Data-driven Discovery of The Quadrotor Equations of Motion Via Sparse Identification of Nonlinear Dynamics

Dynamical systems provide a mathematical framework for understanding complex physical phenomena. The mathematical formulation of these systems plays a crucial role in numerous applications; however, it often proves to be quite intricate. Fortunately, data can be readily available through sensor measurements or numerical simulations. In this study, we employ the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm to extract a mathematical model solely from data. The influence of the hyperparameter $\lambda$ on the sparsity of the identified dynamics is discussed. Additionally, we investigate the impact of data size and the time step between snapshots on the discovered model. To serve as a data source, a ground truth mathematical model was derived from the first principals, we focus on modeling the dynamics of a generic 6 Degrees of Freedom (DOF) quadrotor. For the scope of this initial manuscript and for simplicity and algorithm validation purposes, we specifically consider a sub-case of the 6 DOF system for simulation, restricting the quadrotor's motion to a 2-dimensional plane (i.e. 3 DOF). To evaluate the efficacy of the SINDy algorithm, we simulate three cases employing a Proportional-Derivative (PD) controller for the 3 DOF case including different trajectories. The performance of SINDy model is assessed through the evaluation of absolute error metrics and root mean squared error (RMSE). Interestingly, the predicted states exhibit at most a RMSE of order of magnitude approximately $10^{-4}$, manifestation of the algorithm's effectiveness. This research highlights the application of the SINDy algorithm in extracting the quadrotor mathematical models from data.