Multirobot Symmetric Formations for Gradient and Hessian Estimation With Application to Source Seeking

This paper deals with the problem of estimating in a collaborative way the gradient and the Hessian matrix of an unknown signal via noisy measurements collected by a group of robots. We propose symmetric formations with a reduced number of robots for both the two-dimensional (2-D) and the three-dimensional (3-D) cases, such that the gradient and Hessian of the signal are estimated at the center of the formation via simple computation on local quantities independently of the orientation of the formation. If only gradient information is required, the proposed formations are suitable for mobile robots that need to move in circular motion. We also provide explicit bounds for the approximation error and for the noise perturbation that can be used to optimally scale the formation radius. Numerical simulations illustrate the performance of the proposed strategy for source seeking against alternative solutions available in the literature and show how Hessian estimation can provide faster convergence even in the presence of noisy measurements.