A pair (V, G) is called geometric structure, where V is a vector space and G is a subgroup GL(V), which is a set of transmission matrices. In this thesis we classify structures, which are based on properties of quaternions. Geometric structures based on quaternions are called triple structures. Triple structures are four structures with similar properties as quaternions. Quaternions are generated from real numbers and three complex units. We write quaternions in this shape a+bi+cj+dk.

No description available.

Kalman filtering is a well-established method for fusing sensor data in order to accuratelyestimate unknown variables. Recently, the unscented Kalman filter (UKF) has beenused due to its ability to propagate the first and second moments of the probability distribution of an estimated state through a non-linear transformation. The design of ageneric algorithm which implements this filter occupies the first part of this thesis. The generality and functionality of the filter were tested on a toy example and the results are within machine accuracy when compared to those of an equivalent C++ implementation.Application of this filter to the attitude navigation problem becomes non-trivial when coupled to quaternions. Challenges present include the non-commutation of rotations and the dimensionality difference between quaternions and the degrees of freedom of the motion. The second part of this thesis deals with the formulation of the UKF in the quaternion space. This was achieved by implementing an error-state formulation of the process model, bounding estimation in the infinitesimal space and thus de-coupling rotations from non-commutation and bridging the dimensionality discrepancy of quaternions and their respective covariances.The attitude navigation algorithm was then tested using an IMU and a magnetometer.Results show a bounded estimation error which settles to around 1 degree. A detailed look of the filter mechanization process was also presented showing expected behavior for estimation of the initial attitude with error tolerance of 1 mdeg. The structure and design of the proposed formulation allows for trivially incorporating other sensors inthe estimation process and more intricate modelling of the stochastic processes present,potentially leading to greater estimation accuracy. / Kalman filtrering är en vältablerad metod for att sammanväga sensordata för att erhålla noggranna estimat av okända variabler. Nyligen har den typ av kalman filter som kallas unscented Kalman filter (UKF) ökat i populäritet pa grund av dess förmåga att propagera de första och andra momenten för sannolikhetsfördelningen för ett estimera tillstånd genom en ickelinjär transformation. Designen av en generisk algoritm som implementerar denna typ av filter upptar den första delen av denna avhandling. Generaliteten och funktionaliteten för detta filter testades på ett minimalt exempel och resultaten var identiska med de för en ekvivalent C++-implementation till den noggrannhet som tillåts av den nita maskinprecisionen. Användandet av detta filter för attitydnavigering blir icke-trivialt när det anvands forkvaternioner. De utmaningar som uppstar inkluderar att rotationer inte kommuterar och att de finns en skillnad i dimensionalitet mellan kvaternioner och antalet frihetsgrader i rörelsen. Den andra delen av denna avhandling behandlar formuleringen av ett UKF för ett tillstånd som inkluderar en kvaternion. Detta gjordes genom att implementera en så kallad error state-formulering av processmodellen, vilken begränsar estimeringen till ett innitesimalt tillstånd och därigenom undviker problemen med att kvaternionmultiplikation inte kommuterar och överbryggar skillnaden i dimensionalitet hos kvaternioner och deras motsvarande vinkelosäkerheter.Attitydnavigeringen testades sedan med hjälp av en IMU och en magnetometer.Resultaten visade ett begränsat estimeringsfel som ställer in sig kring 1 grad. Strukturen och designen av den föreslagna formuleringen möjliggör på ett rattframt satt tillägg av andra sensorer i estimeringsprocessen och mer detaljerad modellering av de stokastiska processerna, vilket potentiellt leder till högre estimering noggrannhet.

Location-based services (LBSs) have become pervasive, and the demand for these systems and services is rising. Indoor Positioning Systems (IPSs) are key to extend location-based services indoors where the Global Positioning System (GPS) is not reliable due to low signal strength and complicated signal propagation environment. Most existing IPSs either require the installation of special hardware devices or build a fingerprint map, which is expensive, time-consuming, and labor-intensive. Developments in microelectromechanical systems (MEMS) have resulted in significant advancements in the low-cost compact MARG inertial sensors, making it possible to achieve low-cost and high-accuracy IPSs. This research considers the indoor positioning problem and aims to design and develop an infrastructure-free self-contained indoor positioning and tracking system based on Pedestrian Dead Reckoning (PDR) using MEMS MARG inertial sensors. PDR-based systems rely on MARG inertial sensor measurements to estimate the current position of the object by using a previously determined position without external references. Many issues still exist in developing such systems, such as cumulative errors, high-frequency sensor noises, the gyro drift issue, magnetic distortions, etc. As the MARG sensors are inherently error-prone, the most significant challenge is how to design sensor fusion models and algorithms to accurately extract useful location-based information from individual motion and magnetic sensors. The objective of this thesis is to solve these issues and mitigate the challenges. The proposed positioning system is designed with four main modules at the system level and a dual-mode feature. Specifically, the four main modules are mode detection, step detection and moving distance estimation, heading and orientation estimation, and position estimation. To address the cumulative error issue of using low-cost inertial sensors, signal processing and sensor fusion techniques are utilized for algorithm design. Experimental evaluations show that the proposed position estimation algorithm is able to achieve high positioning accuracy at low costs for the indoor environment. / Thesis / Master of Applied Science (MASc) / With the maturity of microelectromechanical systems (MEMS) technology in recent years, Magnetic, Angular Rate, and Gravity (MARG) sensors are embedded in most smart devices. This research considers the indoor positioning problem and aims to design and develop an infrastructure-free self-contained MEMS MARG inertial sensor-based indoor positioning and tracking system with high precision. The proposed positioning system uses the Pedestrian Dead Reckoning (PDR) approach and includes four main modules at the system level with a dual-mode feature. Specifically, the four main modules are mode detection, step detection and moving distance estimation, heading and orientation estimation, and position estimation. The two modes are static mode and dynamic mode. To address the cumulative error issue of using low-cost inertial sensors, signal processing and sensor fusion techniques are utilized for algorithm design. The detection and estimation algorithms of each module are presented in the system design chapter. Experimental evaluations including trajectory results under five scenarios show that the proposed position estimation algorithm achieves a higher position accuracy than that of conventional estimation methods.

The work is situated within the theory of slice analysis, a generalization of complex analysis for hypercomplex numbers, considering function of both quaternionic and Clifford variables, in both one and several variables. %We first characterize some partial slice sets of The primary focus of the thesis is on the harmonic and polyharmonic properties of slice regular functions. We derive explicit formulas for the iteration of the Laplacian on slice regular functions, proving that their degree of harmonicity increases with the dimension of the algebra. Consequently, we present Almansi-type decompositions for slice functions in several variables. Additionally, using the harmonic properties of the partial spherical derivatives and their connection with the Dirac operator in Clifford analysis, we achieve a generalization of the Fueter and Fueter-Sce theorems in the several variables context. Finally, we establish that regular polynomials of sufficiently low degree are the unique slice regular functions in the kernel of the iteration of the Laplacian, whose power is less than Sce index.

Les travaux présentés s'intéressent au traitement des signaux à valeurs sur des espaces non-commutatifs, en particulier sur le groupe des rotations et les quaternions. Principalement, ce sont les signaux et processus aléatoires qui sont au centre de nos préoccupations, et nous présentons quelques résultats illustrant leur intérêt en physique des ondes polarisées. Nous montrons par exemple comment les processus aléatoires sur le groupe des rotations permettent d'étudier la diffusion multiple des ondes dans les milieux aléatoires et l'apparition de la phase géométrique pour les ondes polarisées dans ces milieux. Les résultats obtenus sont basés sur des notions empruntées à la théorie des groupes et de la représentation, la théorie des processus aléatoires et de l'estimation ainsi qu'à la géométrie différentielle. L'application majeure des résultats présentés est l'étude des ondes élastiques dans les milieux aléatoires.

Nous étudions l'euclidianité des corps de nombres pour la norme et quelques unes de ses généralisations. Nous donnons en particulier un algorithme qui calcule le minimum euclidien d'un corps de nombres de signature quelconque. Cela nous permet de prouver que de nombreux corps sont euclidiens ou non pour la norme. Ensuite, nous appliquons cet algorithme à l'étude des classes euclidiennes pour la norme, ce qui permet d'obtenir de nouveaux exemples de corps de nombres avec une classe euclidienne non principale. Par ailleurs, nous déterminons tous les corps cubiques purs avec une classe euclidienne pour la norme. Enfin, nous nous intéressons aux corps de quaternions euclidiens. Après avoir énoncé les propriétés de base, nous étudions quelques cas particuliers. Nous donnons notamment la liste complète des corps de quaternions euclidiens et totalement définis sur un corps de nombres de degré au plus deux. / We study norm-Euclideanity of number fields and some of its generalizations. In particular, we provide an algorithm to compute the Euclidean minimum of a number field of any signature. This allows us to study the norm-Euclideanity of many number fields. Then, we extend this algorithm to deal with norm-Euclidean classes and we obtain new examples of number fields with a non-principal norm-Euclidean class. Besides, we describe the complete list of pure cubic number fields admitting a norm-Euclidean class. Finally, we study the Euclidean property in quaternion fields. First, we establish its basic properties, then we study some examples. We provide the complete list of Euclidean quaternion fields, which are totally definite over a number field with degree at most two.

The goal of this work is to create an application, which will combine procedural and keyfram animations with subsequent visualization. Composition of this two different animations techniques is used to animate a virtual character. To combine this two techniques one starts with interpolations from keyframe animation and then enchance them by procedural animations to properly fit into the characters surroundings. This procedural part of animation is obtained by using forward and inverse kinematics. Whole application is written in C++, uses GLM math library for computations and OpenGL and GLUT for final visualization.

In the field of robotics and self-navigation, Simultaneous Localization and Mapping (SLAM) is a technique crucial for estimating poses while concurrently creating a map of the environment. Robotics applications often rely on various sensors for pose estimation, including cameras, inertial measurement units (IMUs), and more. Traditional discrete SLAM, utilizing stereo camera pairs and inertial measurement units, faces challenges such as time offsets between sensors. A solution to this issue is the utilization of continuous-time models for pose estimation. This thesis delves into the exploration and implementation of a continuous-time SLAM system, investigating the advantages of multi-modal sensor fusion over discrete stereo vision models. The findings indicate that incorporating an IMU into the system enhances pose estimation, providing greater robustness and accuracy compared to relying solely on visual SLAM. Furthermore, leveraging the continuous model's derivative and smoothness allows for decent pose estimation with fewer measurements, reducing the required quantity of measurements and computational resources.

This thesis presented an overview of the development of a geolocating algorithm as part of a geoprocessor for raw satellite imagery. This algorithm was devised for and limited by the specifications of a state-of-the-art multispectral telescope designed by Aistech Space, hosted onboard the Guardian spacecraft, which will observe Earth through the visible, near infrared, and thermal infrared bands of the electromagnetic spectrum. The geolocation algorithm presented here is composed of the combination of two models. The first is a physical model, which makes use of spacecraft telemetry and external satellite-tracking data to approximate the geographical center of a sensed scene. Secondly, an optical model obtains a reference Landsat image based on the timestamp and approximated location of the sensed scene and utilizes image processing techniques to pinpoint a more precise geographical location of the sensed scene within acceptable limits. This performance was achieved in 77% of the cases considered. To conclude, a roadmap of the subsequent development topics and their relevance was laid out. / Detta examensarbete presenterar en översikt för utvecklingen av en geolokaliseringsalgoritm som en del av en geoprocessor för obearbetade satellitbilder. Algoritmen anpassades för och begränsades av specifikationerna för ett toppmodernt multispektralt teleskop designat av Aistech Space. Teleskopet kommer att finnas ombord på rymdfarkosten Guardian, där den är avsedd att observera jorden i de synliga, nära infraröda och termiska infraröda delarna av det elektromagnetiska spektrumet. Geolokaliseringsalgoritmen som presenteras i detta arbete är sammansatt av en kombination av två modeller. Den första är en fysisk modell, vilken använder sig av rymdfarkostens telemetri och extern satellitspårningsdata för att approximera det geografiska centrumet av en plats. Den andra är en optisk modell, vilken använder sig av en Landsat-referensbild baserad på tidsstämpeln och den ungefärliga positionen av platsen och använder sedan bildbehandlingstekniker för att fastställa en mer exakt geografisk position av platsen inom acceptabla gränser. Denna prestation lyckades uppnås i 77% av de övervägda fallen. Avslutningsvis lades en plan ut för de efterföljande utvecklingsämnena och deras relevans.