{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,18]],"date-time":"2026-05-18T19:47:25Z","timestamp":1779133645713,"version":"3.51.4"},"reference-count":33,"publisher":"MDPI AG","issue":"15","license":[{"start":{"date-parts":[[2022,7,25]],"date-time":"2022-07-25T00:00:00Z","timestamp":1658707200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Cooperation Project of Guizhou Education Department","award":["KY [2021] 297"],"award-info":[{"award-number":["KY [2021] 297"]}]},{"name":"the Cooperation Project of Guizhou Education Department","award":["ZK [2021] 320"],"award-info":[{"award-number":["ZK [2021] 320"]}]},{"name":"the Cooperation Project of Guizhou Education Department","award":["GZNUD [2019] 22"],"award-info":[{"award-number":["GZNUD [2019] 22"]}]},{"name":"the Cooperation Project of Guizhou Education Department","award":["YJSKYJJ [2021] 101"],"award-info":[{"award-number":["YJSKYJJ [2021] 101"]}]},{"DOI":"10.13039\/501100004001","name":"the Science and Technology Foundation of Guizhou Province","doi-asserted-by":"publisher","award":["KY [2021] 297"],"award-info":[{"award-number":["KY [2021] 297"]}],"id":[{"id":"10.13039\/501100004001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004001","name":"the Science and Technology Foundation of Guizhou Province","doi-asserted-by":"publisher","award":["ZK [2021] 320"],"award-info":[{"award-number":["ZK [2021] 320"]}],"id":[{"id":"10.13039\/501100004001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004001","name":"the Science and Technology Foundation of Guizhou Province","doi-asserted-by":"publisher","award":["GZNUD [2019] 22"],"award-info":[{"award-number":["GZNUD [2019] 22"]}],"id":[{"id":"10.13039\/501100004001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004001","name":"the Science and Technology Foundation of Guizhou Province","doi-asserted-by":"publisher","award":["YJSKYJJ [2021] 101"],"award-info":[{"award-number":["YJSKYJJ [2021] 101"]}],"id":[{"id":"10.13039\/501100004001","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the Ph.D. Research Project of Guizhou Normal University","award":["KY [2021] 297"],"award-info":[{"award-number":["KY [2021] 297"]}]},{"name":"the Ph.D. Research Project of Guizhou Normal University","award":["ZK [2021] 320"],"award-info":[{"award-number":["ZK [2021] 320"]}]},{"name":"the Ph.D. Research Project of Guizhou Normal University","award":["GZNUD [2019] 22"],"award-info":[{"award-number":["GZNUD [2019] 22"]}]},{"name":"the Ph.D. Research Project of Guizhou Normal University","award":["YJSKYJJ [2021] 101"],"award-info":[{"award-number":["YJSKYJJ [2021] 101"]}]},{"name":"the M.S. Research Project Foundation of Guizhou Province","award":["KY [2021] 297"],"award-info":[{"award-number":["KY [2021] 297"]}]},{"name":"the M.S. Research Project Foundation of Guizhou Province","award":["ZK [2021] 320"],"award-info":[{"award-number":["ZK [2021] 320"]}]},{"name":"the M.S. Research Project Foundation of Guizhou Province","award":["GZNUD [2019] 22"],"award-info":[{"award-number":["GZNUD [2019] 22"]}]},{"name":"the M.S. Research Project Foundation of Guizhou Province","award":["YJSKYJJ [2021] 101"],"award-info":[{"award-number":["YJSKYJJ [2021] 101"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This paper presents a lateral and longitudinal coupling controller for a trajectory-tracking control system. The proposed controller can simultaneously minimize lateral tracking deviation while tracking the desired trajectory and vehicle speed. Firstly, we propose a hierarchical control structure composed of upper and lower-level controllers. In the upper-level controller, the linear quadratic regulator (LQR) controller is designed to compute the desired front wheel steering angle for minimizing the lateral tracking deviation, and the model-predictive controller is developed to compute the desired acceleration for maintaining the planed vehicle speed. The lower-level controller enables the achievement of the desired steering angle and acceleration via the corresponding component devices. Furthermore, an observer based on the Extended Kalman Filter (EKF) is proposed to update the vehicle driving states, which are sensitive to the trajectory-tracking control and difficult to measure directly using the existing vehicle sensors. Finally, the Co-simulation (CarSim-MATLAB\/Simulink) results demonstrate that the proposed coupling controller is able to robustly realize the trajectory tracking control and can effectively reduce the lateral tracking error.<\/jats:p>","DOI":"10.3390\/s22155556","type":"journal-article","created":{"date-parts":[[2022,7,26]],"date-time":"2022-07-26T00:17:27Z","timestamp":1658794647000},"page":"5556","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":34,"title":["LQR-MPC-Based Trajectory-Tracking Controller of Autonomous Vehicle Subject to Coupling Effects and Driving State Uncertainties"],"prefix":"10.3390","volume":"22","author":[{"given":"Tengfei","family":"Yuan","sequence":"first","affiliation":[{"name":"School of Mechanical and Electrical Engineering, Guizhou Normal University, Guizhou 550025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rongchen","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Mechanical and Electrical Engineering, Guizhou Normal University, Guizhou 550025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/j.conengprac.2019.05.001","article-title":"Coupled longitudinal\/lateral controllers for autonomous vehicles navigation, with experimental validation","volume":"88","author":"Chebly","year":"2019","journal-title":"Control Eng. Pract."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Hedrick, J.K., and Swaroop, D. (2021). Dynamic coupling in vehicles under automatic control. The Dynamics of Vehicles on Roads and on Tracks, CRC Press.","DOI":"10.1201\/9781003210900-17"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"113021","DOI":"10.1109\/ACCESS.2020.3002814","article-title":"Trajectory tracking based on adaptive sliding mode control for agricultural tractor","volume":"8","author":"Yin","year":"2020","journal-title":"IEEE Access"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"33","DOI":"10.3233\/ICA-2007-14104","article-title":"Trajectory planning and sliding-mode control based trajectory-tracking for cybercars","volume":"14","author":"Solea","year":"2007","journal-title":"Integr. Comput.-Aided Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"796","DOI":"10.1109\/TITS.2015.2486815","article-title":"Design and comparison of robust nonlinear controllers for the lateral dynamics of intelligent vehicles","volume":"17","author":"Tagne","year":"2015","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"107238","DOI":"10.1016\/j.ymssp.2020.107238","article-title":"A robust lateral tracking control strategy for autonomous driving vehicles","volume":"150","author":"Zhang","year":"2021","journal-title":"Mech. Syst. Signal Processing"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1016\/j.isatra.2016.04.019","article-title":"T\u2013S fuzzy model predictive speed control of electrical vehicles","volume":"64","author":"Khooban","year":"2016","journal-title":"ISA Trans."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3782","DOI":"10.1109\/TVT.2018.2816936","article-title":"Coordinated longitudinal and lateral motion control for four wheel independent motor-drive electric vehicle","volume":"67","author":"Zhou","year":"2018","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"206559","DOI":"10.1109\/ACCESS.2020.3038050","article-title":"Two-Level MPC Speed Profile Optimization of Autonomous Electric Vehicles Considering Detailed Internal and External Losses","volume":"8","author":"Meshginqalam","year":"2020","journal-title":"IEEE Access"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Awad, N., Lasheen, A., Elnggar, M., and Kamel, A. (2021). Model predictive control with fuzzy logic switching for path tracking of autonomous vehicles. ISA Trans., in press.","DOI":"10.1016\/j.isatra.2021.12.022"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2967","DOI":"10.1016\/j.automatica.2014.10.128","article-title":"Model predictive control: Recent developments and future promise","volume":"50","author":"Mayne","year":"2014","journal-title":"Automatica"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1016\/j.ymssp.2016.03.003","article-title":"A model predictive speed tracking control approach for autonomous ground vehicles","volume":"87","author":"Zhu","year":"2017","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_13","first-page":"189","article-title":"Automatic path control based on integrated steering and external yaw-moment control","volume":"222","author":"Goodarzi","year":"2008","journal-title":"Proc. Inst. Mech. Eng. Part K J. Multi-Body Dyn."},{"key":"ref_14","unstructured":"Snider, J.M. (2009). Automatic Steering Methods for Autonomous Automobile Path Tracking, Robotics Institute. CMU-RITR-09-08."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Yu, H., Zhao, C., Li, S., Wang, Z., and Zhang, Y. (2021). Pre-Work for the Birth of Driver-Less Scraper (LHD) in the Underground Mine: The Path Tracking Control Based on an LQR Controller and Algorithms Comparison. Sensors, 21.","DOI":"10.3390\/s21237839"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Yang, T., Bai, Z., Li, Z., Feng, N., and Chen, L. (2021). Intelligent Vehicle Lateral Control Method Based on Feedforward+ Predictive LQR Algorithm, Multidisciplinary Digital Publishing Institute. Actuators, 2021.","DOI":"10.3390\/act10090228"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Calzolari, D., Sch\u00fcrmann, B., and Althoff, M. (2017, January 16\u201319). Comparison of trajectory tracking controllers for autonomous vehicles. Proceedings of the 2017 IEEE 20th International Conference on Intelligent Transportation Systems (ITSC), Yokohama, Japan.","DOI":"10.1109\/ITSC.2017.8317800"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1006","DOI":"10.1590\/1679-78251365","article-title":"An optimal integrated longitudinal and lateral dynamic controller development for vehicle path tracking","volume":"12","author":"Tavan","year":"2015","journal-title":"Lat. Am. J. Solids Struct."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"542","DOI":"10.1115\/1.2899250","article-title":"A novel PID configuration for speed and position control","volume":"116","year":"1994","journal-title":"J. Dyn. Syst. Meas. Control."},{"key":"ref_20","unstructured":"Lim, E.M. (1998). Lateral and Longitudinal Vehicle Coupling in the Automated Highway System. [Master\u2019s Thesis, University of California]."},{"key":"ref_21","unstructured":"Xiaoping, H., and Yan, W. (2015). Principle and Application of Kalman Filter, Publishing House of Electronics Industry."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1687814015578361","DOI":"10.1177\/1687814015578361","article-title":"A model predictive control approach combined unscented Kalman filter vehicle state estimation in intelligent vehicle trajectory tracking","volume":"7","author":"Yu","year":"2015","journal-title":"Adv. Mech. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"834","DOI":"10.1016\/j.jfranklin.2016.01.005","article-title":"Vehicle state estimation based on minimum model error criterion combining with extended Kalman filter","volume":"353","author":"Liu","year":"2016","journal-title":"J. Frankl. Inst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1017\/S0373463316000655","article-title":"Comparison of Unscented and Extended Kalman filters with application in vehicle navigation","volume":"70","author":"Yang","year":"2017","journal-title":"J. Navig."},{"key":"ref_25","unstructured":"LaViola, J.J. (2003, January 4\u20136). A comparison of Unscented and Extended Kalman filtering for estimating quaternion motion. Proceedings of the 2003 American Control Conference, Denver, CO, USA."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1142","DOI":"10.1017\/S0373463321000370","article-title":"Comparative evaluation of Kalman filters and motion models in vehicular state estimation and path prediction","volume":"74","author":"Tao","year":"2021","journal-title":"J. Navig."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2376","DOI":"10.1109\/TITS.2019.2918176","article-title":"Longitudinal collision avoidance and lateral stability adaptive control system based on MPC of autonomous vehicles","volume":"21","author":"Cheng","year":"2019","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Rajamani, R. (2011). Vehicle Dynamics and Control, Springer Science & Business Media.","DOI":"10.1007\/978-1-4614-1433-9_2"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1109\/TITS.2005.848359","article-title":"Range policy of adaptive cruise control vehicles for improved flow stability and string stability","volume":"6","author":"Zhou","year":"2005","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1687","DOI":"10.1080\/00423114.2015.1055279","article-title":"Design of a feedback-feedforward steering controller for accurate path tracking and stability at the limits of handling","volume":"53","author":"Kapania","year":"2015","journal-title":"Veh. Syst. Dyn."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3354","DOI":"10.1021\/ie300415d","article-title":"How to not make the extended Kalman filter fail","volume":"52","author":"Schneider","year":"2013","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_32","unstructured":"(2022, July 11). CarSim Overview. Available online: https:\/\/www.carsim.com\/products\/carsim\/index.php."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"5180","DOI":"10.1109\/TSMC.2018.2867807","article-title":"Robust tube-based model predictive control for lane change maneuver of tractor-trailer vehicles based on a polynomial trajectory","volume":"50","author":"Yue","year":"2018","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/15\/5556\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:56:19Z","timestamp":1760140579000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/15\/5556"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,25]]},"references-count":33,"journal-issue":{"issue":"15","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["s22155556"],"URL":"https:\/\/doi.org\/10.3390\/s22155556","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7,25]]}}}