准被动双足步行机器人样机研制及运动特性研究
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摘要
被动动力步行的设计思想要求其充分利用被动机器人自身的自然动力学特性,因此步行过程步态自然、能耗低。纯被动步行机器人仅依靠重力的作用就可以在斜坡上稳定行走,而准被动步行机器人能够依靠一定的控制在平地上稳定行走,步态同样自然。
目前对准被动双足步行机器人的研究主要集中在控制方法对步行稳定性的影响及结构参数优化上,对步态的分析较少。为了更深入的研究准被动双足步行机器人的步态特征,本文研制了带躯体的准被动双足步行机器人样机,设计了能够调整刚度的串联弹性驱动单元,解决了膝关节的碰撞使稳定性降低的问题。深入研究并提出了准被动双足步行机器人的位置控制方法及电机目标输出角度的算法,解决了柔性驱动单元弹性元件伸长量计算不准确的问题。最终通过在ADAMS中对Pro/E中建立的样机仿真及实体样机试验来分析准被动双足步行机器人在平地上行走的步态特征,同时验证了结构参数及控制方法的正确性。
首先,本文建立了被动双足步行机器人的动力学模型,对模型进行了结构参数优化。根据参数优化结果在ADAMS中建立了带躯体带膝关节的被动机器人仿真模型,根据仿真结果获得了被动机器人的在平地上行走的步态特征。
其次,详细分析了柔性驱动单元的特点,指出了柔性驱动单元的刚度及平衡位置对准被动双足步行机器人步态特征的影响,并设计出了能够调整刚度的柔性驱动单元。在此基础上建立了带躯体及膝关节的准被动双足步行机器人的样机。
然后,建立了摆动腿的转矩输入模型,并根据被动机器人动力学特性计算出了模型的控制参数。提出了准被动双足步行机器人的位置控制方法,结合驱动器特点提出了控制方法的具体实现方法,同时提出了计算电机目标输出角度的算法,解决了柔性驱动单元弹性元件伸长量计算不准确的问题。
最后,通过Pro/E样机模型在ADAMS中仿真获得了准被动双足步行机器人在平地上行走的步态特征。最终通过实体样机实验,进一步分析了准被动双足步行机器人的步态特征,并验证了被动机器人结构参数、柔性驱动单元刚度及控制参数对步态特征的影响。
The design of passive dynamic walking requires us to fully use of the natural dynamics of passive walkers, so the gait is natural and the energy consumption is low during walking. The pure-passive walking robots can walk on the slope stably only relying on gravity, while the quasi-passive walkers can walk on level ground naturally and stably under certain controls.
Current research on passive biped robot mainly focuses on control strategies for walking stability and structure optimization, but less on gait character analysis. In order to get a better understanding of the quasi-passive biped robot walking gait characters, the paper developed a mechanical prototype of quasi-passive biped robot, and designed a series elastic actuator that can adjust the stiffness of the spring, and resolved the collision problem of the knee which can lead to instability of walking. On this basis, the paper made an intensive study of the control system of the quasi-passive biped robot and put forward the position control law of the DC motor. The motor target output angle algorithm was also put forward at the same time to solve the problem which can not calculate the extension of the spring correctly. At the end the paper used ADAMS to create a simulation of the prototype that was built in Pro/E and analyzed the gait characteristics of quasi-passive biped robot walking on level ground, and proved the correctness of the structural parameters and control strategies by walking experiment.
First, the paper built a dynamic model of passive biped walking robot, and optimized the structural parameters of the model, and made a full analysis of how the different structural parameters can influence the stability of the passive walking robot. On the basis of optimization results, the paper built a simulation model with knee joint and upper body in ADAMS, and fully discussed the gait characters of this model.
Secondly, the paper gave a detailed analysis of the characteristics of the flexible drive, and indicated that the stiffness and equilibrium position of the elastic actuator can influence the gait characteristics of passive walkers, and designed a new serials elastic actuator that can adjust the stiffness. Based on this, the paper then established the mechanical prototype of quasi-passive biped robot with upper body and knees.
Then, the paper put forward the torque input model of the swing leg of quasi-passive biped robot and calculated out of the control parameters of this torque input model. Proposed a position control strategy of the quasi-passive biped walking robot, and proposed the implementation method according to the drive DC motor actuator’s property. In order to solve the problem which can not calculate the extension of the spring correctly, the paper put forward the target output angle algorithm.
Finally, the paper used ADAMS to create a simulation of the prototype that was built in Pro/E and analyzed the gait characteristics of quasi-passive biped robot walking on level ground. Made a further analysis of the gait characteristics of quasi-passive biped robot by real experiment, and proved the influence of the structural parameters, the stiffness of elastic actuator and the control strategies on walking gait characteristics.
目前对准被动双足步行机器人的研究主要集中在控制方法对步行稳定性的影响及结构参数优化上,对步态的分析较少。为了更深入的研究准被动双足步行机器人的步态特征,本文研制了带躯体的准被动双足步行机器人样机,设计了能够调整刚度的串联弹性驱动单元,解决了膝关节的碰撞使稳定性降低的问题。深入研究并提出了准被动双足步行机器人的位置控制方法及电机目标输出角度的算法,解决了柔性驱动单元弹性元件伸长量计算不准确的问题。最终通过在ADAMS中对Pro/E中建立的样机仿真及实体样机试验来分析准被动双足步行机器人在平地上行走的步态特征,同时验证了结构参数及控制方法的正确性。
首先,本文建立了被动双足步行机器人的动力学模型,对模型进行了结构参数优化。根据参数优化结果在ADAMS中建立了带躯体带膝关节的被动机器人仿真模型,根据仿真结果获得了被动机器人的在平地上行走的步态特征。
其次,详细分析了柔性驱动单元的特点,指出了柔性驱动单元的刚度及平衡位置对准被动双足步行机器人步态特征的影响,并设计出了能够调整刚度的柔性驱动单元。在此基础上建立了带躯体及膝关节的准被动双足步行机器人的样机。
然后,建立了摆动腿的转矩输入模型,并根据被动机器人动力学特性计算出了模型的控制参数。提出了准被动双足步行机器人的位置控制方法,结合驱动器特点提出了控制方法的具体实现方法,同时提出了计算电机目标输出角度的算法,解决了柔性驱动单元弹性元件伸长量计算不准确的问题。
最后,通过Pro/E样机模型在ADAMS中仿真获得了准被动双足步行机器人在平地上行走的步态特征。最终通过实体样机实验,进一步分析了准被动双足步行机器人的步态特征,并验证了被动机器人结构参数、柔性驱动单元刚度及控制参数对步态特征的影响。
The design of passive dynamic walking requires us to fully use of the natural dynamics of passive walkers, so the gait is natural and the energy consumption is low during walking. The pure-passive walking robots can walk on the slope stably only relying on gravity, while the quasi-passive walkers can walk on level ground naturally and stably under certain controls.
Current research on passive biped robot mainly focuses on control strategies for walking stability and structure optimization, but less on gait character analysis. In order to get a better understanding of the quasi-passive biped robot walking gait characters, the paper developed a mechanical prototype of quasi-passive biped robot, and designed a series elastic actuator that can adjust the stiffness of the spring, and resolved the collision problem of the knee which can lead to instability of walking. On this basis, the paper made an intensive study of the control system of the quasi-passive biped robot and put forward the position control law of the DC motor. The motor target output angle algorithm was also put forward at the same time to solve the problem which can not calculate the extension of the spring correctly. At the end the paper used ADAMS to create a simulation of the prototype that was built in Pro/E and analyzed the gait characteristics of quasi-passive biped robot walking on level ground, and proved the correctness of the structural parameters and control strategies by walking experiment.
First, the paper built a dynamic model of passive biped walking robot, and optimized the structural parameters of the model, and made a full analysis of how the different structural parameters can influence the stability of the passive walking robot. On the basis of optimization results, the paper built a simulation model with knee joint and upper body in ADAMS, and fully discussed the gait characters of this model.
Secondly, the paper gave a detailed analysis of the characteristics of the flexible drive, and indicated that the stiffness and equilibrium position of the elastic actuator can influence the gait characteristics of passive walkers, and designed a new serials elastic actuator that can adjust the stiffness. Based on this, the paper then established the mechanical prototype of quasi-passive biped robot with upper body and knees.
Then, the paper put forward the torque input model of the swing leg of quasi-passive biped robot and calculated out of the control parameters of this torque input model. Proposed a position control strategy of the quasi-passive biped walking robot, and proposed the implementation method according to the drive DC motor actuator’s property. In order to solve the problem which can not calculate the extension of the spring correctly, the paper put forward the target output angle algorithm.
Finally, the paper used ADAMS to create a simulation of the prototype that was built in Pro/E and analyzed the gait characteristics of quasi-passive biped robot walking on level ground. Made a further analysis of the gait characteristics of quasi-passive biped robot by real experiment, and proved the influence of the structural parameters, the stiffness of elastic actuator and the control strategies on walking gait characteristics.
引文
[1] Y.Sakagami, R.Watanabe, C.Aoyama, S.Matsunaga, N.Higaki, K.Fujimura. The Intelligent Asimo: System Overview and Integration. 2002IEEE/RSJ International Conference on Intelligent Robots and Systems, Lausanne, Switzerland, 2002:2478~2483.
[2] T.McGeer. Passive Dynamic Walking.International Journal of Robotics Research. 1990,9(2):62~82.
[3] M.Vukobratovic, B.Borovac. Zero-Moment Point-Thirty Five Years of Its Life. Int J Hum Robot. 2004,1(1):157~173.
[4] S. Mochon, T. A. McMahon. Ballistic walking [J] .Biomechan, 1980, 13: 49~57.
[5] McGeer T. Passive walking with knees [A]. Proceedings of the IEEE Conference on Robotics and Automation [C]. Piscataway, NJ, USA: IEEE, 1990. 1640~1645.
[6] Kalin Trifonov , Shuji Hashimoto. Active Knee-lock Release for Passive-Dynamic Walking Machines. Proceedings of the 2007 IEEE International Conference onRobotics and Biomimetics 2007:135~141.
[7] S.H.Collins, M.Wisse, A.Ruina. A Three-Dimensional Passive-Dynamic Walking Robot with Two Legs and Knees. International Journal of Robotics Research. 2001,20(7):607~615.
[8] M.Wisse, D.G.E.Hobbelen. Ankle springs instead of arc-shaped feet for passive dynamic walkersIEEE-RAS International Conference on Humanoid Robots. 2006 6th :110– 116.
[9] S.H.Collins, A.Ruina. A Bipedal Walking Robot with Efficient and Human-Like Gait. 2005 IEEE International Conference on Robotics and Automation. Barcelona, Spain, 2005: 1983~1988.
[10] Supporting Online Material: [EB/OL]. http :/ / www. Sciencedaily. Com / releases /2008/04/080411220112.htm, Cornell Robot Sets A Record For Distance Walking.
[11] WisseM, van Frankenhuyzen J. Design and construction of MIKE: a 2D autonomous biped based on passive dynamic walking [A]. Proceedings of the International Symposium on Adaptive Motions of Animals and Machines [C].Tokyo, Japan: Sp ringer- Verlag, 2003. 143~154.
[12] Martijn Wisse, L. Schwab, Richard Q. van der Linde. How to Keep From Falling Forward: Elementary Swing Leg Action for Passive Dynamic Walkers. IEEE TRANSACTIONS ON ROBOTICS. VOL. 21, NO. 3, JUNE 2005.
[13] Wisse M, Schwab A L, van der Helm F C T. Passive dynamic walking model with upper upper body J]. Robotica, 2004, 22 (6): 681~688.
[14] Wisse M. Three additions to passive dynamic walking: actuation, an upper body, and 3D stability [A]. Proceedings of the IEEE International Conference on Humanoid Robots [C]. New York, NJ, USA: IEEE, 2004. 113 - 132.
[15] Schuitema E, Hobbelen D G E, Jonker P P, et al. Using a controller based on reinforcement learning for a passive dynamic walking robot[A]. Proceedings of the IEEE /RAS International Conference on Humanoid Robots [C] . New York , USA: IEEE, 2005. 232~237.
[16] Daan Hobbelen, Tomas de Boer and Martijn Wisse. System overview of bipedal robots Flame and TUlip: tailor-made for Limit Cycle Walking. IEEE/RSJ International Conference on Intelligent Robots and Systems. Acropolis Convention Center Nice, France: IEEE,2008,234-239.
[17] Erik Schuitema, Martijn Wisse, Thijs Ramakers and Pieter Jonker. The Design of LEO: a 2D Bipedal Walking Robot for Online Autonomous Reinforcement Learning. The 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems October 18-22, 2010, T aipei, Taiwan.
[18] R. Tedrake, T.W. Zhang, and H. S. Seung. Learning to walk in 20 minutes. In Proceedingsof the Fourteenth Yale Workshop on Adaptive and Learning Systems, 2005.
[19] A.D.Kuo,J.M.Donelan,A.Ruina.Energetic consequences of walking like an inverted pendulum:Step-to-step transitions. Exercise and Sport Sciences Reviews, 2005,33(2):88~97.
[20] P.Zhang,Y.Tian,Z.Liu.Bisection Method for Evaluation of Attraction Region of Passive Dynamic Walking.ICARA 2009-Proceedings of the 4th International Conference on Autonomous Robots and Agents,Wellington,New zealand, 2009:692~697.
[21]刘振泽.欠驱动步行机器人运动学机理与控制方法研究.吉林大学博士论文. 2007:1~142.
[22]倪修华.双足被动步行机器人性能分析及一种动力输入方法研究.哈尔滨工业大学博士论文.2010:1~95.
[23]毛勇.半被动双足机器人的设计与再励学习控制.清华大学博士论文.2007.
[24]黄岩,谢广明,杨晓华,王启宁,王龙.半被动双足机器人动态行走的位姿估算.北京大学学报(自然科学版).2009,第四期,第45卷.565-571.
[25] M.W.Spong,G.Bhatia.Further results on control of the compass gait biped[C]. Proceedings of IEEE International Conference on Intelligent Robots and Systems,Las Vegas,Nevada,2003:1933-1938.
[26] Endo G, Morimoto J, Nakanishi J, et al. An empirical exploration of a neural oscillator for biped locomotion control.Proceedings of IEEE International Conference on Robotics and Automation, 2004. 3036–3042.
[27]朱因远.非线性振动和运动稳定性.山西:西安交通大学出版社,1992.
[28] M. Garcia, A. Chatterjee, A. Ruina, and M. J. Coleman. The simplest walking model: Stability, complexity, and scaling. ASME J. Biomech. Eng., vol. 120, no. 2, pp. 281–288, Apr. 1998.
[29]刘延柱.高等动力学.高等教育出版社,2000.
[30]文成秀,姚玉玺,闻邦椿.动力系统的点映射—胞映射综合法.振动工程学报.1997,(10):414~417.
[31] BIRKHOFF G.D. Surface transformations and their dyamical applications [J]. Acta Math,1920,43:1~199.
[32] A.L.Schwab,M.Wisse.Basin of Attraction of the Simplest Walking Model. Proceedings of the ASME Design Engineering Technical Conference,Pittsburgh, PA,United States,2001:531~539.
[33] Daan G. E. Hobbelen and Martijn Wisse. Swing-Leg Retraction for Limit Cycle WalkersImproves Disturbance Rejection. IEEE TRANSACTIONS ON ROBOTICS, VOL. 24, NO. 2, APRIL 2008.
[34] Lim H,Setiawan S A, Takanishi A. Balance and impedance control for biped humanoid robot locomotion.Proceedings of IEEEE/RSJ International Conference on Intelligent Robots and Systems, 2001. 494–499.
[35] att,G.A, Williamson, M.M. Series elastic actuators ntelligent Robots and Systems 95. 'Human Robot Interaction and Cooperative Robots', Proceedings. 1995 IEEE/RSJ International Conference on.vol.1399– 406. 1995.Pittsburgh, PA , USA.
[36] Van Ham R, Vanderborght B, Verrelst B, et al. MACCEPA: the mechanically adjustable compliance and controllable equilibrium position actuator used in the’controlled passive walking biped Veronica. Proceedings of International Symposium on Measurement and Control in Robotics, 2005 Symposium on Measurement and Control in Robotics,2005.
[37] Guihard M,Gorce P. Biorobotic foot model applied to BIPMAN robot.Proceedings of IEEE International Conference on Systems, Man and Cybernetics, 2004.6491–6498.
[38] Verrelst B, Van Ham R, Vanderborght B, et al. The pneumatic biped‖lucy‖actuated with pleated pneumatic artificial muscles.Autonomous Robots, 2005, 18(2): 201–213.
[39] Chou C P, Hannaford B. Measurement and modeling of McKibben pneumatic artificial muscle. IEEE Transaction on Robotics and Automation,1996,12(1):90–103.
[40] M.Wisse, D. G. E.Hobbelen, Schwab, A. L. Adding an Upper Body to Passive Dynamic Walking Robots by Means of a Bisecting Hip Mechanism. IEEE Transactions on Robotics. 2007.vol1:112~123.
[41]程勒、程燕平.理论力学第六版.高等教育出版社.1996.
[42]成大先.机械设计手册第五版.化学工业出版社.2008.
[2] T.McGeer. Passive Dynamic Walking.International Journal of Robotics Research. 1990,9(2):62~82.
[3] M.Vukobratovic, B.Borovac. Zero-Moment Point-Thirty Five Years of Its Life. Int J Hum Robot. 2004,1(1):157~173.
[4] S. Mochon, T. A. McMahon. Ballistic walking [J] .Biomechan, 1980, 13: 49~57.
[5] McGeer T. Passive walking with knees [A]. Proceedings of the IEEE Conference on Robotics and Automation [C]. Piscataway, NJ, USA: IEEE, 1990. 1640~1645.
[6] Kalin Trifonov , Shuji Hashimoto. Active Knee-lock Release for Passive-Dynamic Walking Machines. Proceedings of the 2007 IEEE International Conference onRobotics and Biomimetics 2007:135~141.
[7] S.H.Collins, M.Wisse, A.Ruina. A Three-Dimensional Passive-Dynamic Walking Robot with Two Legs and Knees. International Journal of Robotics Research. 2001,20(7):607~615.
[8] M.Wisse, D.G.E.Hobbelen. Ankle springs instead of arc-shaped feet for passive dynamic walkersIEEE-RAS International Conference on Humanoid Robots. 2006 6th :110– 116.
[9] S.H.Collins, A.Ruina. A Bipedal Walking Robot with Efficient and Human-Like Gait. 2005 IEEE International Conference on Robotics and Automation. Barcelona, Spain, 2005: 1983~1988.
[10] Supporting Online Material: [EB/OL]. http :/ / www. Sciencedaily. Com / releases /2008/04/080411220112.htm, Cornell Robot Sets A Record For Distance Walking.
[11] WisseM, van Frankenhuyzen J. Design and construction of MIKE: a 2D autonomous biped based on passive dynamic walking [A]. Proceedings of the International Symposium on Adaptive Motions of Animals and Machines [C].Tokyo, Japan: Sp ringer- Verlag, 2003. 143~154.
[12] Martijn Wisse, L. Schwab, Richard Q. van der Linde. How to Keep From Falling Forward: Elementary Swing Leg Action for Passive Dynamic Walkers. IEEE TRANSACTIONS ON ROBOTICS. VOL. 21, NO. 3, JUNE 2005.
[13] Wisse M, Schwab A L, van der Helm F C T. Passive dynamic walking model with upper upper body J]. Robotica, 2004, 22 (6): 681~688.
[14] Wisse M. Three additions to passive dynamic walking: actuation, an upper body, and 3D stability [A]. Proceedings of the IEEE International Conference on Humanoid Robots [C]. New York, NJ, USA: IEEE, 2004. 113 - 132.
[15] Schuitema E, Hobbelen D G E, Jonker P P, et al. Using a controller based on reinforcement learning for a passive dynamic walking robot[A]. Proceedings of the IEEE /RAS International Conference on Humanoid Robots [C] . New York , USA: IEEE, 2005. 232~237.
[16] Daan Hobbelen, Tomas de Boer and Martijn Wisse. System overview of bipedal robots Flame and TUlip: tailor-made for Limit Cycle Walking. IEEE/RSJ International Conference on Intelligent Robots and Systems. Acropolis Convention Center Nice, France: IEEE,2008,234-239.
[17] Erik Schuitema, Martijn Wisse, Thijs Ramakers and Pieter Jonker. The Design of LEO: a 2D Bipedal Walking Robot for Online Autonomous Reinforcement Learning. The 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems October 18-22, 2010, T aipei, Taiwan.
[18] R. Tedrake, T.W. Zhang, and H. S. Seung. Learning to walk in 20 minutes. In Proceedingsof the Fourteenth Yale Workshop on Adaptive and Learning Systems, 2005.
[19] A.D.Kuo,J.M.Donelan,A.Ruina.Energetic consequences of walking like an inverted pendulum:Step-to-step transitions. Exercise and Sport Sciences Reviews, 2005,33(2):88~97.
[20] P.Zhang,Y.Tian,Z.Liu.Bisection Method for Evaluation of Attraction Region of Passive Dynamic Walking.ICARA 2009-Proceedings of the 4th International Conference on Autonomous Robots and Agents,Wellington,New zealand, 2009:692~697.
[21]刘振泽.欠驱动步行机器人运动学机理与控制方法研究.吉林大学博士论文. 2007:1~142.
[22]倪修华.双足被动步行机器人性能分析及一种动力输入方法研究.哈尔滨工业大学博士论文.2010:1~95.
[23]毛勇.半被动双足机器人的设计与再励学习控制.清华大学博士论文.2007.
[24]黄岩,谢广明,杨晓华,王启宁,王龙.半被动双足机器人动态行走的位姿估算.北京大学学报(自然科学版).2009,第四期,第45卷.565-571.
[25] M.W.Spong,G.Bhatia.Further results on control of the compass gait biped[C]. Proceedings of IEEE International Conference on Intelligent Robots and Systems,Las Vegas,Nevada,2003:1933-1938.
[26] Endo G, Morimoto J, Nakanishi J, et al. An empirical exploration of a neural oscillator for biped locomotion control.Proceedings of IEEE International Conference on Robotics and Automation, 2004. 3036–3042.
[27]朱因远.非线性振动和运动稳定性.山西:西安交通大学出版社,1992.
[28] M. Garcia, A. Chatterjee, A. Ruina, and M. J. Coleman. The simplest walking model: Stability, complexity, and scaling. ASME J. Biomech. Eng., vol. 120, no. 2, pp. 281–288, Apr. 1998.
[29]刘延柱.高等动力学.高等教育出版社,2000.
[30]文成秀,姚玉玺,闻邦椿.动力系统的点映射—胞映射综合法.振动工程学报.1997,(10):414~417.
[31] BIRKHOFF G.D. Surface transformations and their dyamical applications [J]. Acta Math,1920,43:1~199.
[32] A.L.Schwab,M.Wisse.Basin of Attraction of the Simplest Walking Model. Proceedings of the ASME Design Engineering Technical Conference,Pittsburgh, PA,United States,2001:531~539.
[33] Daan G. E. Hobbelen and Martijn Wisse. Swing-Leg Retraction for Limit Cycle WalkersImproves Disturbance Rejection. IEEE TRANSACTIONS ON ROBOTICS, VOL. 24, NO. 2, APRIL 2008.
[34] Lim H,Setiawan S A, Takanishi A. Balance and impedance control for biped humanoid robot locomotion.Proceedings of IEEEE/RSJ International Conference on Intelligent Robots and Systems, 2001. 494–499.
[35] att,G.A, Williamson, M.M. Series elastic actuators ntelligent Robots and Systems 95. 'Human Robot Interaction and Cooperative Robots', Proceedings. 1995 IEEE/RSJ International Conference on.vol.1399– 406. 1995.Pittsburgh, PA , USA.
[36] Van Ham R, Vanderborght B, Verrelst B, et al. MACCEPA: the mechanically adjustable compliance and controllable equilibrium position actuator used in the’controlled passive walking biped Veronica. Proceedings of International Symposium on Measurement and Control in Robotics, 2005 Symposium on Measurement and Control in Robotics,2005.
[37] Guihard M,Gorce P. Biorobotic foot model applied to BIPMAN robot.Proceedings of IEEE International Conference on Systems, Man and Cybernetics, 2004.6491–6498.
[38] Verrelst B, Van Ham R, Vanderborght B, et al. The pneumatic biped‖lucy‖actuated with pleated pneumatic artificial muscles.Autonomous Robots, 2005, 18(2): 201–213.
[39] Chou C P, Hannaford B. Measurement and modeling of McKibben pneumatic artificial muscle. IEEE Transaction on Robotics and Automation,1996,12(1):90–103.
[40] M.Wisse, D. G. E.Hobbelen, Schwab, A. L. Adding an Upper Body to Passive Dynamic Walking Robots by Means of a Bisecting Hip Mechanism. IEEE Transactions on Robotics. 2007.vol1:112~123.
[41]程勒、程燕平.理论力学第六版.高等教育出版社.1996.
[42]成大先.机械设计手册第五版.化学工业出版社.2008.