Analytic Steady-State Accuracy of a Spacecraft Attitude Estimator

Abstract

This paper extends Farrenkopf s analysis of a single-axis spacecraft attitude estimator using gyro and angle sensor data to include the angle output white noise of a rate-integrating gyro. Analytic expressions are derived for the steadystate pre-update and post-update angle and drift bias variances and for the state update equations. It is shown that only part of the state update resulting from the angle sensor measurement is propagated to future times Introduction• to optimally combine star tracker and gyro data for spacecraft There has long been an interest in Kalman filtering 1"2 attitude estimation .3'4Farrenkopffound an analytic solution for the steady-state accuracy of a single-axis Kalman filter combining data from a gyro and an angle sensor, whicl_ has proven very useful for preliminary analysis of spacecraft attitude determination systems? Farrenkopf implicitly assumed a rate gyro model, however, and many missions employ highly accurate rate-integrating gyros (RIGs). The aim of this paper is to modify Farrenkopfs result to be applicable to rate-integrating gyros. Although the intermediate results are significantly more complex than Farrenkopf's, the final expressions for the steady-state covariance are a simple modification to his.We begin by discussing the dynamic models for the spacecraft attitude and the gyro and the models for the gyro and star tracker measurements. Then we derive the equations for gyro and star tracker covariance updates and obtain the steady-state covariance, including a numerical example. The explicit state update equations are derived next; and it is shown that the angle update due to a star tracker measurement can be divided into two parts, only one of which is propagated to future times. The conclusions are stated at the end of the paper. DynamicsThe single-axis spacecraft dynamics are given by (I) 0=0),where 0 is the rotation angle and o_ is the true angular velocity. Because of its own internal dynamics, the RIG does not measure 0 exactly, but instead accumulates its own angle 0. The RIG dynamic equation for this angle is _=_o+b+n,,where b is the gyro drift rate and n_ is a zero mean Gaussian white noise process. The drift rate is assumed to satisfywhere n, is also a zero mean Gaussian white noise process. These processes are assumed to obeyandwhere E{. } denotes the expectation value and fi(.) is the Dirac delta function.(4c)https://ntrs.nasa.gov/search.jsp?R=20000086664 2019-04-30T20:05:18+00:00ZFarrenkopf took 0 to be the measured gyro rate and eliminated w between Eqs.(1) and (2). We will not make this assumption, since it does not provide a convenient procedure for including angle output white noise (also known as readout noise or electronic noise) on the RIG output. 6 The procedure we adopt requires a dynamic model for the angular rate. Since we are interested in applications that use the gyros for rate measurements rather than Eulerian dynamic models, we approximate the rate as a pure random walk, where nw is a third zero mean Gaussian white noise process, unc...

Autonomous Inertial Relative Navigation with Sight-Line-Stabilized Sensors for Spacecraft Rendezvous

“…Much has been written on the attitude estimation in the last few decades with varying degrees of detail and complexity; see [27] for a recent attitude estimator based on the Bortz equation. In this section, however, we generalize a simple, constant-gain, steady-state single-axis technique of gyro measurement update with star tracker measurements developed in [31] to three-axis.…”

“…. ; n) and also with a star tracker at an interval of T n. The index k is reinitialized to 0 when it is equal to n. The three-axis incremental angle k measured by the gyros during the interval t k 1 t t k is k k b k k q k (30) where k is the true change in the spacecraft attitude, b k is the drift rate of the gyro, k is a zero-mean noise in the measurement arising from a 3 1 random-walk rate n v t and a 3 1 drift acceleration n u (see [31]). The variance of k is a 3 3 diagonal matrix for which each element is 2 2 v 2 u 3 =3, where 2 v (rad 2 =s) and 2 u (rad 2 =s 3 ) are power spectral densities of the scalar elements of n v and n u .…”

“…To formulate this update for a three-axis attitude estimator, we first note the steady-state correction of the single-axis gyro-measured attitude with star tracker measurements at k n, following [31]. Let n 0 be the gyro-estimated spacecraft attitude just before the star tracker measurement, and let st be the spacecraft attitude measured by the star tracker at that time.…”

“…Following [31], the spacecraft attitude is measured with gyros at an interval of (5 ms, presently) at t k (k 1; 2; . .…”

Installation Error Dynamic Correction Based on Quasi-Linear Model

“…Installation error correction is essentially non-linear problem, so the solution is more focused on the non-linear methods [1][2][3][4] , there are many research and have achieved some results. In the literature [5][6] using the extended Kalman filter and unscented Kalman filtering method to correct the posture sensor misalignment errors, that can be estimated online real-time, but for large misalignment error it does not work well.…”