Self-contained orbital navigation systems with correlated measurement errors.

Abstract

A self-contained orbital navigation system using earth horizon measurements in the 14-16 ju carbon dioxide absorption band is analyzed using Kalman's linear filter theory. The spatial correlation of the horizon height deviations is converted into time correlation of the sensor measurement errors. The orbital dynamics, horizon phenomenology, and the horizon tracker, star tracker, radar altimeter, and inertial measurement unit instrument errors are represented by linearized differential equations in the Kalman canonical form. A generalized error analysis computer program is used to perform the calculations. The results demonstrate that high sampling rates, which produce marked improvement in performance in the presence of uncorrelated errors, are of no value for reducing the effects of correlated errors for the chosen set of observables. The system performance as a function of time is presented for different values of the error sources. The efficacy of a radar altimeter in reducing the inplane navigation error, especially during the initial transient, is shown. Nomenclature a, b = possible values of ka A = matrix in canonical differential equation B = transformation matrix for horizon height measurements d = const of integration d = horizon deviations reduced to position errors D = transformation matrix for horizon height measurements E = gravitation matrix h = horizon tracker instrument error I = identity matrix kjj -const in autocorrelation function K = Kalman filter matrix M = measurement matrix n = number of states 0 = zero vector O = zero matrix p = covariance matrix of state vector Q = covariance matrix of errors caused by white noise r = perturbation position vector R = nominal position vector R = earth radius matrix R e = radius of earth s = time since last fix S = star tracker instrument error S = covariance matrix of white noise sources t -time u = white noise column vector v = inertial velocity vector w = error caused by white noise column vector We* = acceleration time constant matrix x = state column vector x = estimated state column vector x = error in estimated state column vector y = measurement column vector a = acceleration uncertainty vector p t = horizon tracker measurement angles Presented at the AIAA/ION Guidance and Control Conference, August 16-18, 1965 (no preprint number; published in bound volume of preprints of the meeting); T = gyro drift rate vector 5 = Dirac delta function 8= great circle distance between measurement points 5 e = characteristic distance for horizon deviations €t = horizon height deviations n = IMU misalignment vector from computed local vertical coordinate system 0 = geocentric range angle X = auxiliary variable fj, = gravitational const of earth p = position vector a = standard deviation of error source; subscripts h, s, and e refer to horizon tracker instrument, star tracker, and horizon deviation, respectively T = time const of error source Td = horizon deviation time const as seen by instrument T 6 = horizon deviation time const </> c = autocorrelation function of...