Inverse KKT: Learning cost functions of manipulation tasks from demonstrations

Inverse optimal control (IOC) assumes that demonstrations are the solution to an optimal control problem with unknown underlying costs, and extracts parameters of these underlying costs. We propose the framework of inverse Karush–Kuhn–Tucker (KKT), which assumes that the demonstrations fulfill the KKT conditions of an unknown underlying constrained optimization problem, and extracts parameters of this underlying problem. Using this we can exploit the latter to extract the relevant task spaces and parameters of a cost function for skills that involve contacts. For a typical linear parameterization of cost functions this reduces to a quadratic program, ensuring guaranteed and very efficient convergence, but we can deal also with arbitrary non-linear parameterizations of cost functions. We also present a non-parametric variant of inverse KKT that represents the cost function as a functional in reproducing kernel Hilbert spaces. The aim of our approach is to push learning from demonstration to more complex manipulation scenarios that include the interaction with objects and therefore the realization of contacts/constraints within the motion. We demonstrate the approach on manipulation tasks such as sliding a box, closing a drawer and opening a door.