  Accurate and reliable sensor calibration is essential to fuse LiDAR and
inertial measurements, which are usually available in robotic applications. In
this paper, we propose a novel LiDAR-IMU calibration method within the
continuous-time batch-optimization framework, where the intrinsics of both
sensors and the spatial-temporal extrinsics between sensors are calibrated
without using calibration infrastructure such as fiducial tags. Compared to
discrete-time approaches, the continuous-time formulation has natural
advantages for fusing high rate measurements from LiDAR and IMU sensors. To
improve efficiency and address degenerate motions, two observability-aware
modules are leveraged: (i) The information-theoretic data selection policy
selects only the most informative segments for calibration during data
collection, which significantly improves the calibration efficiency by
processing only the selected informative segments. (ii) The observability-aware
state update mechanism in nonlinear least-squares optimization updates only the
identifiable directions in the state space with truncated singular value
decomposition (TSVD), which enables accurate calibration results even under
degenerate cases where informative data segments are not available. The
proposed LiDAR-IMU calibration approach has been validated extensively in both
simulated and real-world experiments with different robot platforms,
demonstrating its high accuracy and repeatability in commonly-seen human-made
environments. We also open source our codebase to benefit the research
community: {\url{https://github.com/APRIL-ZJU/OA-LICalib}}.
