This work studies the motion of Purcell's three-link microswimmer in viscous flow, by using perturbation expansion of its dynamics under small-amplitude strokes. Explicit leading-order expressions and nextorder correction terms for the displacement of the swimmer are obtained for the cases of a square or circular gait in the plane of joint angles. The correction terms demonstrate the reversal in movement direction for large stroke amplitudes, which has previously only been shown numerically. In addition, asymptotic expressions for Lighthill's energetic efficiency are obtained for both gaits. These approximations enable calculating optimal stroke amplitudes and swimmer's geometry (i.e. ratio of links' lengths) for maximizing either net displacement or Lighthill's efficiency.

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Micron-scale swimmers move in the realm of negligible inertia, dominated by viscous drag forces. In this paper, we formulate the leading-order dynamics of a slender multi-link (N-link) microswimmer assuming small-amplitude undulations about its straight configuration. The energy-optimal stroke to achieve a given prescribed displacement in a given time period is obtained as the largest eigenvalue solution of a constrained optimal control problem. Remarkably, the optimal stroke is an ellipse lying within a two-dimensional plane in the (N -1)-dimensional space of joint angles, where N can be arbitrarily large. For large N, the optimal stroke is a traveling wave of bending, modulo edge effects. If the number of shape variables is small, we can consider the same problem when the prescribed displacement in one time period is large, and not attainable with small variations of the joint angles. The fully nonlinear optimal control problem is solved numerically for the cases N=3 (Purcell's three-link swimmer) and N=5 showing that, as the prescribed displacement becomes small, the optimal solutions obtained using the small-amplitude assumption are recovered. We also show that, when the prescribed displacements become large, the picture is different. For N=3 we recover the non-convex planar loops already known from previous studies. For N=5 we obtain non-planar loops, raising the question of characterizing the geometry of complex high-dimensional loops.deformation. Corresponding biological examples are, respectively, the rotary motion of helical bacterial flagellar bundles, the different shapes of cilia in the power and in the recovery part of one stroke, the beating of a eukaryotic flagellum causing the propagation of bending waves along the length of the flagellum. In fact, bending waves propagating along cilia/flagella and shape modulation during one stroke are used not only for propulsion by micro-organisms, but also for transport inside organs in humans and other higher organisms [17][18][19]. The second example is the one with more connections with the study of minimal artificial swimmers (i.e. swimmers with only two internal degrees of freedom to control shape such as the three-link swimmer of Purcell [3,20], the three-sphere swimmer of Najafi and Golestanian [21], and others). The third example is possibly the most thoroughly exploited paradigm in the fabrication of micro-swimmer prototypes, often through the action of an external magnetic field on a flexible filament [8,10,22,23].Patterns of optimal actuation have been investigated independently, for each of these three examples, with a variety of analytical and numerical methods. For the case of flagellar and ciliary propulsion, these include [2, 24-28] leading to recognizing, for example, helical shapes as the optimal ones for filaments in three dimensions, and smoothed saw-tooth traveling waves as the optimal wave forms for the planar beating of a one-dimensional flagellum or for a planar sheet. In the limit of small amplitudes, the latter reduces to Tayl...

Robotic swimmers are currently a subject of extensive research and development for several underwater applications. Clever design and planning must rely on simple theoretical models that account for the swimmer's hydrodynamics in order to optimize its structure and control inputs. In this work, we study a planar snake-like multi-link swimmer by using the "perfect fluid" model that accounts for inertial hydrodynamic forces while neglecting viscous drag effects. The swimmer's dynamic equations of motion are formulated and reduced into a first-order system due to symmetries and conservation of generalized momentum variables. Focusing on oscillatory inputs of joint angles, we study optimal gaits for 3-link and 5-link swimmers via numerical integration. For the 3-link swimmer, we also provide a small-amplitude asymptotic solution which enables obtaining closed-form approximations for optimal gaits. The theoretical results are then corroborated by experiments and motion measurement of untethered robotic prototypes with 3 and 5 links floating in a water pool, showing a reasonable agreement between experiments and the theoretical model.

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