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Publication# Imitation learning of Globally Stable Non-Linear Point-to-Point Robot Motions using Nonlinear Programming

Abstract

This paper presents a method to learn arbitrary discrete motions from a set of demonstrations. We model a motion as a nonlinear autonomous (i.e. time-invariant) dynamical system, and define the sufficient conditions to make such a system globally asymptotically stable at the target. The convergence of all trajectories is ensured starting from any point in the operational space. Estimation of the model's parameters is done through non-linear programming by formulating the problem as an optimization under non-linear constraints. Being time-invariant and globally stable, the system is able to handle both temporal and spatial perturbations, while performing the motion as close to the demonstrations as possible. The method is evaluated through a set of robotic experiments.

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Nonlinear programming

In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or stationary points) of an objective function over a set of unknown real variables and conditional to the satisfaction of a system of equalities and inequalities, collectively termed constraints. It is the sub-field of mathematical optimization that deals with problems that are not linear.

Lyapunov stability

Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point stay near forever, then is Lyapunov stable. More strongly, if is Lyapunov stable and all solutions that start out near converge to , then is said to be asymptotically stable (see asymptotic analysis).

Linear programming

Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints.

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