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Intuitively speaking, SLAM approaches generally work by comparing incoming sensor data to a map and localizing within that map. Driving forward with only an inclined LIDAR, you won't be able to localize using standard SLAM approaches, as incoming sensor data cannot be compared to existing map information (since the LIDAR always sees a completely new "slice" of the environment that is not part of the map generated so far). It sounds like some sort of odometry estimate would be sufficient for your application, so generating such an estimate would possibly be sufficient. If you have no way to get wheel odometry, other (low cost) options are just using a constant velocity model (i.e. assume your platform moves with fixed velocity, possibly fuse with IMU data) or using one of the visual SLAM/odometry approaches out there such as ORBSLAM2, libviso2 etc.

Of course, another, more expensive option is using a second horizontal LIDAR and using that to do "standard" SLAM.