# Mathematics in Caging of Robotics

## TL;DR

This paper explores the mathematical foundations of the caging problem in robotics, modeling it through Euclidean moves and linking its complexity to combinatorial problems and puzzles.

## Contribution

It provides a mathematical description of caging using Euclidean moves and relates its difficulty to combinatorial and puzzle-like problems, advancing theoretical understanding.

## Key findings

- Caging difficulty is linked to combinatorial complexity.
- Mathematical modeling of caging via Euclidean moves.
- Caging problem relates to wire puzzles and complexity.

## Abstract

It is a crucial problem in robotics field to cage an object using robots like multifingered hand. However the problem what is the caging for general geometrical objects and robots has not been well-described in mathematics though there were many rigorous studies on the methods how to cage an object by certain robots. In this article, we investigate the caging problem more mathematically and describe the problem in terms of recursion of the simple euclidean moves. Using the description, we show that the caging has the degree of difficulty which is closely related to a combinatorial problem and a wire puzzle. It implies that in order to capture an object by caging, from a practical viewpoint the difficulty plays an important role.