This thesis investigates the performance of the invariant extended Kalman filter (IEKF) compared to the multiplicative extended Kalman filter (MEKF) in the context of nonlinear state estimation on matrix Lie groups. The IEKF, a relatively recent variant of the EKF, is particularly suitable for systems with group-affine process models and invariant measurement models. When applied to such systems, the IEKF exhibits guaranteed state-independent error dynamics, which proves advantageous in cases of poor or inaccurate system initialization.

While previous studies have highlighted the benefits of the IEKF in poorly initialized systems, it is unclear whether the IEKF and the multiplicative EKF exhibit significant differences in performance when the system is already accurately initialized. Therefore, this thesis aims to investigate whether the IEKF demonstrates improved performance over the MEKF in 3D pose estimation using inertial measurement units (IMUs).

Specifically, the main research question of this thesis is: How does the estimation accuracy of the invariant EKF compare to the multiplicative EKF in the context of pose estimation? In order to gain insight into this, the investigation focuses on three main questions. Firstly, what are the advantages of utilizing a left-invariant EKF (LIEKF) over an MEKF when dealing with a left-invariant measurement model, and similarly, what are the benefits of employing a rightinvariant IEKF (RIEKF) over an MEKF when dealing with a right-invariant measurement model? Secondly, how does the IMU sensor noise magnitude affect the converging performance
of the filters differently? Thirdly, How does the sensor noise magnitude of the external measurements affect the converging performance of the filters differently?

Additionally to the distinction between the left- and right-IEKF, a similar distinction is made for the MEKF. This thesis distinguishes between an MEKF with orientation deviation states resolved in the body frame (MEKF-b) and an MEKF with orientation deviation states resolved in navigation frame (MEKF-n). This distinction is made since it allows for a more natural comparison between the IEKF and MEKF.

To conduct the evaluation, extensive simulations are performed, allowing for controlled variations in these parameters. The simulation results provide insights into the comparative performance of the IEKF and multiplicative EKF under different conditions, shedding light on their strengths and limitations in 3D pose estimation with IMUs.

It was found that the IEKF and MEKF show very comparable results in a large amount of the applications. The state-independent error dynamics have been shown to be beneficial in situations where the initial information of the state of the system is uncertain. Furthermore, the IEKF has been shown to be beneficial in certain edge cases. Firstly, the IEKF shows to be less sensitive to small process noise covariance matrices Q. Secondly, once the gyroscopic noise becomes very large, the RIEKF showed higher estimation accuracy over the MEKF-n. The LIEKF did also show a marginal improvement in estimation accuracy over the MEKF-b.
Finally, it was found in this thesis there are two ways that the external measurement noise influenced the comparison of the estimation accuracy between the IEKF and MEKF. The MEKF-n showed to be sensitive to a low covariance measurement matrix R and additionally, the MEKF-b and MEKF-n both seemed to be marginally more affected by higher external measurement noise than the LIEKF and RIEKF, respectively.

In conclusion, this thesis provides a comprehensive evaluation of the IEKF and MEKF in 3D pose estimation with IMUs. While the IEKF and MEKF exhibit comparable performance in many cases, the IEKF’s state-independent error dynamics and its advantages in certain scenarios highlight its potential superiority over the MEKF. These findings contribute to the understanding of nonlinear state estimation on matrix Lie groups and offer valuable insights for selecting the appropriate filter for specific applications.