滑雪运动生物力学仿真分析
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摘要
随着国际竞争的日趋激烈,科学技术对于竞技体育的促进作用越来越突出。国内外学者运用运动生物力学的研究方法对各种运动项目进行了大量的研究工作,并取得了丰硕的成果。滑雪运动是一项技术性很强的运动,国内外对于滑雪运动的研究多以实验测试为主,涉及理论建模的研究很少。以往的滑雪模型忽略了过多的细节,没有考虑滑雪杖的作用,不适合多种滑雪运动的研究,不能进行肌肉用力计算等方面的研究。现在仍然缺少能够综合分析处理滑雪运动的平台。我国在滑雪运动项目上的成绩以及研究一直落后于国际先进水平,对该项目规律认识不够全面和深入,原因是起步晚,投入研究较少,研究工具匮乏。
针对上述问题,本文以滑雪者为研究对象,从影响滑雪运动的相关力学问题入手,建立了滑雪运动的生物力学模型以及研发了相应仿真平台。课题研究涉及到多体系统动力学、运动生物力学、机器人动力学与控制、数值优化等领域。主要做了以下研究工作:
第二章建立了三维多刚体滑雪模型并以几种滑雪运动的仿真验证了模型的合理性。多刚体模型包括:雪地路面模型、人体子模型、滑雪板—雪地子模型、空气阻力模型。用函数描述雪地地形。把人体简化为16刚体25自由度的多刚体人体模型,人体各个肢体的惯性参数根据回归公式进行计算。利用函数模拟关节的运动从而达到对人体冗余自由度缩减的目的。建立了一个6自由度单刚体滑雪板模型。用1个矩形和7个圆弧描述滑雪板的正面和侧面几何轮廓。把滑雪板刚性离散化,以浸入力、冲击力、摩擦力描述滑雪板—雪地相互作用力,并乘以载荷分布函数近似滑雪板柔性变形的影响。考虑空气阻力对各个肢体的阻碍作用。最后应用建立的多刚体滑雪模型对双板平行直滑降、双板平行转弯、空中技巧进行仿真,仿真结果与简单模型、经验、实际运动效果非常相像,表明模型是合理的。
第三章建立了变拓扑结构的滑雪杖—雪地模型并利用模型对双滑雪杖支雪起动过程进行了仿真。忽略滑雪杖—雪地的接触碰撞过程,把滑雪杖—雪地的关系用多体系统动力学的单面约束进行描述。根据滑雪杖的拦雪轮与雪地的位置关系按分离、接触两种情况进行处理。把滑雪杖—雪地的缝隙函数与相互作用力关系转换为线性互补关系,作为单面约束动力学系统的补充方程。最后利用模型对使用双滑雪杖支雪起动加速的对称运动进行了仿真分析。
第四章引入Hill三元件肌肉模型建立了含神经、肌肉系统的滑雪运动生物力学模型并应用于探究各种基本滑雪技术的肌肉协调规律。把肌肉力/神经刺激作为设计变量,以完成动作所有肌肉力/神经刺激平方和最小为目标函数,受各肌肉力矩和与所绕关节力矩相等的等式约束,肌肉力/神经刺激非负有上限的不等式约束,建立二次规划问题。通过求解二次规划问题求解肌肉力/神经刺激。使模型不但适用于求解宏观的滑雪运动力学问题,而且能计算完成运动所需要的肌肉肌群用力,方便理解滑雪运动肌肉协调规律,为深入探究运动控制机理提供可能。最后应用该模型对滑降、转弯、支雪几种滑雪运动过程中的肌肉用力进行计算。计算结果为深入探究滑雪运动中肌肉协调控制规律提供可能,为加强平时专项滑雪运动肌肉训练提供依据。
第五章除了从运动学、动力学方面对滑雪运动技术进行评价外,针对特别的运动技术还用能量评价指标、转弯评价指标(转弯同步差异角、转弯同步系数)、稳定性评价指标(静态重心投影稳定性裕量、动态零力矩点稳定裕量)来评价。最后利用这些指标讨论了人体重心位置对转弯的影响,摆臂动作对空中技巧出台效果的影响。
第六章介绍了编写的滑雪运动分析平台并用于仿真各种基本滑雪技术。平台具有以下特点:基于前沿的多体系统动力学理论与计算方法,实现了动力学方程的自动形成;基于先进的计算软件MATLAB,求解计算速度快,代码简单易懂;界面功能明确,操作方便;采用模块化的设计理念,各种功能形成单独的模块,便于修改、维护和扩充;采用统一的数据格式,提高了系统的稳定性和通用性。程序分为前后处理两部分,前处理主要为自动建模、姿态设计、积分求解;后处理主要是关于运动学、动力学计算、能量计算、滑雪稳定性评价、肌肉力计算、动画输出等。本文利用该平台成功地对双板平行直滑降技术、犁式滑降技术、犁式转弯技术、双板平行转弯技术、双滑雪杖支雪动作、空中技巧进行了仿真分析研究。
研究结果表明,本文建立的滑雪模型、滑雪软件适应范围广,能较好地仿真各种滑雪运动,包括滑降、转弯、支雪、空中技巧,体现了滑雪起动、旋转、腾空、落地等基本特点,能够方便快捷的获得相关技术动作的运动学、动力学、生物力学等信息,对于认识、改进和完善动作技术,选择和设计优化的技术方案,有一定的现实意义,编写的软件可以为探寻新技术和论证创新技术的可行性及科学性提供了方便。
本文的研究工作得到了国家自然科学基金No.10472018和10721062的资助。在滑雪运动生物力学方面进行了初步的探索,期望能够对滑雪运动的研究发展有所贡献。
With the increasingly fierce international competition, the effects of science and technology on competitive sports have become increasingly prominent. Domestic and foreign researchers have made a great deal of efforts on sports studies and have achieved fruitful results by sports biomechanical methods. Skiing is a highly skillful technical sport. Most of studies about skiing are based on experiments and a very few skiing models are presented. Furthermore, most of the models are too simple and unsuitable for different types of skiing movements. They cannot compute muscle force and so on. None article is for skiing pole. No software is for skiing simulation due to the complexity of skiing movement. Skiing sports achievements in China have been behind the international advanced level. The reasons are that the studies about skiing started late and lacked of research tools.
Based on the factors influence skiing, this paper builds a biomechanical multibody skiing model and corresponding software. This study involves multibody system dynamics, sports biomechanics, robot dynamics and control, numerical optimization and so on. The main researches are:
The second chapter of this paper builds and tests a multibody skiing model. The model includes: snow ground sub model, skier sub model, ski-snow sub model, air drag sub model. The snow is described by space surface. The skier is simplified as sixteen rigid bodies and connected by twenty-five joints. The human body inertial parameters are obtained by regression equations with body height and mass. The joint motions are approximated by functions. The outline of ski face and side view is described by one rectangle and seven arcs. The ski is discretized as small points but has only six rigid degrees of freedom. A three-element of penetration force, impact force and friction is used to compute the interactions between ski and snow. The pressure distribution function is used to approximate the effects of ski bending. The aerodynamic drag on skier segments is considered. At the end, the skiing model is used to simulate downhill, parallel turning, and aerials. The simulated results agree well with simple model, experimental movement.
The third chapter mainly focuses on the disposal of skiing pole. Neglecting the contact process, the pole-snow is described by unilateral constraint of multibody system dynamics. The relationship of interstitial function and reaction for pole-snow is converted to linear complementary problem that is supplied for dynamics equations. A standard double poling process is simulated by the model at last.
The fourth chapter introduces the famous Hill-type muscle model to the multibody skiing model for the sake of study muscle coordination rules. The muscle forces/neural excitation values are solved by a standard quadratic programming optimization problem, in which muscle force/neural excitation is taken as design variable, the minimal square sum of muscle force/neural excitation is objective function, the non-negative and upper limit muscle force/neural excitation is none-equality constraints, the sum of all muscle moment equaled to joint control torque as equality constraints. Therefore, the skiing model can be used to solve both macroscopical kinetic problems and microcosmic muscle force coordination rules. As applications, muscle force coordination rules for several basic skiing movements are computed.
The fifth chapter presents three evaluation indexes for skiing movement. The indexes are energy evaluation index, turning evaluation index, and stability evaluation index. These indexes are used to discuss the effects of skier center of gravity position on snowplow turning, parallel turning, and arms swing on the take-off status of aerials.
The sixth chapter mainly introduces the skiing movement simulative software and uses it to simulate different skiing techniques. The program includes both parts of pre-processing and post-processing. The pre-processing is for auto building skiing model, posture setting, integral solver. The post-processing is about output of skiing animation, kinematics, dynamics, energy, muscle force, and stability evaluation and so on. At the end, parallel downhill, snowplow downhill, snowplow turn, parallel turning, and aerial skiing are successfully simulated by the program.
All the simulation results show that the skiing model and software can apply to a wide range of simulation: downhill, turning, poling, and aerials. Some skiing characteristics such as start-up, rotation, flying, and landing are shown. The computed kinematic, dynamic, biomechanical information are useful for the understanding and improvement of skiing technology. The software can explore new technologies and demonstrate the feasibility of innovative technology.
This study is financed by the National Natural Science Foundation of China No.10472018 and 10721062. Preliminary exploration about skiing is carried out and hoped to be contributed to ski sports research and development.
针对上述问题,本文以滑雪者为研究对象,从影响滑雪运动的相关力学问题入手,建立了滑雪运动的生物力学模型以及研发了相应仿真平台。课题研究涉及到多体系统动力学、运动生物力学、机器人动力学与控制、数值优化等领域。主要做了以下研究工作:
第二章建立了三维多刚体滑雪模型并以几种滑雪运动的仿真验证了模型的合理性。多刚体模型包括:雪地路面模型、人体子模型、滑雪板—雪地子模型、空气阻力模型。用函数描述雪地地形。把人体简化为16刚体25自由度的多刚体人体模型,人体各个肢体的惯性参数根据回归公式进行计算。利用函数模拟关节的运动从而达到对人体冗余自由度缩减的目的。建立了一个6自由度单刚体滑雪板模型。用1个矩形和7个圆弧描述滑雪板的正面和侧面几何轮廓。把滑雪板刚性离散化,以浸入力、冲击力、摩擦力描述滑雪板—雪地相互作用力,并乘以载荷分布函数近似滑雪板柔性变形的影响。考虑空气阻力对各个肢体的阻碍作用。最后应用建立的多刚体滑雪模型对双板平行直滑降、双板平行转弯、空中技巧进行仿真,仿真结果与简单模型、经验、实际运动效果非常相像,表明模型是合理的。
第三章建立了变拓扑结构的滑雪杖—雪地模型并利用模型对双滑雪杖支雪起动过程进行了仿真。忽略滑雪杖—雪地的接触碰撞过程,把滑雪杖—雪地的关系用多体系统动力学的单面约束进行描述。根据滑雪杖的拦雪轮与雪地的位置关系按分离、接触两种情况进行处理。把滑雪杖—雪地的缝隙函数与相互作用力关系转换为线性互补关系,作为单面约束动力学系统的补充方程。最后利用模型对使用双滑雪杖支雪起动加速的对称运动进行了仿真分析。
第四章引入Hill三元件肌肉模型建立了含神经、肌肉系统的滑雪运动生物力学模型并应用于探究各种基本滑雪技术的肌肉协调规律。把肌肉力/神经刺激作为设计变量,以完成动作所有肌肉力/神经刺激平方和最小为目标函数,受各肌肉力矩和与所绕关节力矩相等的等式约束,肌肉力/神经刺激非负有上限的不等式约束,建立二次规划问题。通过求解二次规划问题求解肌肉力/神经刺激。使模型不但适用于求解宏观的滑雪运动力学问题,而且能计算完成运动所需要的肌肉肌群用力,方便理解滑雪运动肌肉协调规律,为深入探究运动控制机理提供可能。最后应用该模型对滑降、转弯、支雪几种滑雪运动过程中的肌肉用力进行计算。计算结果为深入探究滑雪运动中肌肉协调控制规律提供可能,为加强平时专项滑雪运动肌肉训练提供依据。
第五章除了从运动学、动力学方面对滑雪运动技术进行评价外,针对特别的运动技术还用能量评价指标、转弯评价指标(转弯同步差异角、转弯同步系数)、稳定性评价指标(静态重心投影稳定性裕量、动态零力矩点稳定裕量)来评价。最后利用这些指标讨论了人体重心位置对转弯的影响,摆臂动作对空中技巧出台效果的影响。
第六章介绍了编写的滑雪运动分析平台并用于仿真各种基本滑雪技术。平台具有以下特点:基于前沿的多体系统动力学理论与计算方法,实现了动力学方程的自动形成;基于先进的计算软件MATLAB,求解计算速度快,代码简单易懂;界面功能明确,操作方便;采用模块化的设计理念,各种功能形成单独的模块,便于修改、维护和扩充;采用统一的数据格式,提高了系统的稳定性和通用性。程序分为前后处理两部分,前处理主要为自动建模、姿态设计、积分求解;后处理主要是关于运动学、动力学计算、能量计算、滑雪稳定性评价、肌肉力计算、动画输出等。本文利用该平台成功地对双板平行直滑降技术、犁式滑降技术、犁式转弯技术、双板平行转弯技术、双滑雪杖支雪动作、空中技巧进行了仿真分析研究。
研究结果表明,本文建立的滑雪模型、滑雪软件适应范围广,能较好地仿真各种滑雪运动,包括滑降、转弯、支雪、空中技巧,体现了滑雪起动、旋转、腾空、落地等基本特点,能够方便快捷的获得相关技术动作的运动学、动力学、生物力学等信息,对于认识、改进和完善动作技术,选择和设计优化的技术方案,有一定的现实意义,编写的软件可以为探寻新技术和论证创新技术的可行性及科学性提供了方便。
本文的研究工作得到了国家自然科学基金No.10472018和10721062的资助。在滑雪运动生物力学方面进行了初步的探索,期望能够对滑雪运动的研究发展有所贡献。
With the increasingly fierce international competition, the effects of science and technology on competitive sports have become increasingly prominent. Domestic and foreign researchers have made a great deal of efforts on sports studies and have achieved fruitful results by sports biomechanical methods. Skiing is a highly skillful technical sport. Most of studies about skiing are based on experiments and a very few skiing models are presented. Furthermore, most of the models are too simple and unsuitable for different types of skiing movements. They cannot compute muscle force and so on. None article is for skiing pole. No software is for skiing simulation due to the complexity of skiing movement. Skiing sports achievements in China have been behind the international advanced level. The reasons are that the studies about skiing started late and lacked of research tools.
Based on the factors influence skiing, this paper builds a biomechanical multibody skiing model and corresponding software. This study involves multibody system dynamics, sports biomechanics, robot dynamics and control, numerical optimization and so on. The main researches are:
The second chapter of this paper builds and tests a multibody skiing model. The model includes: snow ground sub model, skier sub model, ski-snow sub model, air drag sub model. The snow is described by space surface. The skier is simplified as sixteen rigid bodies and connected by twenty-five joints. The human body inertial parameters are obtained by regression equations with body height and mass. The joint motions are approximated by functions. The outline of ski face and side view is described by one rectangle and seven arcs. The ski is discretized as small points but has only six rigid degrees of freedom. A three-element of penetration force, impact force and friction is used to compute the interactions between ski and snow. The pressure distribution function is used to approximate the effects of ski bending. The aerodynamic drag on skier segments is considered. At the end, the skiing model is used to simulate downhill, parallel turning, and aerials. The simulated results agree well with simple model, experimental movement.
The third chapter mainly focuses on the disposal of skiing pole. Neglecting the contact process, the pole-snow is described by unilateral constraint of multibody system dynamics. The relationship of interstitial function and reaction for pole-snow is converted to linear complementary problem that is supplied for dynamics equations. A standard double poling process is simulated by the model at last.
The fourth chapter introduces the famous Hill-type muscle model to the multibody skiing model for the sake of study muscle coordination rules. The muscle forces/neural excitation values are solved by a standard quadratic programming optimization problem, in which muscle force/neural excitation is taken as design variable, the minimal square sum of muscle force/neural excitation is objective function, the non-negative and upper limit muscle force/neural excitation is none-equality constraints, the sum of all muscle moment equaled to joint control torque as equality constraints. Therefore, the skiing model can be used to solve both macroscopical kinetic problems and microcosmic muscle force coordination rules. As applications, muscle force coordination rules for several basic skiing movements are computed.
The fifth chapter presents three evaluation indexes for skiing movement. The indexes are energy evaluation index, turning evaluation index, and stability evaluation index. These indexes are used to discuss the effects of skier center of gravity position on snowplow turning, parallel turning, and arms swing on the take-off status of aerials.
The sixth chapter mainly introduces the skiing movement simulative software and uses it to simulate different skiing techniques. The program includes both parts of pre-processing and post-processing. The pre-processing is for auto building skiing model, posture setting, integral solver. The post-processing is about output of skiing animation, kinematics, dynamics, energy, muscle force, and stability evaluation and so on. At the end, parallel downhill, snowplow downhill, snowplow turn, parallel turning, and aerial skiing are successfully simulated by the program.
All the simulation results show that the skiing model and software can apply to a wide range of simulation: downhill, turning, poling, and aerials. Some skiing characteristics such as start-up, rotation, flying, and landing are shown. The computed kinematic, dynamic, biomechanical information are useful for the understanding and improvement of skiing technology. The software can explore new technologies and demonstrate the feasibility of innovative technology.
This study is financed by the National Natural Science Foundation of China No.10472018 and 10721062. Preliminary exploration about skiing is carried out and hoped to be contributed to ski sports research and development.
引文
[1]王庆江.跳板跳水的数值分析与计算机仿真:(硕士学位论文).南京:南京理工大学,2002.
[2]车仁炜,陆念力.多刚体系统动力学建模的一种新方法.汽轮机技术,2003,45(6):364-366.
[3]Roberson R E,Wittenburg J.A dynamical formalism for an arbitrary number of interconnected rigid bodies with reference to the problem of satellite attitude control.Proceedings of the Third International Congress on Automatic Control(London),1967,46:1-8.
[4]Hang E J.Computer-aided kinematics and dynamics of mechanical system.Boston:Allyn and Bacon.1989.
[5]Sayers M W.Symbolic computer methods to automatically formulate vehicle simulation codes:[PhD dissertation].America,Michigan:University of Michigan.1990.
[6]李鹏辉.虚拟样机技术在车辆动力学中的应用.大众科技,2005,11:93-94.
[7]刘贤喜.机械系统虚拟样机仿真软件的实用化研究:(硕士学位论文).北京:中国农业大学,2001.
[8]潘慧炬,李建设,杨丹.运动生物力学学科定义辨正.北京体育大学学报,1999,22(4):63-68.
[9]马南京.运动生物力学中力学和生物学的辩证关系例析.淮北煤师院学报,2003,24(1):47-50.
[10]李建设,王良民.运动生物力学研究技术的发展与存在问题.中国运动医学杂志,2002,21(4):389-391.
[11]马和中.生物力学导论.北京:北京航空学院出版社.1986,23-26.
[12]Radulovic M,Popovic D,Jaukovic N.Control of leg movements driven by electrically stimulated muscles.Journal of Automatic Control,2003,13(2):35-41.
[13]Veg A,Popovic D B,Dosen S.Customizing to user functional electrical stimulation of walking:optimal control.Faculty of Mechanical Engineering,2007,35:135-140.
[14]Peter H V,Howard J C,Patrick E C,Ahmed E B.Nonlinear joint angle control for artificially stimulated muscle.IEEE Transactions on Biomedical Engineering,1992,39(4):368-380.
[15]Ashby B M,Delp S L.Optimal control simulations reveal mechanisms by which arm movement improves standing long jump performance.Journal of Biomechanics,2006,39:1726-1734.
[16]Cheng K B,Hubbard M.Role of arms in somersaulting from compliant surfaces:a simulation study of springboard standing dives.Human Movement Science,2008,27:80-95.
[17]Mills C,Matthew T.G.Pain,Maurice R.Yeadon.,The influence of simulation model complexity on the estimation of internal loading in gymnastics landings.Journal of Biomechanics,2008,41:620-628.
[18]朱长虹.含肌骨特性的人体下肢运动动力学模型的初步研究:(硕士学位论文).北京,清华大学精密仪器与机械学系,1998.
[19]Hatze H.Optimization of human motions.Biomechanics Ⅲ,1973,138-142.
[20]Hatze H.Biomechanical aspects of a successful motion optimization.Biomechanics,1975,V.
[21]Hatze H.The complete optimization of the human motion.Mathematical Biosciences,1976,28:88-135.
[22]Hatze H.A comprehensive model for human motion simulation and its application to the take-off phase of the long jump.Journal of Biomechanics,1981,14(3):135-142.
[23]Pandy M G.Computer modeling and simulation of human movement.Annual Reviews Biomedical Engineering,2001,3:245-273.
[24]Bhargava L J,Pandy M G,Anderson F C.A phenomenological model for estimating metabolic energy consumption in muscle contraction.Journal of Biomechanics,2004,37:81-88.
[25]Pandy M G,Anderson F C,Hull D G.A parameter optimization approach for the optimal control of large-scale musculoskeletal systems.Journal of Biomechanical Engineering,1992,114:450-460.
[26]Pandy M G,Garner B A,Anderson F C.Optimal control of non-ballistic muscular movements:a constraint-based performance criterion for rising from a chair.Journal of biomechanical engineering,1995,117:15-26.
[27]Pandy M G,Zajac F E,Sim E,et al.An optimal control model for maximum-height human jumping.J.Biomechanics,1990,23:1185-1198.
[28]Anderson F C,Pandy M G.Individual muscle contributions to support in normal walking.Gait and Posture,2003,17:159-169.
[29]Anderson F C,Pandy M G.Dynamic optimization of human walking.Journal of biomechanical engineering,2001,123:381-390.
[30]Anderson F C,Pandy M G.Storage and utilization of elastic strain energy during jumping.Journal of Biomechanics,1993,26:1413-1427.
[31]Anderson F C,Pandy M G.Static and dynamic optimization solutions for gait are practically equivalent.Journal of Biomechanics,2001,34:153-161.
[32]Anderson F C,Pandy M G.A dynamic optimization solution for vertical jumping in three dimensions.Computer Methods in Biomechdical Engineering,1999,2:201-231.
[33]Anderson F C.A dynamic optimization solution for a complete cycle of normal gait:An analysis of muscle function and joint contact force:[PhD dissertation].Austin,Texas:University of Texas at Austin,1999.
[34]Zajac F E,Muscle and tendon:properties,models,scaling,and application to biomechanicsand motor control.Critical Reviews in Biomedical Engineering,1989,17:359-411.
[35]Fregly B J,Zajac F E.A state-space analysis of mechanical energy generation,absorption,and transfer during pedaling.Journal of Biomechanics,1996,29:81-90.
[36]Raasch C C,Zajac FE,Ma B,Levine WS.Muscle coordination of maximum-speed pedaling.Journal of Biomechanics,1997,6:595-602.
[37]Neptune R R,Kautz S A,Zajac F E.Muscle contributions to specific biomechanical functions do not change in forward versus backward pedaling.Journal of Biomechanics,2000,33:155-164.
[38]Schutte L M,Rodgers M M,Zajac F E.Improving the efficacy of electrical stimulation-induced leg cycle ergometry:an analysis based on a dynamic musculoskeletal model.IEEE Transactions on Rehabilitation Engineering,1993,1:109-125.
[39]黄勇,忻鼎亮,龚铭新等.对第23届国际运动生物力学会议收录论文述评.上海体育学院学报.
[40]赵焕彬,王海涛.我国运动生物力学的研究现状与发展趋势.河北师范大学学报,2001,25(3):414-416.
[41]Nordt A A.Springer G S,Kollar L P.Simulation of a turn on alpine skis.Sports Engineering,1999,2:181-199.
[42]Nordt,A A.Design of alpine skis:[PhD Thesis].Stanford:Stanford University.1999.
[43]Federolf P A.Finite element simulation of a carving snow ski:[PhD thesis].Zurich:Swiss Federal Institute of Technology,2005.
[44]Kaps P,M(o|¨)ssner M,Nachbauer W,et al.Pressure distribution under a ski during carved turns.2nd International Congress on Skiing and Science,(Christoph am Arlberg,Austria),2000.
[45]Glenne B,Derocco A,Vandergrift J.The modern alpine ski.Cold Regions Science and Technology,1997,26:35-38.
[46]Millet G Y,Hoffman M D,Candau R B,et al.Poling forces during roller skiing:effects of grade.Medicine and Science in Sports and Exercise,1998,30(11):1637-1644.
[47]Nilsson J,Jakobsen V,Tveit P,et al.Pole length and ground reaction forces during maximal double poling in skiing.Sports Biomechanics,2003,2:227-236.
[48]Nilsson J,Tveit P,Eikrehagen O.Effects of speed on temporal patterns in classical style and freestyle cross-country skiing,Sports Biomechanics,2004,3(1):85-107.
[49]Hoffman M D,Clifford P S,Bender F.Effects of velocity on cycle rate and length for three roller skiing techniques.Journal of Applied Biomechanics,1995,11:257-266.
[50]Holmberg H C,Lindinger S,St(o|¨)ggl T,et al.Contribution of the legs to double-poling performance in elite cross-country skiers.Medicine and Science in Sports and Exercise,2006,38(10):1853-1860.
[51]Holmberg H C,Lindinger S,StSggl T,et al.Biomechanical analysis of double poling in elite cross-country skiers.Medicineand Science in Sports and Exercise,2005,37(5):807-818.
[52]Holmberg J,Wagenius P.A biomechanical model of a double-poling skier.2003.
[53]Morawski,J.M.(1973).Control system approach to a ski-turn analysis.Journal of Biomechanics,6:267-279.
[54]Takehiro I,Hitoshi D,Hiroki S,et al.Analysis Method for Biomechanism of Skiing Turn.Transactions of the Japan Society of Mechanical Engineers C,2002,68(666):562-568.
[55]Kawai S,Kato K,Sakata T.Effects of ski model on ski turn.Transactions of the Japan Society of Mechanical Engineers,2002,68:2301-2307.
[56]Kawai S,Otani H,Sakata T.Coupled motion of ski and elastic foundation under ski control.JSME International Journal series C,2003,46:614-621.
[57]Lieu D K,Mote C D.Mechanics of the turning snow ski.Skiing Trauma and Safety:Fifth International Symposium,American society for testing and materials,Philadelphia,USA,1985,117-140.
[58]Tada N,Hirano Y.Simulation of a turning ski using ice cutting data.Sports Engineering,1999,2:55-64.
[59]Tada N,Hirano Y.In search of the mechanics of a turning alpine ski using snow cutting force measurements.Sports Engineering,2002,5:15-22.
[60]Hirano Y,Tada N.Mechanics of a turning snow ski.International Journal of Mechanical Sciences,1994,36:421-429.
[61]Hirano Y,Tada N.Numerical simulation of a turning alpine ski during recreational skiing.Medicine and Science in Sports and Exercise,1996,28:1209-1213.
[62]Gerritsen K G M,Nachbauer W,Van den Bogert A J.Computer simulation of landing movement in downhill skiing:anterior cruciate ligament injuries.Journal of Biomechanics,1996,29(7):845-854.
[63]Bruck F,Lugner P,Schrestter H.A dynamic model for the performance of carving skis.Skiing Trauma and Safety,2003,14:10-23.
[64]佟永典,阎红光,刘平等.自由式滑雪空中技巧运动员落地稳定性主要影响因素的研究.沈阳体育学院学报,2003,22(4):15-17.
[65]闫红光,徐威.国家空中技巧滑雪运动员程爽bFdF水池动作运动特点分析.山东体育学院学报,2006,22(1):85-87.
[66]佟永典,候永民.对自由式滑雪空中技巧落地稳定性的实验研究.冰雪运动,1994,3:44-48,53.
[67]闫红光,佟永典,刘平等.国家自由式滑雪空中技巧运动员徐囡囡bdFF水池动作的运动学分析.辽宁体育科技,2002,24(5):1-3.
[68]郝庆威,门传胜,格永典等.季晓鸥自由式滑雪空中技巧水池动作的运动学研究.沈阳体育学院学报,2003,22(4):18-20,33.
[69]郝庆威,戈炳珠,门传胜.欧小涛、季晓鸥bL FF动作部分技术环节的运动学比较研究.沈阳体育学院学报,1999,3:12-15.
[70]付春明.自由式滑雪空中技巧女子起跳技术的运动学分析:(硕士学位论文).沈阳:东北师范大学.2006.
[71]齐朝晖.多体系统动力学.北京:科学出版社.2008.
[72]洪嘉振.计算多体系统动力学.北京:高等教育出版社.1999.
[73]Hanavan E P.A mathematical model of the human body.Aerospace Medical Research Labs.Technical Report,1964,64-102.
[74]郑秀瑗.现代运动生物力学.北京:国防工业出版社.2002.
[75]Zatsiorsky V M,Selujanov V N.The mass and inertia characteristics of the main segments of the human body.Biomechanics Ⅷ-B,1983,1152-1159.
[76]杨年峰.人体运动协调规律及其参数化描述:(博士学位论文).北京:清华大学,2001.
[77]Van den Bogert A J,Schamhardt H C,Crowe A.Simulation of quadrupedal locomotion using a rigid body model.Journal of Biomechanics,1989,22:33-41.
[78]Winters J M,Stark L.Analysis of fundamental human movements patterns through the use of in-depth antagonistic muscle models.IEEE Transactions on Biomedical Engineering,1985,32:826-839.
[79]Jentschura U D,Fahrbach F.Physics of Skiing:The Ideal-Carving Equation and Its Applications,2003.
[80]Sahashi T,Ichino S.Coefficient of kinetic friction of snow skis during turning descents.Japanese Journal of Applied Physics,1998,37:720-727.
[81]Kaps P,Nachbauer W,M(o|¨)ssner M.Snow frictionand drag in alpine downhill racing.4th World Congress of Biomechanics(Calgary,CA),2002.
[82]Thompson B E,Friess W A,Knapp Ⅱ K N.Aerodynamics of speed skiers.Sports Engineering,2004,4:103-112.
[83]Erdemir A,McLean S,Herzog W,et al.Model-based estimation of muscle forces exerted during movements.Clinical Biomechanics,2007,(22):131-154.
[84]FIS Freestyle Committee.Aerial Site Specifications-ICR 3060.2007.
[85]阎绍泽,贾书惠,吴德隆.含缝隙系统的变拓扑多体系统动力学建模分析.中国机械工程,2000,11(6):624-626.
[86] Glocker C. Formulation of spatial contact situations in rigid multibody systems. Computer Methods in Applied Mechanics and Engineering, 1999, 177: 199-214.
[87] Pfeiffer F. Robots with unilateral constraints. Annual Reviews in Control, 1998, 22: 121-132.
[88] Pfeiffer F. Multibody systems with unilateral constraints. Journal of Applied Mathematics and Mechanics, 2001, 65(4): 665-670.
[89] Cottle R W, Pang J S, Stone R E. The Linear complementarity problem. Academic Press, Inc., San Diego, CA, 1992.
[90] Lemke C E. Bimatrix equilibrium points and mathematical programming. Management Science, 1965, 11: 681-689.
[91] Holmberg L J, Lund A M. A musculoskeletal full-body simulation of cross-country skiing. Journal of Sports Engineering and Technology, 2008, 222: 11-22.
[92] Van Den Bogert A J, Gerritsen K G, Cole G K. Human muscle modelling from a user's perspective. Journal of Electromyography and Kinesiology, 1998, 8: 119-124.
[93] Winters J M. Hill-based muscle models: a systems engineering perspective. Multiple Muscle Systems: Biomechanics and Movement Organization. Springer, New York, USA, 1990, 69-93.
[94] Delp S L, Loan J P. A graphics-based software system to develop and analyze models of musculoskeletal structures. Computers in Biology and Medicine, 1995, 25: 21-34.
[95] Scott S H, Winter D A. A comparison of three muscle pennation assumptions and their effect on isometric and isotonic force. Journal of Biomechanics, 1991, 24: 163-167.
[96] Epstein M, Herzog W. Theoretical Models of Skeletal Muscle. Wiley, New York. 1998.
[97] Zahalak G I. Modeling muscle mechanics (and energetics). Multiple Muscle Systems, Springer-Verlag, 1990, 1-23.
[98] Hatze H. Myocybernetic control models of skeletal muscle. Gutenberg Book Printers, Pretoria, 1981.
[99] Kuo A D. An optimal control model for analyzing human postural balance. IEEE Transactions on Biomedical Engineering, 1995, 42: 87-101.
[100] 陈守良. 动物生理学(第二版) . 北京：北京大学出版社. 1996.
[101] Delp S L. Surgery simulation: a computer graphics system to analyze and design musculoskeletal reconstructions of the lower limb. Stanford: Stanford University, 1990.
[102] Van Soest A J. Jumping from structure to control: [Phd thesis]. Vrije Universiteit Amsterdam, 1992.
[103] Fung Y C. Biomechanics. Springer-Verlag, New York, 1981.
[104] Katz B. The relation between force and speed in muscular contraction. Journal of Physiology, 1939, 96: 45-64.
[105] Nagano A, Gerritsen K G M. Effects of neuromuscular stregth training on vertical jumping performance-a computer simulation study. Journal of Applied Biomechanics, 2001, 17(2): 113-128.
[106] Proske U, Morgan D L. Tendon stiffness: methods of measurement and significance for the control of movement. Journal of Biomechanics, 1987, 20: 75-82.
[107] Van Soest A J, Hui jing P A, Solomonow M. The effect of tendon on muscle force in dynamic isometric contractions: a simulation study. Journal of Biomechanics, 1995, 28: 801-807.
[108] Spagele T. Modellierung, Simulation und Optimierung menschlicher Bewegungen: [PhD thesis]. University at Stuttgart, 1998.
[109] Herzog W. Sensitivity of muscle force estimations to changes in muscle input parameters using nonlinear optimization approaches. Journal of Biomechanical Engineering, 1992, 114: 267-268.
[110] Yoon Y S, Mansour J M. The passive elastic moment at the hip. Journal of Biomechanics, 1982, 15: 905-910.
[111] Audu M L. Optimal control modeling of lower extremity musculoskeletal motion: [Ph. D. Dissertation]. Case Western University, Cleveland, Ohio. 1985.
[112] Ashby B M, Delp S L. Optimal control simulations reveal mechanisms by which arm movement improves standing long jump performance. Journal of Biomechanics, 2006, 39: 1726-1734.
[113] Yamaguchi G T, Zajac F E. Restoring unassisted natural gait to paraplegics via functional neuromuscular stimulation: a computer simulation study. IEEE Transactions on Biomedical Engineering, 1990, 37: 886-902.
[114] Piazza S J, Delp S L The influence of muscles on knee flexion during the swing phase of gait. Journal of Biomechanics, 1996; 29: 723-733.
[115] Gerritsen K GM, Van den Bogert A J, Nigg B M. Direct dynamics simulation of the impact phase in heel-toe running. Journal of Biomechanics, 1995, 28(6): 661-668.
[116] Selbie W S, Caldwell G E. A simulation study of vertical jumping from different starting postures. Journal of Biomechanics, 1996, 29: 1137-1146.
[117] Van Soest A J, Schwab A L, Bobbert M F, et al. The influence of the biarticularity of the gastrocnemius muscle on vertical-jumping performance. Journal of Biomechanics, 1993, 26: 1-8.
[118] Tsirakos D, Baltzopoulos V, Bartlett R. Inverse optimization: functional and physiological considerations related to the force sharing problem. Critical Reviews in Biomedical Engineering, 1997, 25: 371-407.
[119] Seireg, A. , Arvikar, R. J. , The prediction of muscular load sharing and joint forces in the lower extremities during walking. Journal of Biomechanics, 1975, 8: 89-102.
[120]Brand R A,Pedersen D R,Friederich J A.The sensitivity of muscle force predictions to changes in physiologic cross-sectional area.Journal of Biomechanics,1985,19:589-596.
[121]Glitsch,U,Baumann W.The three-dimensional determination of internal loads in the lower extremity.Journal of Biomechanics,1997,30:l123-1131.
[122]王人成,杨年峰,朱长虹等.人体下肢摆动相冗余肌力分析.清华大学学报,1999,39(11):104-106.
[123]Rohrle H,Scholten R,Sigolotto C,et al.Joint forces in the human pelvis-leg skeleton during walking.Journal of Biomechanics,1984,17:409-424.
[124]Pedersen D R,Brand R A,Cheng C,et al.Direct comparison of muscle force predictions using linear and nonlinear programming.Journal of Biomechanical Engineering,1987,109:192-199.
[125]Chow C K,Jacobson D H.Studies of human locomotion via optimal programming.Mathematical Biosciences,1971,10:239-306.
[126]Davy D T,Audu M L.A dynamic optimization technique for predicting muscle forces in the swing phase of gait.Journal of Biomechanics,1987,20:187-201.
[127]Tashman S,Zajac F E,Perkash I.Modeling and simulation of paraplegic ambulation in a reciprocating gait orthoses.Journal of Biomechanical Engineering,1995,117:300-308.
[128]Happee R,Van Der Helm F C.The control of shoulder muscles during goal directed movements,an inverse dynamic analysis.Journal of Biomechanics,1995,28:1179-1191.
[129]Neptune R.Optimization algorithm performance in determining optimal controls in human movement analyses.Journal of Biomechanical Engineering,1999,121:249-252.
[130]程耿东.工程结构优化设计基础.1983,235-248.
[131]单兆鉴.滑雪运动指南.北京:人民体育出版社,2004.
[132]田麦久.运动训练过程.四川体育出版社,1990.
[133]Hatze H,Buys J.Energy-optimal controls in the mammalian neuromuscular system.Biological Cybernetics,1977,27:9-20.
[134]Bhargava L J,Pandy M G,Anderson F C.A phenomenological model for estimating metabolic energy consumption in muscle contraction.Journal of Biomechanics,2004,37:81-88.
[135]Prilutsky B I,Petrova L N,Raitsin L M.Comparison of mechanical energy expenditure of joint moments and muscle forces during human locomotion.Journal of Biomechanics,1996,29(4):405-415.
[136]王宏,聂义勇.20世纪运动稳定性研究的三个重要结果.信息与控制,2000,29(5):393-406.
[137]夏旭峰.基于弹簧-质量模型的仿袋鼠跳跃机器人步态稳定性研究.(硕士学位论文).西安:西北工业大学,2006.
[138]McGhee R B,Frank A A.On the stability properties of quadruped creeping gaits.Mathematical Biosciences,1968,3:331-351.
[139]Vukobratovic M,Juricic D.Contributions to the synthesis of biped gait.IEEE Transaction on Biomedical and Engineering,1969,16(1):1-6.
[140]张永和,王永涛.浅谈自由式滑雪空中技巧起跳环节中的摆臂技术.冰雪运动,2001,(2):22-25.
[141]Alexander R M.Elastic Mechanisms in animal movement.Cambridge,UK:Cambridge University Press,1988.
[142]Farley C T,Glasheen J,McMahon T A.Running Springs:Speed and Animal Size.Journal of Experimental Biology,1993(185):71-86.
[143]Nigg B M,Liu W.The effect of muscle stiffness and damping on simulated impact force peaks during running.Journal of Biomechanics,1999,32:849-856.
[144]Liu W,Nigg B M.A mechanical model to determine the influence of masses and mass distribution on the impact force during running.Journal of Biomechanics,2000,33:219-224.
[145]Ratcliffe R J,Holt K G.Low frequency shock absorption in human walking.Gait & Posture,1997,5:93-100.
[146]Butler R J,Crowell Ⅲ H P,Davis I M.Lower extremity stiffness:implications for performance and injury.Clinical Biomechanics,2003,18:511-517.
[147]Bosco C,Komi P V.Mechanical Characteristics and Fiber Composition of Human Leg Extensor Muscles.European Journal of Applied Physiology and Occupational Physiology,1979,(41):275-284.
[148]Rapoport S,Mizrahi J,Kimmel E,et al.Constant and Variable Stiffness and Damping of the Leg Joints in Human Hopping.Journal of Biomechanical Engineering,2003,8(125):507-514.
[2]车仁炜,陆念力.多刚体系统动力学建模的一种新方法.汽轮机技术,2003,45(6):364-366.
[3]Roberson R E,Wittenburg J.A dynamical formalism for an arbitrary number of interconnected rigid bodies with reference to the problem of satellite attitude control.Proceedings of the Third International Congress on Automatic Control(London),1967,46:1-8.
[4]Hang E J.Computer-aided kinematics and dynamics of mechanical system.Boston:Allyn and Bacon.1989.
[5]Sayers M W.Symbolic computer methods to automatically formulate vehicle simulation codes:[PhD dissertation].America,Michigan:University of Michigan.1990.
[6]李鹏辉.虚拟样机技术在车辆动力学中的应用.大众科技,2005,11:93-94.
[7]刘贤喜.机械系统虚拟样机仿真软件的实用化研究:(硕士学位论文).北京:中国农业大学,2001.
[8]潘慧炬,李建设,杨丹.运动生物力学学科定义辨正.北京体育大学学报,1999,22(4):63-68.
[9]马南京.运动生物力学中力学和生物学的辩证关系例析.淮北煤师院学报,2003,24(1):47-50.
[10]李建设,王良民.运动生物力学研究技术的发展与存在问题.中国运动医学杂志,2002,21(4):389-391.
[11]马和中.生物力学导论.北京:北京航空学院出版社.1986,23-26.
[12]Radulovic M,Popovic D,Jaukovic N.Control of leg movements driven by electrically stimulated muscles.Journal of Automatic Control,2003,13(2):35-41.
[13]Veg A,Popovic D B,Dosen S.Customizing to user functional electrical stimulation of walking:optimal control.Faculty of Mechanical Engineering,2007,35:135-140.
[14]Peter H V,Howard J C,Patrick E C,Ahmed E B.Nonlinear joint angle control for artificially stimulated muscle.IEEE Transactions on Biomedical Engineering,1992,39(4):368-380.
[15]Ashby B M,Delp S L.Optimal control simulations reveal mechanisms by which arm movement improves standing long jump performance.Journal of Biomechanics,2006,39:1726-1734.
[16]Cheng K B,Hubbard M.Role of arms in somersaulting from compliant surfaces:a simulation study of springboard standing dives.Human Movement Science,2008,27:80-95.
[17]Mills C,Matthew T.G.Pain,Maurice R.Yeadon.,The influence of simulation model complexity on the estimation of internal loading in gymnastics landings.Journal of Biomechanics,2008,41:620-628.
[18]朱长虹.含肌骨特性的人体下肢运动动力学模型的初步研究:(硕士学位论文).北京,清华大学精密仪器与机械学系,1998.
[19]Hatze H.Optimization of human motions.Biomechanics Ⅲ,1973,138-142.
[20]Hatze H.Biomechanical aspects of a successful motion optimization.Biomechanics,1975,V.
[21]Hatze H.The complete optimization of the human motion.Mathematical Biosciences,1976,28:88-135.
[22]Hatze H.A comprehensive model for human motion simulation and its application to the take-off phase of the long jump.Journal of Biomechanics,1981,14(3):135-142.
[23]Pandy M G.Computer modeling and simulation of human movement.Annual Reviews Biomedical Engineering,2001,3:245-273.
[24]Bhargava L J,Pandy M G,Anderson F C.A phenomenological model for estimating metabolic energy consumption in muscle contraction.Journal of Biomechanics,2004,37:81-88.
[25]Pandy M G,Anderson F C,Hull D G.A parameter optimization approach for the optimal control of large-scale musculoskeletal systems.Journal of Biomechanical Engineering,1992,114:450-460.
[26]Pandy M G,Garner B A,Anderson F C.Optimal control of non-ballistic muscular movements:a constraint-based performance criterion for rising from a chair.Journal of biomechanical engineering,1995,117:15-26.
[27]Pandy M G,Zajac F E,Sim E,et al.An optimal control model for maximum-height human jumping.J.Biomechanics,1990,23:1185-1198.
[28]Anderson F C,Pandy M G.Individual muscle contributions to support in normal walking.Gait and Posture,2003,17:159-169.
[29]Anderson F C,Pandy M G.Dynamic optimization of human walking.Journal of biomechanical engineering,2001,123:381-390.
[30]Anderson F C,Pandy M G.Storage and utilization of elastic strain energy during jumping.Journal of Biomechanics,1993,26:1413-1427.
[31]Anderson F C,Pandy M G.Static and dynamic optimization solutions for gait are practically equivalent.Journal of Biomechanics,2001,34:153-161.
[32]Anderson F C,Pandy M G.A dynamic optimization solution for vertical jumping in three dimensions.Computer Methods in Biomechdical Engineering,1999,2:201-231.
[33]Anderson F C.A dynamic optimization solution for a complete cycle of normal gait:An analysis of muscle function and joint contact force:[PhD dissertation].Austin,Texas:University of Texas at Austin,1999.
[34]Zajac F E,Muscle and tendon:properties,models,scaling,and application to biomechanicsand motor control.Critical Reviews in Biomedical Engineering,1989,17:359-411.
[35]Fregly B J,Zajac F E.A state-space analysis of mechanical energy generation,absorption,and transfer during pedaling.Journal of Biomechanics,1996,29:81-90.
[36]Raasch C C,Zajac FE,Ma B,Levine WS.Muscle coordination of maximum-speed pedaling.Journal of Biomechanics,1997,6:595-602.
[37]Neptune R R,Kautz S A,Zajac F E.Muscle contributions to specific biomechanical functions do not change in forward versus backward pedaling.Journal of Biomechanics,2000,33:155-164.
[38]Schutte L M,Rodgers M M,Zajac F E.Improving the efficacy of electrical stimulation-induced leg cycle ergometry:an analysis based on a dynamic musculoskeletal model.IEEE Transactions on Rehabilitation Engineering,1993,1:109-125.
[39]黄勇,忻鼎亮,龚铭新等.对第23届国际运动生物力学会议收录论文述评.上海体育学院学报.
[40]赵焕彬,王海涛.我国运动生物力学的研究现状与发展趋势.河北师范大学学报,2001,25(3):414-416.
[41]Nordt A A.Springer G S,Kollar L P.Simulation of a turn on alpine skis.Sports Engineering,1999,2:181-199.
[42]Nordt,A A.Design of alpine skis:[PhD Thesis].Stanford:Stanford University.1999.
[43]Federolf P A.Finite element simulation of a carving snow ski:[PhD thesis].Zurich:Swiss Federal Institute of Technology,2005.
[44]Kaps P,M(o|¨)ssner M,Nachbauer W,et al.Pressure distribution under a ski during carved turns.2nd International Congress on Skiing and Science,(Christoph am Arlberg,Austria),2000.
[45]Glenne B,Derocco A,Vandergrift J.The modern alpine ski.Cold Regions Science and Technology,1997,26:35-38.
[46]Millet G Y,Hoffman M D,Candau R B,et al.Poling forces during roller skiing:effects of grade.Medicine and Science in Sports and Exercise,1998,30(11):1637-1644.
[47]Nilsson J,Jakobsen V,Tveit P,et al.Pole length and ground reaction forces during maximal double poling in skiing.Sports Biomechanics,2003,2:227-236.
[48]Nilsson J,Tveit P,Eikrehagen O.Effects of speed on temporal patterns in classical style and freestyle cross-country skiing,Sports Biomechanics,2004,3(1):85-107.
[49]Hoffman M D,Clifford P S,Bender F.Effects of velocity on cycle rate and length for three roller skiing techniques.Journal of Applied Biomechanics,1995,11:257-266.
[50]Holmberg H C,Lindinger S,St(o|¨)ggl T,et al.Contribution of the legs to double-poling performance in elite cross-country skiers.Medicine and Science in Sports and Exercise,2006,38(10):1853-1860.
[51]Holmberg H C,Lindinger S,StSggl T,et al.Biomechanical analysis of double poling in elite cross-country skiers.Medicineand Science in Sports and Exercise,2005,37(5):807-818.
[52]Holmberg J,Wagenius P.A biomechanical model of a double-poling skier.2003.
[53]Morawski,J.M.(1973).Control system approach to a ski-turn analysis.Journal of Biomechanics,6:267-279.
[54]Takehiro I,Hitoshi D,Hiroki S,et al.Analysis Method for Biomechanism of Skiing Turn.Transactions of the Japan Society of Mechanical Engineers C,2002,68(666):562-568.
[55]Kawai S,Kato K,Sakata T.Effects of ski model on ski turn.Transactions of the Japan Society of Mechanical Engineers,2002,68:2301-2307.
[56]Kawai S,Otani H,Sakata T.Coupled motion of ski and elastic foundation under ski control.JSME International Journal series C,2003,46:614-621.
[57]Lieu D K,Mote C D.Mechanics of the turning snow ski.Skiing Trauma and Safety:Fifth International Symposium,American society for testing and materials,Philadelphia,USA,1985,117-140.
[58]Tada N,Hirano Y.Simulation of a turning ski using ice cutting data.Sports Engineering,1999,2:55-64.
[59]Tada N,Hirano Y.In search of the mechanics of a turning alpine ski using snow cutting force measurements.Sports Engineering,2002,5:15-22.
[60]Hirano Y,Tada N.Mechanics of a turning snow ski.International Journal of Mechanical Sciences,1994,36:421-429.
[61]Hirano Y,Tada N.Numerical simulation of a turning alpine ski during recreational skiing.Medicine and Science in Sports and Exercise,1996,28:1209-1213.
[62]Gerritsen K G M,Nachbauer W,Van den Bogert A J.Computer simulation of landing movement in downhill skiing:anterior cruciate ligament injuries.Journal of Biomechanics,1996,29(7):845-854.
[63]Bruck F,Lugner P,Schrestter H.A dynamic model for the performance of carving skis.Skiing Trauma and Safety,2003,14:10-23.
[64]佟永典,阎红光,刘平等.自由式滑雪空中技巧运动员落地稳定性主要影响因素的研究.沈阳体育学院学报,2003,22(4):15-17.
[65]闫红光,徐威.国家空中技巧滑雪运动员程爽bFdF水池动作运动特点分析.山东体育学院学报,2006,22(1):85-87.
[66]佟永典,候永民.对自由式滑雪空中技巧落地稳定性的实验研究.冰雪运动,1994,3:44-48,53.
[67]闫红光,佟永典,刘平等.国家自由式滑雪空中技巧运动员徐囡囡bdFF水池动作的运动学分析.辽宁体育科技,2002,24(5):1-3.
[68]郝庆威,门传胜,格永典等.季晓鸥自由式滑雪空中技巧水池动作的运动学研究.沈阳体育学院学报,2003,22(4):18-20,33.
[69]郝庆威,戈炳珠,门传胜.欧小涛、季晓鸥bL FF动作部分技术环节的运动学比较研究.沈阳体育学院学报,1999,3:12-15.
[70]付春明.自由式滑雪空中技巧女子起跳技术的运动学分析:(硕士学位论文).沈阳:东北师范大学.2006.
[71]齐朝晖.多体系统动力学.北京:科学出版社.2008.
[72]洪嘉振.计算多体系统动力学.北京:高等教育出版社.1999.
[73]Hanavan E P.A mathematical model of the human body.Aerospace Medical Research Labs.Technical Report,1964,64-102.
[74]郑秀瑗.现代运动生物力学.北京:国防工业出版社.2002.
[75]Zatsiorsky V M,Selujanov V N.The mass and inertia characteristics of the main segments of the human body.Biomechanics Ⅷ-B,1983,1152-1159.
[76]杨年峰.人体运动协调规律及其参数化描述:(博士学位论文).北京:清华大学,2001.
[77]Van den Bogert A J,Schamhardt H C,Crowe A.Simulation of quadrupedal locomotion using a rigid body model.Journal of Biomechanics,1989,22:33-41.
[78]Winters J M,Stark L.Analysis of fundamental human movements patterns through the use of in-depth antagonistic muscle models.IEEE Transactions on Biomedical Engineering,1985,32:826-839.
[79]Jentschura U D,Fahrbach F.Physics of Skiing:The Ideal-Carving Equation and Its Applications,2003.
[80]Sahashi T,Ichino S.Coefficient of kinetic friction of snow skis during turning descents.Japanese Journal of Applied Physics,1998,37:720-727.
[81]Kaps P,Nachbauer W,M(o|¨)ssner M.Snow frictionand drag in alpine downhill racing.4th World Congress of Biomechanics(Calgary,CA),2002.
[82]Thompson B E,Friess W A,Knapp Ⅱ K N.Aerodynamics of speed skiers.Sports Engineering,2004,4:103-112.
[83]Erdemir A,McLean S,Herzog W,et al.Model-based estimation of muscle forces exerted during movements.Clinical Biomechanics,2007,(22):131-154.
[84]FIS Freestyle Committee.Aerial Site Specifications-ICR 3060.2007.
[85]阎绍泽,贾书惠,吴德隆.含缝隙系统的变拓扑多体系统动力学建模分析.中国机械工程,2000,11(6):624-626.
[86] Glocker C. Formulation of spatial contact situations in rigid multibody systems. Computer Methods in Applied Mechanics and Engineering, 1999, 177: 199-214.
[87] Pfeiffer F. Robots with unilateral constraints. Annual Reviews in Control, 1998, 22: 121-132.
[88] Pfeiffer F. Multibody systems with unilateral constraints. Journal of Applied Mathematics and Mechanics, 2001, 65(4): 665-670.
[89] Cottle R W, Pang J S, Stone R E. The Linear complementarity problem. Academic Press, Inc., San Diego, CA, 1992.
[90] Lemke C E. Bimatrix equilibrium points and mathematical programming. Management Science, 1965, 11: 681-689.
[91] Holmberg L J, Lund A M. A musculoskeletal full-body simulation of cross-country skiing. Journal of Sports Engineering and Technology, 2008, 222: 11-22.
[92] Van Den Bogert A J, Gerritsen K G, Cole G K. Human muscle modelling from a user's perspective. Journal of Electromyography and Kinesiology, 1998, 8: 119-124.
[93] Winters J M. Hill-based muscle models: a systems engineering perspective. Multiple Muscle Systems: Biomechanics and Movement Organization. Springer, New York, USA, 1990, 69-93.
[94] Delp S L, Loan J P. A graphics-based software system to develop and analyze models of musculoskeletal structures. Computers in Biology and Medicine, 1995, 25: 21-34.
[95] Scott S H, Winter D A. A comparison of three muscle pennation assumptions and their effect on isometric and isotonic force. Journal of Biomechanics, 1991, 24: 163-167.
[96] Epstein M, Herzog W. Theoretical Models of Skeletal Muscle. Wiley, New York. 1998.
[97] Zahalak G I. Modeling muscle mechanics (and energetics). Multiple Muscle Systems, Springer-Verlag, 1990, 1-23.
[98] Hatze H. Myocybernetic control models of skeletal muscle. Gutenberg Book Printers, Pretoria, 1981.
[99] Kuo A D. An optimal control model for analyzing human postural balance. IEEE Transactions on Biomedical Engineering, 1995, 42: 87-101.
[100] 陈守良. 动物生理学(第二版) . 北京：北京大学出版社. 1996.
[101] Delp S L. Surgery simulation: a computer graphics system to analyze and design musculoskeletal reconstructions of the lower limb. Stanford: Stanford University, 1990.
[102] Van Soest A J. Jumping from structure to control: [Phd thesis]. Vrije Universiteit Amsterdam, 1992.
[103] Fung Y C. Biomechanics. Springer-Verlag, New York, 1981.
[104] Katz B. The relation between force and speed in muscular contraction. Journal of Physiology, 1939, 96: 45-64.
[105] Nagano A, Gerritsen K G M. Effects of neuromuscular stregth training on vertical jumping performance-a computer simulation study. Journal of Applied Biomechanics, 2001, 17(2): 113-128.
[106] Proske U, Morgan D L. Tendon stiffness: methods of measurement and significance for the control of movement. Journal of Biomechanics, 1987, 20: 75-82.
[107] Van Soest A J, Hui jing P A, Solomonow M. The effect of tendon on muscle force in dynamic isometric contractions: a simulation study. Journal of Biomechanics, 1995, 28: 801-807.
[108] Spagele T. Modellierung, Simulation und Optimierung menschlicher Bewegungen: [PhD thesis]. University at Stuttgart, 1998.
[109] Herzog W. Sensitivity of muscle force estimations to changes in muscle input parameters using nonlinear optimization approaches. Journal of Biomechanical Engineering, 1992, 114: 267-268.
[110] Yoon Y S, Mansour J M. The passive elastic moment at the hip. Journal of Biomechanics, 1982, 15: 905-910.
[111] Audu M L. Optimal control modeling of lower extremity musculoskeletal motion: [Ph. D. Dissertation]. Case Western University, Cleveland, Ohio. 1985.
[112] Ashby B M, Delp S L. Optimal control simulations reveal mechanisms by which arm movement improves standing long jump performance. Journal of Biomechanics, 2006, 39: 1726-1734.
[113] Yamaguchi G T, Zajac F E. Restoring unassisted natural gait to paraplegics via functional neuromuscular stimulation: a computer simulation study. IEEE Transactions on Biomedical Engineering, 1990, 37: 886-902.
[114] Piazza S J, Delp S L The influence of muscles on knee flexion during the swing phase of gait. Journal of Biomechanics, 1996; 29: 723-733.
[115] Gerritsen K GM, Van den Bogert A J, Nigg B M. Direct dynamics simulation of the impact phase in heel-toe running. Journal of Biomechanics, 1995, 28(6): 661-668.
[116] Selbie W S, Caldwell G E. A simulation study of vertical jumping from different starting postures. Journal of Biomechanics, 1996, 29: 1137-1146.
[117] Van Soest A J, Schwab A L, Bobbert M F, et al. The influence of the biarticularity of the gastrocnemius muscle on vertical-jumping performance. Journal of Biomechanics, 1993, 26: 1-8.
[118] Tsirakos D, Baltzopoulos V, Bartlett R. Inverse optimization: functional and physiological considerations related to the force sharing problem. Critical Reviews in Biomedical Engineering, 1997, 25: 371-407.
[119] Seireg, A. , Arvikar, R. J. , The prediction of muscular load sharing and joint forces in the lower extremities during walking. Journal of Biomechanics, 1975, 8: 89-102.
[120]Brand R A,Pedersen D R,Friederich J A.The sensitivity of muscle force predictions to changes in physiologic cross-sectional area.Journal of Biomechanics,1985,19:589-596.
[121]Glitsch,U,Baumann W.The three-dimensional determination of internal loads in the lower extremity.Journal of Biomechanics,1997,30:l123-1131.
[122]王人成,杨年峰,朱长虹等.人体下肢摆动相冗余肌力分析.清华大学学报,1999,39(11):104-106.
[123]Rohrle H,Scholten R,Sigolotto C,et al.Joint forces in the human pelvis-leg skeleton during walking.Journal of Biomechanics,1984,17:409-424.
[124]Pedersen D R,Brand R A,Cheng C,et al.Direct comparison of muscle force predictions using linear and nonlinear programming.Journal of Biomechanical Engineering,1987,109:192-199.
[125]Chow C K,Jacobson D H.Studies of human locomotion via optimal programming.Mathematical Biosciences,1971,10:239-306.
[126]Davy D T,Audu M L.A dynamic optimization technique for predicting muscle forces in the swing phase of gait.Journal of Biomechanics,1987,20:187-201.
[127]Tashman S,Zajac F E,Perkash I.Modeling and simulation of paraplegic ambulation in a reciprocating gait orthoses.Journal of Biomechanical Engineering,1995,117:300-308.
[128]Happee R,Van Der Helm F C.The control of shoulder muscles during goal directed movements,an inverse dynamic analysis.Journal of Biomechanics,1995,28:1179-1191.
[129]Neptune R.Optimization algorithm performance in determining optimal controls in human movement analyses.Journal of Biomechanical Engineering,1999,121:249-252.
[130]程耿东.工程结构优化设计基础.1983,235-248.
[131]单兆鉴.滑雪运动指南.北京:人民体育出版社,2004.
[132]田麦久.运动训练过程.四川体育出版社,1990.
[133]Hatze H,Buys J.Energy-optimal controls in the mammalian neuromuscular system.Biological Cybernetics,1977,27:9-20.
[134]Bhargava L J,Pandy M G,Anderson F C.A phenomenological model for estimating metabolic energy consumption in muscle contraction.Journal of Biomechanics,2004,37:81-88.
[135]Prilutsky B I,Petrova L N,Raitsin L M.Comparison of mechanical energy expenditure of joint moments and muscle forces during human locomotion.Journal of Biomechanics,1996,29(4):405-415.
[136]王宏,聂义勇.20世纪运动稳定性研究的三个重要结果.信息与控制,2000,29(5):393-406.
[137]夏旭峰.基于弹簧-质量模型的仿袋鼠跳跃机器人步态稳定性研究.(硕士学位论文).西安:西北工业大学,2006.
[138]McGhee R B,Frank A A.On the stability properties of quadruped creeping gaits.Mathematical Biosciences,1968,3:331-351.
[139]Vukobratovic M,Juricic D.Contributions to the synthesis of biped gait.IEEE Transaction on Biomedical and Engineering,1969,16(1):1-6.
[140]张永和,王永涛.浅谈自由式滑雪空中技巧起跳环节中的摆臂技术.冰雪运动,2001,(2):22-25.
[141]Alexander R M.Elastic Mechanisms in animal movement.Cambridge,UK:Cambridge University Press,1988.
[142]Farley C T,Glasheen J,McMahon T A.Running Springs:Speed and Animal Size.Journal of Experimental Biology,1993(185):71-86.
[143]Nigg B M,Liu W.The effect of muscle stiffness and damping on simulated impact force peaks during running.Journal of Biomechanics,1999,32:849-856.
[144]Liu W,Nigg B M.A mechanical model to determine the influence of masses and mass distribution on the impact force during running.Journal of Biomechanics,2000,33:219-224.
[145]Ratcliffe R J,Holt K G.Low frequency shock absorption in human walking.Gait & Posture,1997,5:93-100.
[146]Butler R J,Crowell Ⅲ H P,Davis I M.Lower extremity stiffness:implications for performance and injury.Clinical Biomechanics,2003,18:511-517.
[147]Bosco C,Komi P V.Mechanical Characteristics and Fiber Composition of Human Leg Extensor Muscles.European Journal of Applied Physiology and Occupational Physiology,1979,(41):275-284.
[148]Rapoport S,Mizrahi J,Kimmel E,et al.Constant and Variable Stiffness and Damping of the Leg Joints in Human Hopping.Journal of Biomechanical Engineering,2003,8(125):507-514.