  Many advanced driver assistance schemes or autonomous vehicle controllers are
based on a motion model of the vehicle behavior, i.e., a function predicting
how the vehicle will react to a given control input. Data-driven models, based
on experimental or simulated data, are very useful, especially for vehicles
difficult to model analytically, for instance, ground vehicles for which the
ground-tire interaction is hard to model from first principles. However,
learning schemes are limited by the difficulty of collecting large amounts of
experimental data or having to rely on high-fidelity simulations. This paper
explores the potential of an approach that uses dimensionless numbers based on
Buckingham's $\pi$ theorem to improve the efficiency of data for learning
models, with the goal of facilitating knowledge sharing between similar
systems. A case study using car-like vehicles compares traditional and
dimensionless models on simulated and experimental data to validate the
benefits of the new dimensionless learning approach. Prediction accuracy
improvements with the dimensionless scheme when using a shared database, that
is, predicting the motion of a vehicle based on data from various different
vehicles was found to be 480\% more accurate for predicting a simple no-slip
maneuver based on simulated data and 11\% more accurate to predict a highly
dynamic braking maneuver based on experimental data. A modified
physics-informed learning scheme with hand-crafted dimensionless features was
also shown to increase the improvement to precision gains of 917\% and 28\%
respectively. A comparative study also shows that using Buckingham's $\pi$
theorem is a much more effective preprocessing step for this task than
principal component analysis (PCA) or simply normalizing the data.
