Unobservable Subspace Evolution and Alignment for Consistent Visual-Inertial Navigation

TL;DR

This work tackles VI-SLAM inconsistency by introducing Unobservable Subspace Evolution (USE), which analyzes how the estimator’s unobservable subspace evolves across all estimation steps, revealing that observability misalignment precedes and causes inconsistency. Building on USE, Unobservable Subspace Alignment (USA) provides targeted interventions—transformation-based and re-evaluation-based—to eliminate misalignment without sacrificing Jacobian optimality. The authors demonstrate, through extensive simulations and real-world datasets, that USA yields competitive or superior consistency and accuracy with modest computational overhead, and they further show that MSCKF consistency differs from EKF-SLAM. The framework offers a principled, scalable approach to diagnosing and correcting inconsistency in practical VINS and holds promise for extension to other SLAM and odometry systems.

Abstract

The inconsistency issue in the Visual-Inertial Navigation System (VINS) is a long-standing and fundamental challenge. While existing studies primarily attribute the inconsistency to observability mismatch, these analyses are often based on simplified theoretical formulations that consider only prediction and SLAM correction. Such formulations fail to cover the non-standard estimation steps, such as MSCKF correction and delayed initialization, which are critical for practical VINS estimators. Furthermore, the lack of a comprehensive understanding of how inconsistency dynamically emerges across estimation steps has hindered the development of precise and efficient solutions. As a result, current approaches often face a trade-off between estimator accuracy, consistency, and implementation complexity. To address these limitations, this paper proposes a novel analysis framework termed Unobservable Subspace Evolution (USE), which systematically characterizes how the unobservable subspace evolves throughout the entire estimation pipeline by explicitly tracking changes in its evaluation points. This perspective sheds new light on how individual estimation steps contribute to inconsistency. Our analysis reveals that observability misalignment induced by certain steps is the antecedent of observability mismatch. Guided by this insight, we propose a simple yet effective solution paradigm, Unobservable Subspace Alignment (USA), which eliminates inconsistency by selectively intervening only in those estimation steps that induce misalignment. We design two USA methods: transformation-based and re-evaluation-based, both offering accurate and computationally lightweight solutions. Extensive simulations and real-world experiments validate the effectiveness of the proposed methods.

Figures (16)

• Figure 1: Status evolutions triggered by prediction and correction in the standard estimator.
• Figure 2: A visual illustration for proof of Theorem \ref{['lamma:1']}. (a) is the state-dependent unobservable subspace where the Aligned unobservable subspaces are evaluated at the current estimates. (b)-(c): After the prediction step, the unobservable subspace is evaluated at the current estimate, i.e., Aligned. (c)-(d): However, after the correction step, the unobservable subspace is still evaluated at the previous estimate, i.e., Misaligned. (d)-(e): The observability misalignment eventually leads to a reduction of the unobservable dimension, i.e., Mismatched. (f): Visualization of nonlinear measurement. (g) and (h): Linearized information evaluated at $\hat{\mathbf{x}}_{k}^{\mathbin{\text{$\ominus$}}}$ and $\hat{\mathbf{x}}_{k}^{\mathbin{\text{$\oplus$}}}$, respectively.
• Figure 3: Status evolutions when USA is integrated into the estimator. Some evolutions originating from the Misaligned and Mismatched are shown as semi-transparent because they do not occur.
• Figure 4: Unobservable subspace alignment through direct and indirect transformations. $\hat{\mathbf{x}}^{\mathbin{\text{$\ominus$}}}$ and $\hat{\mathbf{x}}^{\mathbin{\text{$\oplus$}}}$ are the previous and the current estimates, respectively. The direct transformation matrix has a closed-form solution. The indirect transformation additionally requires the design of an auxiliary matrix to make the unobservable subspace fully constant.
• Figure 5: Different strategies for delayed feature initialization. (a) Applying USA in batch once after the composite step of the initialization. (b) Applying USA separately after both the first and the second substeps.

Theorems & Definitions (10)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Remark 1
• Remark 2
• Remark 3
• Remark 4