Land vehicles need high-precision navigational systems in which multi-sensor integration may be provided. Moreover, land vehicles regularly use Global Navigation Satellite Systems (GNSS) to estimate their position. Unfortunately, several locations, such as tunnels and inside parking garages, where GNSS signals cannot be detected. Several types of research
have been conducted to improve positioning information using multi-sensor integration. Then, the vehicle needs another system for  nding its location in GNSS-denied conditions, such as Inertial Navigation System (INS). Despite the accuracy of INS in short-time period use, inertial navigation systems (INS) are liable to drifts of their positioning solution due to
the inertial sensor errors that are inherent to them; therefore, this problem leads to errors accumulation over time then integration techniques are used to eliminate the resulting errors. Moreover, many  lters are used in the process of integration, such as the Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), Particular Filter (PF) and Invariant Extended Kalman Filter (IEKF). Moreover, this work introduces the left-invariant extended Kalman  lter (LIEKF) as a navigation  lter for a loosely coupled integration to eliminate positioning errors. Furthermore, the LIEKF is based on the symmetry-preserving observer theory, which claims that the estimation error depends on the theory of a Lie group matrix, and the proposed system INS/GPS-based LIEKF converges to constant values, unlike the traditional INS/GPS.
Moreover, the proposed system INS/GPS-based LIEKF depends on State-estimate-independent Jacobians, and the LIEKF is more ecient and has better performance due to results such as the 2D position RMS error due to the INS/GPS-based EKF is 19:43m. However, the 2D position RMS error due to the INS/GPS-based LIEKF is 3:32m with 83% improvement. Moreover, the 2D position errors were enhanced using the INS/GPS-based LIEKF system compared to the INS/GPS-based EKF system.