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NASA NTRS2006

Development of a Nonlinear Probability of Collision Tool for the Earth Observing System

The Earth Observing System (EOS) spacecraft Terra, Aqua, and Aura fly in constellation with several other spacecraft in 705-kilometer mean altitude sun-synchronous orbits. All three spacecraft are operated by the Earth Science Mission Operations (ESMO) Project at Goddard Space Flight Center (GSFC). In 2004, the ESMO project began assessing the probability of collision of the EOS spacecraft with other space objects. In addition to conjunctions with high relative velocities, the collision assessment method for the EOS spacecraft must address conjunctions with low relative velocities during potential collisions between constellation members. Probability of Collision algorithms that are based on assumptions of high relative velocities and linear relative trajectories are not suitable for these situations; therefore an algorithm for handling the nonlinear relative trajectories was developed. This paper describes this algorithm and presents results from its validation for operational use. The probability of collision is typically calculated by integrating a Gaussian probability distribution over the volume swept out by a sphere representing the size of the space objects involved in the conjunction. This sphere is defined as the Hard Body Radius. With the assumption of linear relative trajectories, this volume is a cylinder, which translates into simple limits of integration for the probability calculation. For the case of nonlinear relative trajectories, the volume becomes a complex geometry. However, with an appropriate choice of coordinate systems, the new algorithm breaks down the complex geometry into a series of simple cylinders that have simple limits of integration. This nonlinear algorithm will be discussed in detail in the paper. The nonlinear Probability of Collision algorithm was first verified by showing that, when used in high relative velocity cases, it yields similar answers to existing high relative velocity linear relative trajectory algorithms. The comparison with the existing high velocity/linear theory will also be used to determine at what relative velocity the analysis should use the new nonlinear theory in place of the existing linear theory. The nonlinear algorithm was also compared to a known exact solution for the probability of collision between two objects when the relative motion is strictly circular and the error covariance is spherically symmetric. Figure I shows preliminary results from this comparison by plotting the probabilities calculated from the new algorithm and those from the exact solution versus the Hard Body Radius to Covariance ratio. These results show about 5% error when the Hard Body Radius is equal to one half the spherical covariance magnitude. The algorithm was then combined with a high fidelity orbit state and error covariance propagator into a useful tool for analyzing low relative velocity nonlinear relative trajectories. The high fidelity propagator is capable of using atmospheric drag, central body gravitational, solar radiation, and third body forces to provide accurate prediction of the relative trajectories and covariance evolution. The covariance propagator also includes a process noise model to ensure realistic evolutions of the error covariance. This paper will describe the integration of the nonlinear probability algorithm and the propagators into a useful collision assessment tool. Finally, a hypothetical case study involving a low relative velocity conjunction between members of the Earth Observation System constellation will be presented.

McKinley, David P.↗

NASA NTRS1973

Long and short arc altitude determination for GEOS-C

The accuracy with which the GEOS-C altitude may be estimated over long (7 day) and short (40 minute) orbital arcs is investigated. Over the long are excellent agreement was attained between a simulation of the orbit determination process and a covariance analysis. Both approaches yielded RMS altitude errors of about 1.5 meters over the Caribbean calibration area and approximately 7.5 meters overall. The geopotential was identified as the largest error source. For the short arc, the covariance analysis revealed that the propagated altitude error is linearly dependent upon station survey component errors which are also the largest source of altitude errors. An Appendix contains the mathematics of covariance analysis as applied to orbit determination.

Koch, D.↗

NASA NTRS1985

Systematic estimation of forecast and observation error covariances in four-dimensional data assimilation

A two-part algorithm is presented for reliably computing weather forecast model and observational error covariances during data assimilation. Data errors arise from instrumental inaccuracies and sub-grid scale variability, whereas forecast errors occur because of modeling errors and the propagation of previous analysis errors. A Kalman filter is defined as the primary algorithm for estimating the forecast and analysis error convariance matrices. A second algorithm is described for quantifying the noise covariance matrices of any degree to obtain accurate values for the observational error covariances. Numerical results are provided from a linearized one-dimensional shallow-water model. The results cover observational noise covariances, initial instrumental errors and erroneous model values.

Dee, D. P.↗

NASA NTRS1987

A microwave radiometer weather-correcting sea ice algorithm

A new algorithm for estimating the proportions of the multiyear and first-year sea ice types under variable atmospheric and sea surface conditions is presented, which uses all six channels of the SMMR. The algorithm is specifically tuned to derive sea ice parameters while accepting error in the auxiliary parameters of surface temperature, ocean surface wind speed, atmospheric water vapor, and cloud liquid water content. Not only does the algorithm naturally correct for changes in these weather conditions, but it retrieves sea ice parameters to the extent that gross errors in atmospheric conditions propagate only small errors into the sea ice retrievals. A preliminary evaluation indicates that the weather-correcting algorithm provides a better data product than the 'UMass-AES' algorithm, whose quality has been cross checked with independent surface observations. The algorithm performs best when the sea ice concentration is less than 20 percent.

Walters, J. M.↗

NASA NTRS1994

Comparison of Kalman filter and optimal smoother estimates of spacecraft attitude

Given a valid system model and adequate observability, a Kalman filter will converge toward the true system state with error statistics given by the estimated error covariance matrix. The errors generally do not continue to decrease. Rather, a balance is reached between the gain of information from new measurements and the loss of information during propagation. The errors can be further reduced, however, by a second pass through the data with an optimal smoother. This algorithm obtains the optimally weighted average of forward and backward propagating Kalman filters. It roughly halves the error covariance by including future as well as past measurements in each estimate. This paper investigates whether such benefits actually accrue in the application of an optimal smoother to spacecraft attitude determination. Tests are performed both with actual spacecraft data from the Extreme Ultraviolet Explorer (EUVE) and with simulated data for which the true state vector and noise statistics are exactly known.

Sedlak, J.↗

NASA NTRS1990

Pioneer-Venus radio occultation (ORO) data reduction: Profiles of 13 cm absorptivity

In order to characterize possible variations in the abundance and distribution of subcloud sulfuric acid vapor, 13 cm radio occultation signals from 23 orbits that occurred in late 1986 and 1987 (Season 10) and 7 orbits that occurred in 1979 (Season 1) were processed. The data were inverted via inverse Abel transform to produce 13 cm absorptivity profiles. Pressure and temperature profiles obtained with the Pioneer-Venus night probe and the northern probe were used along with the absorptivity profiles to infer upper limits for vertical profiles of the abundance of gaseous H2SO4. In addition to inverting the data, error bars were placed on the absorptivity profiles and H2SO4 abundance profiles using the standard propagation of errors. These error bars were developed by considering the effects of statistical errors only. The profiles show a distinct pattern with regard to latitude which is consistent with latitude variations observed in data obtained during the occultation seasons nos. 1 and 2. However, when compared with the earlier data, the recent occultation studies suggest that the amount of sulfuric acid vapor occurring at and below the main cloud layer may have decreased between early 1979 and late 1986.

Steffes, Paul G.↗

NASA NTRS2012

Correlation Between Analog Noise Measurements and the Expected Bit Error Rate of a Digital Signal Propagating Through Passive Components

A method of determining the bit error rate (BER) of a digital circuit from the measurement of the analog S-parameters of the circuit has been developed. The method is based on the measurement of the noise and the standard deviation of the noise in the S-parameters. Once the standard deviation and the mean of the S-parameters are known, the BER of the circuit can be calculated using the normal Gaussian function.

Warner, Joseph D.↗

NASA NTRS1999

Massively Parallel Assimilation of TOGA/TAO and Topex/Poseidon Measurements into a Quasi Isopycnal Ocean General Circulation Model Using an Ensemble Kalman Filter

A massively parallel ensemble Kalman filter (EnKF)is used to assimilate temperature data from the TOGA/TAO array and altimetry from TOPEX/POSEIDON into a Pacific basin version of the NASA Seasonal to Interannual Prediction Project (NSIPP)ls quasi-isopycnal ocean general circulation model. The EnKF is an approximate Kalman filter in which the error-covariance propagation step is modeled by the integration of multiple instances of a numerical model. An estimate of the true error covariances is then inferred from the distribution of the ensemble of model state vectors. This inplementation of the filter takes advantage of the inherent parallelism in the EnKF algorithm by running all the model instances concurrently. The Kalman filter update step also occurs in parallel by having each processor process the observations that occur in the region of physical space for which it is responsible. The massively parallel data assimilation system is validated by withholding some of the data and then quantifying the extent to which the withheld information can be inferred from the assimilation of the remaining data. The distributions of the forecast and analysis error covariances predicted by the ENKF are also examined.

Keppenne, Christian L.↗

NASA NTRS2009

Using Covariance Analysis to Assess Pointing Performance

A Pointing Covariance Analysis Tool (PCAT) has been developed for evaluating the expected performance of the pointing control system for NASA s Space Interferometry Mission (SIM). The SIM pointing control system is very complex, consisting of multiple feedback and feedforward loops, and operating with multiple latencies and data rates. The SIM pointing problem is particularly challenging due to the effects of thermomechanical drifts in concert with the long camera exposures needed to image dim stars. Other pointing error sources include sensor noises, mechanical vibrations, and errors in the feedforward signals. PCAT models the effects of finite camera exposures and all other error sources using linear system elements. This allows the pointing analysis to be performed using linear covariance analysis. PCAT propagates the error covariance using a Lyapunov equation associated with time-varying discrete and continuous-time system matrices. Unlike Monte Carlo analysis, which could involve thousands of computational runs for a single assessment, the PCAT analysis performs the same assessment in a single run. This capability facilitates the analysis of parametric studies, design trades, and "what-if" scenarios for quickly evaluating and optimizing the control system architecture and design.

Bayard, David↗

NASA NTRS1977

Skylab water balance error analysis

Estimates of the precision of the net water balance were obtained for the entire Skylab preflight and inflight phases as well as for the first two weeks of flight. Quantitative estimates of both total sampling errors and instrumentation errors were obtained. It was shown that measurement error is minimal in comparison to biological variability and little can be gained from improvement in analytical accuracy. In addition, a propagation of error analysis demonstrated that total water balance error could be accounted for almost entirely by the errors associated with body mass changes. Errors due to interaction between terms in the water balance equation (covariances) represented less than 10% of the total error. Overall, the analysis provides evidence that daily measurements of body water changes obtained from the indirect balance technique are reasonable, precise, and relaible. The method is not biased toward net retention or loss.

Leonard, J. I.↗

NASA NTRS2009

Ares I Static Tests Design

Probabilistic engineering design enhances safety and reduces costs by incorporating risk assessment directly into the design process. In this paper, we assess the format of the quantitative metrics for the vehicle which will replace the Space Shuttle, the Ares I rocket. Specifically, we address the metrics for in-flight measurement error in the vector position of the motor nozzle, dictated by limits on guidance, navigation, and control systems. Analyses include the propagation of error from measured to derived parameters, the time-series of dwell points for the duty cycle during static tests, and commanded versus achieved yaw angle during tests. Based on these analyses, we recommend a probabilistic template for specifying the maximum error in angular displacement and radial offset for the nozzle-position vector. Criteria for evaluating individual tests and risky decisions also are developed.

Carson, William↗

NASA NTRS2012

Impact of Non-Gaussian Error Volumes on Conjunction Assessment Risk Analysis

An understanding of how an initially Gaussian error volume becomes non-Gaussian over time is an important consideration for space-vehicle conjunction assessment. Traditional assumptions applied to the error volume artificially suppress the true non-Gaussian nature of the space-vehicle position uncertainties. For typical conjunction assessment objects, representation of the error volume by a state error covariance matrix in a Cartesian reference frame is a more significant limitation than is the assumption of linearized dynamics for propagating the error volume. In this study, the impact of each assumption is examined and isolated for each point in the volume. Limitations arising from representing the error volume in a Cartesian reference frame is corrected by employing a Monte Carlo approach to probability of collision (Pc), using equinoctial samples from the Cartesian position covariance at the time of closest approach (TCA) between the pair of space objects. A set of actual, higher risk (Pc >= 10 (exp -4)+) conjunction events in various low-Earth orbits using Monte Carlo methods are analyzed. The impact of non-Gaussian error volumes on Pc for these cases is minimal, even when the deviation from a Gaussian distribution is significant.

Ghrist, Richard W.↗

NASA NTRS1987

Experimental validation of a two-dimensional shear-flow model for determining acoustic impedance

Tests were conducted to validate a two-dimensional shear-flow analytical model for determining the acoustic impedance of a liner test specimen in a grazing-incidence, grazing-flow environment. The tests were limited to a test specimen chosen to exhibit minimal effects of grazing flow so that the results obtained by using the shear-flow analytical model would be expected to match those obtained from normal-incidence impedance measurements. Impedances for both downstream and upstream sound propagation were generally consistent with those from normal-incidence measurements. However, sensitivity of the grazing-incidence impedance to small measurement or systematic errors in propagation constant varied dramatically over the range of test frequencies.

Parrott, Tony L.↗

NASA NTRS2008

Identifying Bearing Rotordynamic Coefficients using an Extended Kalman Filter

An Extended Kalman Filter is developed to estimate the linearized direct and indirect stiffness and damping force coefficients for bearings in rotor-dynamic applications from noisy measurements of the shaft displacement in response to imbalance and impact excitation. The bearing properties are modeled as stochastic random variables using a Gauss-Markov model. Noise terms are introduced into the system model to account for all of the estimation error, including modeling errors and uncertainties and the propagation of measurement errors into the parameter estimates. The system model contains two user-defined parameters that can be tuned to improve the filter s performance; these parameters correspond to the covariance of the system and measurement noise variables. The filter is also strongly influenced by the initial values of the states and the error covariance matrix. The filter is demonstrated using numerically simulated data for a rotor-bearing system with two identical bearings, which reduces the number of unknown linear dynamic coefficients to eight. The filter estimates for the direct damping coefficients and all four stiffness coefficients correlated well with actual values, whereas the estimates for the cross-coupled damping coefficients were the least accurate.

Miller, Bard A.↗

NASA NTRS2008

Identifying Bearing Rotordynamic Coefficients using an Extended Kalman Filter

Miller, Brad A.↗

NASA NTRS2008

Identifying Bearing Rotodynamic Coefficients Using an Extended Kalman Filter

An Extended Kalman Filter is developed to estimate the linearized direct and indirect stiffness and damping force coefficients for bearings in rotor dynamic applications from noisy measurements of the shaft displacement in response to imbalance and impact excitation. The bearing properties are modeled as stochastic random variables using a Gauss-Markov model. Noise terms are introduced into the system model to account for all of the estimation error, including modeling errors and uncertainties and the propagation of measurement errors into the parameter estimates. The system model contains two user-defined parameters that can be tuned to improve the filter's performance; these parameters correspond to the covariance of the system and measurement noise variables. The filter is also strongly influenced by the initial values of the states and the error covariance matrix. The filter is demonstrated using numerically simulated data for a rotor bearing system with two identical bearings, which reduces the number of unknown linear dynamic coefficients to eight. The filter estimates for the direct damping coefficients and all four stiffness coefficients correlated well with actual values, whereas the estimates for the cross-coupled damping coefficients were the least accurate.

Miller, Brad A.↗

NASA NTRS1992

Linear regression in astronomy. II

A wide variety of least-squares linear regression procedures used in observational astronomy, particularly investigations of the cosmic distance scale, are presented and discussed. The classes of linear models considered are (1) unweighted regression lines, with bootstrap and jackknife resampling; (2) regression solutions when measurement error, in one or both variables, dominates the scatter; (3) methods to apply a calibration line to new data; (4) truncated regression models, which apply to flux-limited data sets; and (5) censored regression models, which apply when nondetections are present. For the calibration problem we develop two new procedures: a formula for the intercept offset between two parallel data sets, which propagates slope errors from one regression to the other; and a generalization of the Working-Hotelling confidence bands to nonstandard least-squares lines. They can provide improved error analysis for Faber-Jackson, Tully-Fisher, and similar cosmic distance scale relations.

Feigelson, Eric D.↗

NASA NTRSDate not supplied

Uncertainty Propagation in Pre-Flight Prediction of Unmanned Aerial Vehicle Separation Violation

Current forecasts on the future of aeronautics suggest an in- creasing number of unmanned aerial vehicles entering the low- altitude airspace in the next decades (FAA, 2018; Kopardekar et al., 2016). Small vehicles for package delivery as well as larger vehicles for urban air mobility will change the airspace drastically, increasing density of operations both in time, i.e. high number of take-off and landings per unit time, and in space, operating in dense urban environment. This scenario poses challenges to the current approach to air traffic control, and large efforts from academia, industry and regulatory bodies are dedicated to the development of new traffic management strategies that leverage higher computing and simulating capabilities available today. In this paper, we propose a simple look-ahead approach to predict potential minimum separation violations at the strategic level, that is before vehicles start flying, depending on the predefined 4D trajectories and uncertainty affecting the wind acting along those routes. The wind field is extracted from the NOAA North America Mesoscale Forecast System and interpolated using Gaussian process regression, while uncertainty affecting the expected cruise airspeed is propagated through error intervals. The approach allows the prediction of aircraft separation as a function of time, highlighting potential safety violations that would go undetected if uncertainty affecting the expected 4D trajectories is not considered. The paper will also discuss issues related to accuracy and scalability of the approach to multiple vehicle operations.

Trajectory Prediction↗