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Scalar-Measurement Attitude Estimation on $\mathbf{SO}(3)$ with Bias Compensation

Alessandro Melis, Tarek Bouazza, Hassan Alnahhal, Sifeddine Benahmed, Soulaimane Berkane, Tarek Hamel

Abstract

Attitude estimation methods typically rely on full vector measurements from inertial sensors such as accelerometers and magnetometers. This paper shows that reliable estimation can also be achieved using only scalar measurements, which naturally arise either as components of vector readings or as independent constraints from other sensing modalities. We propose nonlinear deterministic observers on $\mathbf{SO}(3)$ that incorporate gyroscope bias compensation and guarantee uniform local exponential stability under suitable observability conditions. A key feature of the framework is its robustness to partial sensing: accurate estimation is maintained even when only a subset of vector components is available. Experimental validation on the BROAD dataset confirms consistent performance across progressively reduced measurement configurations, with estimation errors remaining small even under severe information loss. To the best of our knowledge, this is the first work to establish fundamental observability results showing that two scalar measurements under suitable excitation suffice for attitude estimation, and that three are enough in the static case. These results position scalar-measurement-based observers as a practical and reliable alternative to conventional vector-based approaches.

Scalar-Measurement Attitude Estimation on $\mathbf{SO}(3)$ with Bias Compensation

Abstract

Attitude estimation methods typically rely on full vector measurements from inertial sensors such as accelerometers and magnetometers. This paper shows that reliable estimation can also be achieved using only scalar measurements, which naturally arise either as components of vector readings or as independent constraints from other sensing modalities. We propose nonlinear deterministic observers on that incorporate gyroscope bias compensation and guarantee uniform local exponential stability under suitable observability conditions. A key feature of the framework is its robustness to partial sensing: accurate estimation is maintained even when only a subset of vector components is available. Experimental validation on the BROAD dataset confirms consistent performance across progressively reduced measurement configurations, with estimation errors remaining small even under severe information loss. To the best of our knowledge, this is the first work to establish fundamental observability results showing that two scalar measurements under suitable excitation suffice for attitude estimation, and that three are enough in the static case. These results position scalar-measurement-based observers as a practical and reliable alternative to conventional vector-based approaches.
Paper Structure (14 sections, 5 theorems, 37 equations, 4 figures, 2 tables)

This paper contains 14 sections, 5 theorems, 37 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminary Material
  3. Notation
  4. Observability Principles and Requirements
  5. Riccati Observers for a Class of Nonlinear Systems
  6. System Equations and Measurements
  7. Observer Design
  8. Unbiased case
  9. Biased case
  10. Experimental Results
  11. Results and discussion
  12. Conclusion
  13. Appendix
  14. Proof of Lemma \ref{['lemma:general_condition_lemma_2']}

Key Result

Proposition 1

Consider the system dynamics eq:riccati_dynamics and choose the input signal: with $P(0)$ a positive definite (p.d.) matrix solution to the Continuous Riccati Equation (CRE) with $Q(t)$ and $V(t)$ bounded continuous symmetric positive definite matrix-valued functions. If the pair $(A^\star(t),C^\star(t)):=(A(0,t),C(0,t))$ is uniformly observable, then the origin of eq:riccati_dynamics is locally

Figures (4)

  • Figure 1: Illustration of the proposed estimation approach.
  • Figure 2: Custom 3D-printed rigid body with IMU and five reflective markers (reproduced from laidig2021broad, CC BY 4.0).
  • Figure 3: Evolution of the angular error $\theta$, and the errors on roll, pitch, and yaw for all observers with initial condition $\hat{R}(0)$ set to a rotation of $\pi/4$ around each axis.
  • Figure 4: Evolution of the bias components for all observers with initial condition $\hat{d}(0) = [0.5,\,0.5,\,0.5]^\top$.

Theorems & Definitions (11)

  • Definition 1: Uniform Observability
  • Proposition 1
  • proof
  • Lemma 1
  • Definition 2
  • Corollary 1
  • proof
  • Lemma 2
  • Corollary 2
  • Remark 1
  • ...and 1 more