GNSS-Inertial State Initialization Using Inter-Epoch Baseline Residuals

TL;DR

The paper tackles the challenge of robustly initializing GNSS–inertial state estimates when early measurements provide limited information. It introduces a two-stage approach that first uses inter-epoch GNSS baselines to curb inertial drift and then activates global GNSS constraints once the system becomes sufficiently observable, determined by a Hessian singular-value criterion. Through theoretical observability analysis and empirical validation on EuRoC, GVINS, and MARS-LVIG, the method consistently improves initialization robustness and trajectory accuracy compared with naive full-measurement fusion. The approach also yields better bias and heading estimates, highlighting its practical value for reliable UAV and mobile-robot bootstrap in challenging environments.

Abstract

Initializing the state of a sensorized platform can be challenging, as a limited set of measurements often provide low-informative constraints that are in addition highly non-linear. This may lead to poor initial estimates that may converge to local minima during subsequent non-linear optimization. We propose an adaptive GNSS-inertial initialization strategy that delays the incorporation of global GNSS constraints until they become sufficiently informative. In the initial stage, our method leverages inter-epoch baseline vector residuals between consecutive GNSS fixes to mitigate inertial drift. To determine when to activate global constraints, we introduce a general criterion based on the evolution of the Hessian matrix's singular values, effectively quantifying system observability. Experiments on EuRoC, GVINS and MARS-LVIG datasets show that our approach consistently outperforms the naive strategy of fusing all measurements from the outset, yielding more accurate and robust initializations.

Figures (7)

• Figure 1: Illustration of our GNSS-inertial state initialization. a) Global residuals remain inactive while the system lacks informative measurements. b) At time $k^*$, the GNSS-inertial measurements provide sufficient information to meet our criterion, enabling the inclusion of global constraints for accurate alignment.
• Figure 2: Illustration of the temporal notation for IMU and GNSS measurements.
• Figure 3: Structure of the full observability matrix $\mathcal{O}$, obtained by stacking the Jacobians of all residuals with respect to the state vector $\mathbf{x}$. The Jacobian $\mathbf{J}_r^{-1}(\boldsymbol{\phi})$ denotes the inverse of the right Jacobian of $\mathrm{SO}(3)$, evaluated at the rotational error $\boldsymbol{\phi} = \mathrm{Log} ( \Delta \Tilde{\mathbf{R}}_{ij}^\top \mathbf{R}_i^\top \mathbf{R}_j )$. The last row accounts for the gravitational constraint, where the gravity direction $\hat{\mathbf{g}} \in \mathbb{S}^2$ is parametrized on the unit sphere and its Jacobian is denoted by $\mathbf{J}_{\mathbb{S}^2}$. Accounting for the GNSS–inertial lever arm makes GNSS residuals orientation-sensitive. Absolute terms inform attitude at their node and relative baselines at both endpoints, boosting yaw/bias observability under rotation while reducing to position-only if the arm is negligible.
• Figure 4: Structure of the observability matrix $\mathcal{O}$. Each row corresponds to a residual and each column to a component of the state vector. White cells indicate matrices with zero values, meaning no sensitivity to that state. Gray cells represent general nonzero Jacobians that contribute to observability. Orange cells denote relative constraints that do not improve global observability.
• Figure 5: Initialization with GNSS measurements are shown as green squares, while IMU data are represented as blue circles. Alignment ensures that relative motion inferred from the IMU is consistent with the global trajectory.