Optimal Undulatory Swimming with Constrained Deformation and Actuation Intervals

TL;DR

This work tackles how undulatory microswimmers constrained by local active torque inputs achieve efficient propulsion. The authors model a planar bead-spring filament with total potential $U = U_s + U_p + U_a$ and torques $L_i(t)$ limited by $L_{ m max}$, then apply Proximal Policy Optimization (PPO) to maximize forward displacement, letting beating frequency and shape emerge from the optimization. Two key dimensionless parameters, $k_b^* = k_b/L_{ m max}$ and $T_a^* = T_a/t_0$, govern the emergent beating patterns, which are found to be bang-bang in time with specific phase lags between joints; these patterns depend on the action interval and bending stiffness. The authors establish analytical estimates for the optimal beating period from bending relaxation times, demonstrating qualitative and quantitative agreement with the RL results and linking microscopic viscoelastic dynamics to macroscopic propulsion, with implications for designing artificial elastic microswimmers.

Abstract

In nature, many unicellular organisms are able to swim with the help of beating filaments, where local energy input leads to cooperative undulatory beating motion. Here, we investigate by employing reinforcement learning how undulatory microswimmers modeled as a discretized bead-bend-spring filament actuated by torques which are constrained locally. We show that the competition between actively applied torques and intrinsic bending stiffness leads to various optimal beating patterns characterized by distinct frequencies, amplitudes, and wavelengths. Interestingly, the optimum solutions depend on the action interval, i.e.\ the time scale how fast the microswimmer can \rev{change the applied torques} based on its internal state. We show that optimized stiffness- and action-interval-dependent beating is realized by bang-bang solutions of the applied torques with distinct optimum time-periodicity and phase shift between consecutive joints, which we analyze in detail by a systematic study of possible bang-bang wave solution patterns of applied torques. Our work not only sheds light on how efficient beating patterns of biological microswimmers can emerge based on internal and local constraints, but also offers actuation policies for potential artificial elastic microswimmers.

Figures (10)

• Figure 1: Schematic illustration of the bead-spring swimmer model. Beads of radius $a$ are connected with linear springs with spring constant $k_s$. The swimmer propels by the undulatory motion driven by the active torque $L_i$ assigned to the joints formed by three consecutive beads, acting together with a passive elastic bending energy term with bending stiffness $k_b$, which induces a force in a direction that tends to flatten the joints.
• Figure 2: The optimized motion of microswimmer for (a) long term and (b) approximately half cycle from $t/T = 0$ (dark gray) to $0.45$ (lighter gray), under the dimensionless parameters $k_b^* = 0.5$ and $T_a^* = 0.1$. Note that the beating period is $T^* = 2$ for this motion, and $t^*=900$ corresponds to a displacement after 450 beating cycles.
• Figure 3: (a) Swimming velocities of the optimized swimming $V^*$. (b) Beating period $T^*$, which is the inverse of the beating frequency $f^* = 1/T^*$. (c) Gained distance per unit cycle $V^* T^*$ as a function of the beating frequency $f^*$. (d) Swimming efficiency $\eta^*$. The inset figure shows the power expended by the swimmer $P^*$.
• Figure 4: Swimmer shape for (a) $k_b^* = 0.5$ and (b) $k_b^*=2.0$. Gray connected beads show the swimmer shape, while black dashed lines show the fitting results. The instantaneous amplitude $b^*$ and wavelength $\lambda^*$ are shown in the upper right corner of each figure.
• Figure 5: Geometric features of the optimized swimming mode: (a) amplitude $b^*$ and (b) wavelength $\lambda^*$. The inset figure shows the number of waves in the swimmer's body.