标题： 基于伴随的可收缩弹性体最优控制。 应用于在摩擦性基底上的无肢运动 显示英文标题

标题： Adjoint-based optimal control of contractile elastic bodies. Application to limbless locomotion on frictional substrates

摘要： 在自然界中，无肢运动被各种不同尺度的生物广泛采用。有趣的是，波浪式、爬行和步进/环状步态构成了一类基本的无肢运动，并且在许多物种中经常被观察到，例如毛虫、蚯蚓、水蛭、幼虫以及\emph{线虫}，仅举几例。在本工作中，我们开发了一个计算高效的基于三维有限元（FE）的统一框架，用于无肢生物在柔软底物上的运动。肌肉活动通过变形梯度的乘法分解进行模拟，这使得在摩擦底物上的三维固体中能够模仿广泛的运动模式。特别是，提出了一种基于位置和速度的两场有限元公式。控制偏微分方程被转化为等效的时间连续微分代数方程（DAEs）。接下来，在最优控制理论的框架下研究了最优运动策略。我们采用基于伴随的方法，并推导出一阶最优性条件，这产生了一个具有两点端条件的DAE系统。讨论了隐藏的辛结构和最优性条件的辛欧拉时间积分。所得的离散一阶最优性条件形成一个非线性规划问题，该问题通过前向后向扫视法高效求解。最后，提供了一些数值例子来展示所提出的计算框架的全面性，并研究了无肢生物采用的不同运动模式中的节能最优无肢运动策略。 显示英文摘要

摘要： In nature, limbless locomotion is adopted by a wide range of organisms at various length scales. Interestingly, undulatory, crawling and inching/looping gait constitutes a fundamental class of limbless locomotion and is often observed in many species such as caterpillars, earthworms, leeches, larvae, and \emph{C. elegans}, to name a few. In this work, we developed a computationally efficient 3D Finite Element (FE) based unified framework for the locomotion of limbless organisms on soft substrates. Muscle activity is simulated with a multiplicative decomposition of deformation gradient, which allows mimicking a broad range of locomotion patterns in 3D solids on frictional substrates. In particular, a two-field FE formulation based on positions and velocities is proposed. Governing partial differential equations are transformed into equivalent time-continuous differential-algebraic equations (DAEs). Next, the optimal locomotion strategies are studied in the framework of optimal control theory. We resort to adjoint-based methods and deduce the first-order optimality conditions, that yield a system of DAEs with two-point end conditions. Hidden symplectic structure and Symplectic Euler time integration of optimality conditions have been discussed. The resulting discrete first-order optimality conditions form a non-linear programming problem that is solved efficiently with the Forward Backwards Sweep Method. Finally, some numerical examples are provided to demonstrate the comprehensiveness of the proposed computational framework and investigate the energy-efficient optimal limbless locomotion strategy out of distinct locomotion patterns adopted by limbless organisms.