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Abstract: The super giant slalom (Super-G) is a speed event in alpine skiing, in which the skier trajectory has a significant influence on the athletes’ performances. It is a challenging task to determine an optimal trajectory for the skiers along the entire course because of the complexity and difficulty in the convergence of the optimization model. In this study, a trajectory optimization model for alpine skiers competing in the Super-G is established based on the optimal control theory, in which the objective is to minimize the runtime between the starting point and the finish line. The original trajectory optimization problem is converted into a multi-phase nonlinear optimal control problem solved with a pseudospectral method, and the trajectory parameters are optimized to discover the time-optimal trajectory. Using numerical solution carried out by the MATLAB optimization toolbox, the optimal trajectory is obtained under several equality and inequality constraints. Simulation results reveal the effectiveness and rationality of the trajectory optimization model. A test is carried out to show that our code works properly. In addition, several practical proposals are provided to help alpine skiers improve their training and skiing performance.

基于多阶段非线性最优控制的高山滑雪约束性轨迹优化

[1]Benito J, Johnson BJ, 2016. Trajectory optimization for a Mars ascent vehicle. AIAA/AAS Astrodynamics Specialist Conf, Article 5441.

[2]Betts JT, 1998. Survey of numerical methods for trajectory optimization. J Guid Contr Dynam, 21(2):193-207.

[3]Chen D, Li SQ, Wang JF, et al., 2019. A multi-objective trajectory planning method based on the improved immune clonal selection algorithm. Manufacturing, 59:431-442.

[4]Chen G, Fu Y, Guo JF, 2011. Survey of aircraft trajectory optimization methods. Flight Dynam, 29(4):1-5 (in Chinese).

[5]Chen L, Qi ZH, 2006. Analyses of mechanical characteristics for alpine ski. J Dalian Univ Technol, 46(6):781-784 (in Chinese).

[6]Crain A, Ulrich S, 2019. Experimental validation of pseudospectral-based optimal trajectory planning for free-floating robots. J Guid Contr Dynam, 42(8):1726-1742.

[7]Dębski R, 2014. High-performance simulation-based algorithms for an alpine ski racer’s trajectory optimization in heterogeneous computer systems. Int J Appl Math Comput Sci, 24(3):551-566.

[8]Dębski R, 2016. An adaptive multi-spline refinement algorithm in simulation based sailboat trajectory optimization using onboard multi-core computer systems. Int J Appl Math Comput Sci, 26(2):351-365.

[9]Falck RD, Gray JS, 2019. Optimal control within the context of multidisciplinary design, analysis, and optimization. AIAA Scitech Forum, Article 0976.

[10]Garg D, Patterson MA, Darby CL, et al., 2009. Direct trajectory optimization and costate estimation of general optimal control problems using a Radau pseudospectral method. AIAA Guidance, Navigation, and Control Conf, Article 5989.

[11]Gilgien M, Spörri J, Kröll J, et al., 2014. Mechanics of turning and jumping and skier speed are associated with injury risk in men’s World Cup alpine skiing: a comparison between the competition disciplines. Br J Sport Med, 48(9):742-747.

[12]Graham KF, Rao AV, 2015. Minimum-time trajectory optimization of multiple revolution low-thrust Earth-orbit transfers. J Spacecr Rockets, 52(3):711-727.

[13]Guo Y, Ma JQ, Xiong CF, et al., 2019. Joint optimization of vehicle trajectories and intersection controllers with connected automated vehicles: combined dynamic programming and shooting heuristic approach. Trans Res Part C Emerg Technol, 98:54-72.

[14]Hirano Y, 2006. Quickest descent line during alpine ski racing. Sport Eng, 9(4):221-228.

[15]Hong SM, Seo MG, Shim SW, et al., 2016. Sensitivity analysis on weight and trajectory optimization results for multistage guided missile. IFAC-PapersOnLine, 49(17):23-27.

[16]Huang GQ, Lu YP, Nan Y, 2012. A survey of numerical algorithms for trajectory optimization of flight vehicles. Sci China Technol Sci, 55(9):2538-2560.

[17]Jiang RY, Chao T, Wang SY, et al., 2017. Low-thrust trajectory in interplanetary flight solved by pseudospectral method. J Syst Simul, 29(2):2043-2052, 2058 (in Chinese).

[18]Komissarov S, 2019. Dynamics of carving runs in alpine skiing. I. The basic centrifugal pendulum.

[19]Li TC, 2019. Single-road-constrained positioning based on deterministic trajectory geometry. IEEE Commun Lett, 23(1):80-83.

[20]Li TC, Chen HM, Sun SD, et al., 2019. Joint smoothing and tracking based on continuous-time target trajectory function fitting. IEEE Trans Autom Sci Eng, 16(3):1476-1483.

[21]Lind DA, Sanders SP, 2004. The brachistochrone problem: the path of quickest descent. In: Lind DA, Sanders SP (Eds.), The Physics of Skiing. Springer, New York, NY.

[22]Liu WS, Liang XL, Ma YZ, et al., 2019. Aircraft trajectory optimization for collision avoidance using stochastic optimal control. Asian J Contr, 21(5):2308-2320.

[23]Morbidi F, Bicego D, Ryll M, et al., 2018. Energy-efficient trajectory generation for a hexarotor with dual-tilting propellers. IEEE/RSJ Int Conf on Intelligent Robots and Systems, p.6226-6232.

[24]Paek SW, de Weck O, Polany R, et al., 2016. Asteroid deflection campaign design integrating epistemic uncertainties. Proc IEEE Aerospace Conf, p.1-14.

[25]Patrón RSF, Botez RM, 2015. Flight trajectory optimization through genetic algorithms for lateral and vertical integrated navigation. J Aerosp Inform Syst, 12(8):533-544.

[26]Ranogajec V, Ivanović V, Deur J, et al., 2018. Optimization- based assessment of automatic transmission double- transition shift controls. Contr Eng Pract, 76:155-166.

[27]Rao AV, 2014. Trajectory optimization: a survey. In: Waschl H, Kolmanovsky I, Steinbuch M, et al. (Eds.), Optimization and Optimal Control in Automotive Systems. Springer, Cham, p.3-21.

[28]Rudakov R, Lisovski A, Ilyalov O, et al., 2010. Optimisation of the skiers trajectory in special slalom. Proc Eng, 2(2):3179-3182.

[29]Spörri J, Kröll J, Gilgien M, et al., 2016. Sidecut radius and the mechanics of turning—equipment designed to reduce risk of severe traumatic knee injuries in alpine giant slalom ski racing. Br J Sport Med, 50(1):14-19.

[30]Spörri J, Kröll J, Gilgien M, et al., 2017. How to prevent injuries in alpine ski racing: what do we know and where do we go from here? Sport Med, 47(4):599-614.

[31]Sundström D, Carlsson P, Tinnsten M, 2011. Optimizing pacing strategies on a hilly track in cross-country skiing. Proc Eng, 13:10-16.

[32]Tang XJ, Chen J, 2016. Direct trajectory optimization and costate estimation of infinite-horizon optimal control problems using collocation at the flipped Legendre- Gauss-Radau points. IEEE/CAA J Autom Sin, 3(2):174-183.

[33]von Stryk O, Bulirsch R, 1992. Direct and indirect methods for trajectory optimization. Ann Oper Res, 37(1):357-373.

[34]Wang JB, Cui NG, Wei CZ, 2019. Optimal rocket landing guidance using convex optimization and model predictive control. J Guid Contr Dynam, 42(5):1078-1092.

[35]Youn SH, 2018. Can a skier make a circular turn without any active movement? J Korean Phys Soc, 73(10):1410-1419.

[36]Zhang S, Hou MS, 2016. Trajectory optimization of aerocraft based on shaping and dimension reduction. Acta Armament, 37(6):1125-1130 (in Chinese).

[37]Zheng DK, Wang SY, Meng QW, 2016. Dynamic programming track-before-detect algorithm for radar target detection based on polynomial time series prediction. IET Radar Sonar Nav, 10(8):1327-1336.