基于多阶段非线性最优控制的高山滑雪约束性轨迹优化

蔡聪影; 姚小兰

doi:10.1631/FITEE.1900586

Your Location：

Home >

Browse articles >

基于多阶段非线性最优控制的高山滑雪约束性轨迹优化

常规文章 | Updated：2022-05-19

- 基于多阶段非线性最优控制的高山滑雪约束性轨迹优化
- Trajectory optimization with constraints for alpine skiers based on multi-phase nonlinear optimal control
- 信息与电子工程前沿（英文） 2020年21卷第10期页码：1521-1534
- Affiliations：
  
  School of Automation, Beijing Institute of Technology, Beijing 100081, China
- Author bio：
  
  Xiao-lan YAO, E-mail: yaoxiaolan@bit.edu.cn
- Funds：
  
  supported by the Key Technology Research and Demonstration of National Scientific Training Base Construction of China (No. 2018YFF0300800)
- DOI：10.1631/FITEE.1900586
  中图分类号： O232;TP29
- 收稿：2019-10-30，
  
  修回：2020-;8-28，
  
  纸质出版：2020-10
- Accepted：
Scan QR Code
蔡聪影, 姚小兰. 基于多阶段非线性最优控制的高山滑雪约束性轨迹优化[J]. 信息与电子工程前沿（英文）, 2020,21(10):1521-1534.

CAI Cong-ying, YAO Xiao-lan. Trajectory optimization with constraints for alpine skiers based on multi-phase nonlinear optimal control[J]. Frontiers of Information Technology & Electronic Engineering, 2020, 21(10): 1521-1534.
蔡聪影, 姚小兰. 基于多阶段非线性最优控制的高山滑雪约束性轨迹优化[J]. 信息与电子工程前沿（英文）, 2020,21(10):1521-1534. DOI： 10.1631/FITEE.1900586.

CAI Cong-ying, YAO Xiao-lan. Trajectory optimization with constraints for alpine skiers based on multi-phase nonlinear optimal control[J]. Frontiers of Information Technology & Electronic Engineering, 2020, 21(10): 1521-1534. DOI： 10.1631/FITEE.1900586.

摘要

超级大回转是一项高山滑雪速度系列运动项目，滑雪轨迹对运动员比赛成绩有重要影响。由于优化模型复杂且难以收敛，确定滑雪者全程最优轨迹具有挑战性。本文基于最优控制理论，以滑行时间最小为优化指标，建立高山滑雪运动员在超级大回转中的轨迹优化模型。将轨迹优化问题转化为多阶段非线性最优控制问题，采用伪谱法求解，并优化轨迹参数。MATLAB仿真结果验证了所提轨迹优化模型的有效性和合理性。此外，提出一些切实可行的滑雪策略，帮助高山滑雪运动员提高训练水平和比赛成绩。

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.

关键词

Keywords

references

J Benito , , , BJ Johnson . . Trajectory optimization for a Mars ascent vehicle . . AIAA/AAS Astrodynamics Specialist Conf, Article , , 2016 . . 5441 DOI: 10.2514/6.2016-5441 http://doi.org/10.2514/6.2016-5441 . .

JT Betts . . Survey of numerical methods for trajectory optimization . . J Guid Contr Dynam , , 1998 . . 21 ( ( 2 ): ): 193 - - 207 . . DOI: 10.2514/2.4231 http://doi.org/10.2514/2.4231 . .

D Chen , , , SQ Li , , , JF Wang , , , 等 . . A multi-objective trajectory planning method based on the improved immune clonal selection algorithm . . Manufacturing , , 2019 . . 59 431 - - 442 . . DOI: 10.1016/j.rcim.2019.04.016 http://doi.org/10.1016/j.rcim.2019.04.016 . .

G Chen , , , Y Fu , , , JF Guo . . Survey of aircraft trajectory optimization methods . . Flight Dynam , , 2011 . . 29 ( ( 4 ): ): 1 - - 5 . . DOI: 10.13645/j.cnki.f.d.2011.04.008 http://doi.org/10.13645/j.cnki.f.d.2011.04.008 . .

L Chen , , , ZH Qi . . Analyses of mechanical characteristics for alpine ski . . J Dalian Univ Technol , , 2006 . . 46 ( ( 6 ): ): 781 - - 784 . . DOI: 10.3321/j.issn:1000-8608.2006.06.001 http://doi.org/10.3321/j.issn:1000-8608.2006.06.001 . .

A Crain , , , S Ulrich . . Experimental validation of pseudospectral-based optimal trajectory planning for free-floating robots . . J Guid Contr Dynam , , 2019 . . 42 ( ( 8 ): ): 1726 - - 1742 . . DOI: 10.2514/1.G003528 http://doi.org/10.2514/1.G003528 . .

R Dębski . . High-performance simulation-based algorithms for an alpine ski raceros trajectory optimization in heterogeneous computer systems . . Int J Appl Math Comput Sci , , 2014 . . 24 ( ( 3 ): ): 551 - - 566 . . DOI: 10.2478/amcs-2014-0040 http://doi.org/10.2478/amcs-2014-0040 . .

R Dębski . . An adaptive multi-spline refinement algorithm in simulation based sailboat trajectory optimization using onboard multi-core computer systems . . Int J Appl Math Comput Sci , , 2016 . . 26 ( ( 2 ): ): 351 - - 365 . . DOI: 10.1515/amcs-2016-0025 http://doi.org/10.1515/amcs-2016-0025 . .

RD Falck , , , JS Gray . . Optimal control within the context of multidisciplinary design, analysis, and optimization . . AIAA Scitech Forum, Article , , 2019 . . 0976 DOI: 10.2514/6.2019-0976 http://doi.org/10.2514/6.2019-0976 . .

D Garg , , , MA Patterson , , , CL Darby , , , 等 . . Direct trajectory optimization and costate estimation of general optimal control problems using a Radau pseudospectral method . . AIAA Guidance, Navigation, and Control Conf, Article , , 2009 . . 5989 DOI: 10.2514/6.2009-5989 http://doi.org/10.2514/6.2009-5989 . .

M Gilgien , , , J Spörri , , , J Kröll , , , 等 . . Mechanics of turning and jumping and skier speed are associated with injury risk in menos World Cup alpine skiing: a comparison between the competition disciplines . . Br J Sport Med , , 2014 . . 48 ( ( 9 ): ): 742 - - 747 . . DOI: 10.1136/bjsports-2013-092994 http://doi.org/10.1136/bjsports-2013-092994 . .

KF Graham , , , AV Rao . . Minimum-time trajectory optimization of multiple revolution low-thrust Earth-orbit transfers . . J Spacecr Rockets , , 2015 . . 52 ( ( 3 ): ): 711 - - 727 . . DOI: 10.2514/1.A33187 http://doi.org/10.2514/1.A33187 . .

Y Guo , , , JQ Ma , , , CF Xiong , , , 等 . . 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 , , 2019 . . 98 54 - - 72 . . DOI: 10.1016/j.trc.2018.11.010 http://doi.org/10.1016/j.trc.2018.11.010 . .

Y Hirano . . Quickest descent line during alpine ski racing . . Sport Eng , , 2006 . . 9 ( ( 4 ): ): 221 - - 228 . . DOI: 10.1007/BF02866060 http://doi.org/10.1007/BF02866060 . .

SM Hong , , , MG Seo , , , SW Shim , , , 等 . . Sensitivity analysis on weight and trajectory optimization results for multistage guided missile . . IFAC-PapersOnLine , , 2016 . . 49 ( ( 17 ): ): 23 - - 27 . . DOI: 10.1016/j.ifacol.2016.09.005 http://doi.org/10.1016/j.ifacol.2016.09.005 . .

GQ Huang , , , YP Lu , , , Y Nan . . A survey of numerical algorithms for trajectory optimization of flight vehicles . . Sci China Technol Sci , , 2012 . . 55 ( ( 9 ): ): 2538 - - 2560 . . DOI: 10.1007/s11431-012-4946-y http://doi.org/10.1007/s11431-012-4946-y . .

RY Jiang , , , T Chao , , , SY Wang , , , 等 . . Low-thrust trajectory in interplanetary flight solved by pseudospectral method . . J Syst Simul , , 2017 . . 29 ( ( 2 ): ): 2043 - - 2052, 2058 . . DOI: 10.16182/j.issn1004731x.joss.201709022 http://doi.org/10.16182/j.issn1004731x.joss.201709022 . .

S Komissarov . . Dynamics of carving runs in alpine skiing. I . . The basic centrifugal pendulum , , 2019 . . DOI: 10.31236/osf.io/gp3ef http://doi.org/10.31236/osf.io/gp3ef . .

TC Li . . Single-road-constrained positioning based on deterministic trajectory geometry . . IEEE Commun Lett , , 2019 . . 23 ( ( 1 ): ): 80 - - 83 . . DOI: 10.1109/LCOMM.2018.2879478 http://doi.org/10.1109/LCOMM.2018.2879478 . .

TC Li , , , HM Chen , , , SD Sun , , , 等 . . Joint smoothing and tracking based on continuous-time target trajectory function fitting . . IEEE Trans Autom Sci Eng , , 2019 . . 16 ( ( 3 ): ): 1476 - - 1483 . . DOI: 10.1109/TASE.2018.2882641 http://doi.org/10.1109/TASE.2018.2882641 . .

DA Lind , , , SP Sanders . . The brachistochrone problem: the path of quickest descent , , : : New York, NY In: Lind DA, Sanders SP (Eds.), The Physics of Skiing. Springer , , 2004 . . DOI: 10.1007/978-1-4757-4345-6_19 http://doi.org/10.1007/978-1-4757-4345-6_19 . .

WS Liu , , , XL Liang , , , YZ Ma , , , 等 . . Aircraft trajectory optimization for collision avoidance using stochastic optimal control . . Asian J Contr , , 2019 . . 21 ( ( 5 ): ): 2308 - - 2320 . . DOI: 10.1002/asjc.1855 http://doi.org/10.1002/asjc.1855 . .

F Morbidi , , , D Bicego , , , M Ryll , , , 等 . . Energy-efficient trajectory generation for a hexarotor with dual-tilting propellers . . IEEE/RSJ Int Conf on Intelligent Robots and Systems , , 2018 . . p.6226 - - 6232 . . DOI: 10.1109/IROS.2018.8594419 http://doi.org/10.1109/IROS.2018.8594419 . .

SW Paek , , , O de Weck , , , R Polany , , , 等 . . Asteroid deflection campaign design integrating epistemic uncertainties . . Proc IEEE Aerospace Conf , , 2016 . . p.1 - - 14 . . DOI: 10.1109/AERO.2016.7500905 http://doi.org/10.1109/AERO.2016.7500905 . .

RSF Patrón , , , RM Botez . . Flight trajectory optimization through genetic algorithms for lateral and vertical integrated navigation . . J Aerosp Inform Syst , , 2015 . . 12 ( ( 8 ): ): 533 - - 544 . . DOI: 10.2514/1.I010348 http://doi.org/10.2514/1.I010348 . .

V Ranogajec , , , V Ivanović , , , J Deur , , , 等 . . Optimization-based assessment of automatic transmission double- transition shift controls . . Contr Eng Pract , , 2018 . . 76 155 - - 166 . . DOI: 10.1016/j.conengprac.2018.04.016 http://doi.org/10.1016/j.conengprac.2018.04.016 . .

AV Rao . . Trajectory optimization: a survey . . In: Waschl H, Kolmanovsky I, Steinbuch M, et al. (Eds.), Optimization and Optimal Control in Automotive Systems. Springer, Cham , , 2014 . . p.3 - - 21 . . DOI: 10.1007/978-3-319-05371-4_1 http://doi.org/10.1007/978-3-319-05371-4_1 . .

R Rudakov , , , A Lisovski , , , O Ilyalov , , , 等 . . Optimisation of the skiers trajectory in special slalom . . Proc Eng , , 2010 . . 2 ( ( 2 ): ): 3179 - - 3182 . . DOI: 10.1016/j.proeng.2010.04.129 http://doi.org/10.1016/j.proeng.2010.04.129 . .

J Spörri , , , J Kröll , , , M Gilgien , , , 等 . . Sidecut radius and the mechanics of turningjequipment designed to reduce risk of severe traumatic knee injuries in alpine giant slalom ski racing . . Br J Sport Med , , 2016 . . 50 ( ( 1 ): ): 14 - - 19 . . DOI: 10.1136/bjsports-2015-095737 http://doi.org/10.1136/bjsports-2015-095737 . .

J Spörri , , , J Kröll , , , M Gilgien , , , 等 . . How to prevent injuries in alpine ski racing: what do we know and where do we go from here . . Sport Med , , 2017 . . 47 ( ( 4 ): ): 599 - - 614 . . DOI: 10.1007/s40279-016-0601-2 http://doi.org/10.1007/s40279-016-0601-2 . .

D Sundström , , , P Carlsson , , , M Tinnsten . . Optimizing pacing strategies on a hilly track in cross-country skiing . . Proc Eng , , 2011 . . 13 10 - - 16 . . DOI: 10.1016/j.proeng.2011.05.044 http://doi.org/10.1016/j.proeng.2011.05.044 . .

XJ Tang , , , J Chen . . 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 , , 2016 . . 3 ( ( 2 ): ): 174 - - 183 . . DOI: 10.1109/JAS.2016.7451105 http://doi.org/10.1109/JAS.2016.7451105 . .

O von Stryk , , , R Bulirsch . . Direct and indirect methods for trajectory optimization . . Ann Oper Res , , 1992 . . 37 ( ( 1 ): ): 357 - - 373 . . DOI: 10.1007/BF02071065 http://doi.org/10.1007/BF02071065 . .

JB Wang , , , NG Cui , , , CZ Wei . . Optimal rocket landing guidance using convex optimization and model predictive control . . J Guid Contr Dynam , , 2019 . . 42 ( ( 5 ): ): 1078 - - 1092 . . DOI: 10.2514/1.G003518 http://doi.org/10.2514/1.G003518 . .

SH Youn . . Can a skier make a circular turn without any active movement . . J Korean Phys Soc , , 2018 . . 73 ( ( 10 ): ): 1410 - - 1419 . . DOI: 10.3938/jkps.73.1410 http://doi.org/10.3938/jkps.73.1410 . .

S Zhang , , , MS Hou . . Trajectory optimization of aerocraft based on shaping and dimension reduction . . Acta Armament , , 2016 . . 37 ( ( 6 ): ): 1125 - - 1130 . . DOI: 10.3969/j.issn.1000-1093.2016.06.022 http://doi.org/10.3969/j.issn.1000-1093.2016.06.022 . .

DK Zheng , , , SY Wang , , , QW Meng . . Dynamic programming track-before-detect algorithm for radar target detection based on polynomial time series prediction . . IET Radar Sonar Nav , , 2016 . . 10 ( ( 8 ): ): 1327 - - 1336 . . DOI: 10.1049/iet-rsn.2015.0332 http://doi.org/10.1049/iet-rsn.2015.0332 . .

浏览量

Downloads

CSCD

文章被引用时，请邮件提醒。

Submit

工具集

关联资源

Frequency of arrival-based state estimation and trajectory optimization for the navigation of autonomous marine vehicles

Steering control for underwater gliders

Optimization of formation for multi-agent systems based on LQR

Adaptive dynamic programming for linear impulse systems