Active Sampling based Safe Identification of Dynamical Systems using Extreme Learning Machines and Barrier Certificates

Learning the dynamical system (DS) model from data that preserves dynamical system properties is an important problem in many robot learning applications. Typically, the joint data coming from cyber-physical systems, such as robots have some underlying DS properties associated with it, e.g., convergence, invariance to a set, etc. In this paper, a model learning method is developed such that the trajectories of the DS are invariant in a given compact set. Such invariant DS models can be used to generate trajectories of the robot that will always remain in a prescribed set. In order to achieve invariance to a set, Barrier certificates are employed. The DS is approximated using Extreme Learning Machine (ELM), and a parameter learning problem subject to Barrier certificates enforced at all the points in the prescribed set is solved. To solve an infinite constraint problem for enforcing Barrier Certificates at every point in a given compact set, a modified constraint is developed that is sufficient to hold the Barrier certificates in the entire set. An active sampling strategy is formulated to minimize the number of constraints in learning. Simulation results of ELM learning with and without Barrier certificates are presented which show the invariance property being preserved in the ELM learning when learning procedure involves Barrier constraints. The method is validated using experiments conducted on a robot arm recreating invariant trajectories inside a prescribed set.