Context. Astrometric monitoring of directly imaged exoplanets allows the study of their orbital parameters and system architectures. Because most directly imaged planets have long orbital periods (>20 AU), accurate astrometry is challenging when based on data acquired on timescales of a few years and usually with different instruments. The LMIRCam camera on the Large Binocular Telescope is being used for the LBT Exozodi Exoplanet Common Hunt (LEECH) survey to search for and characterize young and adolescent exoplanets in L' band (3.8 μm), including their system architectures. Aims: We first aim to provide a good astrometric calibration of LMIRCam. Then, we derive new astrometry, test the predictions of the orbital model of 8:4:2:1 mean mot...