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Singularity and Error Analysis of a Simple Quaternion Estimator

Caitong Peng, Daniel Choukroun

TL;DR

This work addresses attitude estimation from two vector observations using a simple, closed-form quaternion estimator derived in four-dimensional quaternion algebra. The authors provide a thorough treatment of singularities via explicit cases and a sequential-rotation remedy to maintain robustness. They develop deterministic and stochastic error analyses, delivering exact and second-order bias and covariance expressions, and extend the Gaussian-noise analysis to exact fourth-order covariance terms. The result is a fast, globally valid estimator with transparent geometric insight and highly accurate error characterization, applicable to nanosatellites with limited sensor suites and to real-time attitude determination tasks.

Abstract

A novel single-frame quaternion estimator processing two vector observations is introduced. The singular cases are examined, and appropriate rotational solutions are provided. Additionally, an alternative method involving sequential rotation is introduced to manage these singularities. The simplicity of the estimator enables clear physical insights and a closed-form expression for the bias as a function of the quaternion error covariance matrix. The covariance could be approximated up to second order with respect to the underlying measurement noise assuming arbitrary probability distribution. The current note relaxes the second-order assumption and provides an expression for the error covariance that is exact to the fourth order, under the assumption of Gaussian distribution. A comprehensive derivation of the individual components of the quaternion additive error covariance matrix is presented. This not only provides increased accuracy but also alleviates issues related to singularity.

Singularity and Error Analysis of a Simple Quaternion Estimator

TL;DR

This work addresses attitude estimation from two vector observations using a simple, closed-form quaternion estimator derived in four-dimensional quaternion algebra. The authors provide a thorough treatment of singularities via explicit cases and a sequential-rotation remedy to maintain robustness. They develop deterministic and stochastic error analyses, delivering exact and second-order bias and covariance expressions, and extend the Gaussian-noise analysis to exact fourth-order covariance terms. The result is a fast, globally valid estimator with transparent geometric insight and highly accurate error characterization, applicable to nanosatellites with limited sensor suites and to real-time attitude determination tasks.

Abstract

A novel single-frame quaternion estimator processing two vector observations is introduced. The singular cases are examined, and appropriate rotational solutions are provided. Additionally, an alternative method involving sequential rotation is introduced to manage these singularities. The simplicity of the estimator enables clear physical insights and a closed-form expression for the bias as a function of the quaternion error covariance matrix. The covariance could be approximated up to second order with respect to the underlying measurement noise assuming arbitrary probability distribution. The current note relaxes the second-order assumption and provides an expression for the error covariance that is exact to the fourth order, under the assumption of Gaussian distribution. A comprehensive derivation of the individual components of the quaternion additive error covariance matrix is presented. This not only provides increased accuracy but also alleviates issues related to singularity.
Paper Structure (17 sections, 63 equations, 4 figures, 2 tables)

This paper contains 17 sections, 63 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Quaternion Estimation using Two Vector Observations
  3. Preliminaries
  4. Closed-form quaternion estimator
  5. Singular Cases
  6. Sequential rotations
  7. Development of the Bias and covariance matrix
  8. Definitions and notations
  9. Deterministic analysis
  10. Exact formulas
  11. Approximation formulas
  12. Random analysis
  13. Biases
  14. The covariance matrix $P_{_{\!\!\!\!{{\bf \Delta}\! \,\bar{\bf q} }}}$
  15. The other covariance matrices
  16. ...and 2 more sections

Figures (4)

  • Figure 1: Case B. Rotation around VM 1.
  • Figure 2: Case B. Rotation around VM 1. View from above.
  • Figure 3: Case D. Rotation that maintains ${\bf d}_{_{1}}$ and ${\bf d}_{_{2}}$ parallel.
  • Figure 4: Rotation where ${\bf d}_{_{1}}$ and ${\bf d}_{_{2}}$ are not parallel.