This paper presents an approximated or ε-saddle point solution for a nonlinear game proposed to solve the coordination of a set of planar robots. The dynamic equation of each individual robot is given by the polynomial approximation of the nonlinear model, this introduces interesting challenges in the coordination control problem. Due to the difficulties in finding an exact solution, we explore the possibilities of the so-called state-dependent Riccati equations to find a near or ε-saddle point solution that effectively drives of the robots to a particular coordinated movement. A pre-designed trajectory is given to one of the robots in order to achieve the coordination by the group of robots. The proposed solution is given as a polynomial Riccati-like state-dependent differential equation which utilizes a p-linear form tensor representation for its polynomial part. A numerical example is presented to illustrate the effectiveness of the approach.

A simple, fast and robust Passive Velocity Field Control (PVFC) scheme for contour tracking in 2D and 3D is presented. A model-free and chattering-free second order sliding mode controller is proposed, which enforces an invariant manifold for all time such that the generalized velocity tracks the spatial velocity field. A constructive simple and intuitive method to design this velocity field is built upon a fuzzy aggregation of orthogonal vectors to guide the end-effector subject to a dissipative map. Stability properties in the sense of Lyapunov establish the fast and yet smooth tracking with typical robustness of sliding modes. In this way, the closed-loop system exhibits the mildness of PVFC but it evades the subtle complexities of the original approach PVFC. A simulation study is discussed, and experimental results highlight the characteristics of the proposed approach.