Controlled fluid transport by the collective motion of microrotors

TL;DR

The paper addresses controlled transport of fluid densities using flow fields generated by groups of microrotors in Stokes flow. It develops a generalized polynomial chaos (gPC) framework to propagate the density and formulates a stochastic optimal control problem, solved with Differential Dynamic Programming (DDP) under two rotor-motion models: direct velocity control and torque-driven advection. Results show that increasing the number of mobile rotors up to about four and choosing appropriate time horizons markedly improves transport efficiency and reduces stretching, with diminishing gains beyond that. The authors further analyze the induced flow with finite-time Lyapunov exponent (FTLE) fields to reveal Lagrangian coherent structures that act as transport barriers guiding particles toward the target, providing a geometric interpretation of the control strategy. This approach offers a scalable method for designing microrotor actuation schedules for targeted fluid transport with potential biomedical applications and can be extended to more realistic 3D swimmer models and uncertain system parameters.

Abstract

Torque-driven microscale swimming robots, or microrotors, hold significant potential in biomedical applications such as targeted drug delivery, minimally invasive surgery, and micromanipulation. This paper addresses the challenge of controlling the transport of fluid volumes using the flow fields generated by interacting groups of microrotors. Our approach uses polynomial chaos expansions to model the time evolution of fluid particle distributions and formulate an optimal control problem, which we solve numerically. We implement this framework in simulation to achieve the controlled transport of an initial fluid particle distribution to a target destination while minimizing undesirable effects such as stretching and mixing. We consider the case where translational velocities of the rotors are directly controlled, as well as the case where only torques are controlled and the rotors move in response to the collective flow fields they generate. We analyze the solution of this optimal control problem by computing the Lagrangian coherent structures of the associated flow field, which reveal the formation of transport barriers that efficiently guide particles toward their target. This analysis provides insights into the underlying mechanisms of controlled transport.

Figures (8)

• Figure 1: Comparison of the optimal control cost for steering a density to a target using mobile microrotors, as described in Sec. \ref{['sec:movingrotors']}, for varying final time (horizontal axis) and varying number of rotors (curves). The minimum point on each curve is shown by a filled circle.
• Figure 2: Streamlines show the time-averaged velocity field for the optimal case for varying number of rotors. Color depicts the particle density at the final time for each case. The red marker and contours depict the mean, 1$\sigma$ and 2$\sigma$ contours for the initial fluid particle distribution. Colored lines indicate the paths of the rotors. The green circle indicates the target at (-1,-1).
• Figure 3: Optimal solution for the case of four rotors, $n_r = 4$ and final time $t_f = 8$ shown as snapshots from the time sequence. Streamlines show the direction of the instantaneous velocity fields at each instant shown. Color depicts the fluid particle density at each time instant. Colored lines show the rotor paths, with circles indicating the rotor position at each instant. Black circle shows the sample mean, white circle shows the gPC-predicted mean.
• Figure 4: Controls as computed from the DDP solution for the case of $n_r=4$ and $t_f=8$. Each curve represents a different rotor, with colors corresponding to the colored trajectories in Fig. \ref{['fig:nr4_traj']}. Top: rotor strengths, $\gamma$. Middle: $x$-component of rotor translational velocity. Bottom: $y$-component of rotor translational velocity. Rotor numbering goes counterclockwise from the rotor directly right of the target in the initial configuration.
• Figure 5: Moment propagation by the polynomial chaos expansion method (labelled gPC) as compared to the Monte Carlo simulation of $10^4$ points sampled from the initial particle distribution (labelled MC) for the optimal solution of the case of $n_r=4$, as shown in Fig. \ref{['fig:nr4_traj']}. Left column: components of the mean, $\mu$. Right column: components of the covariance, $\sigma$.