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Certifiable Alignment of GNSS and Local Frames via Lagrangian Duality

Baoshan Song, Matthew Giamou, Penggao Yan, Chunxi Xia, Li-Ta Hsu

TL;DR

This work introduces a globally optimal solver that transforms raw pseudo-range or Doppler measurements into a convexly relaxed problem and can numerically verify the correctness of the result, filling a gap where existing local optimizers fail.

Abstract

Estimating the absolute orientation of a local system relative to a global navigation satellite system (GNSS) reference often suffers from local minima and high dependency on satellite availability. Existing methods for this alignment task rely on abundant satellites unavailable in GNSS-degraded environments, or use local optimization methods which cannot guarantee the optimality of a solution. This work introduces a globally optimal solver that transforms raw pseudo-range or Doppler measurements into a convexly relaxed problem. The proposed method is certifiable, meaning it can numerically verify the correctness of the result, filling a gap where existing local optimizers fail. We first formulate the original frame alignment problem as a nonconvex quadratically constrained quadratic program (QCQP) problem and relax the QCQP problem to a concave Lagrangian dual problem that provides a lower cost bound for the original problem. Then we perform relaxation tightness and observability analysis to derive criteria for certifiable optimality of the solution. Finally, simulation and real world experiments are conducted to evaluate the proposed method. The experiments show that our method provides certifiably optimal solutions even with only 2 satellites with Doppler measurements and 2D vehicle motion, while the traditional velocity-based VOBA method and the advanced GVINS alignment technique may fail or converge to local optima without notice. To support the development of GNSS-based navigation techniques in robotics, all code and data are open-sourced at https://github.com/Baoshan-Song/Certifiable-Doppler-alignment.

Certifiable Alignment of GNSS and Local Frames via Lagrangian Duality

TL;DR

This work introduces a globally optimal solver that transforms raw pseudo-range or Doppler measurements into a convexly relaxed problem and can numerically verify the correctness of the result, filling a gap where existing local optimizers fail.

Abstract

Estimating the absolute orientation of a local system relative to a global navigation satellite system (GNSS) reference often suffers from local minima and high dependency on satellite availability. Existing methods for this alignment task rely on abundant satellites unavailable in GNSS-degraded environments, or use local optimization methods which cannot guarantee the optimality of a solution. This work introduces a globally optimal solver that transforms raw pseudo-range or Doppler measurements into a convexly relaxed problem. The proposed method is certifiable, meaning it can numerically verify the correctness of the result, filling a gap where existing local optimizers fail. We first formulate the original frame alignment problem as a nonconvex quadratically constrained quadratic program (QCQP) problem and relax the QCQP problem to a concave Lagrangian dual problem that provides a lower cost bound for the original problem. Then we perform relaxation tightness and observability analysis to derive criteria for certifiable optimality of the solution. Finally, simulation and real world experiments are conducted to evaluate the proposed method. The experiments show that our method provides certifiably optimal solutions even with only 2 satellites with Doppler measurements and 2D vehicle motion, while the traditional velocity-based VOBA method and the advanced GVINS alignment technique may fail or converge to local optima without notice. To support the development of GNSS-based navigation techniques in robotics, all code and data are open-sourced at https://github.com/Baoshan-Song/Certifiable-Doppler-alignment.
Paper Structure (19 sections, 2 theorems, 27 equations, 8 figures, 3 tables)

This paper contains 19 sections, 2 theorems, 27 equations, 8 figures, 3 tables.

Table of Contents

  1. INTRODUCTION
  2. Traditional alignment methods: closed-form solution
  3. Advanced alignment methods: raw GNSS measurement based methods
  4. Our porposed approach: convex relaxation with Lagrangian duality
  5. Problem Formulation
  6. Notation
  7. Original problem
  8. Quadratic formulation
  9. Certifiable alignment
  10. Certifiable alignment
  11. Lagrangian duality
  12. Duality gap and observability
  13. Experiments
  14. Simulation setup
  15. Redundant Constraints and Observability
  16. ...and 4 more sections

Key Result

Lemma 1

The unknown variables in problem (equ:dual) with redundant constraints is observable with the Hessian matrix's degree of freedom (DOF) $\ge4$. To obtain instantaneous alignment, we need at least 2 satellites and velocity along 2 axis.

Figures (8)

  • Figure 1: Illustration of alignment between GNSS $e$-frame and local $w$-frame. The key insight is to employ alignment of the local motion and its projection on the GNSS Doppler measurements which implies the global motion.
  • Figure 2: Pipeline of certifiable alignment
  • Figure 3: Optimality success rate in 2D motion. The success is confirmed when the ratio of the smallest eigenvalue to the second-smallest eigenvalue of the Hessian matrix $\mathbf{H}$ is smaller than $10^{-6}$.
  • Figure 4: Optimality success rate in 3D motion. The success is confirmed when the ratio of the smallest eigenvalue to the second-smallest eigenvalue of the Hessian matrix $\mathbf{H}$ is smaller than $10^{-6}$.
  • Figure 5: Doppler noise disturbance test with 3D motion. Noise level denotes the standard deviation of the Gaussian white noise. Alignment error means the rotation error.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Lemma 1: Necessary conditions of observability with redundant constraints
  • proof
  • Lemma 2: Necessary conditions of observability without redundant constraints
  • proof