Rotational Propulsion of Soft Rod-like Robots

By Henry C. Fu and Min Jun Kim

Microrobots—tiny objects that are capable of controllable propulsion—have long been a fantasy in science fiction movies such as 1996’s Fantastic Voyage and 2014’s Big Hero 6. Now, however, this technology is rapidly becoming realizable. Microrobots may find potential applications in microassembly and fabrication, as well as biomedical uses like microsurgery and targeted drug delivery for chemotherapy. The many different types of microrobots utilize various forms of physics-based propulsion mechanisms that range from electromagnetic forces to self-generated chemical or thermal gradients [1]. Here, we discuss a common class of microrobots that are propelled by an external, rotating magnetic field.

Rotationally-propelled microrobots often mimic the movement of the nearly rigid helical bacterial flagellum, which a bacterial flagellar motor rotates like a corkscrew to push the cell forward (see Figure 1a) [7]. Many examples of this type of microrobot hence take helical shapes (see Figure 1b) [1]. However, a key difference separates magnetically-rotated microrobots from bacteria: the magnetic field exerts an external torque on the microrobot, whereas bacteria experience zero net torque since the motor exerts equal and opposite torques on the flagellum and cell body, rotating them in opposite directions. We explored the way in which this difference changes the physical constraints on microrobot propulsion when compared to the swimming patterns of force- and torque-free microorganisms.

The Scallop theorem [10] is a constraint on the type of deformations that a force- and torque-free microswimmer can undergo to achieve translational propulsion. The fluid dynamics of microswimmers fall into the regime of low Reynolds numbers, wherein effects from fluid viscosity dominate effects from fluid inertia. In this case, the equation that describes the fluid’s momentum conservation is simplified from the Navier-Stokes equation—which includes inertial acceleration terms—to the Stokes equation. Flows that obey the Stokes equation have the property of kinematic reversibility. This property states that if a flow velocity field \(\boldsymbol{v}\) in a domain \(D\) is produced at some instant of time by a boundary deformation that is specified by velocities \(\boldsymbol{v}(\partial D)\) at the boundary, then reversal of the velocities at the boundary to \(-\boldsymbol{v}(\partial D)\) will cause the entire flow field to become \(-\boldsymbol{v}\). Kinematic reversibility implies that a swimmer which translates a certain amount while generating a sequence of deformations will translate an equal and opposite amount if it reverses that sequence of deformations. This result forms the Scallop theorem and is often expressed by saying that any swimming stroke which traces the same configurations forward and backward in time (called a “reciprocal” stroke) cannot lead to net translation. Researchers have used the Scallop theorem to understand which types of swimming strokes are effective for numerous biological organisms. One example is the non-reciprocal traveling waves that many microorganisms generate, including rotating bacterial flagella.

However, the Scallop theorem is not a relevant constraint for our magnetically-rotated microrobots, because a rotary magnetic rotation is not kinematically reversible (reversing time reverses the direction of rotation). We are thus attempting to understand the differences in the resulting fundamental constraints on magnetically-rotated microrobots. If they are rigid bodies, such microrobots do not need to be chiral like helical bacterial flagella [3]. Instead, they can have up to two perpendicular planes of mirror symmetry (see Figure 1b). However, in order for propulsion to occur, their rotation axis cannot be normal to a symmetry plane. Rotation can propel even more symmetric rigid bodies in the presence of nonlinearities. For instance, axisymmetric bodies with fore-aft symmetry that rotate along their symmetry axis can be propelled in a nonlinearly viscoelastic fluid [11], even though they cannot be propelled in a Newtonian Stokes fluid.

Here, we describe our recent efforts to examine the way in which nonlinearities that are induced by fluid-structure interactions affect these constraints [6]. Specifically, we investigate whether a rotating magnetic field can propel an axisymmetric and fore-aft symmetric body—a rod-like robot—if the robot is soft and deformable, rather than rigid as assumed in previous studies. The basic idea is that a soft robot deforms into a shape that is determined by the balance between bending stiffness—which works to maintain a straight rod—and magnetic and hydrodynamic forces, which bend the rod (see Figure 2). Since hydrodynamic forces depend on the rotation rate, the shape will as well. Can the fluid-structure interactions create deformed shapes that allow for propulsion even if the undeformed shapes do not?

We fabricated soft, rod-like structures by casting polyacrylamide in pipette tips (see Figure 2a). We then embedded permanent magnets into the ends that point in opposite directions, and placed the structures within a set of Helmholtz coils that generate a time-dependent uniform magnetic field. When we apply a steady magnetic field, the magnetic torques at each end tend to align in the same direction — thus deforming the rod. By observing this deformation, we can deduce the bending stiffness of the rods.

To propel the robots, we rotate a magnetic field with constant magnitude in a plane along a rotation axis that is perpendicular to the plane (see Figure 2b). Within our chosen rotation frequency regime, the soft rod deforms into a steady-state shape that is then bodily rotated about the magnetic field’s rotation axis (see Figure 2b and Animation 1). In addition to sedimentation due to gravity and a rolling motion along the boundary that is perpendicular to the rotation axis, we also observe swimming propulsion along the rotation axis. This swimming propulsion is larger than the rolling propulsion for low frequencies and occurs far away from boundaries (see Figure 3a).

We then created numerical models to understand the observed behavior [6]. These models mathematically describe the flexible structure as a Kirchhoff rod that is coupled to magnetic torques, which arise from the interaction of its embedded magnets and the magnetic field, as well as hydrodynamic forces, which we calculate via the method of regularized Stokeslets [2, 4, 6, 8]. Because the pipette tips were tapered [2], the robots that we fabricated were as well (see Figure 3b). During experiments, we observed that the robots always swim in the direction of their thicker ends. We could also test soft rods without any taper in our numerical models, and found that they do not generate propulsion because untapered rods deform in a fore-aft symmetric manner along the rotation axis. On the other hand, our models did indeed corroborate that tapered rods propel towards their thicker ends; we therefore concluded that, even with deformations, some sort of asymmetry is necessary to enable swimming propulsion. To test this conclusion, we fabricated rods with differing amounts of taper and found that those with more taper experience faster swimming propulsion. We even managed to fabricated robots without any taper that—as predicted by our models—were not capable of swimming propulsion.

Our results indicate that deformations alone do not allow a fore-aft axisymmetric robot to propel itself; some type of asymmetry is required as well. The taper provided this asymmetry in our experiments, but angling the magnets on either end of the robot in different directions could also serve this purpose (see Figure 2a, with \(\theta_i\ne\theta_e\) rather than \(\theta_i=\theta_e\)). More generally, our results suggest that other effective soft robot designs should likewise prioritize asymmetry. Additional ways to generate asymmetry could include placing magnets at only one end of a soft robot, enlarging one end (to carry cargo, for instance), or tailoring the stiffness along the length of the flagellum. Some existing soft robots have already employed these strategies, sometimes in combination [5, 9].

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Henry Fu delivered a minisymposium presentation on this research at the 2023 SIAM Conference on Computational Science and Engineering, which took place in Amsterdam, the Netherlands, earlier this year.

References
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	Henry C. Fu is a professor in the Department of Mechanical Engineering at the University of Utah. His research focuses on microscale fluid dynamics, including the propulsion of microorganisms and microrobots.
	Min Jun Kim is the Robert C. Womack Endowed Chair Professor in the Department of Mechanical Engineering at Southern Methodist University. His scholarly pursuits center around biologically inspired robotics at the micro- and nanoscale, specifically targeting applications in drug delivery and minimally invasive surgery.