{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:07:22Z","timestamp":1760231242773,"version":"build-2065373602"},"reference-count":48,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,8,31]],"date-time":"2022-08-31T00:00:00Z","timestamp":1661904000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000146","name":"NSERC Alliance-AI Advance Program","doi-asserted-by":"publisher","award":["202102595"],"award-info":[{"award-number":["202102595"]}],"id":[{"id":"10.13039\/501100000146","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Robotics"],"abstract":"<jats:p>This paper presents a novel trajectory planning approach for nonlinear dynamical systems; a multi-rotor drone, built on an optimization-based framework proposed by the authors named the Nonlinear Model Predictive Horizon. In the present work, this method is integrated with a Backstepping Control technique. The goal is to remove the non-convexity of the optimization problem in order to provide real-time computation of reference trajectories for the vehicle which respects its dynamics while avoiding sensed static and dynamic obstacles in the environment. Our method is applied to two models of multi-rotor drones to demonstrate its flexibility. Several simulation and hardware flight experiments are presented to validate the proposed design and demonstrate its performance improvement over earlier work.<\/jats:p>","DOI":"10.3390\/robotics11050087","type":"journal-article","created":{"date-parts":[[2022,9,1]],"date-time":"2022-09-01T03:55:38Z","timestamp":1662004538000},"page":"87","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A Backstepping Approach to Nonlinear Model Predictive Horizon for Optimal Trajectory Planning"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8295-8356","authenticated-orcid":false,"given":"Younes","family":"Al Younes","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, University of Alberta, Edmonton, AB T6G 1H9, Canada"}]},{"given":"Martin","family":"Barczyk","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, University of Alberta, Edmonton, AB T6G 1H9, Canada"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"15550","DOI":"10.1016\/j.ifacol.2020.12.2399","article-title":"An optimization-based receding horizon trajectory planning algorithm","volume":"53","author":"Bergman","year":"2020","journal-title":"IFAC-PapersOnLine"},{"key":"ref_2","unstructured":"Manoharan, A., Sharma, R., and Sujit, P.B. (2022). Multi-AAV Cooperative Path Planning using Nonlinear Model Predictive Control with Localization Constraints. arXiv."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1049\/iet-its.2018.5103","article-title":"Fast velocity trajectory planning and control algorithm of intelligent 4WD electric vehicle for energy saving using time-based MPC","volume":"13","author":"Wu","year":"2019","journal-title":"IET Intell. Transp. Syst."},{"key":"ref_4","first-page":"293","article-title":"Path planning and trajectory planning algorithms: A general overview","volume":"Volume 29","author":"Carbone","year":"2015","journal-title":"Motion and Operation Planning of Robotic Systems"},{"key":"ref_5","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_6","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1109\/TSSC.1968.300136","article-title":"A formal basis for the heuristic determination of minimum cost paths","volume":"4","author":"Hart","year":"1968","journal-title":"IEEE Trans. Syst. Sci. Cybern."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1109\/TRO.2004.838026","article-title":"Fast replanning for navigation in unknown terrain","volume":"21","author":"Koenig","year":"2005","journal-title":"IEEE Trans. Robot."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Al-Mutib, K., AlSulaiman, M., Emaduddin, M., Ramdane, H., and Mattar, E. (2011, January 20\u201322). D* lite based real-time multi-agent path planning in dynamic environments. Proceedings of the 2011 Third International Conference on Computational Intelligence, Modelling & Simulation, Langkawi, Malaysia.","DOI":"10.1109\/CIMSim.2011.38"},{"key":"ref_9","first-page":"767","article-title":"ARA*: Anytime A* with provable bounds on sub-optimality","volume":"Volume 16","author":"Thrun","year":"2004","journal-title":"Advances in Neural Information Processing Systems"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1177\/0278364909359210","article-title":"Path planning for autonomous vehicles in unknown semi-structured environments","volume":"29","author":"Dolgov","year":"2010","journal-title":"Int. J. Robot. Res."},{"key":"ref_11","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_12","doi-asserted-by":"crossref","unstructured":"Donald, B.R., Lynch, K.M., and Rus, D. (2001). Rapidly-Exploring Random Trees: Progress and Prospects. Algorithmic and Computational Robotics: New Directions, CRC Press.","DOI":"10.1201\/9781439864135"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"846","DOI":"10.1177\/0278364911406761","article-title":"Sampling-based algorithms for optimal motion planning","volume":"30","author":"Karaman","year":"2011","journal-title":"Int. J. Robot. Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1177\/027836498600500106","article-title":"Real-Time Obstacle Avoidance for Manipulators and Mobile Robots","volume":"5","author":"Khatib","year":"1986","journal-title":"Int. J. Robot. Res."},{"key":"ref_15","first-page":"575","article-title":"Artificial Potential Field Algorithm Implementation for Quadrotor Path Planning","volume":"10","author":"Iswanto","year":"2019","journal-title":"Int. J. Adv. Comput. Sci. Appl."},{"key":"ref_16","unstructured":"Adey, R., Rzevski, G., and Russell, D. (1994). Application Of Artificial Neural Networks To The Robot Path Planning Problem. Applications of Artificial Intelligence in Engineering IX, WIT Press."},{"key":"ref_17","first-page":"143","article-title":"Mobile manipulator path planning by a genetic algorithm","volume":"11","author":"Zhao","year":"1994","journal-title":"J. Field Robot."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1007\/s11804-009-8002-7","article-title":"Research on global path planning based on ant colony optimization for AUV","volume":"8","author":"Wang","year":"2009","journal-title":"J. Mar. Sci. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Qiaorong, Z., and Guochang, G. (2008, January 21\u201324). Path planning based on improved binary particle swarm optimization algorithm. Proceedings of the 2008 IEEE Conference on Robotics, Automation and Mechatronics, Chengdu, China.","DOI":"10.1109\/RAMECH.2008.4681408"},{"key":"ref_20","unstructured":"Martinez-Alfaro, H., and Flugrad, D.R. (1994, January 2\u20135). Collision-free path planning for mobile robots and\/or AGVs using simulated annealing. Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, San Antonio, TX, USA."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Zhang, H.Y., Lin, W.M., and Chen, A.X. (2018). Path planning for the mobile robot: A review. Symmetry, 10.","DOI":"10.3390\/sym10100450"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Sands, T. (2022). Flattening the Curve of Flexible Space Robotics. Appl. Sci., 12.","DOI":"10.20944\/preprints202201.0435.v1"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"LaValle, S.M. (2006). Planning Algorithms, Cambridge University Press.","DOI":"10.1017\/CBO9780511546877"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1164","DOI":"10.1177\/0278364913488805","article-title":"CHOMP: Covariant Hamiltonian optimization for motion planning","volume":"32","author":"Zucker","year":"2013","journal-title":"Int. J. Robot. Res."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Kalakrishnan, M., Chitta, S., Theodorou, E., Pastor, P., and Schaal, S. (2011, January 9\u201313). STOMP: Stochastic trajectory optimization for motion planning. Proceedings of the 2011 IEEE international conference on robotics and automation, Shanghai, China.","DOI":"10.1109\/ICRA.2011.5980280"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Sandberg, A., and Sands, T. (2022). Autonomous Trajectory Generation Algorithms for Spacecraft Slew Maneuvers. Aerospace, 9.","DOI":"10.3390\/aerospace9030135"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Al Younes, Y., and Barczyk, M. (2021). Nonlinear Model Predictive Horizon for Optimal Trajectory Generation. Robotics, 10.","DOI":"10.3390\/robotics10030090"},{"key":"ref_28","unstructured":"Krstic, M., Kokotovic, P.V., and Kanellakopoulos, I. (1995). Nonlinear and Adaptive Control Design, John Wiley & Sons, Inc."},{"key":"ref_29","unstructured":"Vaidyanathan, S., and Azar, A.T. (2020). Backstepping Control of Nonlinear Dynamical Systems, Academic Press."},{"key":"ref_30","unstructured":"Zhou, J., and Wen, C. (2008). Adaptive Backstepping Control of Uncertain Systems: Nonsmooth Nonlinearities, Interactions or Time-Variations, Springer."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Fadhel, F.S., and Noaman, S.F. (2018). The Generalized Backstepping Control Method for Stabilizing and Solving Systems of Multiple Delay Differential Equations. Al-Nahrain J. Sci., 150\u2013156.","DOI":"10.22401\/ANJS.00.1.20"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Ye, Z., and Mohamadian, H. (2009, January 9\u201311). Nonlinear backstepping control of multibody aerodynamic systems with equational modeling. Proceedings of the 2009 IEEE International Conference on Control and Automation, Christchurch, New Zealand.","DOI":"10.1109\/ICCA.2009.5410579"},{"key":"ref_33","unstructured":"Marino, R., and Tomei, P. (1995). Nonlinear Control Design: Geometric, Adaptive, and Robust, Prentice Hall."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Al Younes, Y., and Barczyk, M. (2021). Optimal Motion Planning in GPS-Denied Environments Using Nonlinear Model Predictive Horizon. Sensors, 21.","DOI":"10.3390\/s21165547"},{"key":"ref_35","unstructured":"Murray, R.M., Li, Z., and Sastry, S.S. (1994). A Mathematical Introduction to Robotic Manipulation, CRC Press."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Madani, T., and Benallegue, A. (2006, January 13\u201315). Control of a quadrotor mini-helicopter via full state backstepping technique. Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, USA.","DOI":"10.1109\/CDC.2006.377548"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"98","DOI":"10.4028\/www.scientific.net\/AMM.565.98","article-title":"Quadrotor position Control using cascaded adaptive integral backstepping controllers","volume":"Volume 565","author":"Drak","year":"2014","journal-title":"Applied Mechanics and Materials"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Bouabdallah, S., and Siegwart, R. (November, January 29). Full control of a quadrotor. Proceedings of the 2007 IEEE\/RSJ international conference on intelligent robots and systems, San Diego, CA, USA.","DOI":"10.1109\/IROS.2007.4399042"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1007\/s10846-014-0072-3","article-title":"Enhanced backstepping controller design with application to autonomous quadrotor unmanned aerial vehicle","volume":"79","author":"Basri","year":"2015","journal-title":"J. Intell. Robot. Syst."},{"key":"ref_40","first-page":"5044","article-title":"Robust backstepping sliding-mode control and observer-based fault estimation for a quadrotor UAV","volume":"63","author":"Chen","year":"2016","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Quigley, M., Gerkey, B., Conley, K., Faust, J., Foote, T., Leibs, J., Berger, E., Wheeler, R., and Ng, A.Y. (2009, January 12\u201317). ROS: An open-source Robot Operating System. ICRAWorkshop on Open Source Software in Robotics, Kobe, Japan.","DOI":"10.1109\/MRA.2010.936956"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1002\/oca.939","article-title":"ACADO Toolkit\u2014An Open Source Framework for Automatic Control and Dynamic Optimization","volume":"32","author":"Houska","year":"2011","journal-title":"Optim. Control Appl. Methods"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S0962492900002518","article-title":"Sequential quadratic programming","volume":"4","author":"Boggs","year":"1995","journal-title":"Acta Numer."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1007\/s12532-014-0071-1","article-title":"qpOASES: A parametric active-set algorithm for quadratic programming","volume":"6","author":"Ferreau","year":"2014","journal-title":"Math. Program. Comput."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Hutter, M., and Siegwart, R. (2018). AirSim: High-Fidelity Visual and Physical Simulation for Autonomous Vehicles. Field and Service Robotics, Springer.","DOI":"10.1007\/978-3-319-67361-5"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Meier, L., Honegger, D., and Pollefeys, M. (2015, January 26\u201330). PX4: A node-based multithreaded open source robotics framework for deeply embedded platforms. Proceedings of the 2015 IEEE International Conference on Robotics and Automation (ICRA), Seattle, WA, USA.","DOI":"10.1109\/ICRA.2015.7140074"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Oleynikova, H., Taylor, Z., Fehr, M., Siegwart, R., and Nieto, J. (2017, January 24\u201328). Voxblox: Incremental 3d euclidean signed distance fields for on-board mav planning. Proceedings of the 2017 IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Vancouver, BC, Canada.","DOI":"10.1109\/IROS.2017.8202315"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Dang, T., Mascarich, F., Khattak, S., Papachristos, C., and Alexis, K. (2019, January 4\u20138). Graph-based path planning for autonomous robotic exploration in subterranean environments. Proceedings of the 2019 IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Macau, China.","DOI":"10.1109\/IROS40897.2019.8968151"}],"container-title":["Robotics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2218-6581\/11\/5\/87\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:20:55Z","timestamp":1760142055000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2218-6581\/11\/5\/87"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,31]]},"references-count":48,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["robotics11050087"],"URL":"https:\/\/doi.org\/10.3390\/robotics11050087","relation":{},"ISSN":["2218-6581"],"issn-type":[{"type":"electronic","value":"2218-6581"}],"subject":[],"published":{"date-parts":[[2022,8,31]]}}}