ETD

• neural ode

This thesis investigates a continuous-time learning framework based on Neural Ordinary Differential Equations (Neural ODEs) for the estimation and compensation of Inertial Measurement Units (IMUs) errors. The problem is formulated within the geometric setting of inertial navigation, where orientation evolves on the Lie group SO(3) and velocity and position evolve in Euclidean space.
After introducing the mathematical formulation of the problem, the theory of Neural Differential Equations is presented, leading particular attention to the adjoint sensitivity method involved in the training of Neural ODEs.
Building upon recent Neural ODE approaches for IMU debiasing, a novel continuous time framework is proposed in which the dynamics of gyroscope and accelerometer biases are modeled according to learned differential equations. The resulting system
combines data-controlled Neural ODEs with inertial navigation modeling, allowing the correction process to be interpreted as the evolution of a continuous dynamical system.
A key contribution of this work is the treatment of the Lie group structure underlying the problem, both in the definition of the loss function and in the evolution of rotations, which must be described through local charts. We derive an adjoint formulation that is compatible with the Lie group structure of SO(3). We provide two different implementations using two chart parametrizations of SO(3), namely rotation matrices and non-unit quaternions.
The proposed framework is trained using ground-truth trajectories. The parameters of the Neural ODEs are learned by minimizing the discrepancy between the reconstructed trajectory and the reference one through gradient-based optimization. This allows the model to learn continuous-time corrections for gyroscope and accelerometer measurements directly from data.
Experimental results on the EuRoC MAV dataset demonstrate that the proposed approach effectively reduces inertial sensor errors and improves the accuracy of trajectory reconstruction. Moreover, the numerical experiments highlight the importance of the adjoint formulation, that allow to train the Neural Networks over longer time interval and augmenting the accuracy.