{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T20:45:22Z","timestamp":1775076322107,"version":"3.50.1"},"reference-count":41,"publisher":"MDPI AG","issue":"22","license":[{"start":{"date-parts":[[2020,11,11]],"date-time":"2020-11-11T00:00:00Z","timestamp":1605052800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51875149"],"award-info":[{"award-number":["51875149"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>In the last decade, research studies on parking planning mainly focused on path planning rather than trajectory planning. The results of trajectory planning are more instructive for a practical parking process. Therefore, this paper proposes a trajectory planning method in which the optimal autonomous valet parking (AVP) trajectory is obtained by solving an optimal control problem. Additionally, a vehicle kinematics model is established with the consideration of dynamic obstacle avoidance and terminal constraints. Then the parking trajectory planning problem is modeled as an optimal control problem, while the parking time and driving distance are set as the cost function. The homotopic method is used for the expansion of obstacle boundaries, and the Gauss pseudospectral method (GPM) is utilized to discretize this optimal control problem into a nonlinear programming (NLP) problem. In order to solve this NLP problem, sequential quadratic programming is applied. Considering that the GPM is insensitive to the initial guess, an online calculation method of vertical parking trajectory is proposed. In this approach, the offline vertical parking trajectory, which is calculated and stored in advance, is taken as the initial guess of the online calculation. The selection of an appropriate initial guess is based on the actual starting position of parking. A small parking lot is selected as the verification scenario of the AVP. In the validation of the algorithm, the parking trajectory planning is divided into two phases, which are simulated and analyzed. Simulation results show that the proposed algorithm is efficient in solving a parking trajectory planning problem. The online calculation time of the vertical parking trajectory is less than 2 s, which meets the real-time requirement.<\/jats:p>","DOI":"10.3390\/s20226435","type":"journal-article","created":{"date-parts":[[2020,11,11]],"date-time":"2020-11-11T19:08:28Z","timestamp":1605121708000},"page":"6435","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["A Trajectory Planning Method for Autonomous Valet Parking via Solving an Optimal Control Problem"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8867-3514","authenticated-orcid":false,"given":"Chen","family":"Chen","sequence":"first","affiliation":[{"name":"College of Automotive and Transportation Engineering, Hefei University of Technology, Hefei 230009, China"}]},{"given":"Bing","family":"Wu","sequence":"additional","affiliation":[{"name":"College of Automotive and Transportation Engineering, Hefei University of Technology, Hefei 230009, China"}]},{"given":"Liang","family":"Xuan","sequence":"additional","affiliation":[{"name":"College of Automotive and Transportation Engineering, Hefei University of Technology, Hefei 230009, China"}]},{"given":"Jian","family":"Chen","sequence":"additional","affiliation":[{"name":"College of Automotive and Transportation Engineering, Hefei University of Technology, Hefei 230009, China"}]},{"given":"Tianxiang","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Automotive and Transportation Engineering, Hefei University of Technology, Hefei 230009, China"}]},{"given":"Lijun","family":"Qian","sequence":"additional","affiliation":[{"name":"College of Automotive and Transportation Engineering, Hefei University of Technology, Hefei 230009, China"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3229","DOI":"10.1109\/TITS.2017.2685143","article-title":"A survey of smart parking solutions","volume":"18","author":"Lin","year":"2017","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Dinh, T., and Kim, Y. (2016). A novel location-centric IoT-cloud based on-street car parking violation management system in smart cities. Sensors, 16.","DOI":"10.3390\/s16060810"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Banzhaf, H., Nienh\u00fcser, D., Knoop, S., and Z\u00f6llner, J.M. (2017, January 11\u201314). The future of parking: A survey on automated valet parking with an outlook on high density parking. Proceedings of the IEEE Intelligent Vehicles Symposium (IV), Long Beach, LA, USA.","DOI":"10.1109\/IVS.2017.7995971"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Chirca, M., Roland, C., and Roland, L. (2015, January 15\u201318). Autonomous valet parking system architecture. Proceedings of the IEEE 18th International Conference on Intelligent Transportation Systems, Las Palmas, Spain.","DOI":"10.1109\/ITSC.2015.421"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Huang, K.Y., Chang, S.B., and Tsai, P.R. (2017, January 10\u201313). The advantage of the arduino sensing system on parking guidance information systems. Proceedings of the IEEE International Conference on Industrial Engineering and Engineering Management (IEEM), Singapore, China.","DOI":"10.1109\/IEEM.2017.8290258"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1017\/S0373463318000504","article-title":"Robust ship tracking via multi-view learning and sparse representation","volume":"72","author":"Chen","year":"2019","journal-title":"J. Navigation."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1220","DOI":"10.4218\/etrij.15.0114.0112","article-title":"Parking space recognition for autonomous valet parking using height and salient-line probability maps","volume":"37","author":"Han","year":"2015","journal-title":"Etri. J."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"497","DOI":"10.2307\/2372560","article-title":"On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents","volume":"79","author":"Dubins","year":"1957","journal-title":"Am. J. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"367","DOI":"10.2140\/pjm.1990.145.367","article-title":"Optimal paths for a car that goes both forwards and backwards","volume":"145","author":"Reed","year":"1990","journal-title":"Pac. J. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"656","DOI":"10.3182\/20110828-6-IT-1002.01458","article-title":"Easy path planning and rubust control for automatic parallel parking","volume":"44","author":"Sungwoo","year":"2011","journal-title":"IFAC Pro. Vol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1109\/TITS.2014.2335054","article-title":"Automatic parallel parking in tiny spots: Path planning and control","volume":"16","author":"Vorobieva","year":"2014","journal-title":"IEEE. Trans. Intell. Transp."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1016\/j.cad.2009.12.007","article-title":"Curvature continuous path generation for autonomous vehicle using B-spline curves","volume":"42","author":"Maekawa","year":"2010","journal-title":"Comput. Aided. Design."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Zhang, S., Simkani, M., and Zadeh, M.H. (2011, January 5\u20138). Automatic vehicle parallel parking design using degree polynomial path planning. Proceedings of the IEEE Vehicular Technology Conference (VTC Fall), San Francisco, CA, USA.","DOI":"10.1109\/VETECF.2011.6093275"},{"key":"ref_14","first-page":"1135","article-title":"A review of motion planning techniques for automated vehicles","volume":"17","author":"Nashashibi","year":"2015","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Goldberg, K., Abbeel, P., Bekris, K., and Miller, L. (2020). Asymptotically optimal planning under piecewise-analytic constraints. Algorithmic Foundations of Robotics XII, Springer.","DOI":"10.1007\/978-3-030-43089-4"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1007\/BF01386390","article-title":"A note on two problems in connexion with graphs","volume":"1","author":"Dijkstra","year":"1959","journal-title":"Numer. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1145\/3828.3830","article-title":"Generalized best-first search strategies and the optimality of A*","volume":"32","author":"Dechter","year":"1985","journal-title":"J. ACM"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Kim, C., Suh, J., and Han, J.H. (2020). Development of a hybrid path planning algorithm and a bio-inspired control for an omni-wheel mobile robot. Sensors, 20.","DOI":"10.3390\/s20154258"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1163\/156855300741960","article-title":"Visibility-based probabilistic roadmaps for motion planning","volume":"14","author":"Laumond","year":"2000","journal-title":"Adv. Robot."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1007\/s10846-013-9963-y","article-title":"Spline-based RRT path planner for non-holonomic robots","volume":"73","author":"Yang","year":"2014","journal-title":"J. Intell. Robot. Syst."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Zhang, B., Chen, W., and Fei, M. (2006, January 16\u201318). An optimized method for path planning based on artificial potential field. Proceedings of the sixth International Conference on Intelligent System Design and Applications ISDA, Jinan, China.","DOI":"10.1109\/ISDA.2006.11"},{"key":"ref_22","first-page":"2795","article-title":"A practical and optimal path planning for autonomous parking using fast marching algorithm and support vector machine","volume":"96","author":"Huy","year":"2013","journal-title":"IEICE. Trans. Inf. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1016\/j.robot.2013.12.004","article-title":"Guided autowave pulse coupled neural network (GAPCNN) based real time path planning and an obstacle avoidance scheme for mobile robots","volume":"62","author":"Syed","year":"2014","journal-title":"Robot. Auton. Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1080","DOI":"10.1007\/s40815-016-0224-7","article-title":"Design of path planning and obstacle avoidance for a wheeled mobile robot","volume":"18","author":"Chen","year":"2016","journal-title":"Int. J. Fuzzy. Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"566","DOI":"10.1109\/TEVC.2002.804323","article-title":"A genetic algorithm for shortest path routing problem and the sizing of populations","volume":"6","author":"Ahn","year":"2002","journal-title":"IEEE. Trans. Evolut. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/j.neucom.2012.09.019","article-title":"Robot path planning in uncertain environment using multi-objective particle swarm optimization","volume":"103","author":"Zhang","year":"2013","journal-title":"Neurocomputing"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.robot.2016.12.008","article-title":"Optimal path planning and execution for mobile robots using genetic algorithm and adaptive fuzzy-logic control","volume":"89","author":"Bakdi","year":"2017","journal-title":"Robot. Auton. Syst."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00500-020-04771-5","article-title":"Autonomous mobile robot path planning in unknown dynamic environments using neural dynamics","volume":"24","author":"Chen","year":"2020","journal-title":"Soft Comput."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Jafari, M., Xu, H., and Carrillo, L.R.G. (2018, January 18\u201321). Brain emotional learning-based path planning and intelligent control co-design for unmanned aerial vehicle in presence of system uncertainties and dynamic environment. Proceedings of the IEEE Symposium Series on Computational Intelligence (SSCI), Bangalore, India.","DOI":"10.1109\/SSCI.2018.8628656"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1109\/TITS.2017.2756099","article-title":"Hierarchical trajectory planning of an autonomous car based on the integration of a sampling and an optimization method","volume":"19","author":"Lim","year":"2018","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Bai, G., Liu, L., Meng, Y., Luo, W., Gu, Q., and Ma, B. (2019). Path tracking of mining vehicles based on nonlinear model predictive control. Appl. Sci., 9.","DOI":"10.3390\/app9071372"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.robot.2016.02.004","article-title":"Optimisation based path planning for car parking in narrow environments","volume":"79","author":"Zips","year":"2016","journal-title":"Robot. Auton. Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.knosys.2015.04.016","article-title":"A unified motion planning method for parking an autonomous vehicle in the presence of irregularly placed obstacles","volume":"86","author":"Li","year":"2015","journal-title":"Knowl-Based. Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/j.advengsoft.2015.10.008","article-title":"Precise trajectory optimization for articulated wheeled vehicles in cluttered environments","volume":"92","author":"Li","year":"2016","journal-title":"Adv. Eng. Softw."},{"key":"ref_35","unstructured":"Dariani, I.R., Lobig, T., Lauermann, J., Rieck, J., and Schindler, J. (2019, January 24\u201325). Trajectory planning for cooperative autonomous valet parking. Proceedings of the 14th Magdeburger Maschinenbautage, Magdeburg, Germany."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Chien, C.F., Chen, H.T., and Lin, C.Y. (2020). A low-cost on-street parking management system based on bluetooth beacons. Sensors, 20.","DOI":"10.3390\/s20164559"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"829","DOI":"10.1016\/j.automatica.2011.01.085","article-title":"Pseudospectral methods for solving infinite-horizon optimal control problems","volume":"47","author":"Garg","year":"2011","journal-title":"Automatica"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"689","DOI":"10.2514\/1.31083","article-title":"Optimal reconfiguration of spacecraft formations using the Gauss pseudospectral method","volume":"31","author":"Huntington","year":"2008","journal-title":"J. Guid. Control. Dynam."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2922","DOI":"10.1007\/s10439-016-1591-9","article-title":"Evaluation of direct collocation optimal control problem formulations for solving the muscle redundancy problem","volume":"44","author":"Kinney","year":"2016","journal-title":"Ann. Biomed. Eng."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Li, B., and Shao, Z. (2015, January 6\u20139). An incremental strategy for tractor-trailer vehicle global trajectory optimization in the presence of obstacles. Proceedings of the IEEE International Conference on Robotics and Biomimetics (ROBIO), Zhuhai, China.","DOI":"10.1109\/ROBIO.2015.7418974"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/j.knosys.2016.06.008","article-title":"Spatio-temporal decomposition: A knowledge-based initialization strategy for parallel parking motion optimization","volume":"107","author":"Li","year":"2016","journal-title":"Knowl-Based Syst."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/22\/6435\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:31:59Z","timestamp":1760178719000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/22\/6435"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,11]]},"references-count":41,"journal-issue":{"issue":"22","published-online":{"date-parts":[[2020,11]]}},"alternative-id":["s20226435"],"URL":"https:\/\/doi.org\/10.3390\/s20226435","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,11,11]]}}}