1. Additional Figures Experiment Results2. Movies of Experimental Results3. Appendix3.1. Computing the Magnetic Field for a Single Electromagnet3.2. Measurement of the Ferrofluid Droplet Magnetic Drift Coefficient k’4. ReferencesPlanar Steering of a Single Ferrofluid Drop by Optimal Minimum PowerDynamic Feedback Control of Four Electromagnets at a DistanceR. Probst, J. Lin, A. Komaee, A. Nacev, Z. Cummins, B. Shapiro**Fischell Department of Bio-Engineering, 3178 Martin Hall,University of Maryland, College Park, MD 20742. [email protected] Material 11. Additional Figures Experiment ResultsFigure SM-1: Control of a small 1 L (0.6 mm radius) ferrofluid droplet slowly along a line,square, and spiral path (desired path: gray line; actual path: black line). A quantitative measure of the average error (equation (17)) is noted at the bottom of each column. 210 mmFigure SM-2: Control of a medium size 20 L (1.7 mm radius) ferrofluid droplet faster along a line, square, and spiral path.32. Movies of Experimental ResultsMovie of Figure SM-1 (line)SM-1a FFsmallslow_line.mpgMovie of Figure SM-1 (square)SM-1b FFsmallslow_square.mpgMovie of Figure SM-1 (spiral)SM-1c FFsmallslow_spiral.mpgMovie of Figure SM-2 (line)SM-2a FFmedfast_line.mpgMovie of Figure SM-2 (square)SM-2b FFmedfast_square.mpgMovie of Figure SM-2 (spiral)SM-2c FFmedfast_spiral.mpgMovie of Figure 5 (line)5a FFmedslow_line.mpgMovie of Figure 5 (square)5b FFmedslow_square.mpgMovie of Figure 5 (spiral)5c FFmedslow_spiral.mpgMovie of Figure 6 (line)6a FFsmallfast_line.mpgMovie of Figure 6 (square)6b FFsmallfast_square.mpgMovie of Figure 6 (spiral)6c FFsmallfast_spiral.mpgMovie of Figure 8 (UMD)8 FFmedslow_UMD.mpg43. Appendix 3.1. Computing the Magnetic Field for a Single ElectromagnetHere we derive and compute the magnetic field around a single electromagnet (Figure 2). Weconsider an N-loop solenoid with a constant coil radius a (the thickness of the coil is ignored)and length l, and assume that a current I passes through the solenoid. The solenoid axis isalong the z-axis of a coordinate system such that the solenoid is extended from 0z towardlz . We determine the magnetic field )(rB at any arbitrary point ),,( zyxr  with0z.Let )(rBs be the magnetic field due to a single loop of the solenoid in the xy plane. Then, bythe linearity of Maxwell’s equations, the total magnetic field can be expressed as(A 1.1)),)/(,,()(10NnslNnzyxBrBwhich can be approximated by an integral formulation (A 1.2).),,(),,())/(,,()(10010dvlvzyxBNduuzyxBlNNllNnzyxBlNrBslsNns We use the Biot-Savart law to determine the magnetic field due to a single loop. This lawdescribes the magnetic field due to a differential element dof a wire according to(A 1.3),430ssdIBdswhere 0 is the permeability of free-space, s is the displacement vector from the wiredifferential element to the point at which the field is being computed, and sBd is the differentialcontribution of d to the total magnetic field. Let )0,sin,cos(aaq  be the position vectorof a point on the loop, where  is the angle between qand the x-axis. Then, d and s canbe represented as )0,cos,sin(aad  and ),sin,cos( zayaxqrs.Substituting these vectors into (A 1.3) and integrating over ]2,0[, we get(A 1.4)  .sincos,sin,cos)sin()cos(4)(202/3222yxazzzayaxdIarBsSubstituting this result into (A 1.2), we obtain the total magnetic field(A 1.5)  .sincos,sin)(,cos)()()sin()cos(4)(10202/32220yxalvzlvzlvzayaxdvdNIarBWith some efforts, we can simplify this expression to5(A 1.6)         ,,,,,,,,,,,,,14)(2211110azayaxgazayaxgazaxaygazaxaygazayaxgazayaxgaNIrBwherealand 1g and 2g are defined as(A 1.7)   2/122220222202/12221)sin()cos(1sincos)sin()cos()()sin()cos(cos)(zyxdyxyxzrgzyxdrgFor any droplet position r, the above two integrals are easily computed numerically using thetrapezoidal or Simpson's rule. The magnet length (l = 71.4 mm) and inner coil radius (a = 7 mm) are stated in Section 5.4.However, we do not precisely known N, the number of loops, because we are treating these solidcore magnets as air filled for mathematical simplicity and because the properties of the core arenot stated by the manufacturer. As a result, N is our single free parameter and we choose it tobest match our measured magnetic field data. Figure SM-3 shows a comparison between thepredicted and experimentally measured x-component of the magnetic field for the magnet on theright of the petri dish. Figure SM-3: The predicted versus the measured magnetic field when the right-most magnet is turned on toits maximum value of 28 volts. The vertical axis shows the x-component of the magnetic field (Bx) for (x,y)locations in the petri-dish. The green mesh corresponds to the theory calculated above, the blue dots with thesmall vertical lines show the measured data and its error bars. 6Using the values of a, l, I and N above and plotting the log of the magnetic field intensitysquared gives the coloring of Figure 2. As can be seen both above and on the logarithm scale ofFigure 2, the magnetic field strength drops rapidly with distance from the electromagnet.3.2. Measurement of the Ferrofluid Droplet Magnetic Drift Coefficient k’This section describes our measurements and methods to infer the value of the magnetic driftcoefficient k’ in equation (7) for the ferrofluid droplet. The purpose here is to quantify the bulkbehavior of the droplet, not to infer the detailed physics of magnetic particle-to-particleinteractions (readers interested in that aspect can refer to, for example, [1-5]). Two droplet volumes (5 and 7.5 µL) were placed near the center of the control domain and asingle magnet was turned on to pull the droplet towards it. Time and position data of the dropletwas recorded and compared to the motion predicted by our