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Runge-Kutta Approximations for Direct Coning Compensation Applying Lie Theory

John A. Christian, Michael R. Walker, Wyatt Bridgman, Michael J. Sparapany

TL;DR

This paper addresses coning errors in strapdown IMU gyro integration by framing attitude propagation within a Lie-theoretic context on $SO(3)$. It shows that the Bortz equation can be interpreted via the inverse right Jacobian and develops Runge-Kutta based coning corrections, enabled by modeling angular velocity with polynomial rate models. A key finding is that the classical single-speed coning correction is a special case of RK4, while higher-order corrections require additional gyro measurements to unlock full accuracy. The approach provides a scalable path to higher-order coning compensation, potentially enabling larger integration steps with preserved fidelity.

Abstract

The integration of gyroscope measurements is an essential task for most navigation systems. Modern vehicles typically use strapdown systems, such that gyro integration requires coning compensation to account for the sensor's rotation during the integration. Many coning compensation algorithms have been developed and a few are reviewed. This work introduces a new class of coning correction algorithm built directly from the classical Runge-Kutta integration routines. A simple case is shown to collapse to one of the most popular coning algorithms and a clear procedure for generating higher-order algorithms is presented.

Runge-Kutta Approximations for Direct Coning Compensation Applying Lie Theory

TL;DR

This paper addresses coning errors in strapdown IMU gyro integration by framing attitude propagation within a Lie-theoretic context on . It shows that the Bortz equation can be interpreted via the inverse right Jacobian and develops Runge-Kutta based coning corrections, enabled by modeling angular velocity with polynomial rate models. A key finding is that the classical single-speed coning correction is a special case of RK4, while higher-order corrections require additional gyro measurements to unlock full accuracy. The approach provides a scalable path to higher-order coning compensation, potentially enabling larger integration steps with preserved fidelity.

Abstract

The integration of gyroscope measurements is an essential task for most navigation systems. Modern vehicles typically use strapdown systems, such that gyro integration requires coning compensation to account for the sensor's rotation during the integration. Many coning compensation algorithms have been developed and a few are reviewed. This work introduces a new class of coning correction algorithm built directly from the classical Runge-Kutta integration routines. A simple case is shown to collapse to one of the most popular coning algorithms and a clear procedure for generating higher-order algorithms is presented.

Paper Structure

This paper contains 13 sections, 64 equations, 3 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Attitude Representations and Kinematics
  3. Conventional Aerospace Framework
  4. Lie-Theoretic Framework
  5. Conventional Coning Correction
  6. Simplification of the Coning Correction
  7. Approximate Solutions to the Coning Correction Term
  8. Numerical Methods for Coning Correction
  9. Runge-Kutta methods and sampling requirements
  10. Approximating $\boldsymbol{\omega}$ from $\Delta\boldsymbol{\theta}$
  11. Results
  12. Conclusion
  13. Analytic Derivation of Single-Speed Coning Correction

Figures (3)

  • Figure 1: Illustration of time intervals.
  • Figure 2: Sampling of integrated rate and integration period of interest.
  • Figure 3: Algorithm performance on discrete measurements. ExMid, RK3 and RK4 are employ the Runge-Kutta algorithm \ref{['alg:RK']} with Tableaux \ref{['tbl:expmid']}, \ref{['tbl:rk3']}, and \ref{['tbl:rk4']}, respectively. SingleSpeed $\Theta_2$ (traditional coning compensation) and $\Theta_3$ use Eqs. \ref{['eq:RK4:Theta2']} and \ref{['eq:RK4:Theta3']}, respectively.