A novel methodology is presented for organizing swarms of robots into a formation utilizing artificial potential fields generated from normal and sigmoid functions. These functions construct the surface which swarm members travel on, controlling the overall swarm geometry and the individual member spacing. Nonlinear limiting functions are defined to provide tighter swarm control by modifying and adjusting a set of control variables forcing the swarm to behave according to set constraints, formation and member spacing. The swarm function and limiting functions are combined to control swarm formation, orientation, and swarm movement as a whole. Parameters are chosen based on desired formation as well as user defined constraints. This approach compared to others, is simple, computationally efficient, scales well to different swarm sizes, to heterogeneous systems, and to both centralized and decentralized swarm models. Simulation results are presented for a swarm of four and ten particles following circle, ellipse and wedge formations. Experimental results are also included with a swarm of four unmanned ground vehicles (UGV) as well as UGV swarm and unmanned aerial vehicle (UAV) coordination.