Theory and Applications of Kalman Filtering
Kalman filtering applications in space navigation, underwater operations, surface navigation, and Gauss-Markov processes
C. T. Leondes↗
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CHAMP Magnetic Anomalies of the Antarctic Crust
Regional magnetic signals of the crust are strongly masked by the core field and its secular variations components and hence difficult to isolate in the satellite measurements. In particular, the un-modeled effects of the strong auroral external fields and the complicated- behavior of the core field near the geomagnetic poles conspire to greatly reduce the crustal magnetic signal-to-noise ratio in the polar regions relative to the rest of the Earth. We can, however, use spectral correlation theory to filter the static lithospheric and core field components from the dynamic external field effects. To help isolate regional lithospheric from core field components, the correlations between CHAMP magnetic anomalies and the pseudo magnetic effects inferred from gravity-derived crustal thickness variations can also be exploited.. Employing these procedures, we processed the CHAMP magnetic observations for an improved magnetic anomaly map of the Antarctic crust. Relative to the much higher altitude Orsted and noisier Magsat observations, the CHAMP magnetic anomalies at 400 km altitude reveal new details on the effects of intracrustal magnetic features and crustal thickness variations of the Antarctic.
Kim, Hyung Rae↗
Estimating Antarctic Near-Surface Magnetic Anomalies from Oersted and CHAMP Satellite Magnetometer Observations
Significant improvement in predicting near-surface magnetic anomalies can result from the highly accurate magnetic observations of the CHAMP satellite that is orbiting at about 400 km altitude. In general, regional magnetic signals of the crust are strongly masked by the core field and its secular variations due to wavelength coupling in the spherical harmonic representation and thus are difficult to isolate in the satellite measurements. However, efforts to isolate the regional lithospheric from core field components can exploit the correlations between the CHAMP magnetic anomalies and the pseudo magnetic effects inferred from gravity-derived crustal thickness variations. In addition, we can use spectral correlation theory to filter the static lithospheric field components from the dynamic external field effects. Employing these procedures, we processed the CHAMP magnetic conservations for an improved magnetic anomaly map of the Antarctic crust. Relative to the much higher altitude Oersted and noisier Magsat observations, CHAMP magnetic anomalies at 400 km altitude reveal new details on the effects of intra-crustal magnetic features and crustal thickness variations of the Antarctic. Moreover, these results greatly facilitate predicting magnetic anomalies in the regional coverage gaps of the ADMAP compilation of Antarctic magnetic anomalies from shipborne, airborne and ground surveys. Our analysis suggests that considerable new insights on the magnetic properties of the lithosphere may be revealed by a further order-of-magnitude improvement in the accuracy of the magnetometer.
vonFrese, Ralph R. B.↗
Generalized Theory and Realization of Continuously Loss-Programmable Bandpass Filtering Attenuators
With the increased demand for modern wireless systems in various applications, the need for adjustable radio frequency (RF) systems has dramatically increased. These modern systems rely on operating in the microwave frequency spectrum (1 GHz to 1 THz) without interference from other devices while also retaining the ability to detect very low and very high power signals simultaneously. There is also an ever-increasing demand for reducing the cost, size, weight, and power (C-SWaP) of these modern RF systems, driving the demand for more agile filters in the microwave frequency spectrum. In this paper, a generalized theory of bandpass filtering attenuators (filtenuators) is proposed. A filtenuator is a device that combines the frequency- selective characteristics of a filter and the loss-programmable characteristics of an attenuator into a single component. The loss- programmable aspect of the filtenuator is based on the tuning of a π-network of resistances, which are implemented using PIN diodes to control the individual resistance values electronically. A loss-programmable, third-order Chebyshev bandpass filtenuator is designed, fabricated, and measured to verify the generalized theory. The filtenuator is designed to operate at 1 GHz and have a tunable attenuation range of 2-10 dB. This proposed filtenuator demonstrates the feasibility of a tunable, low C-SWaP solution to increase RF system dynamic range and a design process that allows for future development of filtenuators.
42 ENGINEERING↗
Recent advances in birefringent filters
In the past several years, important developments have been made in the construction and theory of birefringent filters. Tunable filters that operate from 4500 to 7000 A with full widths at half maximum of 0.05 A and maximum transmissions of almost 20 percent have been constructed. Inexpensive large-aperture (at least 15 cm) achromatic waveplates have been fabricated which allow full use of resolution-solid angle products that are 100 times that of Fabry-Perots and 10,000 times that of grating systems.
Title, A. M.↗
Nonlinear estimation theory applied to orbit determination
The development of an approximate nonlinear filter using the Martingale theory and appropriate smoothing properties is considered. Both the first order and the second order moments were estimated. The filter developed can be classified as a modified Gaussian second order filter. Its performance was evaluated in a simulated study of the problem of estimating the state of an interplanetary space vehicle during both a simulated Jupiter flyby and a simulated Jupiter orbiter mission. In addition to the modified Gaussian second order filter, the modified truncated second order filter was also evaluated in the simulated study. Results obtained with each of these filters were compared with numerical results obtained with the extended Kalman filter and the performance of each filter is determined by comparison with the actual estimation errors. The simulations were designed to determine the effects of the second order terms in the dynamic state relations, the observation state relations, and the Kalman gain compensation term. It is shown that the Kalman gain-compensated filter which includes only the Kalman gain compensation term is superior to all of the other filters.
Choe, C. Y.↗
Multifidelity Ensemble Kalman Filtering Using Surrogate Models Defined by Theory-Guided Autoencoders
Data assimilation is a Bayesian inference process that obtains an enhanced understanding of a physical system of interest by fusing information from an inexact physics-based model, and from noisy sparse observations of reality. The multifidelity ensemble Kalman filter (MFEnKF) recently developed by the authors combines a full-order physical model and a hierarchy of reduced order surrogate models in order to increase the computational efficiency of data assimilation. The standard MFEnKF uses linear couplings between models, and is statistically optimal in case of Gaussian probability densities. This work extends the MFEnKF into to make use of a broader class of surrogate model such as those based on machine learning methods such as autoencoders non-linear couplings in between the model hierarchies. We identify the right-invertibility property for autoencoders as being a key predictor of success in the forecasting power of autoencoder-based reduced order models. We propose a methodology that allows us to construct reduced order surrogate models that are more accurate than the ones obtained via conventional linear methods. Numerical experiments with the canonical Lorenz'96 model illustrate that nonlinear surrogates perform better than linear projection-based ones in the context of multifidelity ensemble Kalman filtering. We additionality show a large-scale proof-of-concept result with the quasi-geostrophic equations, showing the competitiveness of the method with a traditional reduced order model-based MFEnKF.
97 MATHEMATICS AND COMPUTING↗
An innovations approach to decoupling of multibody dynamics and control
The problem of hinged multibody dynamics is solved using an extension of the innovations approach of linear filtering and prediction theory to the problem of mechanical system modeling and control. This approach has been used quite effectively to diagonalize the equations for filtering and prediction for linear state space systems. It has similar advantages in the study of dynamics and control of multibody systems. The innovations approach advanced here consists of expressing the equations of motion in terms of two closely related processes: (1) the innovations process e, a sequence of moments, obtained from the applied moments T by means of a spatially recursive Kalman filter that goes from the tip of the manipulator to its base; (2) a residual process, a sequence of velocities, obtained from the joint-angle velocities by means of an outward smoothing operations. The innovations e and the applied moments T are related by means of the relationships e = (I - L)T and T = (I + K)e. The operation (I - L) is a causal lower triangular matrix which is generated by a spatially recursive Kalman filter and the corresponding discrete-step Riccati equation. Hence, the innovations and the applied moments can be obtained from each other by means of a causal operation which is itself casually invertible.
Rodriguez, G.↗
Kalman filtering, smoothing and recursive robot arm forward and inverse dynamics
The inverse and forward dynamics problems for multi-link serial manipulators are solved by using recursive techniques from linear filtering and smoothing theory. The pivotal step is to cast the system dynamics and kinematics as a two-point boundary-value problem. Solution of this problem leads to filtering and smoothing techniques identical to the equations of Kalman filtering and Bryson-Frazier fixed time-interval smoothing. The solutions prescribe an inward filtering recursion to compute a sequence of constraint moments and forces followed by an outward recursion to determine a corresponding sequence of angular and linear accelerations. In addition to providing techniques to compute joint accelerations from applied joint moments (and vice versa), the report provides an approach to evaluate recursively the composite multi-link system inertia matrix and its inverse. The report lays the foundation for the potential use of filtering and smoothing techniques in robot inverse and forward dynamics and in robot control design.
Rodriguez, G.↗
Recursive Robot-Arm Dynamics via Filtering and Smoothing
Forward and inverse dynamics solved using Kalman filtering and Bryson-Frazier smoothing. Dynamics of serial-link robot arm solved by using recursive techniques from linear filtering and smoothing theory. Solutions of dynamical equations give forces, moments, and accelerations at joints between links, and multilink inertia matrix and its inverse. Theoretical developments lay foundation for use of filtering and smoothing techniques in design of robot controls.
Rodriguez, Guillermo↗
An Indirect Mixed-Sensitivity Approach to Microgravity Vibration Isolation: The Exploitation of Kinematic Coupling In Frequency-Weighting Design-Filter Selections
Many space science experiments need an active isolation system to provide them with the requisite microgravity environment. The isolation systems planned for use with the International Space Station have been appropriately modeled using relative position, relative velocity, and acceleration states. In theory, frequency design filters can be applied to these state-space models, in order to develop optimal H, or mixed-norm controllers with desired stability- and performance characteristics. In practice. however, the kinematic coupling among the various states can lead, through the associated frequency-weighting-filters, to conflicting demands on the Riccati design "machinery." The results can be numerically ill-conditioned regulator and estimator Riccati equations and/or reduced intuition in the design process. In addition, kinematic coupling can result in a redundancy in the demands imposed by the frequency weights. Failure properly to account for this type of coupling can lead to an unnecessary increase in controller dimensionality and, in turn, controller complexity. This paper suggests a rational approach to the assignment of frequency weighting design filters, in the presence of the kinematic coupling among states that exists in the microgravity vibration isolation problem.
Hampton, R. David↗
The determination of physical and dynamical parameters of Pluto/Charon and binary asteroids by least-square formation of a matched filter for a time series of images. I - The operational theory
A theory is derived for the determination of the masses, radii, and orbital elements of the Pluto/Charon, or similar, system based on the prediction of an image distribution over space and time and its comparison with observation. The comparison may be ultimately through the theory of least squares or the application of a matched filter to the observations as a three-dimensional signal stream at an initial or intermediate state. The theory is an approximation correct to fifth order in the diameters of celestial bodies. The theory of astronomical seeing that is used is based on Kolmogorov turbulence in the long-exposure limit. The images must be photometric. Linear tracking errors that can be removed are preferable to either automatic or manual guiding, in the collection of candidate observations.
Wildey, R. L.↗
Direct synthesis of optimal linear filter from conditional probability concepts
State of randomly excited dynamical system estimated from noise measurements having finite duration, and optimal linear filter synthesized from conditional probability theory
EXCITED STATE↗
Investigation, development, and application of optimal output feedback theory. Volume 3: The relationship between dynamic compensators and observers and Kalman filters
Relationships between observers, Kalman Filters and dynamic compensators using feedforward control theory are investigated. In particular, the relationship, if any, between the dynamic compensator state and linear functions of a discrete plane state are investigated. It is shown that, in steady state, a dynamic compensator driven by the plant output can be expressed as the sum of two terms. The first term is a linear combination of the plant state. The second term depends on plant and measurement noise, and the plant control. Thus, the state of the dynamic compensator can be expressed as an estimator of the first term with additive error given by the second term. Conditions under which a dynamic compensator is a Kalman filter are presented, and reduced-order optimal estimaters are investigated.
Broussard, John R.↗
Recursive inverse kinematics for robot arms via Kalman filtering and Bryson-Frazier smoothing
This paper applies linear filtering and smoothing theory to solve recursively the inverse kinematics problem for serial multilink manipulators. This problem is to find a set of joint angles that achieve a prescribed tip position and/or orientation. A widely applicable numerical search solution is presented. The approach finds the minimum of a generalized distance between the desired and the actual manipulator tip position and/or orientation. Both a first-order steepest-descent gradient search and a second-order Newton-Raphson search are developed. The optimal relaxation factor required for the steepest descent method is computed recursively using an outward/inward procedure similar to those used typically for recursive inverse dynamics calculations. The second-order search requires evaluation of a gradient and an approximate Hessian. A Gauss-Markov approach is used to approximate the Hessian matrix in terms of products of first-order derivatives. This matrix is inverted recursively using a two-stage process of inward Kalman filtering followed by outward smoothing. This two-stage process is analogous to that recently developed by the author to solve by means of spatial filtering and smoothing the forward dynamics problem for serial manipulators.
Rodriguez, G.↗
Empirical Bayes state estimation in discrete time linear systems
Linear dynamic systems state estimation, using empirical Bayes decision theory to develop filter set
Kamat, S. J.↗
Processor for high-density digital tape-recorded signals
Linear filter and detection theory can bear on problem of reconstructing recorded bit stream. Problem can be taken from realm of nonlinear problems even though basic record process is still recognized as highly nonlinear. Digital tape recorder can be modeled as particular type of linear communication channel with intersymbol interference.
Ashlock, J. C.↗
Experimental boundary integral equation applications in speckle interferometry
A complete data analysis system utilizing laser speckle interferometry and Experimental Integral Equation Method (EBIE) is described. A host computer provides the optical data analysis and serves as an input device to a PDP11/VAX computer for numerical analysis. The basic theory of pointwise filtering and digital correlation is described as experimental data input to the BIE method.
Fraley, J. E.↗