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NASA NTRS1986

A methodology for airplane parameter estimation and confidence interval determination in nonlinear estimation problems

An algorithm for maximum likelihood (ML) estimation is developed with an efficient method for approximating the sensitivities. The ML algorithm relies on a new optimization method referred to as a modified Newton-Raphson with estimated sensitivities (MNRES). MNRES determines sensitivities by using slope information from local surface approximations of each output variable in parameter space. With the fitted surface, sensitivity information can be updated at each iteration with less computational effort than that required by either a finite-difference method or integration of the analytically determined sensitivity equations. MNRES eliminates the need to derive sensitivity equations for each new model, and thus provides flexibility to use model equations in any convenient format. A random search technique for determining the confidence limits of ML parameter estimates is applied to nonlinear estimation problems for airplanes. The confidence intervals obtained by the search are compared with Cramer-Rao (CR) bounds at the same confidence level. The degree of nonlinearity in the estimation problem is an important factor in the relationship between CR bounds and the error bounds determined by the search technique. Beale's measure of nonlinearity is developed in this study for airplane identification problems; it is used to empirically correct confidence levels and to predict the degree of agreement between CR bounds and search estimates.

Murphy, P. C.↗

NASA NTRS1987

Estimation of bias errors in measured airplane responses using maximum likelihood method

A maximum likelihood method is used for estimation of unknown bias errors in measured airplane responses. The mathematical model of an airplane is represented by six-degrees-of-freedom kinematic equations. In these equations the input variables are replaced by their measured values which are assumed to be without random errors. The resulting algorithm is verified with a simulation and flight test data. The maximum likelihood estimates from in-flight measured data are compared with those obtained by using a nonlinear-fixed-interval-smoother and an extended Kalmar filter.

Klein, Vladiaslav↗

NASA NTRS1973

A real-time digital program for estimating aircraft stability and control parameters from flight test data by using the maximum likelihood method

A computer program (Langley program C1123) has been developed for estimating aircraft stability and control parameters from flight test data. These parameters are estimated by the maximum likelihood estimation procedure implemented on a real-time digital simulation system, which uses the Control Data 6600 computer. This system allows the investigator to interact with the program in order to obtain satisfactory results. Part of this system, the control and display capabilities, is described for this program. This report also describes the computer program by presenting the program variables, subroutines, flow charts, listings, and operational features. Program usage is demonstrated with a test case using pseudo or simulated flight data.

Grove, R. D.↗

NASA NTRS1974

Estimation of elastic aircraft parameters using the maximum likelihood method

The application of the maximum likelihood method to estimate the aerodynamic parameters of elastic flight vehicles in a symmetric flight condition is discussed. In this application, particular attention is directed toward the center of mass, elastic deformation, and sensor equations of motion. It is shown that the two major computational problems to be overcome are the inversion of large-sized matrices and the time-wise integration of a large number of linear, ordinary, differential equations.

Schwanz, R. C.↗

NASA NTRS2010

Maximum Likelihood Time-of-Arrival Estimation of Optical Pulses via Photon-Counting Photodetectors

Many optical imaging, ranging, and communications systems rely on the estimation of the arrival time of an optical pulse. Recently, such systems have been increasingly employing photon-counting photodetector technology, which changes the statistics of the observed photocurrent. This requires time-of-arrival estimators to be developed and their performances characterized. The statistics of the output of an ideal photodetector, which are well modeled as a Poisson point process, were considered. An analytical model was developed for the mean-square error of the maximum likelihood (ML) estimator, demonstrating two phenomena that cause deviations from the minimum achievable error at low signal power. An approximation was derived to the threshold at which the ML estimator essentially fails to provide better than a random guess of the pulse arrival time. Comparing the analytic model performance predictions to those obtained via simulations, it was verified that the model accurately predicts the ML performance over all regimes considered. There is little prior art that attempts to understand the fundamental limitations to time-of-arrival estimation from Poisson statistics. This work establishes both a simple mathematical description of the error behavior, and the associated physical processes that yield this behavior. Previous work on mean-square error characterization for ML estimators has predominantly focused on additive Gaussian noise. This work demonstrates that the discrete nature of the Poisson noise process leads to a distinctly different error behavior.

Erkmen, Baris I.↗

NASA NTRS1976

Nonparametric probability density estimation by optimization theoretic techniques

Two nonparametric probability density estimators are considered. The first is the kernel estimator. The problem of choosing the kernel scaling factor based solely on a random sample is addressed. An interactive mode is discussed and an algorithm proposed to choose the scaling factor automatically. The second nonparametric probability estimate uses penalty function techniques with the maximum likelihood criterion. A discrete maximum penalized likelihood estimator is proposed and is shown to be consistent in the mean square error. A numerical implementation technique for the discrete solution is discussed and examples displayed. An extensive simulation study compares the integrated mean square error of the discrete and kernel estimators. The robustness of the discrete estimator is demonstrated graphically.

Scott, D. W.↗

NASA NTRS1985

Time-resolved speckle effects on the estimation of laser-pulse arrival times

A maximum-likelihood (ML) estimator of the pulse arrival in laser ranging and altimetry is derived for the case of a pulse distorted by shot noise and time-resolved speckle. The performance of the estimator is evaluated for pulse reflections from flat diffuse targets and compared with the performance of a suboptimal centroid estimator and a suboptimal Bar-David ML estimator derived under the assumption of no speckle. In the large-signal limit the accuracy of the estimator was found to improve as the width of the receiver observational interval increases. The timing performance of the estimator is expected to be highly sensitive to background noise when the received pulse energy is high and the receiver observational interval is large. Finally, in the speckle-limited regime the ML estimator performs considerably better than the suboptimal estimators.

Tsai, B.-M.↗

NASA NTRS1989

Maximum likelihood techniques in QELS

A framework for the analysis of Quasi Elastic Light Scattering (QELS) experiments designed to be used in microgravity environment is derived. Example calculations of the type to be used to design the QELS system are given. The framework for the analysis is based on the concepts of parameter estimation typified by Maximum Likelihood Estimation methods. These methods not only serve as the template for parameter estimation algorithms, but can also be used for optimal design of the experiments. Optimal design of experiments is facilitated by the fact that these methods not only give procedures for parameter estimation, but also estimates of the errors associated with the parameter estimation.

Edwards, Robert V.↗

NASA NTRS1976

Filtering and smoothing techniques for parameter estimation

In this paper filtering and smoothing criteria for maximum-likelihood parameter estimation are described and compared. It is shown that the parameter estimate using the smoothing criterion may be obtained independently of the smoothed state estimate. Monte Carlo test results are presented comparing the efficiency of the estimation procedures.

Bach, R. E., Jr.↗

NASA NTRS1990

Power system state estimation for a spacecraft power system

An application of the maximum likelihood state estimator to a space-based power system is presented. The state estimator uses current and voltage measurements to generate estimates of node voltages for an electrical power distribution system for the Space Shuttle. Preliminary results on the effect of noisy measurements on estimated parameters are reported. The software used in generating these results is part of an overall package being developed at Louisiana Tech University. Intended applications of this package include the analysis of power systems and real-time parallel processing on the Space Shuttle.

Berry, F. C.↗

NASA NTRS1977

Effects of control inputs on the estimation of stability and control parameters of a light airplane

The maximum likelihood parameter estimation technique was used to determine the values of stability and control derivatives from flight test data for a low-wing, single-engine, light airplane. Several input forms were used during the tests to investigate the consistency of parameter estimates as it relates to inputs. These consistencies were compared by using the ensemble variance and estimated Cramer-Rao lower bound. In addition, the relationship between inputs and parameter correlations was investigated. Results from the stabilator inputs are inconclusive but the sequence of rudder input followed by aileron input or aileron followed by rudder gave more consistent estimates than did rudder or ailerons individually. Also, square-wave inputs appeared to provide slightly improved consistency in the parameter estimates when compared to sine-wave inputs.

Cannaday, R. L.↗

NASA NTRS2004

Path following using visual odometry for Mars Rover in high-slip environments.

An architecture for autonomous operation of Mars rovers in high slip environments has been designed, implemented, and tested. This architecture is composed of several key technologies that enable the rover to accurately follow a designated path, compensate for slippage, and reach intended goals independent of the terrain over which it is traversing (within the mechanical constraints of the mobility system). These technologies include: visual odometry, full vehicle kinematics, a Kalman filter, and a slop compensation/path follower. Visual odometry tracks distinctive scene features in stereo imagery to estimate rover motion between successively acquired stereo image pairs using a maximum likelihood motion estimation algorithm. The full vehicle kinematics for a rocker-bogie suspension system estimates motion, with a no-slip assumption, by measuring wheel, rates, and rocker, bogie, and steering angles. The Kalman filter merges data from an Inertial Measurement Unit (IMU) and visual odometry. This merged estimate is then compared to the kinematic estimate to determining (tracking into account estimate uncertainties) if and how much slippage has occurred then a slip vector is calculated by differencing the current Kalman filter estimate from the kinematic estimate. This slip vector is then used, in conjunction with the inverse kinematics, to determine the necessary wheel velocities and steering angles to compensate for slip and follow the desired path.

Roumeliotis, Stergios I.↗

NASA NTRS1981

The Theory and Practice of Estimating the Accuracy of Dynamic Flight-Determined Coefficients

Means of assessing the accuracy of maximum likelihood parameter estimates obtained from dynamic flight data are discussed. The most commonly used analytical predictors of accuracy are derived and compared from both statistical and simplified geometrics standpoints. The accuracy predictions are evaluated with real and simulated data, with an emphasis on practical considerations, such as modeling error. Improved computations of the Cramer-Rao bound to correct large discrepancies due to colored noise and modeling error are presented. The corrected Cramer-Rao bound is shown to be the best available analytical predictor of accuracy, and several practical examples of the use of the Cramer-Rao bound are given. Engineering judgement, aided by such analytical tools, is the final arbiter of accuracy estimation.

Maine, R. E.↗

NASA NTRS1980

Estimation of the accuracy of dynamic flight-determined coefficients

This paper discusses means of assessing the accuracy of maximum likelihood parameter estimates obtained from dynamic flight data. The commonly used analytical predictors of accuracy are compared from both statistical and simplified geometric standpoints. Emphasizing practical considerations, such as modeling error, the accuracy predictions are evaluated with real and simulated data. Improved computations of the Cramer-Rao bound to correct large discrepancies caused by colored noise and modeling error are presented. This corrected Cramer-Rao bound is the best available analytical predictor of accuracy. Engineering judgement, aided by such analytical tools, is the final arbiter of accuracy estimation.

Maine, R. E.↗

NASA NTRS1989

Summary of longitudinal stability and control parameters as determined from Space Shuttle Challenger flight test data

Estimates of longitudinal stability and control parameters for the space shuttle were determined by applying a maximum likelihood parameter estimation technique to Challenger flight test data. The parameters for pitching moment coefficient, C(m sub alpha), (at different angles of attack), pitching moment coefficient, C(m sub delta e), (at different elevator deflections) and the normal force coefficient, C(z sub alpha), (at different angles of attack) describe 90 percent of the response to longitudinal inputs during Space Shuttle Challenger flights with C(m sub delta e) being the dominant parameter. The values of C(z sub alpha) were found to be input dependent for these tests. However, when C(z sub alpha) was set at preflight predictions, the values determined for C(m sub delta e) changed less than 10 percent from the values obtained when C(z sub alpha) was estimated as well. The preflight predictions for C(z sub alpha) and C(m sub alpha) are acceptable values, while the values of C(z sub delta e) should be about 30 percent less negative than the preflight predictions near Mach 1, and 10 percent less negative, otherwise.

Suit, William T.↗

NASA NTRS1988

Bit error probability calculations for convolutional codes with short constraint lengths on very noisy channels

A technique for estimating convolutional code performance on very noisy channels is considered. Specifically, the performance of short constraint length codes operating near the channel cutoff rate is estimated. Decoding convolutional codes with a sliding window decoder (SWD) are considered. This decoder is an optimal (maximum likelihood) symbol decoder as the window size grows toward infinity, while the Viterbi decoder is the maximum-likelihood sequence estimator. The difference in the decoded BERs (bit error rates) between the two decoders is very small and approaches zero asymptotically as the channel BER decreases. Therefore, an estimate on the decoded BER for the SWD can also be used as an estimate of the decoded BER for Viterbi decoding.

Herro, Mark A.↗

NASA NTRS1991

On the estimation of climatological Z-R relationships

A statistical framework for climatological Z-R parameter estimation is developed and simulation experiments are conducted to examine sampling properties of the estimators. Both parametric and nonparametric models are considered. For parametric models, it is shown that Z-R parameters can be estimated by maximum likelihood, a procedure with optimal large sample properties. A general nonparametric framework for climatological Z-R estimation is also developed. Nonparametric procedures are attractive because of their flexibility in dealing with certain types of measurement errors common to radar data. Simulation experiments show that even under favorable assumptions on error characteristics of radar and raingages, large datasets are required to obtain accurate Z-R parameter estimates. Another important conclusion is that estimation results are generally quite sensitive to radar and raingage measurement thresholds. For fixed sample size, the simulation results can be used to provide quantitative assessments of the accuracy of Z-R model parameter estimates. These results are particularly useful for error analysis of precipitation products that are derived using climatological Z-R relations. One example is the large-area rainfall estimates derived using the height-area rainfall threshold (HART) technique.

Krajewski, Witold F.↗

NASA NTRS1980

Development of advanced acreage estimation methods

The use of the AMOEBA clustering/classification algorithm was investigated as a basis for both a color display generation technique and maximum likelihood proportion estimation procedure. An approach to analyzing large data reduction systems was formulated and an exploratory empirical study of spatial correlation in LANDSAT data was also carried out. Topics addressed include: (1) development of multiimage color images; (2) spectral spatial classification algorithm development; (3) spatial correlation studies; and (4) evaluation of data systems.

Guseman, L. F., Jr.↗