Formation Control Design and Optimization for Uncertain Underactuated Robots Team Using Cooperative Game Theory

This manuscript introduces an innovative approach to address the formation control challenges for a specific category of underactuated robots (UARs). It transforms the formation control problem into a constraint-following problem with three basic constraints: (1) The configuration constraints (CCs), which are to induce behaviors for each UAR. (2) The formation constraints (FCs), which are used to maintain the team formation. (3) The trajectory constraints (TCs), which are imposed to make the team move with a predefined formation. The UAR team is assumed to suffer from time-varying while bounded uncertainty, and the uncertainty bound is unknown. Invoking an interval method, we build a dynamic model for the uncertain UAR team. Considering the three basic constraints in a unified form, we then employ an improved adaptive formation controller based on an extended Udwadia-control (UC) scheme for the uncertain UAR team. The controller can guarantee the UAR team deterministic convergence, which is verified by an explicit Lyapunov analysis. As there are three tunable control gains in the proposed controller, this paper adopts a cooperative game (CG) algorithm to obtain the optimal formation control. Regarding the tunable parameters as three players in a CG with their cost functions representing the tracking performance or the energy cost, the optimal formation control is realized by achieving an equilibrium among players. Simulations on a UAR team consisting of three inverted pendulum robots (IPRs) are performed to test the designed optimal formation controller.