Abstract
The controllability of a fully three-dimensional N-link swimmer is studied. After deriving the equations of motion in a low Reynolds number fluid by means of Resistive Force Theory, the controllability of the minimal 2-link swimmer is tackled using techniques from Geometric Control Theory. The shape of the 2-link swimmer is described by two angle parameters. It is shown that the associated vector fields that govern the dynamics generate, via taking their Lie brackets, all eight linearly independent directions in the combined configuration and shape space, leading to controllability; the swimmer can move from any starting configuration and shape to any target configuration and shape by operating on the two shape variables. The result is subsequently extended to the N-link swimmer. Finally, the minimal time optimal control problem and the minimization of the power expended are addressed and a qualitative description of the optimal strategies is provided.
Highlights
The swimming motion of microorganisms in viscous fluids at low Reynolds number has been studied mathematically since the 1950s [22, 39]
Remark 3.8 By standard results on control theory [12, 40], controllability is ensured with bounded controls, for any final time T < +∞ the 2-link swimmer is controllable by means of absolutely continuous shape parameters ∈ (0, π ) × (−π, π ) for all t ∈ [0, T ] (see the S-component of (3.7))
To provide a dynamical description of the N -link swimmer, we follow the construction of Sect. 2: each link is described by two angles (φt(i), θt(i)) ∈ (0, π ) × (−π, π ) that identify the direction of the link with respect to the co-moving frame
Summary
The swimming motion of microorganisms in viscous fluids at low Reynolds number has been studied mathematically since the 1950s [22, 39]. Once the total viscous force and torque are computed, setting them equal to zero allows us to obtain the equations of motion for the swimmer These are conveniently written in the form of a (nonlinear) control system, so that tools from Geometric Control Theory can be applied. Standard results and methods from Geometric Control Theory are used to prove Theorem 3.6 ensuring controllability of the 2-link swimmer: any given final configuration can be reached starting from any assigned initial configuration by acting on the controls u1, u2 This is obtained by computing the Lie brackets of the vector fields V1 and V2 activated by u1 and u2 and showing that they generate all the possible directions of motion, proving that two linearly independent vectors, the Vi ’s, generate the eight-dimensional space of translations, rotations, and shapes (x, R, (φ, θ)).
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