The generate_training_data function uses a Latin hypercube to randomly select samples that cover the five-dimensional space in the car model (3 states + 2 inputs) to get the X matrix. A more thorough explanation of this is covered in my thesis that I link to in the README file. For each sample we simulate one timestep 50ms forward to get the Y matrix to sample the dynamics in the model. The added noise to the Y matrix simulate measurement noise. Since we only integrate one timestep, the difference between Y and X will be small, especially with samples where the input values (u) are low. We could have decided on a longer sampling period, but then we risk aliasing issues where the GP model will not be able to learn the fast dynamics in the car model.

If you add random noise to the X matrix (that is within the limits of the state space) it would not matter since the X matrix itself is by definition random distributed. It is also possible to have training data of measured/ simulated time-series in the X matrix instead of the one-shot samples, but then we get a lot of redundant data, where we would need quite a lot more data points to pick up all the system dynamics. I think I also have a brief discussion about this in the thesis related to online learning. In a real system, it would not be feasible to use a Latin hypercube to gather training data, but since we have the simulated model we have the option to cheat and get optimal distributed training data.

Thank you very much for your detailed explanation !