AI 中文总结
本研究基于珀塞尔微泳者模型，构建高粘性流体中三连杆游泳机器人变体，应用庞特里亚金最大值原理结合数值方法，求解位移与能量效率最优步态，为机器人最优控制提供方案。
AI 中文摘要
珀塞尔泳者是一种著名的平面游泳微生物模型，由低雷诺数流体动力学控制，包含三个由可驱动旋转关节连接的刚性连杆。该模型已被分析为受一阶非线性动力学控制、具有两个关节角的周期性输入（步态）的机器人运动系统。本研究提出了一种在高粘性流体中运动的三连杆泳者的机器人宏观实现方案，提出了珀塞尔理论模型的简单变体，其连杆非细长，中央有一个刚性球体，用于表示机器人中央浮块的附加阻力，并校准模型参数以匹配实验测量值。接下来，应用基于庞特里亚金最大值原理（PMP）的最优控制公式，以在关节角约束下找到最大化每周期位移的最优步态。采用微分几何方法将问题转化为关节角平面内步态轨迹包围的面积积分，该方法支持可视化解释，可说明关节角约束变化时位移最优步态的拓扑变化。随后，将PMP公式应用于最大化莱特希尔能量效率的问题，以获得边值问题（BVP），其解可得到珀塞尔泳者模型及其带中央球体变体的效率最优步态。最后，利用参数化输入步态为截断傅里叶级数、GPOPS-II求解器等数值方法，为求解BVP生成充分的初始猜测值，以获得效率最优步态。
英文摘要
Purcell's swimmer is a well-known planar model of a swimming microorganism, governed by low Reynolds number hydrodynamics, which is comprised of three rigid links connected by actuated rotary joints. This model has been analyzed as a robotic locomotion system governed by first-order nonlinear dynamics with a periodic input (gait) of the two joint angles. In this work, we present a robotic macro-scale realization of this three-link swimmer moving in a highly viscous fluid. We propose a simple variant of Purcell's theoretical model with non-slender links and a central rigid sphere which represents the added drag of the robot's central flotation block, and calibrate the model's parameters to fit experimental measurements. Next, we apply optimal control formulation based on Pontryagin's Maximum Principle (PMP) in order to find optimal gaits that maximize the displacement per cycle under bounds on the joint angles. Employing a differential geometric method that transforms the problem to area integral enclosed by the gait trajectory in the plane of joint angles, enables visual interpretation which explains topological changes in displacement-optimal gaits upon varying the bound on the joint angles. We then apply PMP formulation to the problem of maximizing Lighthill's energy efficiency in order to obtain a boundary value problem (BVP) whose solution gives efficiency-optimal gaits for Purcell's swimmer model, as well as its variant with a central sphere. Finally, we utilize numerical methods such as parameterizing the input gait as a truncated Fourier series, as well as GPOPS-II solver, to produce sufficient initial guess values for solving the BVPs and obtaining efficiency-optimal gaits.
