Actuation of Ferrobots in a Plane for Independent and Swarm Motion Using a Grid of Electromagnets

We present a robotic platform in which multiple ferromagnetic bots can be moved independently on a plane above a grid of iron-core solenoid electromagnets. Our magnetic robots are iron balls fitted with custom-designed mounts, which we call ferrobots. We chose ferromagnets because when several of them are close together, they can still be separated by turning off the nearest solenoids. A solenoid influences only the ferrobots near it, thus enabling different specified forces applied on them. A significant challenge addressed in this work is calculating the forces on the ferromagnets. This is because the magnetic force on a ferro object is a function of its magnetization and the external field. Analytical calculation of the magnetization in the presence of non-uniform external fields is difficult. Furthermore, large computation is involved if finite element analysis is used. So, in this work, we present an approximate but quick method for calculating the force on a ferromagnetic ball due to a general magnetic field. By assuming that the field inside a ferromagnetic ball can be expressed as a linear combination of Legendre and other polynomials, we show that the magnetization of the ferromagnetic ball is also a polynomial. Furthermore, we express the polynomial coefficients of magnetization in terms of those of the magnetic field. The ferromagnetic force then turns out to be an easily computable analytical expression. The developed force model is used to calculate the current in the solenoids for required forces. The results of our semi-analytical method are compared with finite element analysis with better than 4% error. To further validate our calculation, we developed a two-axis force sensor using an XY-decoupled compliant stage to measure magnetic forces of lower magnitudes in our setup. During the motion of a ferrobots towards a coil, it would build up momentum due to increased force, causing it to overshoot and oscillate at the endpoint. Therefore, we have developed a braking mechanism on the ferrobots mount that responds to the downward magnetic force from the coil that decelerates the ferrobots. We show a reduced overshoot, no oscillation, and fast travel times because of the responsive braking mechanism. Using our platform and force-modelling, we demonstrate the motion of a ferrobots along bespoke trajectories by applying forces as needed. We also demonstrate the swarm motion of multiple ferromagnets and their ability to change the shape of an enclosing elastic band. Together they function as a shape-changing robot that can navigate the workspace, changing shape as required and thereby grasp and manipulate other objects.