Quaternion normalization in additive EKF for spacecraft attitude determination

This work introduces, examines, and compares several quaternion normalization algorithms, which are shown to be an effective stage in the application of the additive extended Kalman filter (EKF) to spacecraft attitude determination, which is based on vector measurements. Two new normalization schemes are introduced. They are compared with one another and with the known brute force normalization scheme, and their efficiency is examined. Simulated satellite data are used to demonstrate the performance of all three schemes. A fourth scheme is suggested for future research. Although the schemes were tested for spacecraft attitude determination, the conclusions are general and hold for attitude determination of any three dimensional body when based on vector measurements, and use an additive EKF for estimation, and the quaternion for specifying the attitude

Quaternion normalization in spacecraft attitude determination

Attitude determination of spacecraft usually utilizes vector measurements such as Sun, center of Earth, star, and magnetic field direction to update the quaternion which determines the spacecraft orientation with respect to some reference coordinates in the three dimensional space. These measurements are usually processed by an extended Kalman filter (EKF) which yields an estimate of the attitude quaternion. Two EKF versions for quaternion estimation were presented in the literature; namely, the multiplicative EKF (MEKF) and the additive EKF (AEKF). In the multiplicative EKF, it is assumed that the error between the correct quaternion and its a-priori estimate is, by itself, a quaternion that represents the rotation necessary to bring the attitude which corresponds to the a-priori estimate of the quaternion into coincidence with the correct attitude. The EKF basically estimates this quotient quaternion and then the updated quaternion estimate is obtained by the product of the a-priori quaternion estimate and the estimate of the difference quaternion. In the additive EKF, it is assumed that the error between the a-priori quaternion estimate and the correct one is an algebraic difference between two four-tuple elements and thus the EKF is set to estimate this difference. The updated quaternion is then computed by adding the estimate of the difference to the a-priori quaternion estimate. If the quaternion estimate converges to the correct quaternion, then, naturally, the quaternion estimate has unity norm. This fact was utilized in the past to obtain superior filter performance by applying normalization to the filter measurement update of the quaternion. It was observed for the AEKF that when the attitude changed very slowly between measurements, normalization merely resulted in a faster convergence; however, when the attitude changed considerably between measurements, without filter tuning or normalization, the quaternion estimate diverged. However, when the quaternion estimate was normalized, the estimate converged faster and to a lower error than with tuning only. In last years, symposium we presented three new AEKF normalization techniques and we compared them to the brute force method presented in the literature. The present paper presents the issue of normalization of the MEKF and examines several MEKF normalization techniques

Methodological progress and substantial insights in the field of internet research

We are pleased to present the first issue of the International Journal of Internet Science (IJIS). As the mission statement indicates the journal is intended to cover two kinds of related research. On the one hand it presents methodological research that shows how the Internet can be used as a tool for data collection. On the other hand, it presents substantial research on the social and psychological implications of the Internet and other new online media. It is our conviction that both streams of research can profit from each other so that the journal includes both. The current issue is giving a first impression on the variety of approaches to be dealt with in this journal while exemplifying the common ground that good empirical research is enhanced by a clear theory-guidance. Many of the articles stem from earlier presentations given at one of the international conferences of the conference series General Online Research (GOR). The selection of articles brings together empirical methodological studies on important determinants of response behaviour in online data collection and empirical research on the social implications of web- and computer use. To the first type of research belong the contribution of Smyth, Dillman, Christian, and Stern, the contribution of Krysan and Couper, the paper of Göritz and the article of Ollesch, Heineken, and Schulte. To the second type belong the articles of Mesch and Talmud as well as the paper of Korupp

Magnetometer-only attitude and rate determination for a gyro-less spacecraft

Attitude determination algorithms that requires only the earth's magnetic field will be useful for contingency conditions. One way to determine attitude is to use the time derivative of the magnetic field as the second vector in the attitude determination process. When no gyros are available, however, attitude determination becomes difficult because the rates must be propagated via integration of Euler's equation, which in turn requires knowledge of the initial rates. The spacecraft state to be determined must then include not only the attitude but also rates. This paper describes a magnetometer-only attitude determination scheme with no a priori knowledge of the spacecraft state, which uses a deterministic algorithm to initialize an extended Kalman filter. The deterministic algorithm uses Euler's equation to relate the time derivatives of the magnetic field in the reference and body frames and solves the resultant transcendental equations for the coarse attitude and rates. An important feature of the filter is that its state vector also includes corrections to the propagated rates, thus enabling it to generate highly accurate solutions. The method was tested using in-flight data from the Solar, Anomalous, and Magnetospheric Particles Explorer (SAMPEX), a Small Explorer spacecraft. SAMPEX data using several eclipse periods were used to simulate conditions that may exist during the failure of the on-board digital sun sensor. The combined algorithm has been found effective, yielding accuracies of 1.5 deg in attitude (within even nominal mission requirements) and 0.01 degree per second (deg/sec) in the rates

Advantages of estimating rate corrections during dynamic propagation of spacecraft rates: Applications to real-time attitude determination of SAMPEX

This paper describes real-time attitude determination results for the Solar, Anomalous, and Magnetospheric Particle Explorer (SAMPEX), a gyroless spacecraft, using a Kalman filter/Euler equation approach denoted the real-time sequential filter (RTSF). The RTSF is an extended Kalman filter whose state vector includes the attitude quaternion and corrections to the rates, which are modeled as Markov processes with small time constants. The rate corrections impart a significant robustness to the RTSF against errors in modeling the environmental and control torques, as well as errors in the initial attitude and rates, while maintaining a small state vector. SAMPLEX flight data from various mission phases are used to demonstrate the robustness of the RTSF against a priori attitude and rate errors of up to 90 deg and 0.5 deg/sec, respectively, as well as a sensitivity of 0.0003 deg/sec in estimating rate corrections in torque computations. In contrast, it is shown that the RTSF attitude estimates without the rate corrections can degrade rapidly. RTSF advantages over single-frame attitude determination algorithms are also demonstrated through (1) substantial improvements in attitude solutions during sun-magnetic field coalignment and (2) magnetic-field-only attitude and rate estimation during the spacecraft's sun-acquisition mode. A robust magnetometer-only attitude-and-rate determination method is also developed to provide for the contingency when both sun data as well as a priori knowledge of the spacecraft state are unavailable. This method includes a deterministic algorithm used to initialize the RTSF with coarse estimates of the spacecraft attitude and rates. The combined algorithm has been found effective, yielding accuracies of 1.5 deg in attitude and 0.01 deg/sec in the rates and convergence times as little as 400 sec

NH3_{3}-SCR over V-W/TiO2_{2} Investigated by Operando X-ray Absorption and Emission Spectroscopy

V–W/TiO$_{2}$-based catalysts, which are used for the removal of NO$_{x}$ from the exhaust of diesel engines and stationary sources via selective catalytic reduction with NH$_{3}$ (NH$_{3}$-SCR), were studied by operando X-ray absorption spectroscopy (XAS) and emerging photon-in/photon-out techniques. In order to minimize the influence of highly X-ray absorbing tungsten and the fluorescence of titanium, we used a high-energy-resolution fluorescence setup that is able to separate efficiently the V Kβ$_{1,3}$ emission lines and additionally allows to record valence-to-core (vtc) X-ray emission lines. High-energy resolution fluorescence-detected XAS (HERFD-XAS) and vtc X-ray emission spectroscopy (vtc-XES) proved to be the only way to perform an operando V K edge X-ray spectroscopic study on industrially relevant V–W/TiO$_{2}$ catalysts so far. The V–W/TiO$_{2}$ and V/TiO$_{2}$ samples synthesized by incipient wetness impregnation and grafting exhibited high activity toward NH$_{3}$-SCR. Raman spectroscopy showed that they mainly contained highly dispersed, isolated, and polymeric V-oxo species. HERFD-XAS and XES identified redox cycling of vanadium species between V$^{4+}$ and V$^{5+}$. With respect to most of the potential NH$_{3}$ adsorption complexes, density functional theory calculations further showed that vtc-XES is more limited than surface-sensitive techniques such as infrared spectroscopy; hence, a combination of X-ray techniques with IR or similar spectroscopies is required to unequivocally identify the mechanism of NH$_{3}$-SCR over vanadia-based catalysts

Competition and Selection Among Conventions

In many domains, a latent competition among different conventions determines which one will come to dominate. One sees such effects in the success of community jargon, of competing frames in political rhetoric, or of terminology in technical contexts. These effects have become widespread in the online domain, where the data offers the potential to study competition among conventions at a fine-grained level. In analyzing the dynamics of conventions over time, however, even with detailed on-line data, one encounters two significant challenges. First, as conventions evolve, the underlying substance of their meaning tends to change as well; and such substantive changes confound investigations of social effects. Second, the selection of a convention takes place through the complex interactions of individuals within a community, and contention between the users of competing conventions plays a key role in the convention's evolution. Any analysis must take place in the presence of these two issues. In this work we study a setting in which we can cleanly track the competition among conventions. Our analysis is based on the spread of low-level authoring conventions in the eprint arXiv over 24 years: by tracking the spread of macros and other author-defined conventions, we are able to study conventions that vary even as the underlying meaning remains constant. We find that the interaction among co-authors over time plays a crucial role in the selection of them; the distinction between more and less experienced members of the community, and the distinction between conventions with visible versus invisible effects, are both central to the underlying processes. Through our analysis we make predictions at the population level about the ultimate success of different synonymous conventions over time--and at the individual level about the outcome of "fights" between people over convention choices.Comment: To appear in Proceedings of WWW 2017, data at https://github.com/CornellNLP/Macro