单腿跳跃机器人的动力学研究
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摘要
目前仿生机器人的研究主要集中在轮式(履带)机器人和爬行机器人上,与这两种机器人相比,腿式跳跃机器人能够跨越较大的障碍物,能够在比较复杂的地形环境中运动,而成为了近年来国内外仿生机器人领域研究的热点对象。
本文总结了单腿跳跃机器人的发展现状,提出了本课题的研究内容。
针对以往单腿跳跃机器人的设计局限性,提出一种新机构模型——仿袋鼠单腿跳跃机器人机构模型,并对其进行动力学建模和运动仿真,仿真结果验证了动力学模型的正确性,为机器人的运动控制和结构优化奠定了理论基础。
针对仿袋鼠单腿跳跃机器人支撑相是二阶非完整约束系统的这一特点,从非线性控制理论出发,以动力学模型为基础,推导出仿袋鼠单腿跳跃机器人在支撑相的平衡点线性化系统,并证明其在支撑相是线性可控的,为机器人的稳定控制提供了理论依据。
针对单腿跳跃机器人系统是一种非线性系统,而周期运动是非线性系统的一类重要解,探讨了搜索周期运动的方法及算法,并以一种简单的单腿跳跃机器人模型为例,采用所述方法及算法找到了其被动周期运动。
最后一部分针对现有的搜索周期运动方法的最大一个缺点:即其初始条件是手动修改,这种手动修改初始条件的方法具有盲目性,即经过无数次的尝试,才能找到所满意的初始条件,这样一来就非常费时,效率低下。因此,如果在确定初始条件前采用优化方法,找到合适的初始条件,那么就可以提高效率,避免了盲目性。
At present, bio-robots research mainly concentrated in the wheel (tracked) robots and crawling robots. Compared with these robots, legged hopping robots can leap relatively large obstacles and locomote on the more complex terrain, and they are the hotspot object of the bio-robots research fields in recent years.
In the paper, the research development of hopping robot is summarized, and the main issue of the paper is illustrated.
For the past one-legged hopping robots design limitations, a new mechanical model——kangaroo-like hopping robot with one leg is proposed, and its dynamics modeling and motion simulation were done, the results of the simulation verify the correctness of the dynamics model, which laid a theoretical basis for the robot motion control and structural optimization.
For the kangaroo-like hopping robot with one leg, the characteristics of its support phase is second-order nonholonomic constraint system, starting from the non-linear control theory, based on dynamics model, deduced the linear system of equilibrium point for the kangaroo-like hopping robot with one leg in the support phase, and proved that it is linearly controllability in the support phase, which provided a theoretical basis for the robot's stability control.
For a one-legged hopping robot system is a nonlinear system, and the periodic motion is an important solution of a class of nonlinear system, explored the searching methods and algorithms of the periodic motion, and a simple model of one-legged hopping robot for example, found its passive periodic motion by using the described methods and algorithms.
In the last part of the research, for the existing searching method of periodic motion, its biggest drawback is to manually modify the initial conditions, the manual modification of initial conditions had blindness, that is, after numerous attempts to find the satisfied initial conditions, so that it is very time-consuming and inefficient. Therefore, if prior to determin initial conditions we used an optimal way to find a suitable initial condition, it can improve efficiency, and avoid blindness.
本文总结了单腿跳跃机器人的发展现状,提出了本课题的研究内容。
针对以往单腿跳跃机器人的设计局限性,提出一种新机构模型——仿袋鼠单腿跳跃机器人机构模型,并对其进行动力学建模和运动仿真,仿真结果验证了动力学模型的正确性,为机器人的运动控制和结构优化奠定了理论基础。
针对仿袋鼠单腿跳跃机器人支撑相是二阶非完整约束系统的这一特点,从非线性控制理论出发,以动力学模型为基础,推导出仿袋鼠单腿跳跃机器人在支撑相的平衡点线性化系统,并证明其在支撑相是线性可控的,为机器人的稳定控制提供了理论依据。
针对单腿跳跃机器人系统是一种非线性系统,而周期运动是非线性系统的一类重要解,探讨了搜索周期运动的方法及算法,并以一种简单的单腿跳跃机器人模型为例,采用所述方法及算法找到了其被动周期运动。
最后一部分针对现有的搜索周期运动方法的最大一个缺点:即其初始条件是手动修改,这种手动修改初始条件的方法具有盲目性,即经过无数次的尝试,才能找到所满意的初始条件,这样一来就非常费时,效率低下。因此,如果在确定初始条件前采用优化方法,找到合适的初始条件,那么就可以提高效率,避免了盲目性。
At present, bio-robots research mainly concentrated in the wheel (tracked) robots and crawling robots. Compared with these robots, legged hopping robots can leap relatively large obstacles and locomote on the more complex terrain, and they are the hotspot object of the bio-robots research fields in recent years.
In the paper, the research development of hopping robot is summarized, and the main issue of the paper is illustrated.
For the past one-legged hopping robots design limitations, a new mechanical model——kangaroo-like hopping robot with one leg is proposed, and its dynamics modeling and motion simulation were done, the results of the simulation verify the correctness of the dynamics model, which laid a theoretical basis for the robot motion control and structural optimization.
For the kangaroo-like hopping robot with one leg, the characteristics of its support phase is second-order nonholonomic constraint system, starting from the non-linear control theory, based on dynamics model, deduced the linear system of equilibrium point for the kangaroo-like hopping robot with one leg in the support phase, and proved that it is linearly controllability in the support phase, which provided a theoretical basis for the robot's stability control.
For a one-legged hopping robot system is a nonlinear system, and the periodic motion is an important solution of a class of nonlinear system, explored the searching methods and algorithms of the periodic motion, and a simple model of one-legged hopping robot for example, found its passive periodic motion by using the described methods and algorithms.
In the last part of the research, for the existing searching method of periodic motion, its biggest drawback is to manually modify the initial conditions, the manual modification of initial conditions had blindness, that is, after numerous attempts to find the satisfied initial conditions, so that it is very time-consuming and inefficient. Therefore, if prior to determin initial conditions we used an optimal way to find a suitable initial condition, it can improve efficiency, and avoid blindness.
引文
[1]Marc H.Raibert.Hopping in Legged Systems-Modeling and Simulation for the Two-Dimensional One-Legged Case.IEEE Transactions on Systems,man and Sybernetics,1984,SMC-14(3):451-463.
[2]Marc H.Raibert,and Francis C.Wimberly.Tabular Control of Balance in a Dynamic Legged System.IEEE Transactions on systems,man and Cybernetics,1984,SMC-14(2):334-339.
[3]Marc H.Raibert,H.Benjamin Brown,Jr.,Michael Chepponis.Experiments in Balance with a 3D One-legged Hopping Machine.The International Journal of Robotics and Research,1984,3(2):75-92.
[4]Jessica K.Hodgins and Marc H.Raibert.Adjusting Step Length for Rough Terrian Locomotion.IEEE Transactions on Robotics and Automation,1991,7(3):289-298.
[5]Hamidreza Dokht Taghirad.Analysis,Design and Control of Hopping Robot.Department of Mechanical Engineering McGill University,Montreal Canada,1993.
[6]Garth Zeglin.The Bow Leg Hopping Robot.The Robotics Institute Carnegie-Mellon University,Pittsburgh,1999.
[7]Garth J Zeglin.Uniroo:A One Legged Dynmic Hopping Robot [D].Bachelor Dissertation,Massachusetts Institute of Technoloy,1991.
[8]S.H.Hyon and T.Mita.Development of a Biologically Inspired Hopping Robot-Kenken [J].In Proceedings of the IEEE International Conference on Robotics and Automtion,2002(4),3984-3991.
[9]Raibert M H.Legged Robots that Balance [M].The MIT Press,1986,Cambridge,MA.
[10]Koditschek D E,B thler M.Analysis of a Simplified Hopping Robot [J].International Journal of Robotics Reseach,1991,10(6):587-605.
[11]Lapshin V V.Motion Control of a Legged Machine in the Supportless Phase of Hopping [J].International Journal of Robotics Reseach,1991,10(4):327-337.
[12]Li Z,Montgomery R.Dynamics and Optimal Control of a Legged Robot in Flight Phase [A].IEEE Conference on Robotics and Automation [C],Cincinnati,OH,May 1990,1816-1821.
[13]M'Closkey R T,Burdick JW.Periodic Motion of a Hopping Robot with Vertical and Forward Motion [J].International Journal of Robotics Reseach,1993,12(3):197-218.
[14]Hodgins J,Raibert M H.Biped Gymnastic [A].4th International Symposium on Robotics Research [C],The MIT Press,1988,5-14,Cambridge,MA.
[15]Berkemeier M D,Fearing R S.Sliding and Hopping Gaits for the Underactuated Acrobat [J].IEEE Transaction on Robotics and Automation,1998,14(4):629-634.
[16]Luca A D,Oriolo G.Stabilization of the Acrobot via Iterative State Steering [J].IEEE Conference on Robotics and Automation,Leuven,Belgium,May 1998:3581-3587.
[17]Hauser J,Murray RM.Nonlinear Contorllers for Non-integrable Systems:the Acrobot Example [A].American Control Conference [C],1990:669-671.
[18]Saito F,Fukuda T,Arai F.Swing and Locomotion Control for a two-link Brachiation Robot [J].IEEE Control System Magazine,February 1994:5-12.
[19]Spong M W.The Swing up Control Problem for the Acrobat [J].IEEE Control System Magazine,February,1995,49-55.
[20]Matsuoka K.A Mechanical Model of Repetitive Hopping Movements [J].Biomechanisms,1979,(5):251-258.
[21]Playter R R,Raibert M H.Control of a Biped Somersault in 3D [J].International Symposium on Theory of Machines and Mechanisms,1992,(2):669-674.
[22]VakakisA F,Burdick J W,Caughey T K.An“interesting”Strange Attactor in the Dynamics of a Hopping Robot [J].The International Journal of Robotics Research,1991,10 (6), 606-618.
[23]Yamafuji K,Mitsuya Y.Development and Motion Control of the Robot Which Jumps And Moves About [J].Transaction of the Japan Society of Mechanical Engineers,1991,57:1616-1620 (in Japanese).
[24]M.W.Spong,P.Corke,and R.Lozano.Nonlinear Control of the Inertial Wheel Pendulum [J].Automatica,2001,37:1845-1851.
[25]H.Sira-Ramirez and S.K.Agrawal.Differentially flat systems.Marcel Dekker,Inc.,New York,2004.
[26]P.Gregorio,M.Ahmadi,and M.Buehler.Design,Control and Energetics of an Electrically Actuated Legged Robot.IEEE Trans.Systems,Man,and Cybernetics,vol.27,no.4,pp.626-634,Aug.1997.
[27]Arai H,Tanie K,Shiroma N.Time-scaling Control of an Underactuated Manipulator [J].Journal of Robotics Systems,1998,15 (9):525-536.
[28]S.H.Hyon,T.Emura,T.Ueta.Detection and Stabilization of Hybrid Periodic Orbits of Passive Running Robots [J].In Proc.Mechatronics and Robotics,vol.4,2004,pp.1309-1314.
[29]Koditschek D and Buehler M.Analysis of a Simplified Hopping Robot.In the int.J.of Robotics Research,vol.10,no.6,pp.587-605,1991.
[30]Chigliazza R M,Altendorfer R,Holmes P and so on.Passively Stable Conservative Locomotion.Submitted to SIAM J.of Applied Dynamical Systems.
[31]Li Z and He J.An Energy Perturbation Approach to Limit Cycle Analysis in Legged Locomotion Systems.Proc.of the IEEE Int.Conf.on Robotics and Automation,pp.1989-1994,1990.
[32]M'Closkey R T,Burdick J,W.and Vakakis A F.On the Periodic Motions of Simple Hopping Robots.Proc.of the IEEE Int.Conf.on Systems,Man and Cybernetics,pp.771-777,1990.
[33]Ostrowski J P and Burdick J W.Designing Feedback Algorithms for Controlling the Periodic Motion of Legged Robots.Proc.of the IEEE Int.Conf.on Robotics and Automation,pp.260-266,1993.
[34]Vakakis A F and Burdick J W.Chaotic Motions in the Dynamics of a Hopping Robot.Proc.of the IEEE Int.Conf.on Robotics and Automation,pp.1464-1469,1990.
[35]Guckenheimer,J.and Holmes P.Nonlinear Oscillations.Dynamical Systems and Bifurcations of Vector Fields,Springer-Verlag,1983.
[36]Khalil H K.Nonlinear Systems.Prentice Hall,2~(nd) ed.,1996.
[37]Kuznetsov Y.Elements of Applied Bifurcation Theory.Springer-Verlag,2~(nd) ed.,1998.
[38]R.W.Brockett.Asymptotic Stability and Feedback Stabilization.In Differential Geometric Control Theory,(R.W.Brockett,R.S.Millman and H.J.Sussmann,Eds.Boston,MA:Brikhauser),pp.181-191,1983.
[39]Arun D.Mahindrakar,Ravi N.Banavar.Controllability Properties of a Planar 3R Underactuated Manipulator.Proceedings of the 2002 IEEE International Conference on Control Applications,Vol.1,pp.489-494,September 2002.
[40]Jonathan Karl Holm.Control of passive Dynamic Robots Using Artificial Potential Energy Fields.In Electrical Engineering in the Graduate College of the University of Illinois at Urbana-Champaign,Master Degree Thesis,2005.
[41]Marc H.Raibert,H.Benjamin Brown,etc.Dynamically Stable Legged Locomotion. The Robotics Institute CarnegieMellon University,Second Report to DARPA,January 1983.
[42]Marc H.Raibert,H.Benjamin Brown,etc.Dynamically Stable Legged Locomotion.The Robotics Institute and Department Computer Science CarnegieMellon University,Progress Report,December 1983.
[43]熊有伦,丁汉,刘恩沧．机器人学．机械工业出版社．1978．
[44]T.Parker,L.Chua.Practical Numerical Algorithms for Chaotic Systems.Springer,1989.
[45]T.Kousaka,M.Matsumoto,and so on.Bifurcation Analysis in Hybrid Nonlinear Dynamical Systems.The 2001 IEEE International Symposium on Circuits and Systems,Vol.3,pp.281-284,May,2001.
[46]T.Kousaka,T.Ueta and H.Kawakarni.Bifurcation of Switched Nonlinear Dynamical Systems.IEEE Trans.Circuits and Systems Ⅱ,vol.CAS-Ⅱ,no.7,pp.787-885,1999.
[47]Sang-Ho Hyon,Takashi Emura.Energy-preserving Control of a Passive One-legged Running Robot.Advanced Robotics,Vol.18,No.4,pp.357-381,2004.
[48]Mojtaba Ahrnadi,Martin Buehler.Stable Control of a Simulated One-legged Running Robot with Hip and Leg Compliance.IEEE Transactions on Robotics and Automation,Vol.13,No.1,pp.96-104,February 1997.
[49]Evangelos Papadopoulos,Nicholas Cherouvin.On Increasing Energy Autonomy for a One-Legged Hopping Robot.Proceedings of the 2004 IEEE Internamtional Conferencd on Robotics & Automation,pp.4645-4650,April 2004.
[50]Steven Collins,Andy Ruina,Russ Tedrake,and Martijin Wisse.Efficient Bipedal Robots Based on Passive Dynamic walkers.Science,2005,307:1082-1085.
[51]Clay M.Thompson,Marc H.Raibert.Passive Dynamic Running.International Symposium of Experimental Robotics,Vol.1,pp.74-83,1989.
[52]Nakano E,Okubo H,Kimura H.Landing of a jumping machine[J].Journal of the Robotics Society of Japan,1991,9(2):169-176(in Japanese)
[53]Okubo H,Nakano E,Kimura H.Design of jumping machine using springs and actuators[J].Journal of the Robotics Society of Japan,1992,10(7):948-954(In Japanese)
[54]Tani K,Shirai M,A jumping machine using the energy stored in springs[J].International Symposium on Theory of Machines and Mechanism,1992(1):265-270
[55]C.Henry Edwards,David E.Penney.常微分方程基础(英文版*原书第5版).机械工业出版社.2006.
[56]刘秉正,彭建华.非线性动力学.高等教育出版社.2004.
[2]Marc H.Raibert,and Francis C.Wimberly.Tabular Control of Balance in a Dynamic Legged System.IEEE Transactions on systems,man and Cybernetics,1984,SMC-14(2):334-339.
[3]Marc H.Raibert,H.Benjamin Brown,Jr.,Michael Chepponis.Experiments in Balance with a 3D One-legged Hopping Machine.The International Journal of Robotics and Research,1984,3(2):75-92.
[4]Jessica K.Hodgins and Marc H.Raibert.Adjusting Step Length for Rough Terrian Locomotion.IEEE Transactions on Robotics and Automation,1991,7(3):289-298.
[5]Hamidreza Dokht Taghirad.Analysis,Design and Control of Hopping Robot.Department of Mechanical Engineering McGill University,Montreal Canada,1993.
[6]Garth Zeglin.The Bow Leg Hopping Robot.The Robotics Institute Carnegie-Mellon University,Pittsburgh,1999.
[7]Garth J Zeglin.Uniroo:A One Legged Dynmic Hopping Robot [D].Bachelor Dissertation,Massachusetts Institute of Technoloy,1991.
[8]S.H.Hyon and T.Mita.Development of a Biologically Inspired Hopping Robot-Kenken [J].In Proceedings of the IEEE International Conference on Robotics and Automtion,2002(4),3984-3991.
[9]Raibert M H.Legged Robots that Balance [M].The MIT Press,1986,Cambridge,MA.
[10]Koditschek D E,B thler M.Analysis of a Simplified Hopping Robot [J].International Journal of Robotics Reseach,1991,10(6):587-605.
[11]Lapshin V V.Motion Control of a Legged Machine in the Supportless Phase of Hopping [J].International Journal of Robotics Reseach,1991,10(4):327-337.
[12]Li Z,Montgomery R.Dynamics and Optimal Control of a Legged Robot in Flight Phase [A].IEEE Conference on Robotics and Automation [C],Cincinnati,OH,May 1990,1816-1821.
[13]M'Closkey R T,Burdick JW.Periodic Motion of a Hopping Robot with Vertical and Forward Motion [J].International Journal of Robotics Reseach,1993,12(3):197-218.
[14]Hodgins J,Raibert M H.Biped Gymnastic [A].4th International Symposium on Robotics Research [C],The MIT Press,1988,5-14,Cambridge,MA.
[15]Berkemeier M D,Fearing R S.Sliding and Hopping Gaits for the Underactuated Acrobat [J].IEEE Transaction on Robotics and Automation,1998,14(4):629-634.
[16]Luca A D,Oriolo G.Stabilization of the Acrobot via Iterative State Steering [J].IEEE Conference on Robotics and Automation,Leuven,Belgium,May 1998:3581-3587.
[17]Hauser J,Murray RM.Nonlinear Contorllers for Non-integrable Systems:the Acrobot Example [A].American Control Conference [C],1990:669-671.
[18]Saito F,Fukuda T,Arai F.Swing and Locomotion Control for a two-link Brachiation Robot [J].IEEE Control System Magazine,February 1994:5-12.
[19]Spong M W.The Swing up Control Problem for the Acrobat [J].IEEE Control System Magazine,February,1995,49-55.
[20]Matsuoka K.A Mechanical Model of Repetitive Hopping Movements [J].Biomechanisms,1979,(5):251-258.
[21]Playter R R,Raibert M H.Control of a Biped Somersault in 3D [J].International Symposium on Theory of Machines and Mechanisms,1992,(2):669-674.
[22]VakakisA F,Burdick J W,Caughey T K.An“interesting”Strange Attactor in the Dynamics of a Hopping Robot [J].The International Journal of Robotics Research,1991,10 (6), 606-618.
[23]Yamafuji K,Mitsuya Y.Development and Motion Control of the Robot Which Jumps And Moves About [J].Transaction of the Japan Society of Mechanical Engineers,1991,57:1616-1620 (in Japanese).
[24]M.W.Spong,P.Corke,and R.Lozano.Nonlinear Control of the Inertial Wheel Pendulum [J].Automatica,2001,37:1845-1851.
[25]H.Sira-Ramirez and S.K.Agrawal.Differentially flat systems.Marcel Dekker,Inc.,New York,2004.
[26]P.Gregorio,M.Ahmadi,and M.Buehler.Design,Control and Energetics of an Electrically Actuated Legged Robot.IEEE Trans.Systems,Man,and Cybernetics,vol.27,no.4,pp.626-634,Aug.1997.
[27]Arai H,Tanie K,Shiroma N.Time-scaling Control of an Underactuated Manipulator [J].Journal of Robotics Systems,1998,15 (9):525-536.
[28]S.H.Hyon,T.Emura,T.Ueta.Detection and Stabilization of Hybrid Periodic Orbits of Passive Running Robots [J].In Proc.Mechatronics and Robotics,vol.4,2004,pp.1309-1314.
[29]Koditschek D and Buehler M.Analysis of a Simplified Hopping Robot.In the int.J.of Robotics Research,vol.10,no.6,pp.587-605,1991.
[30]Chigliazza R M,Altendorfer R,Holmes P and so on.Passively Stable Conservative Locomotion.Submitted to SIAM J.of Applied Dynamical Systems.
[31]Li Z and He J.An Energy Perturbation Approach to Limit Cycle Analysis in Legged Locomotion Systems.Proc.of the IEEE Int.Conf.on Robotics and Automation,pp.1989-1994,1990.
[32]M'Closkey R T,Burdick J,W.and Vakakis A F.On the Periodic Motions of Simple Hopping Robots.Proc.of the IEEE Int.Conf.on Systems,Man and Cybernetics,pp.771-777,1990.
[33]Ostrowski J P and Burdick J W.Designing Feedback Algorithms for Controlling the Periodic Motion of Legged Robots.Proc.of the IEEE Int.Conf.on Robotics and Automation,pp.260-266,1993.
[34]Vakakis A F and Burdick J W.Chaotic Motions in the Dynamics of a Hopping Robot.Proc.of the IEEE Int.Conf.on Robotics and Automation,pp.1464-1469,1990.
[35]Guckenheimer,J.and Holmes P.Nonlinear Oscillations.Dynamical Systems and Bifurcations of Vector Fields,Springer-Verlag,1983.
[36]Khalil H K.Nonlinear Systems.Prentice Hall,2~(nd) ed.,1996.
[37]Kuznetsov Y.Elements of Applied Bifurcation Theory.Springer-Verlag,2~(nd) ed.,1998.
[38]R.W.Brockett.Asymptotic Stability and Feedback Stabilization.In Differential Geometric Control Theory,(R.W.Brockett,R.S.Millman and H.J.Sussmann,Eds.Boston,MA:Brikhauser),pp.181-191,1983.
[39]Arun D.Mahindrakar,Ravi N.Banavar.Controllability Properties of a Planar 3R Underactuated Manipulator.Proceedings of the 2002 IEEE International Conference on Control Applications,Vol.1,pp.489-494,September 2002.
[40]Jonathan Karl Holm.Control of passive Dynamic Robots Using Artificial Potential Energy Fields.In Electrical Engineering in the Graduate College of the University of Illinois at Urbana-Champaign,Master Degree Thesis,2005.
[41]Marc H.Raibert,H.Benjamin Brown,etc.Dynamically Stable Legged Locomotion. The Robotics Institute CarnegieMellon University,Second Report to DARPA,January 1983.
[42]Marc H.Raibert,H.Benjamin Brown,etc.Dynamically Stable Legged Locomotion.The Robotics Institute and Department Computer Science CarnegieMellon University,Progress Report,December 1983.
[43]熊有伦,丁汉,刘恩沧．机器人学．机械工业出版社．1978．
[44]T.Parker,L.Chua.Practical Numerical Algorithms for Chaotic Systems.Springer,1989.
[45]T.Kousaka,M.Matsumoto,and so on.Bifurcation Analysis in Hybrid Nonlinear Dynamical Systems.The 2001 IEEE International Symposium on Circuits and Systems,Vol.3,pp.281-284,May,2001.
[46]T.Kousaka,T.Ueta and H.Kawakarni.Bifurcation of Switched Nonlinear Dynamical Systems.IEEE Trans.Circuits and Systems Ⅱ,vol.CAS-Ⅱ,no.7,pp.787-885,1999.
[47]Sang-Ho Hyon,Takashi Emura.Energy-preserving Control of a Passive One-legged Running Robot.Advanced Robotics,Vol.18,No.4,pp.357-381,2004.
[48]Mojtaba Ahrnadi,Martin Buehler.Stable Control of a Simulated One-legged Running Robot with Hip and Leg Compliance.IEEE Transactions on Robotics and Automation,Vol.13,No.1,pp.96-104,February 1997.
[49]Evangelos Papadopoulos,Nicholas Cherouvin.On Increasing Energy Autonomy for a One-Legged Hopping Robot.Proceedings of the 2004 IEEE Internamtional Conferencd on Robotics & Automation,pp.4645-4650,April 2004.
[50]Steven Collins,Andy Ruina,Russ Tedrake,and Martijin Wisse.Efficient Bipedal Robots Based on Passive Dynamic walkers.Science,2005,307:1082-1085.
[51]Clay M.Thompson,Marc H.Raibert.Passive Dynamic Running.International Symposium of Experimental Robotics,Vol.1,pp.74-83,1989.
[52]Nakano E,Okubo H,Kimura H.Landing of a jumping machine[J].Journal of the Robotics Society of Japan,1991,9(2):169-176(in Japanese)
[53]Okubo H,Nakano E,Kimura H.Design of jumping machine using springs and actuators[J].Journal of the Robotics Society of Japan,1992,10(7):948-954(In Japanese)
[54]Tani K,Shirai M,A jumping machine using the energy stored in springs[J].International Symposium on Theory of Machines and Mechanism,1992(1):265-270
[55]C.Henry Edwards,David E.Penney.常微分方程基础(英文版*原书第5版).机械工业出版社.2006.
[56]刘秉正,彭建华.非线性动力学.高等教育出版社.2004.