Minimal Time-of-Arrival self-localization problems

The m-senders/n-receivers (ms/nr) Time-of-Arrival (ToA) self-localization problem involves determining the relative positions of m senders $\mathbf s_i$ and n receivers $\mathbf r_k$ in 2D or 3D space, given pairwise distance measurements $d_{ik}$ derived from signal propagation times.

Illustration of the 3D 4s/6r ToA self-localization problem

A problem is considered minimal when it admits a finite (but non‑zero) number of solutions for generic distance measurements. This repository contains MATLAB solvers for the following minimal ToA problems:

• 2D 3s/3r
• 3D 4s/6r

The implementations have been tested on

• MATLAB R2019b

The solvers have been generated using the automatic solver generator from https://github.com/martyushev/eliminationTemplates

If you use this code, please cite:

@article{martyushev2026implicitization,
    title={An implicitization-based solution to the minimal 4s/6r ToA problem using Cayley--Menger determinants},
    author={Martyushev, Evgeniy},
    journal={http://arxiv.org/abs/2606.20840 },
    volume={},
    pages={},
    year={2026}
}