Globally Optimal GNSS Multi-Antenna Lever Arm Calibration

TL;DR

The paper tackles GNSS-IMU antenna lever-arm calibration under motion-only measurements by formulating a globally optimal Quadratically Constrained Quadratic Program (QCQP) and solving it via a Lagrangian dual with primal recovery. It decouples motion estimation from lever-arm calibration, accommodates prior knowledge through quadratic constraints, and extends to planar motion to address autonomous-vehicle dynamics. The approach is validated through simulations and KITTI-derived data, showing robust single-antenna performance and significant gains for multi-antenna configurations when priors and regularization are used, with online-ready runtimes. Overall, it provides a certifiably global framework for online extrinsic calibration of GNSS antennas on multi-sensor autonomous systems, including practical planar-motion extensions.

Abstract

Sensor calibration is crucial for autonomous driving, providing the basis for accurate localization and consistent data fusion. Enabling the use of high-accuracy GNSS sensors, this work focuses on the antenna lever arm calibration. We propose a globally optimal multi-antenna lever arm calibration approach based on motion measurements. For this, we derive an optimization method that further allows the integration of a-priori knowledge. Globally optimal solutions are obtained by leveraging the Lagrangian dual problem and a primal recovery strategy. Generally, motion-based calibration for autonomous vehicles is known to be difficult due to cars' predominantly planar motion. Therefore, we first describe the motion requirements for a unique solution and then propose a planar motion extension to overcome this issue and enable a calibration based on the restricted motion of autonomous vehicles. Last we present and discuss the results of our thorough evaluation. Using simulated and augmented real-world data, we achieve accurate calibration results and fast run times that allow online deployment.

Figures (5)

• Figure 1: A calibration scenario for a single time step is illustrated. An IMU sensor, together with two antennas, is moving from station $s_k$ (left) to $s_{k+1}$ (right). The calibration goal is to obtain the transformation $X_i$ for each antenna. The motion of the IMU sensor (cylinder) is represented by $A$ (purple arrow). $X_1$ and $X_2$ (gray arrows) are the transformations from the IMU sensor to the respective antenna. $\boldsymbol{b}_{1}$ and $\boldsymbol{b}_{2}$ (yellow arrows) represent the motion of the respective antenna. As a GNSS sensor does not measure orientation, only translational motions are available.
• Figure 2: Inspired by Briales2017ConvexDuality, the structures of the matrices used to construct the cost functions with three antennas are illustrated. Therefore, the sum of matrices is considered at a singe time step (see \ref{['eq:bigM']}). Fig. \ref{['fig:matricStructure:first']} relates to the matrix $\boldsymbol{M}^i$ of Eq. \ref{['eq:Mkts']}, and Fig. \ref{['fig:matricStructure:second']} relates to the matrix $\widetilde{\boldsymbol{M}}_{k}^{i,j}$ of Eq. \ref{['eq:Mcross']}.
• Figure 3: A simulated path example: The 2D path (a), a sinusoid mixture in this example, is projected onto a rough surface, resulting in a 3D path (b).
• Figure 4: Translation errors of calibrations based on simulated data are plotted over the number of points used for the optimization. A line denotes the $50$th quantile over $10000$ simulation runs for the respective noise level. The gray area behind each line denotes the respective $25$th to $75$th quantile. All curves show a converging behavior dependent on the respective noise level.
• Figure :