In this paper, we present a strategy for organizing swarms of unmanned vehicles into a formation by utilizing artificial potential fields that were generated from normal and sigmoid functions. These functions construct the surface on which swarm members travel, 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 that force the swarm to behave according to set constraints, formation, and member spacing. The artificial potential functions and limiting functions are combined to control swarm formation, orientation, and swarm movement as a whole. Parameters are chosen based on desired formation and user-defined constraints. This approach is computationally efficient and 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 10 and 40 robots that follow circle, ellipse, and wedge formations. Experimental results are included to demonstrate the applicability of the approach on a swarm of four custom-built unmanned ground vehicles (UGVs).