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Robot Positioning Using Torus Packing for Multisets

Chung Shue Chen, Peter Keevash, Sean Kennedy, Élie de Panafieu, Adrian Vetta

TL;DR

The paper tackles robot position reconstruction on a torus by observing multisets of colors in a fixed window, shifting away from the classic full-pattern (de Bruijn) approach. It develops a theoretical framework built around grid colourings and vector sum packings to achieve near-optimal room-size scaling: for fixed dimension $d$ and palette size $k$, the grid size satisfies $n \sim C_k^{1/d}\,m^{k-1}$ with $C_k = \left(\frac{2}{k-1}\right)^{k-1}$, while enabling localization from multisets in a constant number of arithmetic operations. The core contributions are (i) the reduction from grid colouring to vector sum packings, (ii) a explicit near-optimal construction with provable injectivity, and (iii) a constant-time decoding algorithm, together with complexity analysis and a Python implementation. This framework supports low-cost, fast-positioning solutions for indoor and ambient IoT scenarios by using color intensities rather than exact patterns, with potential robustness extensions and new window geometries for practical deployments.

Abstract

We consider the design of a positioning system where a robot determines its position from local observations. This is a well-studied problem of considerable practical importance and mathematical interest. The dominant paradigm derives from the classical theory of de Bruijn sequences, where the robot has access to a window within a larger code and can determine its position if these windows are distinct. We propose an alternative model in which the robot has more limited observational powers, which we argue is more realistic in terms of engineering: the robot does not have access to the full pattern of colours (or letters) in the window, but only to the intensity of each colour (or the number of occurrences of each letter). This leads to a mathematically interesting problem with a different flavour to that arising in the classical paradigm, requiring new construction techniques. The parameters of our construction are optimal up to a constant factor, and computing the position requires only a constant number of arithmetic operations.

Robot Positioning Using Torus Packing for Multisets

TL;DR

The paper tackles robot position reconstruction on a torus by observing multisets of colors in a fixed window, shifting away from the classic full-pattern (de Bruijn) approach. It develops a theoretical framework built around grid colourings and vector sum packings to achieve near-optimal room-size scaling: for fixed dimension and palette size , the grid size satisfies with , while enabling localization from multisets in a constant number of arithmetic operations. The core contributions are (i) the reduction from grid colouring to vector sum packings, (ii) a explicit near-optimal construction with provable injectivity, and (iii) a constant-time decoding algorithm, together with complexity analysis and a Python implementation. This framework supports low-cost, fast-positioning solutions for indoor and ambient IoT scenarios by using color intensities rather than exact patterns, with potential robustness extensions and new window geometries for practical deployments.

Abstract

We consider the design of a positioning system where a robot determines its position from local observations. This is a well-studied problem of considerable practical importance and mathematical interest. The dominant paradigm derives from the classical theory of de Bruijn sequences, where the robot has access to a window within a larger code and can determine its position if these windows are distinct. We propose an alternative model in which the robot has more limited observational powers, which we argue is more realistic in terms of engineering: the robot does not have access to the full pattern of colours (or letters) in the window, but only to the intensity of each colour (or the number of occurrences of each letter). This leads to a mathematically interesting problem with a different flavour to that arising in the classical paradigm, requiring new construction techniques. The parameters of our construction are optimal up to a constant factor, and computing the position requires only a constant number of arithmetic operations.
Paper Structure (18 sections, 10 theorems, 23 equations, 5 figures)

This paper contains 18 sections, 10 theorems, 23 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Definitions and Results
  3. Related Work
  4. Motivation and Engineering Aspects
  5. Overview
  6. From Vector Sum Packing to Grid Colouring
  7. The Separation Property
  8. Separation via Vector Sum Packing
  9. An Example
  10. Vector Sum Packing
  11. Profiles and Duals
  12. Generating a Vector Sum Packing
  13. Summary and Complexity
  14. Conclusion
  15. Engineering Implementation Details
  16. ...and 3 more sections

Key Result

Theorem 9

Fix a dimension $d \geq 2$ and a number of colours $k$ of the form $b d + 1$ for some $b \geq 1$. For any window size $m$ multiple of $2 (k-1)$, there is an $(n, m, d, k)$-grid colouring $W$ (explicitly constructed in the proof) with Furthermore, for any $\bm{x}$ in $\mathds{Z}^d$, given the multiset of colours in $(W_{\bm{x} + \bm{c}})_{\bm{c} \in [0, m-1]^d}$ one can compute $\bm{x}$ mod $n$ wi

Figures (5)

  • Figure 2: Illustration of the steps of the proof of \ref{['th:vector sum packing:to:grid:colouring']} on a grid of dimension $d = 2$, size $n = 8$, window size $m = 4$, number of colours $k = 3$.
  • Figure 3: The recursive construction of the matrix $M^{(b)}$. The indices $j$ for the sets $I_j$ on top of the figure are first increasing even numbers, then decreasing odd numbers.
  • Figure 4: The system schematic and localization method (Side View) OGC23. The width of the projected area is denoted by $v$. Note that the picture is not to scale.
  • Figure 5: Robot moving in a factory hall, activating tags that send their ID via backscattering Nokia23.
  • Figure 6: Grid colouring of size $256$, window size $8$ and number of colours $5$. It corresponds to the parameters $d=2$, $b=2$ and $t=1$.

Theorems & Definitions (21)

  • Example 1
  • Definition 2
  • Example 3
  • Example 4
  • Example 5
  • Definition 6
  • Definition 7
  • Example 8
  • Theorem 9
  • Definition 11
  • ...and 11 more