Conduction Electrohydrodynamics with Mobile Electrodes: A Novel Actuation System for Untethered Robots

Electrohydrodynamics (EHD) refers to the direct conversion of electrical energy into mechanical energy of a fluid. Through the use of mobile electrodes, this principle is exploited in a novel fashion for designing and testing a millimeter‐scale untethered robot, which is powered harvesting the energy from an external electric field. The robot is designed as an inverted sail‐boat, with the thrust generated on the sail submerged in the liquid. The diffusion constant of the robot is experimentally computed, proving that its movement is not driven by thermal fluctuations, and then its kinematic and dynamic responses are characterized for different applied voltages. The results show the feasibility of using EHD with mobile electrodes for powering untethered robots and provide new evidences for the further development of this actuation system for both mobile robots and compliant actuators in soft robotics.


Diffusion Constant
Our experiment for computing the Diffusion Constant of the robot consisted in placing the specimen still in the middle of the tank, free to move passively in the liquid under thermal fluctuations for 5 seconds (V 0 condition) and then applying a voltage of 1250 V for another 5 seconds (V I condition). We tracked the x and y positions of the Center Of Mass (COM) of the robot relative to the initial position. We repeated the experiment 20 times to obtain statistically significant data and check the repeatability and robustness of the phenomenon. With these data, we computed the diffusion constant for both the V I and V 0 conditions as where r(t) 2 was obtained from a linear fitting of the mean square displacement r(t) 2 , this latter computed as r(t) 2 = Σ n i=1 (x(t) 2 i + y(t) 2 i )/n. [1] 2 Thrust In order to compute the thrust generated by EHD on the robot submerged in the fluid, we assumed the following dynamical model where F T is the thrust we want to estimate, m is the mass of the specimen, m a is the added mass of the fluid, F D is the drag force. [2] We computed for the cylinder and electrode sail an added mass m a = 5.3 g, while we neglected the contribution of the stabilizing fins (Section 2.1). As for the drag force, we took into account both the pressure and friction drags, obtaining (Section 2.
Putting together the results, the thrust is

Added Mass
As far as the computation of the added mass is concerned, we considered the specimen as constituted by a cylinder (the main body) and a square plate (the electrode sail). We instead neglected the contribution of the stabilizing wings because they exposed a very small surface to the flow. We considered the cylinder as fully submerged in the NaCl solution and the plate as fully submerged in Novec 7000 fluid. We relied on the solutions obtained through potential theory [3,2] for the estimation of the added mass m a . Using the fluid properties summarized in table 1 and the geometrical parameters in  table 2, for a square plate moving perpendicularly to its surface we have m a plate = 0.478πρ s l 3 /4 = 4.2g.
For the cylinder, we approximated it with a parallelepiped of height h and squared base of side d and we have m a cyl = Kρd 2 h.
Values for K are provided in tables in the literature, and for h/d = 3.75 we interpolated K = 0.47, obtaining m a cyl = 1.1 g. [3] So the total added mass is m a = m a plate + m a cyl = 5.3 g.

Drag
In order to compute the drag F D (v(t)), we first of all computed the Reynolds number (Re) governing the fluid flow around the robot. We obtained two different values for the cylinder Re cyl = v * 6.9 · 10 3 and for the electrode fin  Re plate = v * 62.2 · 10 3 , using as characteristic dimensions d for the cylinder and l for the fin. In the ranges of speeds involved in our experiment we can consider always the flow as laminar (1 < Re < 10 4 ) because Re plate ∈ [62, 2500], Re cyl ∈ [6.9, 276] for v ∈ [0.1, 4] cm s −1 . In this laminar flow regime, both the friction drag F DF and pressure drag F DP should be taken into account. Given that both the cylinder and the plate electrode have sharp edges, according to Hoerner [5] it is possible to consider the drag coefficient approximately constant for our Re values: C D ≈ 2. As far as the frictional drag is concerned, we will use the model proposed by Blasius [6] for flat surfaces parallel to the flow: C F = 1.328 * √ Re. With these coefficients, the drag forces will be F DP = 0.5 * C D ρSv 2 and F DF = 0.5 * C F ρS w v 2 , where S is the surface perpendicular to the flow, while S w is the one parallel to the flow. We took into account F DP for the cylinder and the electrode sail and the F DF for the cylinder and the lateral wings. We neglected the pressure drag for the wings (S small) and the friction drag for the electrode sail (S w small). With these considerations, we obtained    F DP (v(t)) = [0.5 * C D * (ρ ws * (πd 2 /4) + ρ N 7 * l 2 ] * v(t) 2 F DF (v(t)) = [0.5 * 1.328 * ρ ws * µ ws /d * (4ab + πdh)] * v(t) 3/2 F D (v(t)) = F DP (v(t)) + F DF (v(t)) (6)