Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A compositional framework for classical kinematic systems

Andrea Abeje-Stine, David Weisbart

TL;DR

The work develops a compositional, category-theoretic framework for open classical kinematic systems, modeling subsystems as ACM-diagrams and their embeddings via a rigid inclusion category $\mathsf{Kin}(\mathcal{F})$. The core contributions include new existence criteria for global configuration spaces as $\mathcal{F}$-limits over ACM-diagrams, the introduction of weldings and reductions to build complex ACM-diagrams, and the establishment of a rigid-inclusion formalism that makes open systems composable. The Newton Daemon and $\mathsf{SE}(n)$-based linkage analyses demonstrate the framework's applicability to time-varying constraints and planar/spatial linkages, including no-go results for two-actor universal joints and a positive construction for a spatial cylindrical joint. Overall, the paper provides a rigorous, scalable basis for modeling, composing, and analyzing open CMK systems with feedback and geometric constraints, with clear implications for both static structure and dynamic evolution of mechanisms.

Abstract

Our aim is to introduce a category-theoretic framework sufficiently general to describe a wide variety of open kinematic systems in classical mechanics while uniquely characterizing systems with specified simplest components. The framework models open systems as morphisms in a category $\mathsf{Kin}(\mathcal{F})$, where composition encodes relationships between subsystems and their embedding into larger systems. Unlike previous approaches, the framework supports a precise treatment of geometric constraints, enabling the characterization of systems with feedback. A consequence is a structural formulation of lower kinematic pairs that clarifies system interactions.

A compositional framework for classical kinematic systems

TL;DR

The work develops a compositional, category-theoretic framework for open classical kinematic systems, modeling subsystems as ACM-diagrams and their embeddings via a rigid inclusion category . The core contributions include new existence criteria for global configuration spaces as -limits over ACM-diagrams, the introduction of weldings and reductions to build complex ACM-diagrams, and the establishment of a rigid-inclusion formalism that makes open systems composable. The Newton Daemon and -based linkage analyses demonstrate the framework's applicability to time-varying constraints and planar/spatial linkages, including no-go results for two-actor universal joints and a positive construction for a spatial cylindrical joint. Overall, the paper provides a rigorous, scalable basis for modeling, composing, and analyzing open CMK systems with feedback and geometric constraints, with clear implications for both static structure and dynamic evolution of mechanisms.

Abstract

Our aim is to introduce a category-theoretic framework sufficiently general to describe a wide variety of open kinematic systems in classical mechanics while uniquely characterizing systems with specified simplest components. The framework models open systems as morphisms in a category , where composition encodes relationships between subsystems and their embedding into larger systems. Unlike previous approaches, the framework supports a precise treatment of geometric constraints, enabling the characterization of systems with feedback. A consequence is a structural formulation of lower kinematic pairs that clarifies system interactions.
Paper Structure (15 sections, 28 theorems, 313 equations)

This paper contains 15 sections, 28 theorems, 313 equations.

Table of Contents

  1. Introduction
  2. Composition of systems
  3. Constraint skeletons of systems
  4. Compositional description of linearly-ordered systems
  5. ACM-diagrams and their $\mathcal{F}$-limits
  6. ACM-diagrams and weldings
  7. $\mathcal{F}$-limits of ACM-diagrams
  8. The rigid inclusion category
  9. The rigid inclusion category
  10. Open CMK systems as rigid inclusions
  11. CMK systems and open CMK systems
  12. The Newton Daemon
  13. Planar and spatial linkages
  14. The universal joint.
  15. Sliding hinge motion set.

Key Result

Theorem 2.1

If ${{\mathsf{C}}}$ has $\mathcal{F}$-pullbacks and is span-tight, then $\circ$ defines composition in a category $\mathsf{Span}(\mathcal{F})$ whose objects are the objects of ${{\mathsf{C}}}$, whose morphisms are isomorphism classes of spans in ${{\mathsf{C}}}$, and whose identity morphism at any o

Theorems & Definitions (89)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.1
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3: Constrained actor
  • Definition 3.4
  • ...and 79 more