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Planar Steering of a Single Ferrofluid Drop



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Planar Steering of a Single Ferrofluid Drop by Optimal Minimum Power Dynamic Feedback Control of Four Electromagnets at a Distance R 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 benshap umd edu Supplementary Material 1 1 Additional Figures Experiment Results 10 mm Figure 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 2 Figure SM 2 Control of a medium size 20 L 1 7 mm radius ferrofluid droplet faster along a line square and spiral path 3 2 Movies of Experimental Results Movie of Figure SM 1 line SM 1a FFsmallslow line mpg Movie of Figure SM 1 square SM 1b FFsmallslow square mpg Movie of Figure SM 1 spiral SM 1c FFsmallslow spiral mpg Movie of Figure SM 2 line SM 2a FFmedfast line mpg Movie of Figure SM 2 square SM 2b FFmedfast square mpg Movie of Figure SM 2 spiral SM 2c FFmedfast spiral mpg Movie of Figure 5 line 5a FFmedslow line mpg Movie of Figure 5 square 5b FFmedslow square mpg Movie of Figure 5 spiral 5c FFmedslow spiral mpg Movie of Figure 6 line 6a FFsmallfast line mpg Movie of Figure 6 square 6b FFsmallfast square mpg Movie of Figure 6 spiral 6c FFsmallfast spiral mpg Movie of Figure 8 UMD 8 FFmedslow UMD mpg 4 3 Appendix 3 1 Computing the Magnetic Field for a Single Electromagnet Here we derive and compute the magnetic field around a single electromagnet Figure 2 We consider 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 is along the z axis of a coordinate system such that the solenoid is extended from z 0 toward z l We determine the magnetic field B r at any arbitrary point r x y z with z 0 Let Bs r be the magnetic field due to a



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