Abstract
Recently, more and more research has been conducted to develop Connected Autonomous Vehicles (CAVs) applications that ensures the safety driving of CAVs under some extreme situations. This brief presents a robust control strategy for CAVs to preserve a precise tracking performance and maintain the stability of lateral dynamics when passing a sharp curve with uncertain road friction coefficient changes. In the proposed robust lateral dynamics control, robust optimization-based lateral dynamics controller is designed to achieve the stability of the lateral dynamics with the consideration of the road friction coefficient uncertainty. Simulation validations are carried out to evaluate the proposed control strategy. The results show that the robust optimization-based lateral dynamics can improve the robustness even with the uncertainty of the road friction coefficient.
1 Introduction
Autonomous vehicles will meet more emergency scenarios when leaving the research laboratory and entering public roads (Kritayakirana and Gerdes, 2012; Shen and Raksincharoensak, 2021). Vehicle stabilization under uncertain scenarios is one of the most important issues in the control of autonomous vehicles (Yue et al., 2019; Shen et al., 2020a; Guo et al., 2020). Recently, Model Predictive Control (MPC) has been used to improve the vehicle dynamics stability (Yuan et al., 2019). In (Taghavifar, 2019), neural network autoregressive with exogenous input system has been applied to obtain an accurate and explicit model in order to contribute to the control of the system over the prediction horizon. (Weiskircher et al., 2017). proposed a MPC-based predictive trajectory guidance and tracking control framework for autonomous and semiautonomous vehicles in dynamic public traffic. Moreover, a data-driven predictive control is proposed in (Li and Schutter, 2021) which is model-free predictive control method.
However, the normal MPC without considering the uncertainty is not able to address the problem caused by environment uncertainty. The state space model-based prediction has large variance and even mean bias if there are any uncertainties in disturbance or the system parameters (Shen et al., 2020b). If there is uncertain road friction changes when passing a sharp curve and the model used in MPC cannot reflect the uncertainty, MPC will lose some precise on the lateral dynamics control. To improve the robustness against uncertainty, it is necessary to design a robust controller. In (Heshmati-Alamdari et al., 2020), a robust predictive controller is designed for underwater robotic vehicles which forms a high robust closed-loop system against parameter uncertainties. Besides, (Gao et al., 2021), proposed a robust lateral trajectory following control for autonomous vehicles. Robust model predictive control is a potential solution to the issue caused by uncertain road friction in this research. In the problem formulation of robust model predictive control, the road friction is regarded as a uncertain variable. For all possible realizations of uncertain variable, a fixed control law has a cost. We focused on finding a control law that minimize the upper bound of the cost for all possible realizations of uncertain variable. In this way, the robustness of the control strategy is able to be attained. To achieve robust model predictive controller, it is essentially to solve a robust optimization problem or a chance constrained optimization problem in every time step (Nemirovski and Shapiro, 2006; Shen et al., 2019). Although it is NP-hard to solve a robust optimization problem or a chance constrained optimization problem (Hong et al., 2011; Geletu et al., 2017; Pena-Ordieres et al., 2020), the approximate solution can be obtained by formulating a solvable approximate problem of the original one (Luedtke and Ahmed, 2008; Shen et al., 2021; Campi and Garatti, 2019, 2011). Robust model predictive control was widely applied in water qulity management (Takyi and Lence, 1999) and other process control applications (Henrion and Moller, 2003). Recently, robust model predictive control has been applied to the automotive powertrain control to optimize the fuel efficiency with stochastic constraint on the knock (Shen et al., 2017; Shen and Shen, 2017) and the energy management system in hybrid electric vehicle (Shen et al., 2016). Robust model predictive control can also be applied to ensure the robustness for an autonomous vehicle when it passes a sharp curve with uncertain road condition.
This paper presents a novel robust model predictive control strategy for automated vehicles to preserve a precise tracking performance and maintain the stability of lateral dynamics. The optimal feedback control input is obtained in every step by solving a robust optimization problem. The robust optimization problem is solved by scenario approach introduced in (Calariore and Campi, 2006). Simulation validations are carried out to evaluate the proposed control strategy.
2 Proposed Method
2.1 Background and Problem Description
In Figure 1, the vehicle passed a sharp curve with water-covered surface. The water-covered surface is the area with orange color. The single track model of vehicle dynamics can be described by the following equations:
FIGURE 1
Here, yc is the lateral distance. is the yaw rate. m is the mass of the vehicle. δf is the steer angle. V is the vehicle speed.
In order to apply MPC, the vehicle lateral dynamics model is transformed to the lateral deviation from the reference model. The used linear model is asWhere
Here, ycr is the lateral deviation from the reference trajectory. is the yaw rate. R is the radius of the curve. m is the mass of the vehicle. δf is the steer angle. V is the vehicle speed.
Notice that Cf and Cr are both decided by the road friction coefficient. Since the road friction coefficient is uncertain, Cf and Cr are both uncertain variable as well.
Equation 5 is a continuous differential equation and can be transformed to a discrete state-space model by Euler method. Since at every time step, the state variable is decided by the input δf and the state variable in the previous step. The state variable at k + 1 can be expressed by the previous input sequence δf (0), …, δf(k) and the state variable at the initial step. Since the objective is to minimize the difference between the actual trajectory and the reference one, the cost function is a function of the input sequence and known state variable at initial step. To obtain the optimal input, a robust optimization problem should be solved. The problem can be formulated generally by
Here, if we consider K steps forward. δ is the uncertain variable. In our problem, it includes Cf and Cr. J(u) = E and h (u, δ) is defined as
2.2 Scenario Approach
In scenario approach, independent samples δ(i), i = 1, … , N is identically extracted from Δ randomly, a deterministic convex optimization problem can be formed as (Calariore, 2017; Campi et al., 2018; Campi and Garatti, 2018)which is a standard finitely constrained optimization problem. The optimal solution of the program Eq. 11 is called as the scenario solution for program Eq. 9 generally. Moreover, since the extractions δ(i), i = 1, … , N is randomly chosen, the optimal solution is random variable. If is expected to satisfythen, N should have a lower limitation Nl
Note that β is an important factor and choosing β = 0 makes Nl = ∞. Namely, if the number of chosen samples gets larger, the probability of satisfying the original probabilistic constraints approaches 1. Actually, when number of chosen samples becomes infinity, the samples cover the whole sample space. The feasible area determined by probabilistic constraints is only a subset of whole sample space. Then, it becomes a problem which requires total robustness. Therefore, the scenario approach conducts to a solution with total robustness which is more conservative than the probabilistic constraints require.
2.3 Implementation of Robust Model Predictive Control
The implementation of robust MPC is shown in Figure 2. At time step k + 1, it uses the first element of u calculated in time step k as the input. Namely, δf(k) = u (1). x(k) denotes the state variable vector at time step k. Moreover, since the LMPC controller takes relative variable calculation as feedback, there will be a relative variable calculation. In the relative variable calculation, the relative variable is calculated based on the feedback state variable from plant model or real vehicle and the information of curve, for example, radius value R.
FIGURE 2
3 Validation Results and Conclustion
The validation is implemented by simulation. Since the real vehicle is not available, a plant model is established and used instead of the real vehicle. The plant model adopts the single track nonlinear model described by
The magic formula is used to model the friction forces which refers to (Yuan et al., 2019).
For the simulation conditions, the radius has six options: 100, 110, 120, 130, 140, and 150 m. For each R, three coefficients of friction for the wet road is randomly generated from (0.4,0.6). For each pair of a value of R and a value of coefficients of friction, the following longitudinal velocity values have be tested:
Figure 3 shows one example of the validation results. The friction coefficient of dry road is μdry = 0.8 which the one of wet road is μwet = 0.5. The radius of the curve is 100 m. The middle part of the road is wet. The longitudinal velocity for passing the curve is V = 65 km/h. If MPC is used by setting Cr and Cf according to μdry = 0.8, the tracking error increases during the wet road. However, by considering μ ∈ [0.4, 0.9], the robust MPC keeps the tracking performance stable during the wet road.
FIGURE 3
Figure 4 shows a comprehensive statistical validation results of all cases. Obviously, the robust MPC succeeded to decrease the maximal deviation into the error bound. However, the normal MPC failed in most cases since the model has a very large bias compared to the real dynamics due to the uncertain friction coefficient.
FIGURE 4
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
Author contributions
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Funding
The authors appreciate the supports of Foundation of Natural Science Research in Colleges and Universities of Jiangsu Province (Grant: 18KJB510043), National Key R& D Program of China (Grant: 2018YFB0105201), Natural Science Foundation of China (Grant: 51975394; 61903269) and Natural Science Foundation of Jiangsu Province (Contract: BK20200271).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
References
1
CalarioreG. C.CampiM. C. (2006). The Scenario Approach Torobust Control Design. IEEE Trans. Automatic Control.51 (5), 742–753. 10.1109/tac.2006.880951
2
CalarioreG. (2017). Repetitive Scenario Design. IEEE Trans. Automatic Control.62 (3), 1125–1137. 10.1109/tac.2017.2788139
3
CampiM. C.GarattiS. (2011). A Sampling-And-Discarding Approach to Chance-Constrained Optimization: Feasibility and Optimality. J. Optimization Theor. Appl.148 (2), 257–280. 10.1007/s10957-010-9754-6
4
CampiM. C.GarattiS. (2019). Introduction to the Scenario Approach. Philadelphia: MOS-SIAM Series on Optimization.
5
CampiM. C.GarattiS.RamponiF. A. (2018). A General Scenario Theory for Nonconvex Optimization and Decision Making. IEEE Trans. Automat. Contr.63 (12), 4067–4078. 10.1109/tac.2018.2808446
6
CampiM. C.GarattiS. (2018). Wait-and-Judge Scenario Optimization. Math. Program167, 155–189. 10.1007/s10107-016-1056-9
7
GaoH.KanZ.LiK. (2021). Robust Lateral Trajectory Following Control of Unmanned Vehicle Based on Model Predictive Control. Ieee/asme Trans. Mechatron., early access. 10.1109/TMECH.2021.3087605
8
GeletuA.HoffmannA.KlöppelM.LiP. (2017). An Inner-Outer Approximation Approach to Chance Constrained Optimization. SIAM J. Optim.27, 1834–1857. 10.1137/15m1049750
9
GuoN.ZhangX.ZouY.LenzoB.ZhangT. (2020). A Computationally Efficient Path-Following Control Strategy of Autonomous Electric Vehicles with Yaw Motion Stabilization. IEEE Trans. Transp. Electrific.6 (2), 728–739. 10.1109/tte.2020.2993862
10
HenrionR.MollerA. (2003). Optimization of a Continuous Distillation Process under Random Inflow Rate. Comput. Math. Appli.45, 13–30. 10.1016/s0898-1221(03)80017-2
11
Heshmati-AlamdariS.KarrasG. C.MarantosP.KyriakopoulosK. J.KyriakopoulosK. J. (2020). A Robust Predictive Control Approach for Underwater Robotic Vehicles. IEEE Trans. Contr. Syst. Technol.28 (6), 2352–2363. 10.1109/tcst.2019.2939248
12
HongL. J.YangY.ZhangL. (2011). Sequential Convex Approximations to Joint Chance Constrained Programs: A Monte Carlo Approach. Operations Res.59, 617–630. 10.1287/opre.1100.0910
13
KritayakiranaK.GerdesJ. C. (2012). Autonomous Vehicle Control at the Limits of Handling. Ijvas10 (4), 271–296. 10.1504/ijvas.2012.051270
14
LiD.De SchutterB. (2021). Distributed Model-Free Adaptive Predictive Control for Urban Traffic Networks. IEEE Trans. Contr. Syst. Technol., 1–13. early access. 10.1109/TCST.2021.3059460
15
LuedtkeJ.AhmedS. (2008). A Sample Approximation Approach for Optimization with Probabilistic Constraints. SIAM J. Optim.19 (2), 674–699. 10.1137/070702928
16
NemirovskiA.ShapiroA. (2006). Convex Approximations of Chance Constrained Programs. SIAM J. Optimization17, 969–996. 10.1137/050622328
17
Peña-OrdieresA.LuedtkeJ. R.WächterA. (2020). Solving Chance-Constrained Problems via a Smooth Sample-Based Nonlinear Approximation. SIAM J. Optim.30 (3), 2221–2250. 10.1137/19m1261985
18
ShenX.OuyangT.YangN.ZhuangJ. (2021). Sample-Based Neural Approximation Approach for Probabilistic Constrained Programs. IEEE Trans. Neural Netw. Learn. Syst., 1–8. early access. 10.1109/TNNLS.2021.3102323
19
ShenX.OuyangT.ZhangY.ZhangX. (2020a). Computing Probabilistic Bounds on State Trajectories for Uncertain Systems. IEEE Trans. Emerg. Top. Comput. Intell., 1–6. early access. 10.1109/TETCI.2020.3019040
20
ShenX.RaksincharoensakP. (2021). Pedestrian-Aware Statistical Risk Assessment. IEEE Trans. Intell. Transport. Syst.22, 1–9. early access. 10.1109/TITS.2021.3074522
21
ShenX.ShenT. (2017). Real-Time Statistical Learning-Based Stochastic Knock Limit Control for Spark-Ignition Engines. Appl. Therm. Eng.127, 1518–1529. 10.1016/j.applthermaleng.2017.08.150
22
ShenX.ZhangJ.ShenT. (2016). “Real-time Scenario-Based Stochastic Optimal Energy Management Strategy for HEVs,” in Proceedings of 2016 European Control Conference, Aalborg, Denmark, 29 June–1 July 2016, 631–636. 10.1109/ecc.2016.7810359
23
ShenX.ZhangX.RaksincharoensakP. (2020b). Probabilistic Bounds on Vehicle Trajectory Prediction Using Scenario Approach. IFAC-PapersOnline53 (2), 2385–2390. 10.1016/j.ifacol.2020.12.038
24
ShenX.ZhangY.ShenT.KhajorntraidetC. (2017). Spark advance Self-Optimization with Knock Probability Threshold for Lean-Burn Operation Mode of SI Engine. Energy122, 1–10. 10.1016/j.energy.2017.01.065
25
ShenX.ZhuangJ.ZhangX. (2019). Approximate Uncertain Program. IEEE Access7, 182357–182365. 10.1109/access.2019.2958621
26
TaghavifarH. (2019). Neural Network Autoregressive with Exogenous Input Assisted Multi-Constraint Nonlinear Predictive Control of Autonomous Vehicles. IEEE Trans. Veh. Technol.68 (7), 6293–6304. 10.1109/tvt.2019.2914027
27
TakyiA. K.LenceB. J. (1999). Surface Water Quality Management Using a Multiple-Realization Chance Constraint Method. Water Resour. Res.35, 1657–1670. 10.1029/98wr02771
28
WeiskircherT.WangQ.AyalewB. (2017). Predictive Guidance and Control Framework for (Semi-)Autonomous Vehicles in Public Traffic. IEEE Trans. Contr. Syst. Technol.25 (6), 2034–2046. 10.1109/tcst.2016.2642164
29
YuanH.SunX.GordonT. (2019). Unified Decision-Making and Control for Highway Collision Avoidance Using Active Front Steer and Individual Wheel Torque Control. Vehicle Syst. Dyn.57 (8), 1188–1205. 10.1080/00423114.2019.169745610.1080/00423114.2018.1535125
30
YueM.NingY.ZhaoX.ZongG. (2019). Point Stabilization Control Method for WIP Vehicles Based on Motion Planning. IEEE Trans. Ind. Inf.15 (6), 3368–3378. 10.1109/tii.2018.2875048
Summary
Keywords
model predictive conrol, robust optimisation, vehicle dynamic, uncertainty, stability
Citation
Miao J, Dai Y, Xie O, Chen H, Niu F, Shen Y, Wu YZ, Sun H, Niu X, Zhu Q and Shen W (2021) Robust Vehicle Dynamics Control for a Sharp Curve With Uncertain Road Condition. Front. Energy Res. 9:762246. doi: 10.3389/fenrg.2021.762246
Received
21 August 2021
Accepted
08 September 2021
Published
14 October 2021
Volume
9 - 2021
Edited by
Yahui Zhang, Yanshan University, China
Reviewed by
Xun Shen, Tokyo University of Agriculture and Technology, Japan
Yang Tian, Yanshan University, China
Updates
Copyright
© 2021 Miao, Dai, Xie, Chen, Niu, Shen, Wu, Sun, Niu, Zhu and Shen.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Jing Miao, jmiao@mail.usts.edu.cn
This article was submitted to Wind Energy, a section of the journal Frontiers in Energy Research
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.