: The proposed eco approach controller can enable the CAV truck to automatically overtake the slow-moving vehicles when they impede the normal operation of the ego truck. The lane-changing and overtaking maneuvers reduce the chance of being obstructed and provides the opportunity for the ego truck to further improve fuel saving and mobility. It also enables the passengers on board to have a more comfortable experience.
• Highly efficient and optimal control for the articulated truck: One feature of the research is to solve the computational efficiency and optimality challenge that the nonlinear dynamic model adds to the control of the articulated truck. The research utilized data-driven-based Koopman operator theory to globally linearize the dynamic model of the articulated truck. The linearized dynamic model can support the highly efficient receding horizon control while ensuring the accuracy of the original model.
• Ecological but not slow: The proposed controller has the ability to balance fuel saving and travel speed. The controller is designed to optimize travel time together with fuel cost. Weights are ascribed to both optimized objectives. Therefore, the amount of mobility sacrificed to save fuel can be adjusted according to the user’s preference.

The logic of the proposed eco approach system is shown in Figure 1. It contains two components: (I) offline system modeling and (II) online control. In terms of offline system modeling, it is focused on collecting data offline according to the original truck dynamic model, and then the data-driven-based Koopman operator theory was utilized to linearize the truck model. The linearized model can be stored in the eco approach system in advance. In terms of online control, it is divided into three modules. Module 1 addresses the communication between vehicles and infrastructures. Module 2 and module 3 achieve the functions of planning and control. The details of the various modules are provided as follows:

• Module 1: As indicated in Figure 1, the controlled area is defined as the communication range of roadside units. In this area, the information of signal timing and traffic conditions is collected by the roadside units and transmitted to the ego truck through vehicle-to-infrastructure (V2I) communication. The proposed eco approach system requires to be equipped with the following devices: (i) onboard GPS; (ii) on-board communication device; (iii) communication devices traffic lights; and (iv) roadside units supporting perception and V2I communication.
• Module 2: This module is activated when the central controller detects that the ego truck has just entered the controlled area. The state of the ego truck and the information collected in Module 1 are fed to the controller. Then, combining the stored dynamic model and the input information, the controller can optimize the travel duration and the vehicle trajectory for the improvement of sustainability and mobility.
• Module 3: The optimal trajectory generated from Module 2 is transmitted to the CAD truck as the control commands. Then, the CAV truck receives commands and adjusts its states accordingly.

3. Problem Formulation

The formulation of the proposed eco approach system is presented in this section. A longitudinal and lateral coupled dynamic model of articulated truck was adopted to enable the ego truck to have automatic overtaking capability.

In order to reduce the computational burden and achieve real-time control, the data-driven-based Koopman operator theory was adopted to linearize the truck dynamic model. The globally linearized system dynamics are able to support the receding horizon control both in longitudinal and lateral directions while ensuring the accuracy of the original model.

In order to optimize travel time and achieve the trade-off between fuel saving and mobility, the proposed eco approach controller was formulated as an optimal control problem in the space domain instead of the time domain. For the time domain formulation, the optimization time horizon was a variable, and the free-time optimal control problem was hard to be solved efficiently. By contrast, in the space domain, the optimization space horizon is fixed since the distance from the ego truck’s current position to the stop bar is known, and the time can be taken as a state. In addition, the signal timing constraints were considered in the optimal control problem in order to ensure driving safety.

Table 1. Indices and parameters.

Indices and Parameters	Meaning
$\mathit{x}$	The state vector of the system dynamics
$t$	The time variable (s)
$\mathit{w}$	Ego truck’s slowness (s/m)
$\dot{y}$	Ego truck’s lateral speed (m/s)
$\dot{{ϵ}_r}$	Relative yaw angle rate of the tractor from the road centerline (rad/s)
$\dot{{ϵ}_f}$	Relative yaw angle rate of the tractor and the trailer (rad/s)
$y$	Lateral position of the tractor C.G. from the road centerline (m)
${ϵ}_r$	Relative yaw angle of the tractor from the road centerline (rad)
${ϵ}_f$	Relative yaw angle of the tractor and the trailer (rad)
$s$	Ego truck’s longitudinal position (m)
$v$	Ego truck’s longitudinal speed (m/s)
$a$	Ego truck’s longitudinal acceleration (m/s2)
$u$	The control input vector of the system dynamics
$b$	Ego vehicle’s moderation (s/m2)
${\delta }_f$	Ego vehicle’s steering angle (rad)
$\mathcal{K}$	Koopman operator
$\psi$	The real-value observable (a nonlinear base function)
$\mathit{A}$	The system matrix of the Koopman system dynamics
$\mathit{B}$	The control matrix of the Koopman system dynamics
$t_0$	The current time detected by the on-board device (s)
$t_f$	The terminal time when the ego truck passes through the stop line (s)
$s_0$	The current longitudinal position of the ego truck (m)
$∆\mathrm{L}$	The longitudinal distance from the vehicle’s current position to the stop line (m)
$y_{des}$	Ego vehicle’s desired lateral position (the preferred lane) (m)
$C$	The cycle of the signal light (s)
$G$	The duration of the green light (s)
$R$	The duration of the red light (s)

lists the indices and parameters utilized hereafter.

3.1. State and Control Input Definition

Definition 1:

In the space domain, the longitudinal position $s$ of the ego truck is taken as the independent variable and the time is taken as a component of the state vector. Detailedly, the state vector and the control input vector are respectively defined as follows:

$$\mathit{x}={\left(t,w,\dot{y},\dot{{ϵ}_r,}\dot{{ϵ}_f},y,{ϵ}_r,{ϵ}_f\right)}^T$$

(1)

$$\mathit{u}={\left(b,{\delta }_f\right)}^T$$

(2)

where, in terms of the state vector, $w$ is the longitudinal variable that is defined as slowness [47], $w=\frac{dt}{ds}=\frac{1}{v}$. In terms of the control input vector, $b$ is the longitudinal control input that is defined as moderation [47], $b=\frac{dw}{ds}=-a{w}^3$.

In addition, for a clearer illustration of the problem, the state and control input definition of the articulated truck is depicted in Figure 2.

3.2. Koopman System Dynamics

In this subsection, the dynamic model of the articulated truck is globally linearized through Koopman operator theory. This theory is briefly described as follows:

For any autonomous dynamic system $\dot{\mathit{x}}=f\left(\mathit{x}\right),$ $\mathit{x}\in {ℝ}^n$, the Koopman operator is defined as the following equation [35]:

$$\mathcal{K}\psi \left(\mathit{x}\right)=\psi \left(f\left(\mathit{x}\right)\right)$$

(3)

where $\psi :{ℝ}^n\mapsto ℝ$ is a real-value observable (often referred to as a nonlinear base function) of the dynamic system. A linear vector space is spanned by the collection of all such observables. The Koopman operator is denoted as $\mathcal{K}$, which is a linear transformation on this linear vector space. It serves as a lifting of the dynamics from the state space to the linear space of the observables. The advantage of this lifting is that the evolution of the dynamics is provided as a linear rule, but the disadvantage is that this linear space of the observable is infinite-dimensional.

With the rise of data science, researchers have focused on utilizing rich data to achieve the finite-dimensional approximation for the Koopman operator. Among these studies, the extended dynamic mode decomposition (EDMD) [41] has gained considerable attention due to its great interpretability and few computation requirements. The authors of [39] further generalized the EDMD algorithm from the autonomous system to the controlled system.

Theorem 1.

Based on the EDMD algorithm for the controlled system [39], the coefficient matrices of the linearized system dynamics of the articulated truck can be obtained via the following equation:

$$\left[\mathit{A},\mathit{B}\right]={\mathit{Y}}_{lift}{\left[{\mathit{X}}_{lift}, \mathit{U}\right]}^{†}$$

(4)

where $†$ represents the Moore–Penrose pseudoinverse of a matrix.

$\mathit{A}$ and $\mathit{B},$ respectively, denote the system matrix and control matrix of the linearized system dynamics.

${\mathit{X}}_{\mathrm{lift}}$ is the data in form of space series generated by the known observables:

$${\mathit{X}}_{lift}=\left[\mathit{\psi }\left({\mathit{x}}_1\right),\mathit{\psi }\left({\mathit{x}}_2\right),\dots ,\mathit{\psi }\left({\mathit{x}}_K\right)\right]$$

(5)

where $\mathit{\psi }\left(\mathit{x}\right)={\left[{\psi }_1\left(\mathit{x}\right),{\psi }_2\left(\mathit{x}\right),\dots .,{\psi }_N\left(\mathit{x}\right)\right]}^T$ is a vector in the lifted linear space of observables. ${\psi }_i\left(\mathit{x}\right)$ (i = 1, 2, …, N) is a given basis of the nonlinear function.

${\mathit{Y}}_{lift}$ is a series of predicted data generated by the known observables and the original dynamic model of the articulated truck:

$${\mathit{Y}}_{lift}=\left[\mathit{\psi }\left({\mathit{y}}_1\right),\mathit{\psi }\left({\mathit{y}}_2\right),\dots ,\mathit{\psi }\left({\mathit{y}}_K\right)\right]$$

(6)

where ${\mathit{y}}_k=g\left({\mathit{x}}_k,{\mathit{u}}_k\right)$. $g\left({\mathit{x}}_k,{\mathit{u}}_k\right)$ is the original articulated truck dynamic model that is transformed from the time domain to the space domain. The detailed mathematic formulation of the original truck model could be found in [48].

$\mathit{U}$ is a series of control inputs in the space domain:

$$\mathit{U}=\left[{\mathit{u}}_1,{\mathit{u}}_2,\dots ,{\mathit{u}}_{\mathit{K}}\right]$$

(7)

Proof of Theorem 1.

Based on the Koopman operator theory, the state space could be lifted to a linear function space spanned by a series of observables. Given the nonlinear base function vector (observables) $\psi \left(\mathit{x}\right)$, the linearized system dynamics in the discrete form are formulated as follows:

$${\mathit{z}}_{\mathit{k}+1}=\mathit{A}{\mathit{z}}_{\mathit{k}}+\mathit{B}{\mathit{u}}_{\mathit{k}}$$

(8)

where $z_k=\mathit{\psi }\left({\mathit{x}}_k\right)={\left[{\psi }_1\left({\mathit{x}}_k\right),{\psi }_2\left({\mathit{x}}_k\right),\dots .,{\psi }_N\left({\mathit{x}}_k\right)\right]}^T$. The undetermined matrices $\mathit{A}$ and $\mathit{B}$ form the Koopman operator in this lifted space.

It was assumed that a set of data in a space series $\mathit{X}=\left[{\mathit{x}}_1,{\mathit{x}}_2,\dots ,{\mathit{x}}_K\right],\mathit{Y}=\left[{\mathit{y}}_1,{\mathit{y}}_2,\dots ,{\mathit{y}}_K\right],\mathit{U}=\left[{\mathit{u}}_1,{\mathit{u}}_2,\dots ,{\mathit{u}}_K\right]$ is available via experiment or simulation, and the data satisfy the relation ${\mathit{y}}_k=g\left({\mathit{x}}_k,{\mathit{u}}_k\right)$. The given base function vector $\mathit{\psi }\left({\mathit{x}}_k\right)$ lifts the data from the state space to the linear space of observables, and the dataset ${\mathit{X}}_{lift}$, ${\mathit{Y}}_{lift}$ can be achieved. Combining Equation (8), the lifted data satisfy the following relation:

$${\mathit{Y}}_{lift}=\mathit{A}{\mathit{X}}_{lift}+\mathit{B}\mathit{U}$$

(9)

The solution to matrices $\mathit{A}$ and $\mathit{B}$ can be obtained according to the following least squares problem:

$$\left[\mathit{A},\mathit{B}\right]=\underset{\mathit{A},\mathit{B}}{argmin}||{\mathit{Y}}_{lift}-\mathit{A}{\mathit{X}}_{lift}-\mathit{B}\mathit{U}||$$

(10)

Equation (4) can be achieved by analytically solving this least squares problem and, thus, Theorem 1 is proven. □

3.3. Cost Function

The objective of the optimization not only focuses on fuel saving, but also on the improvement of mobility. The cost function in the space domain is defined as follows:

$$\underset{u}{\mathit{min}}J=\underset{u}{\mathit{min}}{\beta }_0{\left(t_f-t_0\right)}^2+{{\displaystyle \int }}_{s_0}^{s_o+∆L}\left[{\beta }_{1}bw+{\beta }_2{\left(y-y_{des}\right)}^2+{\beta }_3{ϵ}_r{}^2+{\beta }_4{ϵ}_f{}^2\right]ds$$

(11)

where ${\beta }_0$, ${\beta }_1$, ${\beta }_2$, ${\beta }_3$, and ${\beta }_4$ are the weighting factors that could be tuned according to the user’s preference. $t_0$ is the current moment detected by the onboard equipment. $t_f$ is the terminal time that is a decision variable. $t_f-t_0$ represents the travel duration that the truck takes to make through the signal light. It reflects the mobility of the ego truck. $s_0$ is the current longitudinal position detected by onboard GPS. $∆L$ is the longitudinal distance from the vehicle’s current position to the stop line. The first term in the integral represents the power request that indirectly reflects fuel consumption. The second term represents the lane-changing demand of the truck. The last two terms in the integral denote the stability of the truck pose.

3.4. Constraints

The optimal control problem is constrained by system dynamics, speed limit, time constraints, control input range, collision avoidance, and initial condition. The system dynamics constraints are presented in the Section ‘Koopman System dynamics’. The information of signal timing is considered in the time constraints.

(1)

Speed limit: The speed limit constrains both the longitudinal speed (referred to as the slowness in the space domain) and the lateral speed. The longitudinal speed will never exceed the road speed limit and is greater than zero. The lateral speed should be constrained within a range to guarantee safety. The speed limit constraints are specified as:

$$w_{min}\le w\le w_{max}$$

(12)

$${\dot{y}}_{min}\le \dot{y}\le {\dot{y}}_{max}$$

(13)

(2)

Time constraints: As shown in Figure 3, time-related parameters are defined. C is the cycle length, R is the red light duration, and G is the green light duration. The amber light is considered as a part of the green light. $\tau$ is defined as the minimum travel duration for the ego truck to reach the stop bar from its current position. This time constraint guarantees that the terminal time $t_f$ is constrained by signal timing so that the ego truck is able to pass through the stop line before the end of the green light. Therefore, it would make a difference whether the ego truck passes the intersection during this current green or the next green.

If the ego truck enters the controlled area when the signal light is in red duration, the constraint is formulated as follows:

$$R_{rest}\le t_f-t_0\le R_{rest}+G$$

(14)

where $R_{rest}$ refers to the rest red duration until the light turns green.

If the ego truck enters the controlled area when the signal is green, this constraint relies on whether the ego truck has the capability to pass through the stop bar before the light turns red. The constraint is specified as follows:

$$\{\begin{array}{c}{t}_f-t_0\le G_{rest}, if G_{rest}\ge \tau \\ t_f-t_0\ge G_{rest}+R, if G_{rest}\le \tau \end{array}$$

(15)

where $G_{rest}$ represents the rest of green time until the red light starts.

The minimal travel time $\tau$ is calculated as follows. It is determined by the time that the ego truck spends during acceleration ($t^{*})$ and the distance that the ego truck travels during the acceleration $\left(L^{*}\right)$:

$$\tau =t^{*}-t_0+\frac{∆L-L^{*}}{v_{max}}$$

(16)

$$t^{*}=\frac{v_{max}-v_0}{a_{max}}+t_0$$

(17)

$$L^{*}=\frac{v_{max}{}^2-v_0{}^2}{2a_{max}}$$

(18)

where $t^{*}$ is the travel time that the ego truck spends in the acceleration phase; $L^{*}$ is the longitudinal distance driven by the ego truck during the acceleration phase; $v_{max}$ is the road speed limit and $a_{max}$ is the maximum acceleration.

(3)

Control input range: The control input, including the moderation and the steering angle, are in a reasonable value range where both the comfort and the vehicle performance are taken into consideration. This constraint is specified as:

$$b_{min}\le b\le b_{max}$$

(19)

$${\delta }_{f,min}\le {\delta }_f\le {\delta }_{f,max}$$

(20)

(4)

Collision avoidance: This constraint ensures that the ego truck keeps a certain safe distance from the preceding vehicles on the road. In the space domain, headway is utilized to formulate this safety constraint. The detailed formulation of this constraint is shown as follows:

$$t-t_{pre}\ge h_{safe}$$

(21)

where $t$ and $t_{pre}$, respectively, are referred to the time that the ego truck and the preceding vehicle spend on reaching the same position; and $h_{safe}$ is the pre-setting value of the safe time headway.

(5)

Initial condition: The initial condition of the optimal control problem is formulated as follows:

$${\mathit{z}}_0=\mathit{\psi }\left({\mathit{x}}_0\right)={\left[{\psi }_1\left({\mathit{x}}_0\right),{\psi }_2\left({\mathit{x}}_0\right),\dots .,{\psi }_N\left({\mathit{x}}_0\right)\right]}^T$$

(22)

$${\mathit{x}}_0={\left(t_0,w_0,\dot{y_0},\dot{{ϵ}_r{}_0,}\dot{{ϵ}_f{}_0},y_0,{ϵ}_r{}_0,{ϵ}_f{}_0\right)}^T$$

(23)

The information of the initial state ${\mathit{x}}_0$ can be collected by the onboard and roadside detection devices. The initial condition ${\mathit{z}}_0$ in the lifted linear space can be achieved by operating the given observables on ${\mathit{x}}_0$.

4. Solution Method

The offline linearization of the system dynamics is solved via the EDMD algorithm, which was introduced in detail in Section 3.2. The solution algorithm is summarized as follows (Algorithm 1):

Algorithm 1: EDMD for the Koopman system dynamics

Input: a set of data $\mathit{X}=\left[{\mathit{x}}_1,{\mathit{x}}_2,\dots ,{\mathit{x}}_K\right],\mathit{U}=\left[{\mathit{u}}_1,{\mathit{u}}_2,\dots ,{\mathit{u}}_K\right]$ collected by the simulation experiment; the observable $\psi (∆)$; the original model of the articulated truck g(xk, uk)

Output: the coefficient matrices $\mathit{A}$ and $\mathit{B}$ in the linearized system dynamics

1.    Use dataset $\mathit{X},\mathit{U},$ and the truck model to achieve predicted data $\mathit{Y}=\left[{\mathit{y}}_1,{\mathit{y}}_2,\dots ,{\mathit{y}}_K\right]$

2.    Use the observable to lift X, Y to ${\mathit{X}}_{lift}$, ${\mathit{Y}}_{lift}$, where ${\mathit{X}}_{lift}=\left[\mathit{\psi }\left({\mathit{x}}_1\right),\mathit{\psi }\left({\mathit{x}}_2\right),\dots ,\mathit{\psi }\left({\mathit{x}}_K\right)\right]$, ${\mathit{Y}}_{lift}=\left[\mathit{\psi }\left({\mathit{y}}_1\right),\mathit{\psi }\left({\mathit{y}}_2\right),\dots ,\mathit{\psi }\left({\mathit{y}}_K\right)\right]$

3.    Solve the least square problem: $\left[\mathit{A},\mathit{B}\right]=\underset{\mathit{A},\mathit{B}}{\mathrm{argmin}}{\mathit{Y}}_{lift}-\mathit{A}{\mathit{X}}_{lift}-\mathit{B}\mathit{U}$

4.    Obtain $\left[\mathit{A},\mathit{B}\right]$ by calculating ${\mathit{Y}}_{lift}{\left[{\mathit{X}}_{lift}, \mathit{U}\right]}^{†}$

The optimal control problem is solved by Dynamic Programing [49]. The complexity of this algorithm is O(n). The algorithm is summarized as follows (Algorithm 2). Furthermore, MPC is adopted for the implementation of the proposed controller. In the MPC mechanism, the purpose of the proposed controller is to find the cost-minimizing control strategy over the entire optimization horizon. Only the control strategy within the update horizon was applied to the ego truck. Then, the state of the traffic (vehicles and signal timing information) was sampled again and the calculation was repeated using the new current state, yielding a new control and new predicted state path. This mechanism enables the proposed controller to be implemented in a closed loop and forms feedback control.

Algorithm 2: Dynamic Programing for the optimal control problem

Input: linearized system dynamics ${\mathit{z}}_{k+1}=\mathit{A}{\mathit{z}}_k+\mathit{B}{\mathit{u}}_k$, the weighting factors ${\beta }_0$, ${\beta }_1$, ${\beta }_2$, ${\beta }_3$, β4, the initial condition ${\mathit{z}}_0,$ the constraint-related parameters

Output: control ${\mathit{u}}_k$ and state ${\mathit{x}}_k$ on each control step

1.    Discretize the optimal control problem, $k=0,1,\dots .,K$

2.    Calculate the weighting matrix ${\mathit{Q}}_k$ and ${\mathit{S}}_k$ according to ${\beta }_0$, ${\beta }_1$, ${\beta }_2$, ${\beta }_3$ and β4. ${\mathit{Q}}_k$ is the weight of the quadratic term of states, ${\mathit{S}}_k$ is the weight of the coupling term of the states and control inputs:

3.    ${\mathit{Q}}_k=diag\left(0,0,0,0,0,{\beta }_2,{\beta }_3,{\beta }_4\right), \left(k=0,\dots .,K-1\right)$,

4.    ${\mathit{Q}}_K=diag\left({\beta }_0,0,0,0,0,0,0,0\right)$,

5.    ${\mathit{S}}_k={\left[\begin{array}{cccccccc}0& {\beta }_1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]}^T, \left(k=0,\dots .,K\right)$

6.    Initialization: set ${\mathit{P}}_K={\mathit{Q}}_K$

Backward compute${\mathit{P}}_{\mathit{k}}$

7.    For k$\leftarrow$K-1 to 0 do

8.          ${\mathit{P}}_k\leftarrow {\mathit{A}}^T{\mathit{P}}_{k+1}\mathit{A}+{\mathit{Q}}_k-({\mathit{A}}^T{\mathit{P}}_{k+1}\mathit{B}+{\mathit{S}}_k){\left({\mathit{B}}^T{\mathit{P}}_{k+1}\mathit{B}\right)}^{-1}\left({\mathit{B}}^T{\mathit{P}}_{k+1}\mathit{A}+{\mathit{S}}_k{}^T\right)$

9.    End for

Forward compute the control law

10.    For k$\leftarrow$0 to K-1 do

11.    ${\mathit{u}}_k\leftarrow -{\left({\mathit{B}}^T{\mathit{P}}_{k+1}\mathit{B}\right)}^{-1}({\mathit{B}}^T{\mathit{P}}_{k+1}\mathit{A}+{\mathit{S}}_k{}^T)*{\mathit{z}}_k$

12.    ${\mathit{z}}_{k+1}\leftarrow \mathit{A}{\mathit{z}}_k+\mathit{B}{\mathit{u}}_k$

13.    If any constraint is activated on the state at the k + 1 step, then

14.    Set the control input at the k + 1 state as the feasible control ufeasible

15.    Reset the control input ${\mathit{u}}_k\leftarrow min\left[max\left({\mathit{u}}_{min},{\mathit{u}}_{feasible}\right),{\mathit{u}}_{max}\right]$

16.    End if

17.    End for

5. Verification

In this section, the performance of the proposed eco approach system is evaluated. The simulation evaluation contained two experiments. One is the numerical experiment that verifies the accuracy of the Koopman system dynamics. The other experiment is to validate the performance of the controller and to conduct a sensitivity analysis of the arrival time of the ego truck.

5.1. Evaluation for Koopman System Dynamics

In this subsection, the accuracy of the Koopman system dynamics is verified. It was compared to the original nonlinear dynamics of the articulated truck. In order to lift the dynamics to a linear space of observables, the original system dynamics were discretized with space step $∆s=1m$, and 100 trajectories over 500 steps were sampled in the simulation experiment. The control input of each trajectory was a random signal with uniform distribution over the control input range. Each trajectory started from the initial state, which was randomly generated from the uniform distribution. The base functions ${\psi }_i$ (i = 1, …, N) were chosen to be the state itself (${\psi }_1\left(\mathit{x}\right)=t,{\psi }_2\left(\mathit{x}\right)=w,{\psi }_3\left(\mathit{x}\right)=\dot{y},{\psi }_4\left(\mathit{x}\right)=\dot{{ϵ}_r},{\psi }_5\left(\mathit{x}\right)=\dot{{ϵ}_f},{\psi }_6\left(\mathit{x}\right)=y,{\psi }_7\left(\mathit{x}\right)={ϵ}_r,{\psi }_8\left(\mathit{x}\right)={ϵ}_f,$) and 50 thin plate spline radial basis functions with centers ${\mathit{x}}_c$ were selected randomly with uniform distribution (${\psi }_i\left(\mathit{x}\right)=||\mathit{x}-{\mathit{x}}_c{||}^2\log ||\mathit{x}-{\mathit{x}}_c||,i=9,\dots N$). Hence, the dimension of the state space was lifted to N = 58.

Figure 4 shows the evaluation result of Koopman system dynamics. The result was transformed from the space domain to the time domain for easier observation. It compares the trajectory respectively generated by the original dynamics (${\xi }_{original}$) and the Koopman dynamics (${\xi }_{Koopman}$) starting from the same initial state with a random control signal.

The relative root-mean-square error (RMSE) was used to measure the evaluation results. The mathematical formulation of RMSE was specified as Equation (24). As depicted in Figure 4a–b, it is observed that the longitudinal trajectory of Koopman dynamics almost overlaps the one from the original dynamics and the relative RMSE is about 0.002%. This means that the features of the dynamics in the longitudinal direction are hardly lost when the dynamics are lifted to the linear space of observables. The reason is that, as in the case of as any other vehicle, the longitudinal system dynamics of the articulated truck are also in linearity. The Koopman operator cannot exert an adverse impact on the linear system dynamics. The features of the original linear dynamics can be fully reproduced in the lifted space. In addition, Figure 4c–d indicate the evaluation results of the lateral system dynamics. The relative RMSE associated with the lateral position is approximately 18.9% and that on lateral speed is about 2.8%. Due to the high nonlinearity of the lateral truck dynamic model, a portion of the system dynamic features might be lost in the process of lifting, thus the Koopman operator cannot fully reproduce the original dynamics. However, as the error is small and in an acceptable range, the lateral Koopman dynamics are able to support the eco approach controller design for the articulated truck.

$$RMSE=\frac{||{\xi }_{Koopman}-{\xi }_{original}||}{||{\xi }_{original}||}$$

(24)

5.2. Evaluation for the Eco Approach Controller

In this subsection, the performance of the proposed eco approach controller is validated. For the verification, a simulation experiment was designed, including Measurements of Effectiveness (MOE), controller types, and parameter setting. In addition, a sensitivity analysis on arrival time was conducted in the experiment to verify how the fuel consumption and mobility change with respect to the arrival time of the ego truck.

5.2.1. Experiment Design

(1)

Measurements of Effectiveness (MOE): Fuel consumption, average speed, and fuel efficiency were adopted as the MOE. Fuel consumption was calculated by the VT-micro model [50]. Averaged speed was utilized for measuring the mobility level of the ego truck. In addition, a new MOE, named fuel efficiency, was proposed to standardize the fuel cost over mobility. It was defined as the fuel consumption per unit averaged speed:

$$\eta =\frac{c_{fuel}}{\overline{v}}$$

(25)

where $\eta$ is the fuel efficiency, $c_{fuel}$ denotes the fuel consumption, and $\overline{v}$ represents the average speed of the ego truck.

(2)

Controller types: The proposed eco approach controller was evaluated against two baseline controllers to demonstrate its advantage over human drivers and the benefit brought by the automated lane-changing and overtaking capability:

• Baseline: The ego truck is controlled by the human driver. In this case, a human-driven articulated truck cruises at a constant desired speed (20 m/s). The behavior of the human driver is simulated by the microscopic simulation software VISSIM. The VISSIM has a whole package of decision maker and controller that is able to mimic the behavior of human drivers (Widemann driver model). The parameters in the model adopt the default ones that can show the regular human driver behavior.
• Baseline optimal controller: In this case, only longitudinal automation is considered. Neither lane-changing nor overtaking maneuver exist. This optimal controller is borrowed from [51]. The parameters in the cost function are tuned in order for the controller to have the same performance as the proposed eco approach controller in the scenario where no lane changing occurs.
• Proposed controller: Both longitudinal and lateral automation are considered. The ego articulated truck has automated lane-changing and overtaking capabilities.

(3)

Parameter settings: The following settings were adopted in the simulation experiment, and the partial parameters are shown in Figure 5:

• The controlled area considers the DSRC communication range; thus, the controller is activated when the ego truck arrives at 300 m upstream of the signalized intersection.
• The control step in the space domain is 1 m.
• The cycle of the signal is 90 s.
• The road speed limit is 45 mph (about 20 m/s). and the initial speed of the ego truck is 10 m/s.

The initial distance between the ego truck and the preceding vehicle is 20 m.

• The acceleration range is [−5,2] m/s² [5].
• The steering angle range is [−0.7,0.7] rad.
• The longitudinal safe headway is 2 s.
• The lane width is 3.5 m.
• The parameters in the truck dynamic model are those referred to in [48].
• ${\beta }_0=1$, ${\beta }_1=100$, ${\beta }_2=50$, ${\beta }_3=10$, ${\beta }_4=10$.

5.2.2. Sensitivity Analysis and Results

The performance of the proposed controller was demonstrated in the analytic test. The simulation experiment was designed as follows (As shown in Figure 5): An articulated truck cruises in the controlled area, and a general vehicle travels at a low speed in front of the ego truck. The desired speed of this impeding vehicle is 6.5 m/s. In addition, it is expected that the moment when the ego truck enters the controlled area has a significant effect on the results of the experiment, and thus a sensitivity analysis associated with the arrival time of the ego truck was conducted. Therefore, six arrival scenarios were tested:

• Scenario A: the truck enters the controlled area at the beginning of the red signal (5 s after the red light starts).
• Scenario B: the truck enters the controlled area in the middle of the red signal (25 s after the red light starts).
• Scenario C: the truck enters the controlled area at the end of the red signal (35 s after the red light starts).
• Scenario D: the truck enters the controlled area at the beginning of the green signal (5 s after the green light starts).
• Scenario E: the truck enters the controlled area in the middle of the green signal (20 s after the green light starts).
• Scenario F: the truck enters the controlled area at the end of the green signal (30 s after the green light starts).

The program was coded in MATLAB 2020a (This software is provided by MathWorks, and is run in Tongji University, Shanghai, China) and ran on an i5-8250U 1.80 GHz processor with 8.00 GB RAM. The computation time for each optimal control was about 30 ms. This high-speed computation efficiency indicates that the proposed controller has great potential for supporting receding horizon control.

Figure 6, Figure 7 and Figure 8 depict the results of the simulation experiment. The evaluation results demonstrate that the proposed eco approach controller can outperform the regular human driver by up to 32.65% in terms of fuel saving. Compared against the baseline optimal controller, the proposed controller can generate additional 17.50% fuel saving benefits. In addition, in the aspect of mobility, the proposed controller has a great impact on mobility. The largest mobility improvement can reach 160.36%. Fuel efficiency (fuel consumption per unit mobility) was improved by up to 67.15% owing to the proposed eco approach controller. These benefits confirm that the proposed controller can achieve the aforementioned optimization objectives: fuel saving and mobility improvement.

Figure 6a,b, respectively, shows the fuel consumption and the benefit percentage under different arrival scenarios. It is observed that, in the duration of the red light, the significant fuel saving brought by the proposed controller could be achieved. The reason is that, in the red duration, especially in the middle and at the end of the red light, even though the regular human driver could overtake the preceding slow-moving vehicle for the purpose of fuel saving, it has to decelerate or stop due to the sight of the red signal and re-accelerate when the light turns green. The unnecessary deceleration and sudden acceleration waste more fossil fuels. By contrast, the proposed controller can integrate the signal timing information to plan the optimal trajectory, and thus the ego truck can moderately decelerate in the red light duration and decide its best time to pass through the stop bar. In the middle and at the end of the green light, significant fuel consumption benefits could also be observed. Under these two arrival scenarios, the rest green time is too short to be achieved; thus, the proposed controller decides to slightly slow down to make the next green signal. This decision could allow the truck to avoid a full stop and save fuel. On the contrary, the human-driven vehicle could immediately accelerate to pass through the stop bar, but fails and has to stop. Both acceleration and stopping result in more fuel consumption. Furthermore, there is an interesting phenomenon that, at the beginning of the green light, the fuel consumption generated by the proposed controller is higher than that obtained by the baseline optimal controller. This occurs because the proposed controller decided to sacrifice fuel in order to achieve a better overall fuel efficiency.

Figure 7a,b confirms that the proposed controller has a significant performance with respect to mobility improvement. In all test scenarios, since the conventional eco approach controller (baseline optimal controller) does not have automated lane-changing and overtaking capabilities, the ego truck has to follow the impeding vehicle and its mobility suffers a more serious loss than that of the human driver and the proposed controller. For the human-driven vehicle, as it could overtake the obstacle vehicle in front, its mobility improvement is significant and even higher than that obtained by the proposed eco approach controller in certain test scenarios. Nevertheless, the fuel consumption of the human-driven vehicle increases due to the human driver’s inappropriate operation. In other words, the great mobility improvement in the human-driven vehicle is based on the sacrifice of fuel saving. This result is confirmed in Figure 6.

The performance of the proposed control in terms of fuel efficiency is demonstrated in Figure 8a,b. It is confirmed that, in any scenario, under the same mobility level, the eco approach controller with lane-changing and overking capabilities has the ability to save the most fuel. The most fuel efficiency benefits are observed when the arrival time is in the middle or at the end of the red light duration. This occurs because, under these two test scenarios, the proposed controller represents a great performance both on fuel consumption and mobility. Furthermore, it is worth noting that, in scenario D (at the beginning of the green light), the performance of the proposed controller is worse than that of the conventional controller in terms of fuel saving, as shown in Figure 6, but for fuel efficiency, the benefit is significant. This phenomenon is due to the mobility benefits offered by the proposed controller compensating, and even exceeding, the fuel cost, and thus the overall fuel efficiency can be improved. This result also demonstrates that the conventional eco approach controller has serious shortcomings and is unable to achieve optimal fuel savings.

6. Conclusions and Future Research

This research paper presented an enhanced eco approach system for connected and automated articulated trucks, which are able to: (i) overtake slow-moving vehicles for sustainability and mobility; (ii) efficiently optimize the travel duration approaching a signalized intersection; (iii) achieve the trade-off between fuel saving and vehicle mobility; and (iv) improve computational efficiency and optimality for the articulated truck control. To achieve these features, the problem was formulated as an optimal control problem. A longitudinal and lateral coupled truck dynamic model was utilized to allow the truck to have automatic overtaking capability. The data-driven-based Koopman operator theory was adopted to globally linearize the truck dynamic model to reduce the computational burden while ensuring optimality. The optimal control problem was transformed from the time domain to the space domain in order to optimize travel duration and considering the signal timing constraint. A quantitative evaluation was conducted to validate the performance of the Koopman system dynamics. In addition, a simulation experiment was designed to compare the proposed controller against human drivers and the conventional eco approach, which only has longitudinal automation. The results demonstrate that the proposed controller improves the fuel efficiency by 5.12–67.15%, and outperforms the two baseline controllers by 9.09–32.65% in terms of fuel saving. This range is caused by the different arrival time of the ego articulated truck. The detailed investigation on the experiment revealed that:

• Koopman system dynamics is able to support the receding horizon optimization for the state of the articulated truck.
• The conventional eco approach system is disabled when a slow-moving vehicle impedes the ego vehicle. Under such circumstances, its performance might be worse than a regular human driver.
• Arrival time has a significant impact on the performance of the proposed controller.
• Most fuel savings are observed when the ego truck enters the controlled area in the middle or at the end of the red light duration.
• The benefits are the least when the ego truck’s arrival time is at the beginning of the green signal.
• The proposed controller might sacrifice fuel to achieve a better overall fuel efficiency.

The proposed eco approach controller can be packaged into a decision-making and control module of the ADAS system, and can be applied to automated driving (automation level ≥ L2). However, the proposed eco approach system only considers the optimization at the vehicle level and does not consider the impact of traffic flow. Future studies will focus on integrated optimization both at the vehicle and the powertrain levels. Furthermore, the algorithm used to solve the proposed optimal control problem is solely applicable to the cost function with a quadratic form. It is necessary to develop a more universal solution method in the future.