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The Invariant Rauch-Tung-Striebel Smoother

Niels van der Laan, Mitchell Cohen, Jonathan Arsenault, James Richard Forbes

TL;DR

An invariant Rauch-Tung-Striebel (IRTS) smoother applicable to systems with states that are an element of a matrix Lie group and compared to invariant and multiplicative versions of the Gauss-Newton approach to solving the batch state estimation problem.

Abstract

This paper presents an invariant Rauch-Tung- Striebel (IRTS) smoother applicable to systems with states that are an element of a matrix Lie group. In particular, the extended Rauch-Tung-Striebel (RTS) smoother is adapted to work within a matrix Lie group framework. The main advantage of the invariant RTS (IRTS) smoother is that the linearization of the process and measurement models is independent of the state estimate resulting in state-estimate-independent Jacobians when certain technical requirements are met. A sample problem is considered that involves estimation of the three dimensional pose of a rigid body on SE(3), along with sensor biases. The multiplicative RTS (MRTS) smoother is also reviewed and is used as a direct comparison to the proposed IRTS smoother using experimental data. Both smoothing methods are also compared to invariant and multiplicative versions of the Gauss-Newton approach to solving the batch state estimation problem.

The Invariant Rauch-Tung-Striebel Smoother

TL;DR

An invariant Rauch-Tung-Striebel (IRTS) smoother applicable to systems with states that are an element of a matrix Lie group and compared to invariant and multiplicative versions of the Gauss-Newton approach to solving the batch state estimation problem.

Abstract

This paper presents an invariant Rauch-Tung- Striebel (IRTS) smoother applicable to systems with states that are an element of a matrix Lie group. In particular, the extended Rauch-Tung-Striebel (RTS) smoother is adapted to work within a matrix Lie group framework. The main advantage of the invariant RTS (IRTS) smoother is that the linearization of the process and measurement models is independent of the state estimate resulting in state-estimate-independent Jacobians when certain technical requirements are met. A sample problem is considered that involves estimation of the three dimensional pose of a rigid body on SE(3), along with sensor biases. The multiplicative RTS (MRTS) smoother is also reviewed and is used as a direct comparison to the proposed IRTS smoother using experimental data. Both smoothing methods are also compared to invariant and multiplicative versions of the Gauss-Newton approach to solving the batch state estimation problem.
Paper Structure (17 sections, 1 theorem, 51 equations, 3 figures)

This paper contains 17 sections, 1 theorem, 51 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matrix Lie Groups
  4. RTS Smoothing
  5. Multiplicative RTS Smoothing
  6. Invariant RTS Smoothing
  7. L-IRTS Smoothing
  8. R-IRTS Smoothing
  9. Sample Problem: $SE(3)$ with Bias
  10. Problem Formulation
  11. Using both Left- and Right-Invariant Measurements
  12. Error Propagation
  13. Linearization of the Measurement Model
  14. Jacobians using MRTS smoothing
  15. Experimental Results Using the Starry Night Dataset
  16. ...and 2 more sections

Key Result

Theorem 1

If the function $\mbf{F}\left(\mbf{X}(t),\mbf{u}(t)\right)$ is group affine and the error is either left- or right-invariant, then the error propagation will be state independent.

Figures (3)

  • Figure 1: Mean RMSEs for the IRTS and MRTS smoothers on experimental data with low initialization error.
  • Figure 2: Mean RMSEs for the IRTS and MRTS smoothers on experimental data with high initialization error.
  • Figure 3: Mean RMSEs for each smoother and Gauss-Newton algorithms for each iteration.

Theorems & Definitions (1)

  • Theorem 1: State-independent error dynamics MyPaper:Barrau