弹性接触对被动行走多体系统动力学与稳定性的影响研究
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摘要
被动行走是双足仿生行走的新概念。相比于传统的双足行走，其优点为能量消耗率很低。被动行走的仿真研究一般采用倒立摆模型假设，认为支撑足与地面铰接，足在地面上不发生弹起以及相对滑动。这种简化模型忽略了足与地面接触的过程。在各种多体系统中，碰撞与接触问题对于系统的动力学行为与稳定性有很重要的影响，实际的被动行走多体系统中足与地面接触也是行走过程的重要环节。
为研究上述接触过程对行走的影响，本文首先建立了考虑足与地面弹性接触的二维被动行走模型，利用Hertz接触（弹簧阻尼）模型以及经过改进的Coulomb摩擦模型分别描述足与地面的法向接触力与切向摩擦力。通过动力学仿真，在寻找到周期行走步态的同时，得到了行走过程的接触力变化。通过改变模型的接触刚度系数、阻尼系数以及摩擦系数，发现接触阻尼系数与摩擦系数对于行走步态基本无影响，接触刚度系数对于行走的步态影响最大。接触刚度系数越大，行走的平均速度和碰撞力峰值越大。
通过增加仿真模型的斜面倾角，可得到单步周期、二步周期、多步周期等周期加倍的行走结果，这与Gacia等人的倒立摆模型结果类似。改变模型参数及接触参数，可得到参数对步态分岔点的影响。仿真结果表明接触刚度系数越大，步态分岔发生的倾角越小，其它接触参数对分岔点没有影响。
进一步利用Floquet乘子以及胞映射方法对行走的稳定性研究。通过简单假设使动力学系统的状态空间降维，得到了3维胞空间下的被动行走吸引域。通过改变接触参数，发现接触刚度系数与接触阻尼系数对吸引域大小没有影响；摩擦系数越大，行走的吸引域越大。这说明摩擦力是考虑弹性接触的被动行走稳定性的决定因素。
为验证仿真模型结果，建立了含有测试系统的简单直腿被动行走实验模型。利用陀螺仪测量双腿摆角，利用薄膜力传感器测量足地接触的法向力。在不同行走介质上测试的结果，说明了摩擦力对行走稳定性的决定作用。利用测试得到的行走步态以及接触法向力曲线，与仿真模型结果进行了对比，二者吻合较好。这说明考虑足与地面弹性接触的被动行走模型是更加接近真实物理模型的。
Passive dynamic walking is a new concept of walking in contrast to the traditionalbiped walking, and its advantage is the low energy dissipation. The passive walkerperforms like an inverted pendulum on the assumption that the stance leg stick on theslope. The contact procedure is neglected in this simple walking model. As we know,the impaction and the contact procedure have great influences on the dynamiccharacters and stability in the multi-body system, so the contact procedure is obviouslyimportant in the passive walking gait.
A model with elastic contact between the feet and the ground is established inorder to concentrate on the procedure mentioned above. The Hertz contact law and theapproximate Coulomb friction law is introduced into the walking model as the normaland tangential contact model. The period-one gait is found in the simulation, and thecontact forces are derived from the equations. By changing the contact parameters inthis model, we find the contact damping and the friction coefficients have nothing to dowith the walking gaits, and the contact stiffness is the most significant parameter onwalking gaits. Larger contact stiffness implies the higher walking speed and the largercontact force peak.
With this passive walking model, the period-one, the period-two and the chaos ofthe walking gait are found as the walking slope increases. This phenomenon is the sameas Gacia’s models. When the parameters change, the influences on bifurcation is found.If the contact stiffness is larger, the bifurcation point advances as the walking slopeincreases. The contact damping and the friction coefficients have nothing to do with thebifurcation point.
With Floquet multiplier and the cell mapping method, the stability of this walkingmodel is discussed. Some assumptions are introduced in order to reduce the state spacedimensions to three. The basin of attraction of this model is obtained from the specialassumption in the cell mapping method. The basin of attraction of the model changes asthe contact parameters change. The contact stiffness and damping have nothing to dowith the basin of attraction, while the large contact friction coefficients is beneficial forthe stability of this model.
The testing model is established in order to validate the simulation model. Twogyros are used for testing the swing angles of the legs, and the FlexiForce sensors areused for testing the normal contact forces of the model. The testing data of walking ondifferent surfaces indicates the friction coefficients are beneficial for the walking. At thesame time, the comparison of the walking gait and the contact forces between thetesting data and the simulation implies that the model with elastic contact is moreapproached to the real walking model.
为研究上述接触过程对行走的影响，本文首先建立了考虑足与地面弹性接触的二维被动行走模型，利用Hertz接触（弹簧阻尼）模型以及经过改进的Coulomb摩擦模型分别描述足与地面的法向接触力与切向摩擦力。通过动力学仿真，在寻找到周期行走步态的同时，得到了行走过程的接触力变化。通过改变模型的接触刚度系数、阻尼系数以及摩擦系数，发现接触阻尼系数与摩擦系数对于行走步态基本无影响，接触刚度系数对于行走的步态影响最大。接触刚度系数越大，行走的平均速度和碰撞力峰值越大。
通过增加仿真模型的斜面倾角，可得到单步周期、二步周期、多步周期等周期加倍的行走结果，这与Gacia等人的倒立摆模型结果类似。改变模型参数及接触参数，可得到参数对步态分岔点的影响。仿真结果表明接触刚度系数越大，步态分岔发生的倾角越小，其它接触参数对分岔点没有影响。
进一步利用Floquet乘子以及胞映射方法对行走的稳定性研究。通过简单假设使动力学系统的状态空间降维，得到了3维胞空间下的被动行走吸引域。通过改变接触参数，发现接触刚度系数与接触阻尼系数对吸引域大小没有影响；摩擦系数越大，行走的吸引域越大。这说明摩擦力是考虑弹性接触的被动行走稳定性的决定因素。
为验证仿真模型结果，建立了含有测试系统的简单直腿被动行走实验模型。利用陀螺仪测量双腿摆角，利用薄膜力传感器测量足地接触的法向力。在不同行走介质上测试的结果，说明了摩擦力对行走稳定性的决定作用。利用测试得到的行走步态以及接触法向力曲线，与仿真模型结果进行了对比，二者吻合较好。这说明考虑足与地面弹性接触的被动行走模型是更加接近真实物理模型的。
Passive dynamic walking is a new concept of walking in contrast to the traditionalbiped walking, and its advantage is the low energy dissipation. The passive walkerperforms like an inverted pendulum on the assumption that the stance leg stick on theslope. The contact procedure is neglected in this simple walking model. As we know,the impaction and the contact procedure have great influences on the dynamiccharacters and stability in the multi-body system, so the contact procedure is obviouslyimportant in the passive walking gait.
A model with elastic contact between the feet and the ground is established inorder to concentrate on the procedure mentioned above. The Hertz contact law and theapproximate Coulomb friction law is introduced into the walking model as the normaland tangential contact model. The period-one gait is found in the simulation, and thecontact forces are derived from the equations. By changing the contact parameters inthis model, we find the contact damping and the friction coefficients have nothing to dowith the walking gaits, and the contact stiffness is the most significant parameter onwalking gaits. Larger contact stiffness implies the higher walking speed and the largercontact force peak.
With this passive walking model, the period-one, the period-two and the chaos ofthe walking gait are found as the walking slope increases. This phenomenon is the sameas Gacia’s models. When the parameters change, the influences on bifurcation is found.If the contact stiffness is larger, the bifurcation point advances as the walking slopeincreases. The contact damping and the friction coefficients have nothing to do with thebifurcation point.
With Floquet multiplier and the cell mapping method, the stability of this walkingmodel is discussed. Some assumptions are introduced in order to reduce the state spacedimensions to three. The basin of attraction of this model is obtained from the specialassumption in the cell mapping method. The basin of attraction of the model changes asthe contact parameters change. The contact stiffness and damping have nothing to dowith the basin of attraction, while the large contact friction coefficients is beneficial forthe stability of this model.
The testing model is established in order to validate the simulation model. Twogyros are used for testing the swing angles of the legs, and the FlexiForce sensors areused for testing the normal contact forces of the model. The testing data of walking ondifferent surfaces indicates the friction coefficients are beneficial for the walking. At thesame time, the comparison of the walking gait and the contact forces between thetesting data and the simulation implies that the model with elastic contact is moreapproached to the real walking model.
引文
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[3] McGeer T. Passive dynamic walking. The International Journal of Robotics Research,1990,9:62-82
[4] Arthur D Kuo. The six determinants of gait and the inverted pendulum analogy: A dynamicwalking perspective. Human Movement Science,2007,26:617–656
[5] Carlotta Mummolo, Joo H. Kim. Passive and dynamic gait measures for biped mechanism:formulation and simulation analysis. Robotica, doi:10.1017/S0263574712000586
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[8] Mariano Garcia, Anindya Chatterjee, Andy Ruina, Michael Coleman. The simplest walkingmodel: stability, complexity, and scaling. Journal of Biomechanical Engineering,1998,120(2):281-288
[9] Michael J, Coleman, Anindya Chatterjee, Andy Ruina. Motions of a rimless spoked wheel: asimple3D system with impacts. Dynamics and Stability of Systems,1997,12(3):139-160
[10] Coleman M, A Ruina. An Uncontrolled Walking Toy That Cannot Stand Still. PhysicalReview Letters,1998,80(16):3658
[11] Coleman M. A stability study of a three dimensional passive dynamic model of humangait[Ph.D. Thesis]. New York: Cornell University,1998
[12] Garcia M. Stability, Chaos, and Scaling Laws: Passive–Dynamic Gait Models [Ph D Thesis].USA: Cornell University,1998
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