Dynamics and Optimal Actuation of a Three-Sphere Low-Reynolds-Number Swimmer with Muscle-Like Arms

• Published in: Acta applicandae mathematicae Vol. 149; no. 1; pp. 53 - 86
• Main Authors: Montino, Alessandro, DeSimone, Antonio
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.06.2017; Springer Nature B.V
• Subjects: Actuation; Applications of Mathematics; Arms; Calculus of Variations and Optimal Control; Optimization; Computational Mathematics and Numerical Analysis; Control systems; Control theory; Differential equations; Mathematical models; Mathematics; Mathematics and Statistics; Optimization; Ordinary differential equations; Partial Differential Equations; Probability Theory and Stochastic Processes; Qualitative analysis; Microswimmers; Hill’s muscle model; Low-Reynolds-number hydrodynamics; Optimal control
• ISSN: 0167-8019; 1572-9036
• DOI: 10.1007/s10440-016-0087-9

The three-sphere swimmer by Najafi and Golestanian is composed of three spheres connected by two arms. The case in which the swimmer can control the lengths of the two arms has been studied in detail. Here we study a variation of the model in which the swimmer’s arms are constructed according to Hill’s model of muscular contraction. The swimmer is able to control the tension developed in the active components of the arms. The two shape parameters and the tensions acting on the two arms are then obtained by solving a system of ordinary differential equations. We study the qualitative properties of the solutions, compute analytically their leading order approximation and compare them with numerical simulations. We also formulate and solve some optimisation problems, aimed at finding the actuation strategies maximising performance, for various performance measures. Finally, we discuss the structure of the governing equations of our microswimmers from the point of view of control theory. We show that our systems are control affine systems with drift.