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

Modeling Elementary Heterogeneous Chemistry and Electrochemistry in Solid-Oxide Fuel Cells

This paper presents a new computational framework for modeling chemically reacting flow in anode-supported solid-oxide fuel cells (SOFC). Depending on materials and operating conditions, SOFC anodes afford a possibility for internal reforming or catalytic partial oxidation of hydrocarbon fuels. An important new element of the model is the capability to represent elementary heterogeneous chemical kinetics in the form of multistep reaction mechanisms. Porous-media transport in the electrodes is represented with a dusty-gas model. Charge-transfer chemistry is represented in a modified Butler-Volmer setting that is derived from elementary reactions, but assuming a single rate-limiting step. The model is discussed in terms of systems with defined flow channels and planar membrane-electrode assemblies. However, the underlying theory is independent of the particular geometry. Examples are given to illustrate the model

An Evaluation of Attitude-Independent Magnetometer-Bias Determination Methods

Although several algorithms now exist for determining three-axis magnetometer (TAM) biases without the use of attitude data, there are few studies on the effectiveness of these methods, especially in comparison with attitude dependent methods. This paper presents the results of a comparison of three attitude independent methods and an attitude dependent method for computing TAM biases. The comparisons are based on in-flight data from the Extreme Ultraviolet Explorer (EUVE), the Upper Atmosphere Research Satellite (UARS), and the Compton Gamma Ray Observatory (GRO). The effectiveness of an algorithm is measured by the accuracy of attitudes computed using biases determined with that algorithm. The attitude accuracies are determined by comparison with known, extremely accurate, star-tracker-based attitudes. In addition, the effect of knowledge of calibration parameters other than the biases on the effectiveness of all bias determination methods is examined

Extending differential optical absorption spectroscopy for limb measurements in the UV

Methods of UV/VIS absorption spectroscopy to determine the constituents in the Earth's atmosphere from measurements of scattered light are often based on the Beer-Lambert law, like e.g. Differential Optical Absorption Spectroscopy (DOAS). While the Beer-Lambert law is strictly valid for a single light path only, the relation between the optical depth and the concentration of any absorber can be approximated as linear also for scattered light observations at a single wavelength if the absorption is weak. If the light path distribution is approximated not to vary with wavelength, also linearity between the optical depth and the product of the cross-section and the concentration of an absorber can be assumed. These assumptions are widely made for DOAS applications for scattered light observations. <br><br> For medium and strong absorption of scattered light (e.g. along very long light-paths like in limb geometry) the relation between the optical depth and the concentration of an absorber is no longer linear. In addition, for broad wavelength intervals the differences in the travelled light-paths at different wavelengths become important, especially in the UV, where the probability for scattering increases strongly with decreasing wavelength. <br><br> However, the DOAS method can be extended to cases with medium to strong absorptions and for broader wavelength intervals by the so called air mass factor modified (or extended) DOAS and the weighting function modified DOAS. These approaches take into account the wavelength dependency of the slant column densities (SCDs), but also require a priori knowledge for the air mass factor or the weighting function from radiative transfer modelling. <br><br> We describe an approach that considers the fitting results obtained from DOAS, the SCDs, as a function of wavelength and vertical optical depth and expands this function into a Taylor series of both quantities. The Taylor coefficients are then applied as additional fitting parameters in the DOAS analysis. Thus the variability of the SCD in the fit window is determined by the retrieval itself. <br><br> This new approach provides a description of the SCD the exactness of which depends on the order of the Taylor expansion, and is independent from any assumptions or a priori knowledge of the considered absorbers. <br><br> In case studies of simulated and measured spectra in the UV range (332–357 nm), we demonstrate the improvement by this approach for the retrieval of vertical profiles of BrO from the SCIAMACHY limb observations. The results for BrO obtained from the simulated spectra are closer to the true profiles, when applying the new method for the SCDs of ozone, than when the standard DOAS approach is used. For the measured spectra the agreement with validation measurements is also improved significantly, especially for cases with strong ozone absorption. <br><br> While the focus of this article is on the improvement of the BrO profile retrieval from the SCIAMACHY limb measurements, the novel approach may be applied to a wide range of DOAS retrievals

Preferences of diabetes patients and physicians: A feasibility study to identify the key indicators for appraisal of health care values

<p>Abstract</p> <p>Background</p> <p>Evidence-based medicine, the Institute of Medicine (IOM) and the German Institute for Quality and Efficiency in Health Care (IQWiG), support the inclusion of patients' preferences in health care decisions. In fact there are not many trials which include an assessment of patient's preferences. The aim of this study is to demonstrate that preferences of physicians and of patients can be assessed and that this information may be helpful for medical decision making.</p> <p>Method</p> <p>One of the established methods for assessment of preferences is the conjoint analysis. Conjoint analysis, in combination with a computer assisted telephone interview (CATI), was used to collect data from 827 diabetes patients and 60 physicians, which describe the preferences expressed as levels of four factors in the management and outcome of the disease. The first factor described the main treatment effect (reduction of elevated Hb_A1c, improved well-being, absence of side effects, and no limitations of daily life). The second factor described the effect on the body weight (gain, no change, reduction). The third factor analyzed the mode of application (linked to meals or flexible application). The fourth factor addressed the type of product (original brand or generic product). Utility values were scaled and normalized in a way that the sum of utility points across all levels is equal to the number of attributes (factors) times 100.</p> <p>Results</p> <p>The preference weights confirm that the reduction of body weight is at least as important for patients - especially obese patients - and physicians as the reduction of an elevated Hb_A1c. Original products were preferred by patients while general practitioners preferred generic products.</p> <p>Conclusion</p> <p>Using the example of diabetes, the difference between patients' and physicians' preferences can be assessed. The use of a conjoint analysis in combination with CATI seems to be an effective approach for generation of data which are needed for policy and medical decision making in health care.</p

Accounting for the effect of horizontal gradients in limb measurements of scattered sunlight

Limb measurements provided by the SCanning Imaging Absorption spectrometer for Atmospheric CHartographY (SCIAMACHY) on the ENVISAT satellite allow retrieving stratospheric profiles of various trace gases on a global scale, among them BrO for the first time. For limb observations in the UV/VIS spectral region the instrument measures scattered light with a complex distribution of light paths: the light is measured at different tangent heights and can be scattered or absorbed in the atmosphere or reflected by the ground. By means of spectroscopy and radiative transfer modelling these measurements can be inverted to retrieve the vertical distribution of stratospheric trace gases. &lt;br&gt;&lt;/br&gt; The fully spherical 3-D Monte Carlo radiative transfer model &quot;Tracy-II&quot; is applied in this study. The Monte Carlo method benefits from conceptual simplicity and allows realizing the concept of full spherical geometry of the atmosphere and also its 3-D properties, which is important for a realistic description of the limb geometry. Furthermore it allows accounting for horizontal gradients in the distribution of trace gases. &lt;br&gt;&lt;/br&gt; In this study the effect of horizontally inhomogeneous distributions of trace gases along flight/viewing direction on the retrieval of profiles is investigated. We introduce a tomographic method to correct for this effect by combining consecutive limb scanning sequences and utilizing the overlap in their measurement sensitivity regions. It is found that if horizontal inhomogenity is not properly accounted for, typical errors of 20% for NO&lt;sub&gt;2&lt;/sub&gt; and up to 50% for OClO around the altitude of the profile peak can arise for measurements close to the Arctic polar vortex boundary in boreal winter

A PC-based magnetometer-only attitude and rate determination system for gyroless spacecraft

This paper describes a prototype PC-based system that uses measurements from a three-axis magnetometer (TAM) to estimate the state (three-axis attitude and rates) of a spacecraft given no a priori information other than the mass properties. The system uses two algorithms that estimate the spacecraft's state - a deterministic magnetic-field only algorithm and a Kalman filter for gyroless spacecraft. The algorithms are combined by invoking the deterministic algorithm to generate the spacecraft state at epoch using a small batch of data and then using this deterministic epoch solution as the initial condition for the Kalman filter during the production run. System input comprises processed data that includes TAM and reference magnetic field data. Additional information, such as control system data and measurements from line-of-sight sensors, can be input to the system if available. Test results are presented using in-flight data from two three-axis stabilized spacecraft: Solar, Anomalous, and Magnetospheric Particle Explorer (SAMPEX) (gyroless, Sun-pointing) and Earth Radiation Budget Satellite (ERBS) (gyro-based, Earth-pointing). The results show that, using as little as 700 s of data, the system is capable of accuracies of 1.5 deg in attitude and 0.01 deg/s in rates; i.e., within SAMPEX mission requirements

Automated Generation of Microkinetics for Heterogeneously Catalyzed Reactions Considering Correlated Uncertainties

The study presents an ab-initio based framework for the automated construction of microkinetic mechanisms considering correlated uncertainties in all energetic parameters and estimation routines. 2000 unique microkinetic models were generated within the uncertainty space of the BEEF-vdW functional for the oxidation reactions of representative exhaust gas emissions from stoichiometric combustion engines over Pt(111) and compared to experiments through multiscale modeling. The ensemble of simulations stresses the importance of considering uncertainties. Within this set of first-principles-based models, it is possible to identify a microkinetic mechanism that agrees with experimental data. This mechanism can be traced back to a single exchange-correlation functional, and it suggests that Pt(111) could be the active site for the oxidation of light hydrocarbons. The study provides a universal framework for the automated construction of reaction mechanisms with correlated uncertainty quantification, enabling a DFT-constrained microkinetic model optimization for other heterogeneously catalyzed systems

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