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A proposal for adaptive cruise control balancing followability and comfortability through reinforcement learning
ROBOMECH Journal volume 9, Article number: 22 (2022)
Abstract
Adaptive cruise control (ACC), which is an extension of conventional cruise control, has been applied in many commercial vehicles. Traditional ACC is controlled by proportionalintegralderivative control or linear quadratic regulation (LQR), which can provide sufficient performance to follow a preceding vehicle. However, they can also cause excessive acceleration and jerk. To avoid these excessive behaviors, we propose reinforcement learning (RL), which can consider various objectives to determine control inputs, as an ACC controller. To balance the performance of following a preceding vehicle and reducing jerk, RL rewards are designed using unique thresholds. Additionally, to balance performance and robustness to the zeroorder delay (dead time) of the controlled system, dead time is also considered by scattering it randomly in the learning phase. As a result of this study, an RL agent trained using the proposed RL method and two LQR units specialized for followability and comfortability were simulated using Simulink® (MATLAB®).
Introduction
In recent years, advanced driverassistance systems (ADAS) have been studied to enhance driving comfort, reduce driving fatigue and stress, reduce accident risks, increase traffic capacity, and reduce fuel consumption [1, 2]. In particular, from the perspectives of driving fatigue and stress reduction, the adaptive cruise control (ACC) system is an important type of ADAS that has been deployed in commercial vehicles since 1995 in Japan [3]. ACC is an extension of the conventional cruise control system, which could only maintain a vehicle’s preset velocity, whereas ACC maintains a vehicle’s preset distance from a preceding vehicle [4, 5]. ACC performs acceleration and deceleration for a driver based on information observed by various devices such as radar, sensors, and cameras. Even conventional control methods such as proportionalintegralderivative control and linear quadratic regulation (LQR) [6], which have been used commonly for decades, provide sufficient performance to follow a preceding vehicle. However, they can cause excessive acceleration and jerk, which can reduce driving comfort and increase fuel consumption [7]. To avoid these excessive behaviors, ACC should be improved to consider not only followability, but also comfortability (reducing jerk).
In addition, it is also known that these conventional control methods cannot consider the constraint saturated by performance’s limits, and it makes the control performance worse. To avoid this deterioration, the model predictive control (MPC) is usually used [8,9,10]. However, the MPC’s control performance depends on the accuracy of modeling the controlled plant (the MPC determines control input by calculating with a modeled plant every sampling time) [11].
To consider the robustness against the modeling accuracy and the disturbance, the fuzzy logic control (FLC) has been also studied for decades. This controller can deal the plant state fuzzily based on the linguistic control rules (LCR). However, this method cannot decide control input smoothly due to the LCR, so the control input keeps constantly on increasing and decreasing [12].
To solve this problem, in this study, reinforcement learning (RL), which can consider various objectives to determine control inputs, was considered for improving ACC. The RL can also consider the constraints. Additionally, to balance the performance of followability and comfortability, an RL reward function was designed using unique thresholds. These thresholds vary depending on the speed of the host vehicle. When the difference between the target distance and measured distance from the host vehicle to the preceding vehicle becomes less than a certain threshold, the RL agent collects rewards depending on the difference in distance. Another threshold for reducing jerk is set to help passengers feel “comfortable.” If jerk becomes less than a certain threshold, the RL agent collects rewards.
Additionally, it is assumed that there is zeroorder delay (dead time) in the controlled system. Therefore, one of the goals of the proposed RL agent is to enhance robustness to dead time by learning random variations during the learning phase of RL.
Definition of ACC and the controlled vehicle model
Definition of ACC objectives and conditions
For this research, the ACC objectives were defined as listed below.

O1)
To achieve the required performance to follow the preceding vehicle, the error between the reference distance and measured distance to the preceding vehicle should converge to zero.

O2)
To achieve the required performance for following a preceding vehicle, the relative velocity between the host vehicle and preceding vehicle should converge to zero (this object is related to O1, so servo balance between O1 and O2 should be tuned experimentally.

O3)
To achieve the required performance for driving comfort and fuel consumption, the acceleration and jerk of the host vehicle should converge to zero.
Additionally, to consider an ACC system implemented into an actual vehicle, the following conditions are defined.

C1)
The host vehicle has dead time in the controlled system. Therefore, the host vehicle’s acceleration responds to a control command from the ACC system with a firstorder delay and dead time.

C2)
The reference distance which is the ACC system’s target is defined based on the policy of time headway (THW). The THW is the time the host vehicle takes to reach a point on the road passed by the preceding vehicle. This parameter is one of factor used by the driver to perceive the risk to close to the preceding vehicle, and its magnitude influences the stress level of the driver when following a preceding vehicle [12].

C3)
Acceleration and deceleration are saturated by performance’s limits of powertrain, brake and road friction.

C4)
States (vector) of the controlled plant, which is defined in “Definition of a vehicle model controlled by ACC” section, are observable.
A scenario in which a host vehicle follows preceding vehicle using ACC is illustrated in Fig. 1
Definition of a vehicle model controlled by ACC
According to C1, the host vehicle’s acceleration is defined under the firstorder delay as follow:
Here \(a_{h}\) is the host vehicle’s acceleration, \(u\) is the acceleration control command from the ACC system, and \(\tau\) is the time constant of the firstorder delay system. By the Laplace transform, this equation expressed as a time function is transformed to (1) with the Laplace operator \(s\):
According to C2, the reference distance to preceding vehicle of the ACC system based on the THW policy is defined as (2)
where \(d_{r}\) is the reference distance to preceding vehicle, \(d_{s}\) is the safety distance based on the THW policy, \(t_{hw}\) is the THW, and \(v_{h}\) is the velocity of the host vehicle.
Based on the calculated \(d_{r}\) value, the error between the reference distance and measured distance to preceding vehicle are defined in (3). Additionally, the error between the host and preceding vehicle velocity is defined in (4).
Here, \(d_{e}\) is the error between the reference distance and measured distance, \(d\) is the measured distance to preceding vehicle, \(v_{e}\) is the error between the host vehicle velocity and preceding vehicle velocity, and \(v_{p}\) is the velocity of the preceding vehicle.
The controlled plant state vector \({\varvec{x}}\) and output vector \({\varvec{y}}\) are defined in (5) and (6), respectively.
Based on the defined state vector \({\varvec{x}}\) and output vector \({\varvec{y}}\), the statespace representation is defined in (7a) and (7b).
Here, \(a_{p}\) is the preceding vehicle’s acceleration. The coefficient matrices \({\varvec{A}},\user2{ B}\), \({\varvec{G}}\), \({\varvec{C}}\), \({\varvec{D}}\), and \({\varvec{Z}}\) in (7a) and (7b) are expressed as follows:
The reinforcement learning applied to an ACC controller
Algorithm for the reinforcement learning
In this study, the deep deterministic policy gradient (DDPG) [13] was used for learning ACC driving behavior to archive the objectives O1 to O3 under the conditions C1 to C4. The DDPG is a common RL method and deep learning is adopted in the actorcritic method. The controller implemented into host vehicle to determine ACC driving behaviors based on learning results is called an “agent.” To design the agent and evaluate its performance, the RL toolbox (MATLAB®) was used to implement its RL as a controller.
Design of the actor network and critic network
For the DDPG, there are two networks called the actor network and critic network, each of which have different purposes. The agent observes the state (vector) from the environment surrounding the host vehicle and the actor network determines the best action based on the observed state. The critic network then receives the state from the agent and action from the actor, and returns the expected value of the total reward. Finally, after executing the action from the actor, the actor network is updated based on the calculated reward. A block diagram for an ACC system controlled by this logic is presented in Fig. 2. L in the figure is the dead time and e^{−Ls} is the dead time factor in the controlled system.
The structures of the actor network and critic network are listed in Tables 1 and 2, respectively.
In the input and output layers, the number of neurons for the state input is four and the number of neurons for the action output is one. Additionally, each fully connected layer contains 48 neurons. There are seven neurons in the input layer on the state path, and four neurons of them are represented as the plant’s output from vector \({\varvec{y}}\) defined in (6) and three neurons are added to consider the environment precisely. In addition, there is one neuron in the input layer on action path, and it is represented as the acceleration command \(u\) defined in (1).
Design of the reward function
To archive O1 to O3, the reward function r_{t} at time t is defined by (8).
Here, \(w_{1}\) to \(w_{5}\) are weights, \(M_{t}\) is a reward value, and \(j_{t}\) is the jerk.
The reward value \(M_{t}\) can receive a reward only when the error between the reference distance and measured distance to preceding vehicle or jerk become less than the corresponding thresholds, which are defined as follows. As shown below definition, the value of \(M_{t}\) depends on how much the measurement values exceed the referential values. The reward value of \(M_{t}\) is defined in (9).
Here, \(M_{1t}\) is a reward value that reduces the error between the reference distance and measured distance to preceding vehicle, and \(M_{2t}\) is a reward value for reducing jerk. For these reward values, the weights are defined as \(w_{6}\) and \(w_{7}\), respectively.
The purpose of \(M_{1t}\) is to determine the control target depending on the host vehicle velocity. When the host vehicle velocity is high, the ACC controller wants to avoid approaching the distance to the preceding vehicle to reduce collision risk. In contrast, when the host vehicle velocity is low, the ACC controller’s target is to close the distance to the preceding vehicle to increase traffic capacity. To achieve the purpose of \(M_{1t}\), \(M_{1t}\) should be calculated depends on host vehicle velocity. Thus, in this case, \(d_{rt}\) is used to calculate \(M_{1t}\) because \(d_{rt}\) includes host vehicle velocity as one of factors. The logic of \(M_{1t}\) is illustrated in Fig. 3. \(M_{1t}\) is calculated only when the absolute value of \(d_{et}\) becomes less than 10% of the absolute value of \(d_{rt}\); In other case when the absolute value of \(d_{rt}\) becomes more than 10% of the absolute value \(d_{rt}\), \(M_{1t}\) becomes zero. The criteria value of 10% is tunable and it is defined experimentally in advance. The maximum value of \(M_{1t}\) is \(w_{6}\).
The purpose of \(M_{2t}\) is to control the ACC comfortably. To achieve the purpose of \(M_{2t}\), \(M_{2t}\) becomes \(w_{7}\) only when the absolute value of the jerk becomes less than the threshold of 2.5 m/s^{3}. This value was selected because passengers feel uncomfortable if the jerk exceeds 2.5 m/s^{3} [7].
If the jerk is calculated from the measured acceleration, then there is a huge delay between the actuator and sensor due to sensing delay and the zeroorder delay of data communication. Therefore, in this study, jerk was calculated as the rate of change of the acceleration input during one sampling period, as shown in (10).
Here, \(T_{s}\) is the sampling period.
Definition of the linear quadratic regulation to be compared to the reinforcement learning
To evaluate the advantageous effect of applying an RL agent to an ACC controller, the results obtained using LQR is considered for comparison. The system in (7a) is expanded by defining jerk as the control input \(u^{\prime}\) [14] because the LQR should also consider to reduce jerk to be compared with the RL from the perspective of performance and comfortability fairly. The model definition of (7a) is transformed into (11) as follows:
where the state vector \(\user2{x^{\prime}}\) and coefficient matrices \(\user2{A^{\prime}}\), \(\user2{B^{\prime}}\), \(\user2{G^{\prime}}\) are expressed as shown below.
The optimal control input for state feedback control is defined below. First, the evaluation function for state feedback control is defined in (12).
Here, \(J\) is the evaluation function, \({\varvec{Q}}\) is the weight vector for the state, and \(r\) is the weight vector for the input. To minimize this evaluation function, the optimal control input is defined in (13).
Here, the matrix P is a symmetric matrix based on the algebraic Riccati equation.
For comparison to the RL control method, two weight combinations “Q_{d} and r_{d}” and “Q_{j} and r_{j},” which are specialized for followability and comfortability, respectively, are defined as shown below.

1.
Weight combination specialized in followability:
$${\varvec{Q}}_{{\varvec{d}}} {\text{ = diag}}[\begin{array}{*{20}c} 5 & {100} & 0 & {40} & 0 \\ \end{array} ],\;r_{d} = 60$$ 
2.
Weight combination specialized in comfortability:
$${\varvec{Q}}_{{\varvec{j}}} {\text{ = diag}}[\begin{array}{*{20}c} 5 & {100} & 0 & {50} & {50} \\ \end{array} ],\;r_{j} = 3000$$
Note that these weight combinations are chosen by parameter studies. Some of ACC simulation results which are studied to decide weight combinations are shown in “Appendix”.
Usually, the zeroorder delay on communication can be changed fluently, so it is very difficult to consider it. In addition, the zeroorder delay is very small value. Thus, the zeroorder delay is often considered as one of a disturbance that can be deal with controller’s robustness. That’s why LQR doesn’t consider the zeroorder delay on communication here.
Evaluation of followability and comfortability
Through comparisons to the two controllers for LQR defined in “Definition of LQR to be compared to RL” section, the controller for RL is evaluated based on simulations in this section.
Conditions for simulations
The initial conditions for the host vehicle and preceding vehicle are listed in Table 3. In this simulation, the sampling time was 0.1 s, simulation continued for 60 s, and the preceding vehicle performed repeated accelerations and decelerations during the simulation period.
Regarding C3, the acceleration and deceleration of the host vehicle were saturated between 2 and − 3 m/s^{2}. The other conditions for conducting simulations are listed in Table 4.
The weights for the reward function for RL were tuned in advance and are listed in Table 5.
To improve the RL controller’s followability, the initial velocity of the preceding vehicle was randomly set between 10 and 30 m/s in intervals of 1 m/s during the learning phase. In addition to improving followability, the controller should be robust to the dead time of the controlled system. Therefore, the dead time of the controlled system was also scattered between 0.01 and 0.10 s in intervals of 0.01 s randomly during the learning phase. By this method, the zeroorder delay can be considered even if it is changed fluently, so this method requires engineers not to consider the value of the zeroorder delay but also the range of the zeroorder delay preliminarily. During the learning phase, rewards were returned in every sampling period (0.1 s) and the total rewards were considered as the accumulated results. The threshold for the total reward to stop learning is a tunable parameter because there is a tradeoff between performance and the time required for learning. Typically, after many learning episodes are completed, the total rewards converge and performance decreases with overfitting and overtraining. Additionally, the value of the total reward depends on the definition of the reward function. Hence, the threshold to stop the learning is also tuned in advance depending to the definition of the reward function (for defined reward function, the criteria of the total reward to stop the learning is 1400). For RL, learning was completed when the total reward for an episode with 600 steps reached 1614.3 and the total number of learning steps required to reach the goal was 446,819.
Simulations with one condition of preceding vehicle velocity and dead time
The simulation results when the zeroorder delay is 0.02 s and the other conditions are as discussed in “Conditions for simulations” section are presented in Fig. 4. The results for LQR controllers prioritizing followability (to reduce error between the reference distance and measured distance to preceding vehicle) and prioritizing comfortability (to reduce jerk) are plotted as “LQR for followability” and “LQR for comfortability,” respectively. The graphs show the (a) error between the reference distance and measured distance to preceding vehicle, (b) relative velocity, (c) host vehicle acceleration, and (d) host vehicle jerk. As mentioned above, jerk is considered as the change rate of the acceleration input during one sampling period in the controller. On the other hand, actual jerk which driver feels should be calculated as the change rate of measured acceleration, so jerk discussed in the graphs is defined (15).
\(j_{t measured}\) is measured jerk which is calculated by measured acceleration, and it is represented as “Jerk” in below graphs only.
According to the simulation results, our considerations are separated in two perspectives of followability and comfortability.
Followability perspective
The error between the reference distance and measured distance to preceding vehicle controlled by RL converges faster than that controlled by LQR. Additionally, in the steady state [after the host vehicle speed reaches the preceding vehicle speed at approximately 16 s in graph (b)], the relative velocity’s overshoot controlled by RL is equal to or better than the overshoot controlled by LQR. The main reason why RL controls with less relative velocity’s overshoot than LQR is that RL learns to handle the dead time properly during the learning phase.
Comfortability perspective
An RL agent with a reward function for suppressing the absolute value of jerk to less than 2.5 m/s^{3} can reduce overshoot more effectively than LQR. In fact, the jerk controlled by LQR exceeds 2.5 m/s^{3}. However, the followability controlled by LQR is clearly worse than RL’s followability even though the jerk is almost same. Graph (a) shows that RL can converge \(d_{e}\) more rapidly than LQR for followability and control jerk to be less than 2.5 m/s^{3}. From these results, it seems that RL can consider the balance between followability and comfortability.
From these two perspectives, it can be concluded that RL can control with higher performance than LQR by balancing followability and controllability. The LQR is well known that it provides the optimal state feedback gain considering the infinite future prediction. However, it can work as the linear controller, so it is also known that it can work without considering the constraint. This cause deterioration of controller’s performance because the saturation isn’t a scenario which is considered by the LQR controller.
Simulations with scattered initial preceding vehicle velocities
In this section, the RL agent’s behavior is evaluated when the initial preceding vehicle velocity is scattered. The purpose of this confirmation is to check whether unexpected behaviors are caused if the initial preceding vehicle velocity is outside the learned conditions. ACC using RL was simulated when the initial preceding vehicle velocity was scattered from 15 to 55 m/s in 5 m/s intervals and the results are presented in Fig. 5. The other conditions are same as simulation conducted in “Simulations with one condition of preceding vehicle velocity and dead time” section.
In the simulation results, it appears that when the initial preceding vehicle velocity increases, followability performance decreases. The reason for this performance decrease is that the acceleration and deceleration of the host vehicle are saturated between 2 and − 3 m/s^{2}, as defined in “Conditions for simulations” section. Therefore, even if RL controls with high performance by balancing followability and controllability, followability performance still decreases. From a comfortability perspective, even when the error between the reference distance and measured distance to preceding vehicle is large [e.g., when the initial preceding vehicle velocity is 45 m/s or 55 m/s in graph (a)], jerk is controlled under the threshold of 2.5 m/s^{3} mostly. As a result, it seems that RL can control for various initial preceding vehicle velocities properly and there is no unexpected behavior (e.g., accelerating unexpectedly or crashing to preceding vehicle), even if the initial preceding vehicle velocity exceeds the learned conditions.
Simulations with scattered dead time variably
In this section, the RL agent’s behavior is evaluated when the dead time is scattered. Similar to “Simulations with scattered initial preceding vehicle velocities” section, the purpose of this confirmation is to check whether unexpected behaviors occur if the dead time is not within the learned conditions. ACC using RL was simulated when the dead time in the controlled system was scattered from 0.02 to 0.20 s. The other conditions are same as simulation conducted in “Simulations with one condition of preceding vehicle velocity and dead time” section. The results are presented in Fig. 6.
In the simulation results, it appears that there is a very small impact on followability if the dead time is scattered over the learned range from 0.01 to 0.10 s. When the error between the reference distance and measured distance to preceding vehicle is overshot at approximately 2 s, the maximum gap is 1.0 m (3%). In contrast, from a comfortability perspective, more dead time increases jerk. When the jerk is overshot, the maximum value of jerk is 2.1 m/s^{3}. These gaps are caused by the balance of the tuned weights in the reward function for RL. It can be concluded that there is no excessive behavior in terms of followability, even if the dead time exceeds the learned conditions. Additionally, when the dead time is doubled from the maximum value in the learned conditions, the comfortability (value of jerk) is almost the same as the result achieved by LQR for comfortability with the dead time of 0.02 s. Additionally, after the second overshoot of the jerk at approximately 6 s, the ACC results controlled by RL are almost the same, even when the dead time is scattered.
Conclusions
In this study, RL was applied to ACC and a unique reward function with reward value was defined to consider the balance between followability and comfortability.
To evaluate RL performance, the simulation of ACC was conducted using a trained RL agent and two types of LQR controllers (for followability and comfortability). The simulation results revealed that the LQR controllers can control ACC with high performance for either followability or comfortability through proper tuning. However, it is difficult for LQR to balance followability and comfort. In contrast, RL can balance followability and comfortability because RL considers the balance between followability and comfortability, as well as dead time. Additionally, when the initial preceding vehicle velocity and dead time in the controlled system were scattered, RL performance was evaluated. The simulation results confirmed that there is robustness to the initial preceding vehicle velocity and dead time in the controlled system. Based on these results, it can be concluded that the RL method with a unique threshold can ideally control followability and comfortability.
Availability of data and materials
The datasets analyzed during the current study available from the corresponding author on reasonable request.
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MN initiated this research, designed and simulated. MN analyzed the simulation data, interpreted the results and wrote the manuscript with the help and review from MH. Both authors read and approved the final manuscript.
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Appendix: Simulation results with the linear quadratic regulation weight combinations
Appendix: Simulation results with the linear quadratic regulation weight combinations
As mentioned above, to decide LQR weight combinations, parameter studies are conducted. Here, some of simulation results with weight combinations are shown. The simulation results when the zeroorder delay is 0.02 s and the other conditions are as discussed in “Conditions for simulations” section are presented. The Fig. 7 shows tuning results specialize in followability and Fig. 8 shows tuning results specialize in comfortability. The marked result as “Result of represented weight combination” in each is chosen for comparison with RL above.
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Maruyama, N., Mouri, H. A proposal for adaptive cruise control balancing followability and comfortability through reinforcement learning. Robomech J 9, 22 (2022). https://doi.org/10.1186/s40648022002357
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DOI: https://doi.org/10.1186/s40648022002357
Keywords
 Vehicle trajectory
 Autonomous driving
 Adaptive cruise control
 Reinforcement learning