Batch Least-squares Research Articles

Mouse embryos cultured in bovine amniotic fluid

Classical Orbit Determination algorithms provide an estimation of the state vector and its associated uncertainty assuming Gaussian processes and linear dynamics. However, a common problem of Orbit Determination processes is the misrepresentation of the Resident Space Object uncertainty through the estimated and predicted covariance. Ultimately, this causes a great impact in the quality and accuracy of Space Surveillance and Tracking products as the covariance is overly optimistic (too small) and the true uncertainty of the object is not properly captured. One of the causes for the unrealism of the covariance is found in the uncertainty of the dynamical and measurement models used to describe the motion of the objects.The aim of this work is to devise a novel methodology to improve the covariance realism of Orbit Determination and Orbit Propagation processes through the classical theory of consider parameters in batch least-squares estimators. The devised methodology uses the theory of consider parameters to add the uncertainty of dynamic and measurement models to the estimated covariance.The magnitude of the consider parameters is weighted using statistical inference techniques, proposing an innovative method to derive its contribution to the covariance. The conceived methodology is suitable for any type of measurement or object, although its primary goal is to correct unrealistic covariance of non-cooperative targets.Among the wide variety of uncertainty sources affecting covariance realism, the influence and effect of two relevant modelling uncertainties, the atmospheric drag force and the range bias, is discussed during this publication. The proposed methodology is first applied to a simulated scenario of tracking measurements, using a Monte Carlo approach. A case involving real radar observations is presented as well, where the uncertainty realism of the Sentinel-3A orbit determination and propagation is assessed and improved. Using the novel methodology of Covariance Determination, covariance realism improvement of a real tracking campaign is achieved and the uncertainty of the atmospheric drag force model quantified.

Approaches of FIR system identification with noisy data using higher order statistics

The problem of estimating the parameters of a moving average model from the cumulant statistics of the noisy observations of the system output is discussed. The system is driven by an independently identically distributed nonGaussian sequence that is not observed. The noise is additive and may be colored and nonGaussian. Following some existing linear parametric approaches to this problem, a linear method and a modification to an existing linear method are proposed for consistent parameter estimation in measurement noise under the assumption that the system order is known. Both recursive closed-form and batch least-squares versions of the parameter estimators are presented. The existing and the proposed linear methods utilize only a partial set of the relevant output statistics (this restriction being necessary to obtain a linear estimator), whereas there exist nonlinear method that exploit a much larger set of output statistics. A simulation example where two existing linear methods and the two new methods are compared to two existing nonlinear methods is presented. >

High-Accuracy GPS Semikinematic Positioning: Modeling and Results

The concept of GPS semikinematic positioning is reviewed, with an emphasis on accuracies at the centimeter level. Details of the mathematical model used herein are given. A two-step approach combines a Kalman filter to process kinematic carrier phase, phase rate, and pseudorange data, while a batch least-squares adjustment processes static carrier phase data. A semikinematic test performed over a well-controlled traverse in a mountainous region near Calgary is used to assess the performance of the algorithm. Several runs of the traverse were made with the station occupation time at 2–3 min. The analysis shows that centimeter-level accuracies can be achieved when five satellites are observed in good geometery, even when frequent cycle slips occur. An accuracy improvement technique of reverse processing is also investigated. Recommendations for improving semi-kinematic surveying are made.

Choice of least-squares algorithms for the identification of ARM AX dynamic systems

A comparative analysis of three identification algorithms, which minimise the sum of the squared errors between the model output and the observations of the output, is presented. Batch least-squares and extended recursive least-squares algorithms can be used to identify the model structure goodness of ARMAX application of each algorithm are investigated.

Sequential Orbit Determination with Auto-Correlated Gravity Modeling Errors

A sequential orbit determination algorithm is given which models auto-correlated gravity errors for nearcircular orbits. Using results from geodesy theory it is demonstrated that gravity modeling error auto-correlation cannot be approximated with the Kalman white process noise model. The resulting auto-correlated orbit error process is transformed from the status of non-Markov to Markov at the expense of an assumption which imposes symmetries on certain covariance functions. It is emphasized that the character of the trajectory estimation problem is significantly changed by recognizing and using the auto-correlation property associated with gravity modeling errors. I. Introduction S EQUENTIAL orbit determination procedures enjoy significant operational advantages over the older methods1 of batch least-squares differential corrections: ephemeris integration across overlapping measurement time intervals is eliminated thereby significantly reducing computation time, force modeling discontinuities are easily bridged with a significant reduction in the time required to remove transients due to uncertainties associated with the discontinuities, and the extended Kalman filter provides a structure to formally account for stochastic force modeling errors. With the richer structure associated with sequential procedures, one would have hoped to say that they enjoy a consistent and significant accuracy improvement over the< older batch methods. In the author's experience this has not been realized. Herein the author will attempt to show that a fundamental reason for this disappointment is to be found in the failure of stochastic force modeling errors to satisfy the white process noise hypothesis of the Kalman filter. Particularly we will show in Sec. V that gravity modeling errors are autocorrelated with respect to time so that any white noise approximation will yield a nonoptimal procedure. The problem of interest is stated in Sec. Ill, using the notation and definitions introduced in Sec. II. An approximate solution which accounts for gravity modeling error auto-correlation is given in Sec. IV. The remaining sections (V-IX) are devoted to the derivation of the solution given in Sec. IV. Equation (6), given in the solution, follows Eq. (26) in Sec. IX from a derivational point of view.

A nonlinear counterexamle for batch and extended sequential estimation algorithms

A simple nonlinear example is presented which shows the well-known iterated batch least-squares and extended sequential estimation algorithms may converge to different estimates. For this example one may even say the extended sequential algorithms converge to the wrong value.