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A Generic Observer Design for Inertial Navigation Systems Using an LTV Framework

Sifeddine Benahmed, Soulaimane Berkane, Tarek Hamel

TL;DR

The paper presents a generic observer framework for inertial navigation that recasts INS estimation into a linear time-varying (LTV) system, enabling a continuous-time Kalman-filter-like approach for joint pose and attitude estimation using diverse exteroceptive measurements. By introducing a 15-dimensional body-frame state and deriving time-varying system and output matrices, it achieves uniform observability under practical sensing configurations (stereo landmarks and GPS with optional magnetometer/velocity). The authors prove equivalence between the full-system observability and a reduced-dimensional version, provide conditions for both time-varying and constant gains, and validate the method with simulations on stereo-aided and GPS-aided INS scenarios. The work demonstrates robust convergence and flexibility, while noting that enforcing the $RR^{\top}=I_3$ constraint can relax observability requirements and suggesting future work on biases and sensor imperfections.

Abstract

This paper addresses the problem of accurate pose estimation-position, velocity, and orientation-of a rigid body using an Inertial Measurement Unit (IMU) in combination with generic exteroceptive measurements. By reformulating the vehicle's dynamics and measurement models within a linear time-varying (LTV) framework, we enable the application of a linear Kalman filter, significantly simplifying observer design for inertial navigation systems (INS). A key strength of this approach lies in its generality: rather than relying on specific measurement modalities, our framework accommodates a broad class of exteroceptive measurements. To illustrate its effectiveness, we conduct a uniform observability (UO) analysis for two fundamental benchmark cases-GPS-aided INS and landmark-aided INS-deriving sufficient conditions that guarantee the global uniform exponential stability of the proposed filter. Simulations for both applications confirm the versatility and robustness of our approach.

A Generic Observer Design for Inertial Navigation Systems Using an LTV Framework

TL;DR

The paper presents a generic observer framework for inertial navigation that recasts INS estimation into a linear time-varying (LTV) system, enabling a continuous-time Kalman-filter-like approach for joint pose and attitude estimation using diverse exteroceptive measurements. By introducing a 15-dimensional body-frame state and deriving time-varying system and output matrices, it achieves uniform observability under practical sensing configurations (stereo landmarks and GPS with optional magnetometer/velocity). The authors prove equivalence between the full-system observability and a reduced-dimensional version, provide conditions for both time-varying and constant gains, and validate the method with simulations on stereo-aided and GPS-aided INS scenarios. The work demonstrates robust convergence and flexibility, while noting that enforcing the constraint can relax observability requirements and suggesting future work on biases and sensor imperfections.

Abstract

This paper addresses the problem of accurate pose estimation-position, velocity, and orientation-of a rigid body using an Inertial Measurement Unit (IMU) in combination with generic exteroceptive measurements. By reformulating the vehicle's dynamics and measurement models within a linear time-varying (LTV) framework, we enable the application of a linear Kalman filter, significantly simplifying observer design for inertial navigation systems (INS). A key strength of this approach lies in its generality: rather than relying on specific measurement modalities, our framework accommodates a broad class of exteroceptive measurements. To illustrate its effectiveness, we conduct a uniform observability (UO) analysis for two fundamental benchmark cases-GPS-aided INS and landmark-aided INS-deriving sufficient conditions that guarantee the global uniform exponential stability of the proposed filter. Simulations for both applications confirm the versatility and robustness of our approach.
Paper Structure (15 sections, 4 theorems, 42 equations, 3 figures)

This paper contains 15 sections, 4 theorems, 42 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem Formulation
  4. Main Results
  5. Pre-Processing: Linear Time-Varying Model
  6. Observer Design: Continuous-Time Kalman Filter and Attitude Reconstruction
  7. Uniform Observability
  8. State Estimation For Stereo-Aided INS
  9. State Estimation For GPS-Aided INS
  10. Simulation Results
  11. Conclusion
  12. Proof of Lemma \ref{['lemma_equivalance_of_UO_between_A_C_and_Abar_Cbar']}
  13. Proof of Lemma \ref{['Lemma:sufficient_condition_for_UO_stereo_INS']}
  14. Proof of Lemma \ref{['Lemma:constant_gain_stereo_INS']}
  15. Proof of Lemma \ref{['Lemma:sufficient_condition_for_UO_GPS_INS']}

Key Result

Lemma 1

The pair $(A(\cdot),C(\cdot))$ is uniformly observable if and only if the pair $(\bar{A},\bar{C}(\cdot))$ is uniformly observable.

Figures (3)

  • Figure 1: Illustration of the proposed state estimation approach.
  • Figure 2: Estimation errors and trajectories for Stereo-aided INS.
  • Figure 3: Estimation errors and trajectories for GPS-aided INS.

Theorems & Definitions (5)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4