Mathworks, MA USA.
Purdue University, West Lafayette, IN USA.
J Theor Biol. 2018 Aug 14;451:57-66. doi: 10.1016/j.jtbi.2018.04.020. Epub 2018 Apr 13.
The relative leg stiffness of most running animals falls in a small range between 7 and 27. Here we present a theoretical study of an established running model, an actuated Spring Loaded Inverted Pendulum model, to determine if the energetic cost and stability of running might be co-optimized over this range of leg stiffness values. The energetic cost of the model is quantified as the energy spent to move a unit mass a unit distance. The stability of the model is based on the system response to perturbations with respect to periodic locomotion solutions, and uses the linearized dynamics of Poincaré return maps and the resulting maximum eigenvalue and singular value decomposition in order to analyze asymptotic stability and the overall system response to perturbations, respectively. We find that there exists a tradeoff between stability and energetic cost in the model with respect to variation in forcing (actuation) level: For a given leg stiffness, the energetic cost tends to be more optimal with smaller forcing, and the opposite for stability. We find that intermediate levels of forcing can achieve near asymptotic stability or complete asymptotic stability while remaining small enough to yield a relatively low energetic cost consistent with human-like values. We demonstrate that this outcome can be achieved in the model with a simple optimization function that balances stability and energetic cost. We then investigate the stability and energetic cost when both leg stiffness and forcing are varied. Overall, the analysis shows that leg stiffness values in or near the biological range offers a good chance of simultaneously achieving both reasonable energetic cost and stability in the model. The results of this study suggest that stability and energetic cost may be interacting factors that have a combined influence on the effective leg stiffness and actuation (forcing) used by running animals.
大多数奔跑动物的相对腿部僵硬度在 7 到 27 之间。在这里，我们对一个已建立的奔跑模型，即受激弹簧加载倒立摆模型进行了理论研究，以确定在这个腿部僵硬度范围内，奔跑的能量成本和稳定性是否可以共同优化。该模型的能量成本被量化为移动单位质量单位距离所消耗的能量。该模型的稳定性基于系统对周期性运动解的扰动的响应，并使用 Poincaré 回归映射的线性化动力学以及由此产生的最大特征值和奇异值分解来分别分析渐近稳定性和系统对扰动的整体响应。我们发现，在模型中，随着强制（激励）水平的变化，稳定性和能量成本之间存在权衡：对于给定的腿部僵硬度，能量成本随着强制的减小而变得更加优化，而稳定性则相反。我们发现，中间水平的强制可以实现接近渐近稳定性或完全渐近稳定性，同时仍然足够小，从而产生与人类相似的值相对较低的能量成本。我们证明，通过平衡稳定性和能量成本的简单优化函数，可以在模型中实现这种结果。然后，我们研究了腿部僵硬度和强制都变化时的稳定性和能量成本。总的来说，分析表明，在生物学范围内的腿部僵硬度值为同时在模型中实现合理的能量成本和稳定性提供了很好的机会。这项研究的结果表明，稳定性和能量成本可能是相互作用的因素，它们对奔跑动物的有效腿部僵硬度和激励（强制）有共同的影响。
