A performance-centered design method for adaptive cruise control system
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Abstract
The adaptive cruise control (ACC) system that can take into account safety andriding comfort has attracted widespread attention. The challenges lie in designing an optimal car-following system with some predefined performance constraints based on the nonlinear dynamics, including being safe with the given certain system constraints, such as safe driving within the system input and output boundaries, also comprising the control stability. In this paper, a novel ACC design approach is proposed by transforming performance boundaries into control input and output constraints, taking into account the need for safe operation. Firstly, a nonlinear dynamics system is modeled for the ACC system based on the vehicle longitudinal dynamics. Then, a performance-centered ACC system is established based on a control barrier function and a control Lyapunov function for safety and stability concerns, respectively. Subsequently, an optimal control strategy with performance constraints is formulated and recast into a standard quadratic programming problem considering the need for stability and reliability. To validate the effectiveness of the proposed method, a real-world experiment is performed, whose results illustrate the safety performance and practical application of the ACC system.
Keywords
1. INTRODUCTION
The adaptive cruise control (ACC) system has been an optional or standard equipment in commercial vehicles [1]. Its key performance is longitudinal collision avoidance and riding comfort, especially for autonomous driving vehicles [2,3]. The challenge is the potential conflicts between comfort, safety and stability, which was usually eliminated by specifying the reasonable performance function and constraint [4-6]. To guarantee driving comfort and safety, the control method with input, state or output constraints is the preferred one, such as the model predictive control (MPC) [7] and control barrier functions (CBFs) [8].
The performance balance and assurance is realized by the optimization objective and constraint condition in the MPC framework [9]. Similarly, the full state and output constraints can be actualized by the predefined CBFs or control Lyapunov functions (CLFs) of the closed-loop control system [10]. The CBF-based method also proves to be effective in the stabilization with safety problems [11], especially for the safety-critical control [12]. A safety-critical control scheme was investigated for unknown structured systems by using the CBF method [13]. An output-dependent universal barrier function was established for the consensus tracking control problem for multiagent systems with prescribed performance constraints [14]. To simultaneously consider the safety, stability, and some other performances, the CBFs and CLFs are integrated together in general, such as the quadratic programming (QP) with CBF and CLF constraints [15,16]. These attempts can balance different kinds of control performances well [17,18]. It has been shown that the optimization problem subject to the control constraints and state convergence for affine control systems can be reduced to a sequence of QPs by using CBF and CLF [15]. For example, a QP was constructed with the unification of CLF for the control objectives and CBF for the admissible state conditions, which were demonstrated on ACC and lane-keeping control problems [4,19]. Then, the optimal control input was incorporated with the performance constraints. An adaptive control scheme was developed based on the barrier Lyapunov function (BLF) for nonlinear stochastic systems with full state constraints [20]. In [21], the MPC and CBF were integrated into a complete stabilizing iteration scheme for linear discrete-time systems subject to polytopic input and state constraints. However, its calculation burden for the practical application is a huge challenge. Moreover, for the specific performance requirements of ACC systems, the safety may conflict with other performance limitations. Therefore, a performance-centered ACC strategy should be considered further for these various performances.
Effectively balancing the potential conflicts among multiple control performances, such as distance tracking, car-following stability, driving comfort, and safety of the ACC system, remains a huge challenge given system nonlinearity [22]. Furthermore, reducing computational requirements and ensuring the desired safety performance is the last important issue of its application. Thus, we explore a safe and reliable control algorithm that can guarantee safety and stability performances, which can be divided into four key parts as shown in Figure 1 and it can be constructed through four corresponding steps: dynamics modeling, CBF for the predefined safety requirement, CLF for the stability performance, and QP problem formation to solve the optimal control input. Therefore, the main contribution here is to provide a performance-centered controller considering the predefined performance requirements for the ACC systems, which lie in the following: (1) Considering the vehicle longitudinal actuator model and car-following strategy, the ACC model is characterized as a nonlinear dynamics system. For its specific model form, a CBF is designed to indirectly establish the relationship between the input and each performance constraint; and (2) An optimal control strategy with various predefined performance constraints is formulated as a standard QP problem with the CBF and CLF constraints for the safety and stability requirements, respectively. Then, a potential practical solution is obtained to enhance the control performance for ACC closed-loop systems.
Figure 1. Frame diagram of performance-centered ACC method.
The outline of this work is as follows. Section 2 establishes the ACC system model considering the input and output constraints. Section 3 describes the proposed method of the performance-centered controller. The experimental results are expressed in section 4. Finally, the main conclusions are summarized in section 5.
2. ACC SYSTEM MODELING
2.1. Actuator model
The actuator of the ACC system is the driving/braking-by-wire subsystem equipped on autonomous driving vehicles. The longitudinal dynamics model is to characterize the steady-state relation between the longitudinal motion and the longitudinal force of the vehicle. When the lateral and yaw motions of a vehicle are ignored, then the vehicle's longitudinal dynamics is given as
where
where
Consider the dynamics of the powertrain system as a first-order process
where
Assume that the road slope and friction coefficient are slow-varying, and according to (4), the induced dynamics of acceleration can be described as:
Generally, the comfort performance is decided by the acceleration and jerk together, which can be indirectly judged by the input force as given in (5). Hence, the comfort constraint can be transformed into the input constraint. Then, the comfort constraint on the input should be considered and the set of control bounds is designated as follows:
where the positive coefficients
To introduce the input regulation effect, we define
thus,
Then, the input constraint (6) is equivalent to
where
2.2. Car-following model
The main objective of the car-following mode of ACC systems is to follow the lead car with a desired safety distance which is usually established as the following headway spacing policy [25-27]:
where
Define the spacing state for the car-following process as
where
where
where
Assume that the system state trajectory is predictable within the neighborhood of its desired state
2.3. Performance-centered design requirement
The safety constraint is the primary performance evaluation. Generally, a driver cannot easily drive on a narrow road at high speed or adjacent to an obstacle. Similarly, autonomous driving requires a speed reduction on a narrow road to improve the accuracy of the vehicle control. To drive more safely, the target speed should be determined by considering the environment information, such as the proximity of obstacles and collision probabilities. Moreover, the ego vehicle should maintain a safe car-following distance to avoid any potential collisions.
Hence, we define the controlled output as the speed and safe distance evaluations for the safety consideration:
and the output limitations are considered as
Remark 1 If the vehicle is driving on a curved road, its safe speed satisfies the following dynamics constraint [30]:
where
Then, vehicle safety during a curved road is also guaranteed based on the performance constraint (14).
Definition 1 The set
Lemma 1 If there exits a continuously differentiable CBF
Hence, the ACC system aims to employ an input
where
Remark 2 It is difficult to establish the direct mapping relationship between the input constraint and safety boundaries (defined as
If the vehicle is safe at the initial sampling, i.e.,
Assumption 1 The initial state of the vehicle is safe when the ACC system is just activated, that is,
Then, the set (14) is forward invariant for (12) based on the definition 1, Lemma 1 and Assumption 1. According to the norm definition
where
According to (18) and (19), it yields
To avoid that the denominator of the differential function
3. PERFORMANCE-CENTERED CONTROLLER
3.1. Convergence analysis
Lemma 2 A continuously differentiable function
and
We define a CLF for the exponential stability of the ACC system (12) according to the Lemma 2
then, there exist positive numbers
Thus, there exist
where
According to (23) and (25), it yields
Remark 3 Define
3.2. Performance-centered controller design
The objective of ACC is to design an optimal controller with the input constraint
where the three cost functions represent the energy expenditure of control input and riding comfort during the deceleration and acceleration process, the punishment of the relaxation variable, and the car-following performance trend cost, respectively.
where
where
Therefore, the optimal control strategy for the ACC system at each sampling time can be solved by the following QP problem:
In some unusual circumstances, the car-following stability, comfort and safety performances may conflict, leading to a null set due to various system limitations, as given by
To avoid the QP problem (30) becoming infeasible and to guarantee its feasibility, a new relaxation variable
which can be transformed into the standard form as
where
The QP problem (33) can be solved using the interior-point method by the existing solver or toolbox [34], where the real-time solving process is omitted here.
Remark 4 The feasibility of the proposed method with various performance constraints is guaranteed based on the penalty variable
Remark 5 A single vehicle ACC model (12) is considered in this paper, which can be extended to the consensus tracking control problem of cooperative adaptive cruise control (CACC) by modifying the ACC model (12) to a CACC model. For some other multiagent systems similar to [6,14], it is also available by using the reconstructed dynamic model and prescribed performance constraints.
4. EXPERIMENTAL VERIFICATION
The whole parameters and their effects are listed in Table 1. After plenty of attempts and comparative analysis, and also according to practical experiences, the ACC performance parameters and controller parameters are given as Table 1. Then, the QP problem (33) can be solved for its real-time application. According to the magnitudes of the cost function (27), we can select the weights
Parameters and their effects of ACC system
| Parameters | Effects description | Assignment strategy |
| Standstill distance | Statistics of car-following distance | |
| Time headway | Reaction time of driver refereed to RSS model [35] | |
| Referenced acceleration | Empirical value of a comfortable start process | |
| | Maximal acceleration and deceleration | Acceptable comfort and safety |
| Deceleration and acceleration toleration | Acceptable comfort and safety | |
| Time constant of chassis system | Measured value | |
| Constant of CBF to avoid zero denominator | Small enough positive number | |
| Constant of CBF to guarantee safety | Very small positive number after some attempts | |
| Constant of CBF condition to guarantee safety | Safety condition after some attempts | |
| Constant of CLF to guarantee convergence | Sufficiently large to guarantee tracking performance | |
| | Weight coefficients of optimization objective | Magnitudes and significance of cost function |
To verify the performance of the proposed method, the field tests are conducted by a modified HAVAL SUV with drive/brake/steer-by-wire systems as shown in Figure 2, and the specification of the vehicle is given in Table 2. The vehicle is equipped with sensors such as a global navigation satellite system (GNSS)/inertial navigation system (INS) with two antennas, a yaw rate gyro equipped near the vehicle centroid, a Mobileye 560 camera installed behind the windshield, a Delphi Electronically Scanning Radar (ESR) and five Rear and Side Detection System (RSDS) Radars equipped around the car. The decision and control platform is established based on a Dspace Micro Autobox Ⅱ placed in the trunk. Both signals are transmitted by the CAN-bus. The Dspace Micro Autobox Ⅱ acts as the real-time controller, which receives the feedback information from the sensors and vehicle, and sends the control command to the chassis system. Then, a closed-loop ACC system is established. Because the whole perception, decision, planning and control algorithms (including the proposed method) are implemented in one embedded controller with the sampling period as 10 ms, the real-time performance is implied by the following real car experiments. To make a more intuitive comparison, three test scenarios are adopted and they are implemented in the same microprocessor with the same sampling period of 10 ms, where the first two experiments are performed by the proposed method and the third one is executed by the baseline method.
Figure 2. Test vehicle and scene. (A) Experimental vehicle; (B) Test field with lane markers.
Experimental vehicle specifications
| Item | Description |
| Total mass | 1,700 kg |
| Scale factor of rotating mass | 1.1 |
| Distance from center of gravity to front axle | 1.2 m |
| Distance from center of gravity to rear axle | 1.48 m |
| Vehicle width | 1.86 m |
| Vehicle length | 4.6 m |
| Drag coefficient | 0.389 |
| Front face area | 2.86 m |
4.1. Test scenario Ⅰ
To demonstrate the performances of the proposed method, a typical urban test scenario is first performed under a dynamic environment to verify the controller capabilities, which can present the results for the most relevant scenarios. To achieve a convenient and reasonable verification and enhance the applicability to real-world scenarios, the experiment is performed in a real road environment. The relative distance and safe distance of the preceding vehicle are shown in Figure 3. At the beginning (near 60 s), the car ahead is fast decelerated with the deceleration -5 m/s
Figure 3. Actual distance and safe distance of scenario Ⅰ.
In this scenario, the speed limitation is
Figure 4. ACC system input and responses of scenario Ⅰ.
The CLF and its relaxation are shown in Figure 5, which implies that the QP problem (33) can be solved in real time according to the Figure 4 and Figure 5. Moreover, convergence is obtained during every steady car-following process, although there will be some drastic fluctuations when a cut-in or cut-out object occurs. On the contrary, the CLF will be convergent after the car-following process is steady.
Figure 5.
The controlled outputs and their CBFs are shown in Figure 6 and Figure 7. Then, the safety performance evaluations
Figure 6.
Figure 7.
4.2. Test scenario Ⅱ
The adopted second test scene is a similar but more complex urban environment with a higher traffic flow to further verify the effectiveness of the proposed method. There exist more curve driving, lane changing and cut-in or cut-out behaviors. The actual distance and safe distance of the preceding vehicle are shown in Figure 8. At the time 167s, the car ahead is fast decelerated and the controlled vehicle is decelerated with the maximal deceleration -1.57 m/s
Figure 8. Actual distance and safe distance of scenario Ⅱ.
The input, acceleration and speed of this scenario are shown in Figure 9. It can be seen that there is a provisional parking in front of the red light and a Stop & Go driving behavior. It is obvious that the input constraint (6) is satisfied, where the comfort performance will be obtained. In addition, the vehicle can remain stationary while maintaining a safe distance during the Stop & Go condition. Moreover, the vehicle speed is much lower than its safe speed at the two obvious curve driving moments 222 and 340 s. According to the results in Figure 8 and Figure 9, the safe distance and speed are ensured during the car-following process and curve scene.
Figure 9. ACC system input and responses of scenario Ⅱ.
The CLF and its relaxation are shown in Figure 10, which can ensure that the QP problem (33) with the CBF and CLF constraints is solvable in real time. The controlled outputs and their CBFs are shown in Figure 11 and Figure 12. Then, the safe distance
Figure 10.
Figure 11.
Figure 12.
4.3. Test scenario Ⅲ
To make a further horizontal comparison, another well-known method is adopted in this test scene to validate the effectiveness of the proposed method: the intelligent driver model (IDM) method. We know that the IDM approach is a widely used method in vehicle ACC systems due to its stability and excellent calculation efficiency [36], which imitates the driving behavior habits of human drivers and will be employed as a baseline for comparison. The IDM method is a simple empirical model only with the same uncomplicated numerical calculation, which is defined as follows:
where the desired headway is given as
The IDM method possesses excellent computational efficiency in its practical applications as exhibited as (34). After several attempts, the proposed method and IDM can be executed with the same sampling period as 10 ms in the real microprocessor as shown in Figure 12, which implies that the real-time calculation efficiency of the proposed method is approximate to that of the IDM method and it is satisfying for the practical application as well.
Because the experiment is carried out in real road scenarios, the surrounding vehicles are usually driven by human drivers. It is very hard to repeat the same scenarios 1 and 2 for making a reasonable comparison. Then, we adopt another similar scenario 3 on a real urban road to verify the proposed method. The relative distance and safe distance of the preceding vehicle obtained by the IDM method are shown in Figure 13, where the corresponding input, acceleration and speed are shown in Figure 14. We can see that the car-following distance is closer to its safe distance, even less than the safe distance at 216 s, where safety is hard to guarantee. The maximal acceleration 1.89 m/s
Figure 13. Actual distance and safe distance of scenario Ⅲ.
Figure 14. ACC system input and responses of scenario Ⅲ.
The controlled output is shown in Figure 15. It is clear that the output
Figure 15.
To make a more intuitive comparison, some quantitative analysis of the performance evaluation indicators is listed in Table 3. The scenarios 1 and 2 are performed by the proposed method and the scenario 3 is performed by IDM method [36]. To further evaluate the car-following performance of the proposed controller, the root mean square (RMS) value of the acceleration
Experimental results comparison
| Scenarios | 1 | 2 | 3 | |
| 1 RMS of | ||||
| Methods | Proposed method in this paper | IDM | ||
| Safety indexes | 0.5456 m | 0.12 m | -0.546 m | |
| 0.6276 m/s | 0.9227 m/s | 5.053 m/s | ||
| Comfort indexes | 1.68 m/s | 1.575 m/s | 1.89 m/s | |
| RMS1 | 0.2934 m/s | 0.3073 m/s | 0.409 m/s | |
where
It is worth mentioning that the proposed method is finally transformed into a QP-based optimization problem, where its essence is a kind of optimization-based approach. There are two other typical methods: linear quadratic regulator (LQR) and MPC. The LQR is an optimization method without several predefined performance constraints; then, it is hard to ensure the performance requirements. Correspondingly, the MPC is a similar optimization practice, which is usually transformed into a standard QP problem with some performance constraints. However, the MPC method is implemented by the predictive model and iterative optimization, where the calculation burden of the model predictive procedure is a huge challenge that prevents its practical application. Therefore, the proposed method can well balance the potential conflict between the multiple performance requirements, such as safety, comfort, stability and real-time performances. It can outdo the IDM, LQR and MPC methods and be used in practical applications. Then, the effectiveness of the proposed performance-centered ACC method is verified.
5. CONCLUSIONS AND DISCUSSION
In this work, the CBF and CLF are introduced and deployed on the ACC system, which provides a potential practical solution to enhance the control performance for ACC systems. Firstly, the ACC system model is modeled as a nonlinear dynamics system based on the vehicle longitudinal dynamics model and the car-following headway spacing policy. Then, the performance-centered optimal controller is established with the CBF and CLF constraints induced from various predefined performance constraints. The effectiveness of the proposed method is verified by a real road experimental scenario.
Due to the performance-centered design philosophy of the proposed method, it can also be applied in some other autonomous driving systems, such as lane-keeping or other path-tracking systems with similar nonlinear dynamics. For some other vehicle control applications, a similar approach can be adopted step-by-step: dynamics modeling, CBF established for the safety and comfort requirement, CLF established for the stability performance, and QP problem formation to solve its control input. On the other hand, future studies will focus on the whole autonomous driving system with the safety-centered requirements for the vehicle lateral-longitudinal dynamics integrated control within more traffic scenarios such as traffic jams, also including the vehicle stability control. Moreover, the consensus tracking control and string stable problems will be considered for the CACC system or some other nonlinear multiagent systems.
DECLARATIONS
Authors' contributions
Performed data acquisition and analysis and provided administrative, technical, and material editing: Zhan S
Made substantial contributions to conception and design of the study: Liu C
Availability of data and materials
Not applicable.
Financial support and sponsorship
The authors greatly appreciate the National Science Fund of the People's Republic of China (Grant No. 52102444), the Open Foundation of the National Key Laboratory of Multi-perch Vehicle Driving Systems (Grant No. QDXT-WY-202407-16), the Zhiyuan Laboratory (Grant No. ZYL2024019) and the Central Guidance on Local Science and Technology Development Fund of Hebei Province (Grant No. 226Z2204G).
Conflicts of interest
Liu C is a Junior Editorial Board Member of the journal Complex Engineering Systems and the guest editor of the Special Issue of "Generalized Dynamics Modelingand Dynamics Control of Autonomous Driving Vehicle", while the other author has declared that he has no conflicts of interest.
Ethical approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Copyright
© The Author(s) 2024.
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Zhan, S.; Liu, C. A performance-centered design method for adaptive cruise control system. Complex Eng. Syst. 2024, 4, 20. http://dx.doi.org/10.20517/ces.2024.60
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