MPC-based path tracking with PID speed control for high-speed autonomous vehicles considering time-optimal travel

Abstract

In order to track the desired path as fast as possible, a novel autonomous vehicle path tracking based on model predictive control (MPC) and PID speed control was proposed for high-speed automated vehicles considering the constraints of vehicle physical limits, in which a forward-backward integration scheme was introduced to generate a time-optimal speed profile subject to the tire-road friction limit. Moreover, this scheme was further extended along one moving prediction window. In the MPC controller, the prediction model was an 8-degree-of-freedom (DOF) vehicle model, while the plant was a 14-DOF vehicle model. For lateral control, a sequence of optimal wheel steering angles was generated from the MPC controller; for longitudinal control, the total wheel torque was generated from the PID speed controller embedded in the MPC framework. The proposed controller was implemented in MATLAB considering arbitrary curves of continuously varying curvature as the reference trajectory. The simulation test results show that the tracking errors are small for vehicle lateral and longitudinal positions and the tracking performances for trajectory and speed are good using the proposed controller. Additionally, the case of extended implementation in one moving prediction window requires shorter travel time than the case implemented along the entire path.

Keywords

model predictive control / path tracking / minimum-time speed profile / vehicle dynamics / arbitrary path

References

[1] LIU Chang, LEE S, VARNHAGEN S, TSENG H E. Path planning for autonomous vehicles using model predictive control [C]// IEEE Intelligent Vehicles Symposium. IEEE, 2017: 174–179. DOI: https://doi.org/10.1109/IVS.2017.7995716.

[2] YurtseverE, LambertJ, CarballoA, TakedaK. A survey of autonomous driving: Common practices and emerging technologies. IEEE Access, 2020, 8: 58443-58469

[3] BrownM, FunkeJ, ErlienS, GerdesJ C. Safe driving envelopes for path tracking in autonomous vehicles. Control Engineering Practice, 2017, 61: 307-316

[4] CUI Qing-jia, DING Rong-jun, WEI Chong-feng, ZHOU Bing. Path-tracking and lateral stabilisation for autonomous vehicles by using the steering angle envelope [J]. Vehicle System Dynamics, 2020: 1–25. DOI: https://doi.org/10.1080/00423114.2020.1776344

[5] HwangJ Y, KimJ S, LimS S, ParkK H. A fast path planning by path graph optimization. IEEE Transactions on Systems, Man, and Cybernetics, 2003, 33(1): 121-128

[6] LikhachevM, FergusonD, GordonG, StentzA, ThrunS. Anytime search in dynamic graphs. Artificial Intelligence, 2008, 172: 1613-1643

[7] DolgovD, ThrunS, MontemerloM, DiebelJ. Path planning for autonomous vehicles in unknown semi-structured environments. International Journal of Robotics Research, 2010, 29(5): 485-501

[8] PivtoraikoM, KnepperR A, KellyA. Differentially constrained mobile robot motion planning in state lattices. Journal of Field Robotics, 2009, 26(3): 308-333

[9] KavrakiL E, KolountzakisM N, LatombeJ C. Analysis of probabilistic roadmaps for path planning. IEEE Transactions on Robotics and Automation, 1998, 14(1): 166-171

[10] LavalleS M, KuffnerJ J. Randomized kinodynamic planning. The International Journal of Robotics Research, 2001, 20(5): 378-400

[11] ZIEGLER J, BENDER P, DANG T, STILLER C. Trajectory planning for bertha-a local, continuous method [C]// 2014 IEEE Intelligent Vehicles Symposium Proceedings. IEEE, 2014: 450–457. DOI: https://doi.org/10.1109/IVS.2014.6856581.

[12] GonzálezD, PérezJ, MilanésV, NashashibiF. A review of motion planning techniques for automated vehicles. IEEE Transactions on Intelligent Transportation Systems, 2016, 17(4): 1135-1145

[13] HuangY-J, WangH, KhajepourA, DingH-T, YuanK, QinY-C. A novel local motion planning framework for autonomous vehicles based on resistance network and model predictive control. IEEE Transactions on Vehicular Technology, 2020, 69(1): 55-66

[14] BealC E, GerdesJ C. Model predictive control for vehicle stabilization at the limits of handling. IEEE Transactions on Control Systems Technology, 2013, 21(4): 1258-1269

[15] MarinoR, ScalziS, NettoM. Nested PID steering control for lane keeping in autonomous vehicles. Control Engineering Practice, 2011, 19: 1459-1467

[16] WangS-T, YinX-H, LiP, ZhangM-Z, WangX. Trajectory tracking control for mobile robots using reinforcement learning and PID. Iranian Journal of Science and Technology, Transactions of Electrical Engineering, 2020, 44(3): 1059-1068

[17] HuC, WangR-R, YanF-J, ChenN. Output constraint control on path following of four-wheel independently actuated autonomous ground vehicles. IEEE Transactions on Vehicular Technology, 2016, 65(6): 4033-4043

[18] LeeK, JeonS, KimH, KumD. Optimal path tracking control of autonomous vehicle: Adaptive full-state linear quadratic Gaussian (LQG) control. IEEE Access, 2019, 7: 109120-109133

[19] KangC M, KimW, ChungC C. Observer-based backstepping control method using reduced lateral dynamics for autonomous lane-keeping system. ISA Transactions, 2018, 83: 214-226

[20] NorouziA, MasoumiM, BarariA, SaniS F. Lateral control of an autonomous vehicle using integrated backstepping and sliding mode controller. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 2019, 233(1): 141-151

[21] GongJ-W, XuW, JiangY, LiuK, GuoH-F, SunY-J. Multi-constrained model predictive control for autonomous ground vehicle trajectory tracking. Journal of Beijing Institute of Technology, 2015, 24(4): 441-448

[22] LiuK, GongJ-W, KurtA, ChenH-Y, OzgunerU. Dynamic modeling and control of high-speed automated vehicles for lane change maneuver. IEEE Transactions on Intelligent Vehicles, 2018, 3(3): 329-339

[23] NamH, ChoiW, AhnC. Model predictive control for evasive steering of an autonomous vehicle. International Journal of Automotive Technology, 2019, 20(5): 1033-1042

[24] LiuK, GongJ-W, ChenS-P, ZhangY, ChenH-Y. Dynamic modeling analysis of optimal motion planning and control for high-speed self-driving vehicles. Journal of Mechanical Engineering, 2018, 54(14): 141-151

[25] FalconeP, TsengH E, BorrelliF, AsgariJ, HrovatD. MPC-based yaw and lateral stabilisation via active front steering and braking. Vehicle System Dynamics, 2008, 46: 611-628

[26] MING Ting-you, DENG Wei-wen, ZHANG Su-min, ZHU Bing. MPC-based trajectory tracking control for intelligent vehicles [J]. SAE Technical Papers, 2016: 2016-01-0452. DOI: https://doi.org/10.4271/2016-01-0452.

[27] GuoH-Y, LiuJ, CaoD-P, ChenH, YuR, LvC. Dual-envelop-oriented moving horizon path tracking control for fully automated vehicles. Mechatronics, 2018, 50: 422-433

[28] TangL-Q, YanF-W, ZouB, WangK-W, LvC. An improved kinematic model predictive control for high-speed path tracking of autonomous vehicles. IEEE Access, 2020, 8: 51400-51413

[29] AttiaR, OrjuelaR, BassetM. Combined longitudinal and lateral control for automated vehicle guidance. Vehicle System Dynamics, 2014, 52(2): 261-279

[30] LinF, ZhangY-W, ZhaoY-Q, YinG-D, ZhangH-Q, WangK-Z. Trajectory tracking of autonomous vehicle with the fusion of DYC and longitudinal-lateral control. Chinese Journal of Mechanical Engineering, 2019, 32: 1-16

[31] LippT, BoydS. Minimum-time speed optimisation over a fixed path. International Journal of Control, 2014, 87(6): 1297-1311

[32] AlcaláE, PuigV, QuevedoJ, RosoliaU. Autonomous racing using linear parameter varying-model predictive control (LPV-MPC). Control Engineering Practice, 2020, 95: 104270

[33] BobrowJ E, DubowskyS, GibsonJ S. Time-optimal control of robotic manipulators along specified paths. The International Journal of Robotics Research, 1985, 4(3): 3-17

[34] VerscheureD, DemeulenaereB, SweversJ, SchutterJ D, DiehlM. Time-optimal path tracking for robots: A convex optimization approach. IEEE Transactions on Automatic Control, 2009, 54(10): 2318-2327

[35] DINH Q T, DIEHL M. An application of sequential convex programming to time optimal trajectory planning for a car motion [C]// Proceedings of the IEEE Conference on Decision and Control. IEEE, 2009: 4366–4371. DOI: https://doi.org/10.1109/CDC.2009.5399823.

[36] FAULWASSER T, HAGENMEYER V, FINDEISEN R. Optimal exact path-following for constrained differentially flat systems[C]// Proceedings of the 18th IFAC World Congress. Elsevier, 2011: 9875–9880. DOI: https://doi.org/10.3182/20110828-6-IT-1002.03132.

[37] CasanovaD, SharpR S, SymondsP. Minimum time manoeuvring: The significance of yaw inertia. Vehicle System Dynamics, 2000, 3477-115

[38] VELENIS E, TSIOTRAS P. Minimum time vs maximum exit velocity path optimization during cornering [C]// IEEE International Symposium on Industrial Electronics. IEEE, 2005: 355–360. DOI: https://doi.org/10.1109/ISIE.2005.1528936

[39] HendrikxJ P M, MeijlinkT J J, KriensR F C. Application of optimal control theory to inverse simulation of car handling. Vehicle System Dynamics, 1996, 26: 449-461

[40] CasanovaDOn minimum time vehicle manoeuvring: The theoretical optimal lap, 2000, Cranfield, Cranfield University

[41] KellyD PLap time simulation with transient vehicle and tyre dynamics, 2008, Cranfield, Cranfield University

[42] THEODOSIS P A, GERDES J C. Generating a racing line for an autonomous racecar using professional driving techniques [C]// ASME 2011 Dynamic Systems and Control Conference and Bath/ASME Symposium on Fluid Power and Motion Control, ASME, 2012: 853–860. DOI: https://doi.org/10.1115/DSCC2011-6097.

[43] GerdtsM, KarrenbergS, Müller-BeßlerB, StockG. Generating locally optimal trajectories for an automatically driven car. Optimization and Engineering, 2009, 10: 439-463

[44] TimingsJ P, ColeD J. Minimum maneuver time calculation using convex optimization. Journal of Dynamic Systems, Measurement and Control-Transactions of the ASME, 2013, 135(3): 031015

[45] PerantoniG, LimebeerD J N. Optimal control for a Formula One car with variable parameters. Vehicle System Dynamics, 2014, 525653-678

[46] KapaniaN R, SubositsJ, GerdesJ C. A sequential two-step algorithm for fast generation of vehicle racing trajectories. Journal of Dynamic Systems, Measurement, and Control-Transactions of the ASME, 2016, 138(9): 091005

[47] SpenkoM, KurodaY, DubowskyS, IagnemmaK. Hazard avoidance for high-speed mobile robots in rough terrain. Journal of Field Robotics, 2006, 235311-331

[48] HellewellJ S, PopovA A, BurnettG E. Hierarchical control for trajectory generation and tracking via active front steering. Journal of Dynamic Systems, Measurement, and Control, 2020, 142: 011002

[49] XuS-B, PengH. Design, analysis, and experiments of preview path tracking control for autonomous vehicles. IEEE Transactions on Intelligent Transportation Systems, 2020, 21(1): 48-58

[50] VelenisE, TsiotrasP. Minimum-time travel for a vehicle with acceleration limits: Theoretical analysis and receding-horizon implementation. Journal of Optimization Theory and Applications, 2008, 138275-296

[51] LepetičM, KlančarG, ŠkrjancI, MatkoD, PotočnikB. Time optimal path planning considering acceleration limits. Robotics and Autonomous Systems, 2003, 45: 199-210

[52] KantK, ZuckerSW. Toward efficient trajectory planning: The path-velocity decomposition. The International Journal of Robotics Research, 1986, 5(3): 72-89

[53] BiancoC G L, PiazziA, RomanoM. Smooth motion generation for unicycle mobile robots via dynamic path inversion. IEEE Transactions on Robotics, 2004, 20(5): 884-891

[54] LiniG, PiazziA, ConsoliniL. Algebraic solution to minimum-time velocity planning. International Journal of Control, Automation and Systems, 2013, 11(4): 805-814

[55] ConsoliniL, LocatelliM, MinariA, PiazziA. An optimal complexity algorithm for minimum-time velocity planning. Systems and Control Letters, 2017, 103: 50-57

[56] CHEN Shu-ping, CHEN Hui-yan. MPC-based path tracking with PID speed control for autonomous vehicles [C]// IOP Conference Series: Materials Science and Engineering. IOP, 2020: 012034. DOI: https://doi.org/10.1088/1757-899X/892/1/012034.

[57] BorrelliF, FalconeP, KeviczkyT, AsgariJ, HrovatD. MPC-based approach to active steering for autonomous vehicle systems. International Journal of Vehicle Autonomous Systems, 2005, 3: 265-291

[58] ShimT, GhikeC. Understanding the limitations of different vehicle models for roll dynamics studies. Vehicle System Dynamics, 2007, 45(3): 191-216

[59] HE Jun-jie, CROLLA D A, LEVESLEY M C, MANNING W J. Integrated active steering and variable torque distribution control for improving vehicle handling and stability [J]. SAE Technical Papers, 2004: 2004-01-1071. DOI: https://doi.org/10.4271/2004-01-1071.

[60] GAO Yi-qi, GRAY A, FRASCH J V, LIN T, TSENG E H, HEDRICK J K, BORRELLI F. Spatial predictive control for agile semi-autonomous ground vehicles [C]// Proceedings of the 11th International Symposium on Advanced Vehicle Control, 2012. https://borrelli.me.berkeley.edu/pdfpub/pub-1054.pdf.

[61] ZhangC-Y, ChuD-F, LiuS-D, DengZ-J, WuC-Z. Trajectory planning and tracking for autonomous vehicle based on state lattice and model predictive control. IEEE Intelligent Transportation Systems Magazine, 2019, 11(2): 29-40

[62] GU Tian-yu, SNIDER J, DOLAN J M, LEE J W. Focused trajectory planning for autonomous on-road driving [C]//IEEE Intelligent Vehicles Symposium. IEEE, 2013: 547–552. DOI: https://doi.org/10.1109/IVS.2013.6629524.

[63] CAPORALE D, SETTIMI A, MASSA F, AMEROTTI F, CORTI A, FAGIOLINI A, GUIGGIANI M, BICCHI A, PALLOTTINO L. Towards the design of robotic drivers for full-scale self-driving racing cars [C]// IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2019: 5643–5649. DOI: https://doi.org/10.1109/ICRA.2019.8793882.

[64] MengY, WuY-M, GuQ, LiuL. A decoupled trajectory planning framework based on the integration of lattice searching and convex optimization. IEEE Access, 2019, 7: 130530-130551

[65] SUBOSITS J, GERDES J C. Autonomous vehicle control for emergency maneuvers: The effect of topography[C]// 2015 American Control Conference, IEEE, 2015: 1405–1410. DOI: https://doi.org/10.1109/ACC.2015.7170930.

[66] LiS-B, WangJ-Q, LiK-Q. Stabilization of linear predictive control systems with softening constraints. Journal of Tsinghua University (Science & Technology), 2010, 50(11): 1848-1852