Mortar Trajectory Estimation by a Deep Error-State Kalman Filter in a GNSS-Denied Environment

This paper presents a Deep Error State Kalman Filter (Deep ES-KF) to estimate a projectile trajectory using only the embedded Inertial Measurement Unit (IMU) composed by a triaxial accelerometer, gyrometer and magnetometer. The ES-KF evolution model is evaluated by a Long Short-Term Memory (LSTM) to predict the projectile trajectory, i.e. projectile positions, velocities and Euler angles. Then, the ES-KF prediction step evaluates the trajectory errors from accelerometer and gyrometer measurements and corrects them during the update step using magnetometer measurements and the reference magnetic field. Finally, the LSTM predicted trajectory is corrected by the ES-KF updates. The LSTM input data are the embedded IMU, the reference magnetic field, flight parameters specific to the projectile considered and a time vector. A transfer learning method is used to ensure an optimal accuracy. The Deep ES-KF accuracy is compared to a classical Dead-Reckoning, to the ES-KF without neural network and to the pre-trained LSTM on mortar fire simulations. The Root Mean Square Error (RMSE), the Mean Absolute Error (MAE) and the success rate are evaluated. The LSTM contribution is clearly highlighted as the Deep ES-KF and the pre-trained LSTM exhibit considerably less relevant errors than the ES-KF and the Dead-Reckoning to estimate mortar positions and velocities. However, deep learning-based methods are not the optimal solution to estimate the projectile orientation. However, among the four compared methods, the Deep ES-KF presents the best position and velocity estimation and provides an accurate long-term navigation solution.