Using optimal control to obtain maximum displacement gait for Purcell's three-link swimmer

Wiezel, Oren; Or, Yizhar

doi:10.1109/cdc.2016.7798947

Cited by 17 publications

(29 citation statements)

References 40 publications

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“…Explicit expressions for the derivatives are given in the supplementary document. The first-order derivatives at the origin are zero since G 31 , G 32 are even functions in (φ 1 , φ 2 ) and soθ (2) = 0 in (38), implying that θ (2) (t) = 0 is zero (under zero initial conditions). Now, using the expression for θ(t) we can calculate expansions of the net motion q(t).…”

Section: Expansion Of Swimmer Displacementmentioning

confidence: 99%

Optimization and small-amplitude analysis of Purcell's three-link microswimmer model

Wiezel¹,

Or²

2016

Proc. R. Soc. A.

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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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Section: Expansion Of Swimmer Displacementmentioning

confidence: 99%

Optimization and small-amplitude analysis of Purcell's three-link microswimmer model

Wiezel¹,

Or²

2016

Proc. R. Soc. A.

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“…The feet interact with the environment through a linear isotropic drag with viscous drag constant of 100 in both the translation and rotation directions. many others to which this work is applicable, such as the Purcell three link swimmer parameterized by amplitude and phase as in [19], or a kinematic Ackerman car.…”

Section: Discussionmentioning

confidence: 99%

“…The local connection A(r) can be rewritten in terms of the new shape basis parameters [4]. But, a gait with two shape variables can also be parameterized by a reciprocating phase and an amplitude [19]. For the serpenoid swimmer, we can therefore reparameterize the gait Eq.…”

Section: Snake-like Swimmer Example 51 Model and Methodsmentioning

confidence: 99%

Geometric Motion Planning for Systems With Toroidal and Cylindrical Shape Spaces

Gong¹,

Ren²,

Whitman

et al. 2018

Volume 3: Modeling and Validation; Multi-Agent and Networked Systems; Path Planning and Motion Control; Tracking Control System

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Kinematic motion planning using geometric mechanics tends to prescribe a trajectory in a parameterization of a shape space and determine its displacement in a position space. Often this trajectory is called a gait. Previous works assumed that the shape space is Euclidean when often it is not, either because the robotic joints can spin around forever (i.e., has an S 1 configuration space component, or its parameterization has an S 1 dimension). Consider a shape space that is a torus; gaits that "wrap" around the full range of a shape variable and return to its starting configuration are valid gaits in the shape space yet appear as line segments in the parameterization. Since such a gait does not form a closed loop in the parameterization, existing geometric mechanics methods cannot properly consider them. By explicitly analyzing the topology of the underlying shape space, we derive geometric tools to consider systems with toroidal and cylindrical shape spaces.

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“…For the study of optimal strokes for minimal artificial swimmers the reader is referred, e.g. to [20,[29][30][31][32][33][34].More recently, it has been recognized that, in the presence of external forces or torques, the scallop paradigm has to be reconsidered. This is a consequence of the fact that, in this case, the governing equations can no longer be cast as an affine control systems without drift.…”

mentioning

confidence: 99%

“…For the study of optimal strokes for minimal artificial swimmers the reader is referred, e.g. to [20,[29][30][31][32][33][34].…”

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confidence: 99%

Energy-optimal strokes for multi-link microswimmers: Purcell's loops and Taylor's waves reconciled

et al. 2019

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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...

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Using optimal control to obtain maximum displacement gait for Purcell's three-link swimmer

Cited by 17 publications

References 40 publications

Optimization and small-amplitude analysis of Purcell's three-link microswimmer model

Optimization and small-amplitude analysis of Purcell's three-link microswimmer model

Geometric Motion Planning for Systems With Toroidal and Cylindrical Shape Spaces

Energy-optimal strokes for multi-link microswimmers: Purcell's loops and Taylor's waves reconciled

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