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State Estimation Using Single Body-Frame Bearing Measurements

Sifeddine Benahmed, Soulaimane Berkane

TL;DR

The paper tackles the problem of jointly estimating a rigid body's inertial position $p^{\mathcal{I}}$, velocity $v^{\mathcal{I}}$, and attitude $R$ using IMU data plus a single body-frame bearing to a known landmark and a body-frame vector measurement. It introduces a Riccati observer for a linear time-varying model expressed in the body frame with state $x=[p^{\mathcal{B}}; v^{\mathcal{B}}; g^{\mathcal{B}}; m^{\mathcal{B}}]^T$, where $A(t)$, $B$, and $C(t)$ are modulated by the angular velocity $\omega(t)$ and the bearing $\eta^{\mathcal{B}}(t)$, and the estimator gain $K(t)$ is obtained from a Riccati equation. The authors establish a uniform observability (and thus global exponential convergence) result under a persistency-of-excitation condition on the bearing-relative motion, and they provide an algebraic method to reconstruct the attitude from the estimated gravity $\hat{g}^{\mathcal{B}}$ and body-frame vector $\hat{m}^{\mathcal{B}}$, plus reduced-order variants that estimate gravity direction with only the bearing and IMU data. Simulation results on a rich 3D trajectory validate the approach, showing convergence of position, velocity, and attitude estimates even in the presence of magnetometer noise. The work offers a theoretically grounded alternative to stochastic filters for GPS-denied navigation and enables robust attitude/pose estimation with minimal sensing.

Abstract

This paper addresses the problem of simultaneous estimation of the position, linear velocity and orientation of a rigid body using single bearing measurements. We introduce a Riccati observer-based estimator that fuses measurements from a 3-axis accelerometer, a 3-axis gyroscope, a single body-frame vector observation (e.g., magnetometer), and a single bearing-to-landmark measurement to obtain the full vehicle's state (position, velocity, orientation). The proposed observer guarantees global exponential convergence under some persistency of excitation (PE) condition on the vehicle's motion. Simulation results are presented to show the effectiveness of the proposed approach.

State Estimation Using Single Body-Frame Bearing Measurements

TL;DR

The paper tackles the problem of jointly estimating a rigid body's inertial position , velocity , and attitude using IMU data plus a single body-frame bearing to a known landmark and a body-frame vector measurement. It introduces a Riccati observer for a linear time-varying model expressed in the body frame with state , where , , and are modulated by the angular velocity and the bearing , and the estimator gain is obtained from a Riccati equation. The authors establish a uniform observability (and thus global exponential convergence) result under a persistency-of-excitation condition on the bearing-relative motion, and they provide an algebraic method to reconstruct the attitude from the estimated gravity and body-frame vector , plus reduced-order variants that estimate gravity direction with only the bearing and IMU data. Simulation results on a rich 3D trajectory validate the approach, showing convergence of position, velocity, and attitude estimates even in the presence of magnetometer noise. The work offers a theoretically grounded alternative to stochastic filters for GPS-denied navigation and enables robust attitude/pose estimation with minimal sensing.

Abstract

This paper addresses the problem of simultaneous estimation of the position, linear velocity and orientation of a rigid body using single bearing measurements. We introduce a Riccati observer-based estimator that fuses measurements from a 3-axis accelerometer, a 3-axis gyroscope, a single body-frame vector observation (e.g., magnetometer), and a single bearing-to-landmark measurement to obtain the full vehicle's state (position, velocity, orientation). The proposed observer guarantees global exponential convergence under some persistency of excitation (PE) condition on the vehicle's motion. Simulation results are presented to show the effectiveness of the proposed approach.
Paper Structure (7 sections, 1 theorem, 39 equations, 7 figures)

This paper contains 7 sections, 1 theorem, 39 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. State Estimation for Linear Time-Varying (LTV) Systems
  4. Problem Formulation
  5. Main Results
  6. Simulation Results
  7. Conclusion

Key Result

Lemma 1

Let $\eta^{\mathcal{I}}:=R\eta^{\mathcal{B}}$ be the bearing-to-landmark expressed in the inertial frame. If there exist $\delta,\mu>0$ such that then the pair $(A(\cdot),C(\cdot))$ in equation:LTV_state_model is uniformly observable.

Figures (7)

  • Figure 1: Illustration of the bearing measurement $\eta^{\mathcal{B}}$, the position of the rigid body $p^{\mathcal{I}}$ and the landmark $p_{\ell}^{\mathcal{I}}$.
  • Figure 2: Illustration of the proposed state estimation approach.
  • Figure 3: Time behaviour of the components of the real and estimated position.
  • Figure 4: Position and velocity estimation errors.
  • Figure 5: True and estimated trajectory in the inertial frame.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Definition 1
  • Lemma 1
  • Remark 1: Attitude Estimation on $\mathds{SO}(3)$
  • Remark 2: Decoupled Observer
  • Remark 3: Reduced-Order Observer