The search for invariants is a fundamental aim of scientific endeavors. These invariants, such as Newton's laws of motion, allow us to model and predict the behavior of systems across many different problems. In the nascent field of Human-Swarm Interaction (HSI), a systematic identification of fundamental invariants is still lacking. Discovering and formalizing these invariants will provide a foundation for developing, and better understanding, effective methods for HSI. We propose two invariants underlying HSI for geometric-based swarms: (1) collective state is the fundamental percept associated with a bio-inspired swarm, and (2) a human's ability to influence and understand the collective state of a swarm is determined by the balance between the span and persistence. We provide evidence of these invariants by synthesizing much of our previous work in the area of HSI with several new results, including a novel user study where users manage multiple swarms simultaneously. We also discuss how these invariants can be applied to enable more efficient and successful teaming between humans and bio-inspired collectives and identify several promising directions for future research into the invariants of HSI.

Leveraging the abilities of multiple affordable robots as a swarm is enticing because of the resulting robustness and emergent behaviors of a swarm. However, because swarms are composed of many different agents, it is difficult for a human to influence the swarm by managing individual agents. Instead, we propose that human influence should focus on (a)~managing the higher level attractors of the swarm system and (b)~managing trade-offs that appear in mission-relevant performance. We claim that managing attractors theoretically allows a human to abstract the details of individual agents and focus on managing the collective as a whole. Using a swarm model with two attractors, we demonstrate this concept by showing how limited human influence can cause the swarm to switch between attractors. We further claim that using quorum sensing allows a human to manage trade-offs between the scalability of interactions and mitigating the vulnerability of the swarm to agent failures.