Magnetometers are widely equipped in smartphones. They measure the direction and the magnitude of the magnetic field of the environment. Since the measurements are not transition data, there is no drift when estimating position and orientation using a magnetometer. Furthermore, magnetic field localization using magnetometers requires no extra devices set in the environment, and this indicates the cost of localization using a magnetometer can be lower than other localization methods that need multiple devices set in the localizing area. Therefore, magnetic field localization is an interesting method for indoor localization. However, there exists a research gap in the algorithms that have been applied to magnetic field localization. In the current research, the Extended Kalman filter (EKF) and the Particle filter (PF) are applied to magnetic field localization. The EKF is more efficient than the PF, but has low accuracy when the distribution is multimodal. On the other hand, the PF is more computationally costly compared to the EKF but is more robust to the multimodality. As a result, a survey of the possible solutions to the current research gap was carried out. From this survey, Gaussian sum filter (GSF) was found to be a promising candidate as the solution to the research gap. To test the performance and assumptions of the GSF, the GSF was applied to a fully simulated magnetic field localization system and a localization system with the measurements obtained from a real-world magnetometer. The results from these simulations show that the GSF is more suitable for multimodality than the EKF. Besides, the computational cost of the GSF is found to be lower than the PF while the GSF has an equivalent or even better accuracy than the PF.

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.

Nowadays, machine learning (ML) methods rapidly evolve for their use in model-based control applications. Model-based control requires an accurate model description of the dynamical system to reassure the performance of the controller. Conventionally, this model description is retrieved from first-principles modelling which can be problematic if the system consists of high-order and/or time-varying dynamics. In these applications, ML methods may benefit because of their promising potential to model complicated system behaviour from data. In the field of ML methods for the modelling and control of dynamical systems, Gaussian processes (GPs) form an interesting opportunity. The ability of GPs to directly learn nonlinear system dynamics prevents huge costs and/or efforts when modelling complex systems. GP dynamical models are capable of accurately predicting the behaviour of dynamical systems while also measuring the confidence level of the prediction. For making predictions, GPs do not require huge amounts of data which benefits them over other ML methods.

The powerful model predictive control (MPC) and the fully data-driven dynamical modelling capabilities of GPs makes their combination an interesting candidate for sophisticated control systems. MPC has advantages over other control methods since the controller allows operational constraints that provide freedom in controller design or prevent the system to be steered in an infeasible direction, and, the controller is easily extendable to nonlinear and multivariable control. Next to this, an MPC controller is naturally modified to incorporate GP dynamical models (GP-MPC). The GP-MPC controller exploits the GP dynamical model for making predictions over the prediction horizon while it is also possible to incorporate the confidence of the predictions for increased robustness of the controller. Whereas GP-MPC is studied extensively as an augmented model for other modelling techniques, fully data-driven GP-MPC approaches are also deemed to be promising.

This thesis considers the use of GPs for learning and modelling dynamic systems for incorporation in a realtime MPC application. The dynamic system studied in this thesis is a double pendulum system in both a simulation and a real-world environment. The simulation provides initial insights into the problem and allows the rapid development of an algorithm. Eventually, the proposed realtime GP-MPC algorithm is tested on a physical laboratory-scale setup. The results show that the realtime data-driven controller is able to track a reference with high accuracy, while also being robust to disturbances. These promising results on GPs for realtime nonlinear control might be a step for GPs to be incorporated into future control systems.

Underwater position estimation is challenging due to the absence of Global Navigation Satellite System (GNSS) signals. Underwater vehicles are typically equipped with a Doppler Velocity Log (DVL) that measures the velocity relative to the seafloor. Aside from the velocity, the DVL also measures the range of each of the four beams. When compared against a bathymetry height map, these measured ranges provide additional information enabling improving position estimates. Unfortunately, surveys often occur in areas where detailed bathymetry maps are unavailable. In these cases, bathymetry Simultaneous Localization and Mapping (SLAM) could be used to improve the position estimates compared to the velocity integration position. With SLAM, the map is being estimated during the mission while at the same time using the map for position determination.

In this thesis, reduced rank Gaussian processes (GPs) are used as map representation for SLAM. The downside of regular GPs is a time complexity of O(n^3) , reduced rank GPs improve the computation performance. GPs provide Gaussian distributions at any point, leading to a neat integration with probabilistic SLAM algorithms. To the best of the author’s knowledge, this has not been used in bathymetry SLAM. This report investigates how reduced rank approximated GPs, representing the bathymetry, can be integrated into a SLAM algorithm to improve the position estimates of an underwater vehicle equipped with a DVL and low-quality gyroscopes.

A squared exponential kernel is used as GPs model of the bathymetry. The reduced rank approximation is vital for real-time SLAM performance. This map representation is integrated with a Rao Blackwellized particle filter (RBPF) that estimates both the underwater vehicle’s trajectory and the bathymetry map.

The SLAM algorithm is evaluated using data from an underwater vehicle operated at the surface such that a GNSS reference position is available. Experiments of the SLAM algorithm show a reduced position error compared to the GNSS reference. The resulting algorithm has a computation time of up to 30 times faster than the Autonomous Underwater Vehicle (AUV) collects data while improving position estimates. This concludes that the RBPF using reduced rank GPs is capable of onboard improved position estimation on underwater vehicles.