Autonomous motion in changing environment, fibrations and reaction mechanisms

TL;DR

The paper addresses autonomous motion in changing environments by formulating a fibrational framework of configuration spaces $p:E\to B$, where $B$ encodes external conditions and each fibre $E_b$ represents admissible configurations under $b$. It introduces infinitesimal reaction mechanisms via infinitesimal lifting functions $\mathcal{L}: E\times_B TB\to TE$ (a nonlinear generalisation of Ehresmann connections) to generate incremental, real-time responses to environmental changes, and distinguishes scenarios with known-environment trajectories from those with unknown dynamics measured on the fly. The authors analyze complexity via parametrised topological complexity ${\mathsf{TC}}[p:E\to B]$, showing that when environmental variation is known in advance the extended motion algorithm has the same complexity as the corresponding stationary-environment problem, while uncertain changes are handled by reaction mechanisms that adapt incrementally. Through concrete examples, including a two-robot plane model and both linear and nonlinear reaction schemes, they reveal when linear mechanisms suffice for safety and when nonlinear reaction forms become necessary, providing a foundation for robust, boundary-aware autonomous navigation in dynamic fields.

Abstract

In this paper we develop further the formalism of fibrations of configuration spaces as a tool for modelling motion of autonomous systems in variable environments. We analyse the situations when the external conditions may change during the motion of the system and analyse two possibilities: (a) when the behaviour of the external conditions is known in advance; and (b) when the future changes of 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 the case (a) the complexity of the motion algorithm is the same as in the case of constant external conditions; this generalises the result of \cite{FGY}. In case (b) we introduce a new concept of a reaction mechanism which allows to take into account unexpected and unpredictable changes in the environment. A reaction mechanism is mathematically an infinitesimal lifting function on a fibre bundle, a nonlinear generalisation 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, are inevitable and ubiquitous.

Figures (8)

• Figure 1: Parametrized motion planning algorithm
• Figure 2: The dotted line represents the path $L(e', \gamma)$ and $\tilde{\gamma} =\tilde{s}(\gamma, e, e')$.
• Figure 3: Vector field $N$ near $\partial X$.
• Figure 4: A configuration of two robots $M$ and $N$.
• Figure 5: Motion of $c_M$ and $c_N$ under the infinitesimal reaction mechanism 1.

Theorems & Definitions (23)

• Definition 1.1
• Definition 2.1
• Corollary 2.2
• Theorem 2.3: See Sp, Chapter 2, §7, Theorem 8
• Theorem 2.4: See Sp, chapter 2, § 7, Theorem 13
• Theorem 3.1
• proof
• Definition 4.1
• Definition 4.2
• Theorem 4.3