Gauss's Principle With Inequality Constraints for Multiagent Navigation and Control

Multiagent navigation systems present opportunities for many applications due to their agility and cooperation. In any multiagent navigation system, it is critical that actual interagent collisions are strictly prevented. In this article, we present a solution to the 2-D multiagent navigation problem with collision avoidance. Our solution to this problem is based on a novel extension to Gauss's principle of least constraint (GPLC), in which a fixed set of strict equality constraints is replaced by time-varying sets of active inequality constraints. To the best of our knowledge, this is the first instance that extends GPLC with dynamic incorporation and stabilization of active inequality constraints and with actuator delay and saturation. Herein, the dynamics of a collision-free multiagent system satisfies the Karush-Kuhn-Tucker conditions. Active inequality constraints enforce collision avoidance, leader following, and agglomeration behaviors, and they are stabilized using Baumgarte's error stabilization approach. We show that in dense configurations, the positional arrangement of the agents can lead to linearly dependent constraints, and we propose specialized solutions involving QR decomposition and regularization. The efficacy and efficiency of the proposed method are demonstrated by a dimensional analysis of a worst-case scenario and numerical studies of up to 100 agents tracking a prescribed virtual leader.