Autonomous Motion in Changing Environments: Fibrations and Reaction Mechanisms

Abstract.

In this paper, we develop further the formalism of fibrations of configuration spaces as a tool for modeling the motion of autonomous systems in variable environments. We analyze the situations when the external conditions may change during the motion of the system and analyze two possibilities: (a) the behavior of the external conditions is known in advance; and (b) the future changes in the external conditions are unknown, but we can measure the current state and the current velocity of the external conditions at every moment of time. We prove that in case (a), the complexity of the motion algorithm is the same as in the case of constant external conditions. In case (b), we introduce a new concept of a reaction mechanism that allows for taking into account unexpected and unpredictable changes in the environment. A reaction mechanism is mathematically an infinitesimal lifting function on a fiber bundle, a nonlinear generalization of the classical concept of an Ehresmann connection. We illustrate these notions by examples which show that nonlinear infinitesimal lifting function (reaction mechanisms) appear naturally and are inevitable and ubiquitous.