Recently, the multi-modal fusion with 3D LiDAR, camera, and IMU has shown great potential in applications of automation-related fields. Yet a prerequisite for a successful fusion is that the geometric relationships among the sensors are accurately determined, which is called an extrinsic calibration problem. To date, the existing targetbased approaches to deal with this problem rely on sophisticated calibration objects (sites) and well-trained operators, which is timeconsuming and inflexible in

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... practical applications. Contrarily, a few target-free methods can overcome these shortcomings, while they only focus on the calibrations of two types of the sensors. Although it is possible to obtain LiDAR-visual-inertial extrinsics by chained calibrations, problems such as cumbersome operations, large cumulative errors, and weak geometric consistency still exist. To this end, we propose LVI-ExC, an integrated LiDAR-Visual-Inertial Extrinsic Calibration framework, which takes natural multi-modal data as input and yields sensor-to-sensor extrinsics end-to-end without any auxiliary object (site) or manual assistance. To fuse multi-modal data, we formulate the LiDAR-visual-inertial extrinsic calibration as a continuous-time simultaneous localization and mapping problem, in which the extrinsics, trajectories, time differences, and map points are jointly estimated by establishing sensor-tosensor and sensor-to-trajectory constraints. Extensive experiments show that LVI-ExC can produce precise results. With LVI-ExC's outputs, the LiDAR-visual reprojection results and the reconstructed environment map are all highly consistent with the actual natural scenes, demonstrating LVI-ExC's outstanding performance. To ensure that our results are fully reproducible, all the relevant data and codes have been released publicly 1 .