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Global Positioning on Earth

Mireille Boutin, Rob Eggermont, Gregor Kemper

TL;DR

The paper challenges the assumption of a unique GP solution for users on Earth by showing that, under a sphere model, three satellites can yield up to four solutions, and that with four or more satellites, satellite configurations lying on a hyperboloid sheet can make two positions indistinguishable, producing two GP solutions. It characterizes the geometry via hyperboloids of revolution and focal points, and provides methods to count and locate solutions on/near a sphere, including a quadratic-in-$t$ approach and linear-algebraic reductions for $m\ge4$ satellites. The authors propose practical, numerically stable strategies (SoS, RSoS, TLS-based approaches) to obtain the correct number of solutions near or on a sphere, supported by numerical experiments that compare solution methods and demonstrate robustness near problematic configurations. The work has implications for GPS integrity and solver design, highlighting non-uniqueness in realistic satellite configurations and offering concrete algorithms to determine the full solution set in near-spherical geometries.

Abstract

Contrary to popular belief, the global positioning problem on earth may have more than one solutions even if the user position is restricted to a sphere. With 3 satellites, we show that there can be up to 4 solutions on a sphere. With 4 or more satellites, we show that, for any pair of points on a sphere, there is a family of hyperboloids of revolution such that if the satellites are placed on one sheet of one of these hyperboloid, then the global positioning problem has both points as solutions. We give solution methods that yield the correct number of solutions on/near a sphere.

Global Positioning on Earth

TL;DR

The paper challenges the assumption of a unique GP solution for users on Earth by showing that, under a sphere model, three satellites can yield up to four solutions, and that with four or more satellites, satellite configurations lying on a hyperboloid sheet can make two positions indistinguishable, producing two GP solutions. It characterizes the geometry via hyperboloids of revolution and focal points, and provides methods to count and locate solutions on/near a sphere, including a quadratic-in- approach and linear-algebraic reductions for satellites. The authors propose practical, numerically stable strategies (SoS, RSoS, TLS-based approaches) to obtain the correct number of solutions near or on a sphere, supported by numerical experiments that compare solution methods and demonstrate robustness near problematic configurations. The work has implications for GPS integrity and solver design, highlighting non-uniqueness in realistic satellite configurations and offering concrete algorithms to determine the full solution set in near-spherical geometries.

Abstract

Contrary to popular belief, the global positioning problem on earth may have more than one solutions even if the user position is restricted to a sphere. With 3 satellites, we show that there can be up to 4 solutions on a sphere. With 4 or more satellites, we show that, for any pair of points on a sphere, there is a family of hyperboloids of revolution such that if the satellites are placed on one sheet of one of these hyperboloid, then the global positioning problem has both points as solutions. We give solution methods that yield the correct number of solutions on/near a sphere.
Paper Structure (8 sections, 2 theorems, 10 equations, 3 figures)

This paper contains 8 sections, 2 theorems, 10 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Two solutions on a sphere
  4. Global positioning on or near a sphere
  5. On a Sphere With Three satellites
  6. On a Sphere With Four or More Satellites
  7. Near a Sphere
  8. Numerical experiments

Key Result

Theorem 1

Let $\mathbf{x} \neq \mathbf{x}' \in \mathop{\mathrm{\mathbb{R}}}\nolimits^3$. Let $\mathbf{m} = \frac{1}{2}(\mathbf{x} + \mathbf{x}')$, let $c = \| \mathbf{x} - \mathbf{x}'\|$ and let $\tilde{\mathbf{u}} = \frac{1}{c}(\mathbf{x} - \mathbf{x}')$. Then for every $0 < a < c$, the hyperboloid of revolu

Figures (3)

  • Figure 1: (Left) Two solutions (red dots) to the GP problem on a sphere and one possible locus of forbidden satellite positions. (Right) Visibly non-circular loci.
  • Figure 3: Mean distance to user position for ILS, SoS and RSoS when user is on the earth sphere (left) and $\sim 6$km above the earth (right).
  • Figure 4: Mean distance between the two solutions of SoS and RSoS when the user is on the earth sphere (left) and $\sim 6$km above the earth sphere (right).

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • proof
  • Remark 3
  • Example 4