Keywords

1 Introduction

Autonomous vehicles (AVs) have been advancing rapidly in recent years, primarily attributed to their potential in enhancing road safety and transport efficiency. In the meantime, vehicles are becoming increasingly electrified, making over-actuation technologies more feasible and advantageous.

There have been extensive studies exploring torque vectoring for enhancing vehicle safety, e.g., [1, 2], and energy efficiency, e.g., [3,4,5]. It is clear that combining autonomous driving with over-actuation technologies allows for the effective use of their respective strengths. For example, previous studies [6,7,8] have explored active camber, four-wheel steering and their integrations for improving the active safety of over-actuated AVs.

To model AVs, several vehicle coordinate systems have been used, including Cartesian, Frenet and spatial coordinates [9,10,11,12]. The spatial transformation allows for the natural consideration of obstacle constraints on the state vector [12], and the Frenet transformation facilitates a singularity-free formulation [13, 14]. Still, these transformations and formulations are either not implemented with dynamic vehicle models or not for over-actuated applications.

Based on the reviewed literature, the present study aims to achieve energy and time optimal control of autonomous vehicles by using Frenet frame modelling and over-actuation. This study enhances the existing Frenet-based modeling by incorporating double-track dynamic vehicle models and torque vectoring. This integration aims to leverage the advantages of both Frenet frame modelling and advanced models. Utilising this modelling method, this study focuses on optimising energy consumption and travel time simultaneously.

2 Vehicle Dynamics Modelling

By using the Frenet frame, the vehicle dynamics are modelled and presented in this section. The model includes a double-track dynamic vehicle model, road geometry, the Dugoff tyre model [15], load transfer as well as the possibility for controlling the individual wheel torques, i.e., torque vectoring. The vehicle motions in the longitudinal, lateral and yaw directions, the vehicle positions in the Frenet frame [13], as well as the wheel dynamics are described with the following equations:

$$\begin{aligned} m \dot{v}_{x} = & \,m v_{y} \omega _{z}-\left( F_{y f l}+F_{y f r}\right) \sin \delta _{f}+\left( F_{x f l}+F_{x f r}\right) \cos \delta _{f} \nonumber \\ & + F_{x r l}+F_{x r r} - C_{d}A_{f}\frac{D_{a}v_{x}^2}{2} \end{aligned}$$
(1)
$$\begin{aligned} m \dot{v}_{y} = &\, -m v_{x} \omega _{z} + \left( F_{y f l}+F_{y f r}\right) \cos \delta _{f}+F_{y r l}+F_{y r r} + \left( F_{x f l}+F_{x f r}\right) \sin \delta _{f} \end{aligned}$$
(2)
$$\begin{aligned} I_{z} \dot{\omega }_{z} = & \,l_{f}\left( F_{y f l}+F_{y f r}\right) \cos \delta _{f}-l_{r}\left( F_{y r l}+F_{y r r}\right) + \tfrac{B_{f}}{2}\left( F_{y f l}-F_{y f r}\right) \sin \delta _{f} \nonumber \\ & + \tfrac{B_{f}}{2}\left( F_{x f r}-F_{x f l}\right) \cos \delta _{f} + \tfrac{B_{r}}{2}\left( F_{x r r}-F_{x r l}\right) +l_{f}\left( F_{x f l}+F_{x f r}\right) \sin \delta _{f} \end{aligned}$$
(3)
$$\begin{aligned} \dot{\alpha } = & \omega _{z} - \kappa _{c}(s) \frac{v_{x}\cos \alpha - v_{y} \sin \alpha }{1 - n \kappa _{c}(s)} \end{aligned}$$
(4)
$$\begin{aligned} \dot{s} = & \frac{v_{x}\cos \alpha - v_{y} \sin \alpha }{1 - n \kappa _{c}(s)} \end{aligned}$$
(5)
$$\begin{aligned} \dot{n} = & v_{x} \sin \alpha + v_{y} \cos \alpha \end{aligned}$$
(6)
$$\begin{aligned} I_{wi}\dot{\omega }_{i} = &\, T_{i} - r_{e}F_{xi}. \end{aligned}$$
(7)

\({v_x}\) and \({v_y}\) are vehicle velocities at the centre of gravity (CoG), \({\omega _z}\) is the yaw rate, and \(\omega _{i}\) is the angular velocity of the wheels. \(\alpha \) denotes the yaw deviation with respect to the centreline of the road, and n and s are the lateral position deviation from the road centreline and the distance travelled along the road centreline, respectively. \(F_{xi}\) and \(F_{yi}\) are the longitudinal and lateral tyre forces, respectively. \(\delta _{f}\) and \(T_{i}\) are the steering angles for the front wheels and the drive/braking torque on the wheel, respectively. m, \(I_z\), \(I_{wi}\), \(B_f\), \(B_r\), \(l_f\), \(l_r\), \(A_{f}\), \(C_{d}\), and \(r_{e}\) are the vehicle and tyre parameters, and \(D_{a}\) is the air density. Detailed explanations on these parameters, as well as tyre modelling, load transfer and torque vectoring can be found in [2].

The state-space form of the modelled vehicle dynamics is used in Sect. 3 and given as follows:

$$\begin{aligned} x_{k+1} = f(x_{k},u_{k}). \end{aligned}$$
(8)

3 Problem Formulation

Optimal control problems (OCPs) are used to formulate the energy and time optimal control of AVs. To achieve the objective of reducing energy consumption and optimising travel time, the cost function is carefully designed. Specifically, the power consumption of the wheels is minimised for energy efficiency. Moreover, the travel time is directly optimised in the cost function. Additionally, a term for actuator rate is included, to improve control efficiency and avoid acutator oscillations.

The OCPs are formulated based on the detailed vehicle dynamics as described in Sect. 2, with the capability of exploiting torque vectoring for energy and time optimal control. Moreover, various constraints are applied for considering factors such as road boundary, velocity, active safety and vehicle physical limits.

The OCP problem for jointly minimising energy consumption and travel time is given as follows:

$$\begin{aligned} & \underset{\boldsymbol{x},\boldsymbol{u}, \boldsymbol{\varDelta u},\boldsymbol{s},T}{{\text {min}}} \; & Q_{e} \sum _{k=0}^{N_{t}-1} \sum _{i=0}^{4} T_{k,i} \omega _{k,i} + \sum _{k=0}^{N_{t}-1} ||\varDelta u_{k}||_{R_{du}}^{2} + ||s_{k}||_{Q_{s}} + Q_{T} T \end{aligned}$$
(9a)
$$\begin{aligned} & {\text {s.t.}} & x_{k+1} = f(x_{k},u_{k}), \; k \in \{0,1,\cdots ,N_{t}-1\} \end{aligned}$$
(9b)
$$\begin{aligned} & g(x_{k},s_{k}) \le 0 \end{aligned}$$
(9c)
$$\begin{aligned} & x_{0} = \tilde{x}_{0} \quad x_{f} = \tilde{x}_{f} \end{aligned}$$
(9d)
$$\begin{aligned} & u_{min} \le u_{k,i} \le u_{max} \end{aligned}$$
(9e)
$$\begin{aligned} & \varDelta u_{min} \le \varDelta u_{k,i} \le \varDelta u_{max} \end{aligned}$$
(9f)

where \(\boldsymbol{x}=[x_{1}, \cdots , x_{N_{t}}]\), \(\boldsymbol{u}=[u_{0}, \cdots , u_{N_{t}-1}]\), \(\varDelta \boldsymbol{u}=[\varDelta u_{0}, \cdots , \varDelta u_{N_{t}-1}]\) and \(\boldsymbol{s}=[s_{0}, \cdots , s_{N_{t}-1}]\) are the sequences of states, control actions, variation of control actions, and slack vectors, respectively. T denotes the travel time to be optimised. \(x_{0}\) and \(x_{f}\) are the initial and final states, respectively. \(Q_{e}\), \(Q_{T}\), \(R_{du}\) and \(Q_{s}\) are the weights on the energy consumption, travel time, control variation, slack vector, respectively.

Table 1. Comparison of energy usage and travel time with torque vectoring: Optimising travel time vs. Jointly optimising energy usage and travel time.
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Fig. 1.
figure 1

Comparison of vehicle trajectories with torque vectoring: optimising travel time versus jointly optimising energy usage and travel time. “Centre” refers to the centerline of the road, and “Track” indicates the boundary of the road.

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Two sets of cost function terms are formulated to investigate two strategies: one focused on minimising travel time and the other aimed at simultaneously optimising energy consumption and travel time. For the second strategy, the cost function described in Eq. (9a) is employed. For the first strategy, the component related to energy usage is excluded from Eq. (9a). The results for both strategies are discussed in Sect. 4.

4 Results and Discussion

This section presents results for the two strategies examined: one for minimising travel time and the other for jointly optimising energy consumption and travel time. For the second strategy, two cases are analysed: one with free travel time and the other with a fixed travel time. In addition, these strategies are evaluated for the cases with and without torque vectoring. The evaluation is carried out based on a section of a handling test track. Implementation details regarding the strategies can be found in [2, 16, 17].

Table 1 presents a comparison of the results for purely optimising travel time versus jointly optimising both energy usage and travel time, utilising torque vectoring. As can be seen, by considering purely time in the formulation, the vehicle consumes \(0.5609 \, {\text {kWh}}\) of energy and takes \(23.01 \, {\text {s}}\) to complete the test. In contrast, by jointly considering energy usage and travel time in the formulation, it consumes \(10.4\%\) less energy but only travels \(1.1\%\) slower. Moreover, when the travel time is relaxed to \(28 \, {\text {s}}\), the energy consumption is significantly reduced.

Table 2. Comparison of energy usage and travel time without torque vectoring.
Full size table

Figure 1 shows the vehicle trajectories for the two investigated strategies while considering torque vectoring. Starting at the outer border of the track, the vehicle accelerates into the first corner with an initial velocity (\(v_{x}\)) around \(15 \, {\text {m/s}}\). To achieve minimum travel time, the vehicle consistently attempts to reach the maximum possible velocity, particularly during the first third and the final section of the track. Nevertheless, the vehicle at times decelerates to negotiate the cornering manoeuvres. When energy is considered in the problem formulation, the vehicle achieves lower peak velocities. Additionally, the vehicle aims to maintain a smoother trajectory to conserve energy, as evidenced by the larger lateral deviations around positions of 100 and \(400 \, {\text {m}}\) in the strategy for optimising both energy and time. When the travel time is allowed to be larger, i.e., 28 s, the vehicle travels closer to the centreline of the track, as indicated by the smaller yaw and lateral deviations compared to the other two cases. However, the vehicle does not travel strictly along the centreline to maintain a smoother trajectory.

Table 2 presents the results for purely optimising travel time and jointly optimising energy usage and travel time without utilising torque vectoring. By comparing Table 2 with Table 1, it is evident that torque vectoring contributes to faster lap times at the expense of higher energy consumption. On the other hand, when completing the manoeuvre slower, i.e., \(28 \, {\text {s}}\), torque vectoring consumes a similar amount of energy. This is likely because the required energy to overcome the resistance forces is similar for both cases in such a low-dynamic manoeuvre.

5 Conclusions

This study investigated energy and time optimal control of autonomous vehicles using Frenet frame modelling and over-actuation. Enhancements have been made to the existing Frenet-based modelling, including the incorporation of a double-track dynamic vehicle model, an advanced tyre model and torque vectoring. The problem was formulated within an optimal control framework, featuring carefully designed cost function terms and constraints. Two control strategies were examined: one for minimising travel time and the other for jointly optimising energy consumption and travel time. The findings in the studied driving scenarios are summarised as follows. Firstly, considering both energy and time in the formulation can reduce energy consumption by \(10.4 \%\) with only a slight increase in travel time. Secondly, maintaining a smoother trajectory contributes to energy conservation. Finally, torque vectoring results in faster lap times but at the cost of higher energy consumption. Additionally, torque vectoring consumes a similar amount of energy when traveling within the same time frame.

In the next step, the plan is to further study energy and time optimal control by exploring torque vectoring energy reduction and examining additional driving scenarios.