Richard R. Harman

SATELLITE ANGULAR RATE ESTIMATION FROM VECTOR MEASUREMENTS

Analgorithm ispresented forestimating theangularratevectorofa satellitethat isbased on the timederivatives of vector measurements expressed in a reference and in body coordinates. The computed derivatives are fed into a special Kalman e lter, which yields an estimateof the spacecraft angularvelocity. Thise lter, an extended interlaced Kalman e lter (EIKF), is an extension of the interlaced Kalman e lter (IKF) presented in the literature. Like the IKF, the EIKF is a suboptimal Kalman e lter that, although being linear, estimates the stateof a nonlineardynamic system. It consists of two or three parallel Kalman e lters whose individual estimates are fed to one another and are considered as known inputs by the other parallel e lter (s). The nonlinear dynamics stem from the nonlinear differential equation that describes the rotation of a three-dimensional body. Tests using simulated as well as real Rossi X-Ray Timing Explore satellite data indicate that the algorithm works satisfactorily.

Optimized TRIAD Algorithm for Attitude Determination

TRIAD is a well known simple algorithm that generates the attitude matrix between two coordinate systems when the components of two abstract vectors are given in the two systems. TRIAD however, is sensitive to the order in which the algorithm handles the vectors, such that the resulting attitude matrix is influenced more by the vector processed first. In this work we present a new algorithm, which we call Optimized TRIAD, that blends in a specified manner the two matrices generated by TRIAD when processing one vector first, and then when processing the other vector first. On the average, Optimized TRIAD yields a matrix which is better than either one of the two matrices in that is ti the closest to the correct matrix. This result is demonstrated through simulation.

Angular-Rate Estimation Using Delayed Quaternion Measurements

This paper presents algorithms for estimating the angular-rate vector of satellites using quaternion measurements. Two approaches are compared one that uses differentiated quaternion measurements to yield coarse rate measurements, which are then fed into two different estimators. In the other approach the raw quaternion measurements themselves are fed directly into the two estimators. The two estimators rely on the ability to decompose the non-linear part of the rotas rotational dynamics equation of a body into a product of an angular-rate dependent matrix and the angular-rate vector itself. This non unique decomposition, enables the treatment of the nonlinear spacecraft (SC) dynamics model as a linear one and, thus, the application of a PseudoLinear Kalman Filter (PSELIKA). It also enables the application of a special Kalman filter which is based on the use of the solution of the State Dependent Algebraic Riccati Equation (SDARE) in order to compute the gain matrix and thus eliminates the need to compute recursively the filter covariance matrix. The replacement of the rotational dynamics by a simple Markov model is also examined. In this paper special consideration is given to the problem of delayed quaternion measurements. Two solutions to this problem are suggested and tested. Real Rossi X-Ray Timing Explorer (RXTE) data is used to test these algorithms, and results are presented.

In-Space Calibration of a Skewed Gyro Quadruplet

Anewapproachtogyrocalibrationispresented,wherethespacecraftdynamicsequation,attitudemeasurements, and the gyro outputs are used in a pseudolinear Kalman e lter that estimates the calibration parameters. Also an algorithm is presented for calibrating a skewed quadruplet rather than the customary triad gyro set aligned along the body coordinate axes. In particular, a new misalignment error model is derived for this case. The new calibration algorithm is applied to the EOS-AQUA satellite gyros. The effectiveness of the new algorithm is demonstrated through simulations. YRO calibration, as well as calibration of other instruments, includes two stages. During the e rst stage, the instrument er- ror parameters are estimated, and in the second stage, those errors are continuously removed from the gyro readings. In the classical approach to gyro calibration, the gyro outputs are used to maintain or compute body orientation rather than used as measurements in the context of e ltering. In inertial navigation, for example, 1 gyro errors cause erroneous computation of velocity and position, and then when the latter are compared to measured velocity and posi- tion, a great portion of the computed velocity and position errors can be determined. The latter errors are then fed into a Kalman e lter (KF) that uses the inertial navigation system error model to infer the gyro errors. Similarly, when applying the classical approach to spacecraft(SC) attitude determination, the gyro outputs are used to compute theattitude, andthentheattitude measurements 2;3 areused to determine the attitude errors, which again using a KF indicates what the gyro errors are. Several ways of treating gyro calibration have been presented in the literature. 4i6 In the approach adopted in this work, the gyro outputs are used as angular rate measurements and are compared to estimated angular rates. However,thisapproachrequires the knowl- edge of the angular rate. In the past, 7 the estimated angular rate was computed in a rather simplistic way, assuming basically that the rate was constant. In the present work, the estimated angular rate is derived using a KF whose input can be any kind of attitude mea- surement; therefore, the angular rate experienced by the SC can be continuously changing, and yet a good estimate of the rate, neces- sary for calibration, can be obtained. The calibration algorithm presented in this work was derived for a set ofquadrupletgyros. This required the derivation of anewerror model,particularly forthe gyro misalignments.Thenewcalibration algorithmwas applied to thegyro package ofthe EOS-AQUA satel- lite. The latter consists of four gyros, which are given the task of measuring the three components of the SC angular velocity vector resolved in the body Cartesian coordinates.

State-Dependent Pseudo-Linear Filter for Spacecraft Attitude and Rate Estimation

This paper presents the development and performance of a special algorithm for estimating the attitude and angular rate of a spacecraft. The algorithm is a pseudo-linear Kalman filter, which is an ordinary linear Kalman filter that operates on a linear model whose matrices are current state estimate dependent. The nonlinear rotational dynamics equation of the spacecraft is presented in the state space as a state-dependent linear system. Two types of measurements are considered. One type is a measurement of the quaternion of rotation, which is obtained from a newly introduced star tracker based apparatus. The other type of measurement is that of vectors, which permits the use of a variety of vector measuring sensors like sun sensors and magnetometers. While quaternion measurements are related linearly to the state vector, vector measurements constitute a nonlinear function of the state vector. Therefore, in this paper, a state-dependent linear measurement equation is developed for the vector measurement case. The state-dependent pseudo linear filter is applied to simulated spacecraft rotations and adequate estimates of the spacecraft attitude and rate are obtained for the case of quaternion measurements as well as of vector measurements.

RESULTS OF THE MAGNETOMETER NAVIGATION (MAGNAV) INFLIGHT EXPERIMENT

The Magnetometer Navigation (MAGNAV) algorithm is currently running as a flight experiment as part of the Wide Field Infrared Explorer (WIRE) Post-Science Engineering Testbed. Initialization of MAGNAV occurred on September 4, 2003. MAGNAV is designed to autonomously estimate the spacecraft orbit, attitude, and rate using magnetometer and sun sensor data. Since the Earths magnetic field is a function of time and position, and since time is known quite precisely, the differences between the computed magnetic field and measured magnetic field components, as measured by the magnetometer throughout the entire spacecraft orbit, are a function of the spacecraft trajectory and attitude errors. Therefore, these errors are used to estimate both trajectory and attitude. In addition, the time rate of change of the magnetic field vector is used to estimate the spacecraft rotation rate. The estimation of the attitude and trajectory is augmented with the rate estimation into an Extended Kalman filter blended with a pseudo-linear Kalman filter. Sun sensor data is also used to improve the accuracy and observability of the attitude and rate estimates. This test serves to validate MAGNAV as a single low cost navigation system which utilizes reliable, flight qualified sensors. MAGNAV is intended as a backup algorithm, an initialization algorithm, or possibly a prime navigation algorithm for a mission with coarse requirements. Results from the first six months of operation are presented.

Implicit and Explicit Spacecraft Gyro Calibration

This paper presents a comparison between two approaches to sensor calibration. According to one approach, called explicit, an estimator compares the sensor readings to reference readings, and uses the difference between the two to estimate the calibration parameters. According to the other approach, called implicit, the sensor error is integrated to form a different entity, which is then compared with a reference quantity of this entity, and the calibration parameters are inferred from the difference. In particular this paper presents the comparison between these approaches when applied to in-flight spacecraft gyro calibration. Reference spacecraft rate is needed for gyro calibration when using the explicit approach; however, such reference rates are not readily available for in-flight calibration. Therefore the calibration parameter-estimator is expanded to include the estimation of that reference rate, which is based on attitude measurements in the form of attitude-quaternion. A comparison between the two approaches is made using simulated data. It is concluded that the performances of the two approaches are basically comparable. Sensitivity tests indicate that the explicit filter results are essentially insensitive to variations in given spacecraft dynamics model parameters.

Pseudo Linear Attitude Determination of Spinning Spacecraft

This paper presents the overall mathematical model and results from pseudo linear recursive estimators of attitude and rate for a spinning spacecraft. The measurements considered are vector measurements obtained by sun-sensors, fixed head star trackers, horizon sensors, and three axis magnetometers. Two filters are proposed for estimating the attitude as well as the angular rate vector. One filter, called the q-Filter, yields the attitude estimate as a quaternion estimate, and the other filter, called the D-Filter, yields the estimated direction cosine matrix. Because the spacecraft is gyro-less, Euler s equation of angular motion of rigid bodies is used to enable the estimation of the angular velocity. A simpler Markov model is suggested as a replacement for Eulers equation in the case where the vector measurements are obtained at high rates relative to the spacecraft angular rate. The performance of the two filters is examined using simulated data.

The Effect of Sensor Failure on the Attitude and Rate Estimation of MAP Spacecraft

This work describes two algorithms for computing the angular rate and attitude in case of a gyro and a Star Tracker failure in the Microwave Anisotropy Probe (MAP) satellite, which was placed in the L2 parking point from where it collects data to determine the origin of the universe. The nature of the problem is described, two algorithms are suggested, an observability study is carried out and real MAP data are used to determine the merit of the algorithms. It is shown that one of the algorithms yields a good estimate of the rates but not of the attitude whereas the other algorithm yields a good estimate of the rate as well as two of the three attitude angles. The estimation of the third angle depends on the initial state estimate. There is a contradiction between this result and the outcome of the observability analysis. An explanation of this contradiction is given in the paper. Although this work treats a particular spacecraft, its conclusions are more general.

Rigid Body Rate Inference from Attitude Variation

In this paper we research the extraction of the angular rate vector from attitude information without differentiation, in particular from quaternion measurements. We show that instead of using a Kalman filter of some kind, it is possible to obtain good rate estimates, suitable for spacecraft attitude control loop damping, using simple feedback loops, thereby eliminating the need for recurrent covariance computation performed when a Kalman filter is used. This considerably simplifies the computations required for rate estimation in gyro-less spacecraft. Some interesting qualities of the Kalman filter gain are explored, proven and utilized. We examine two kinds of feedback loops, one with varying gain that is proportional to the well known Q matrix, which is computed using the measured quaternion, and the other type of feedback loop is one with constant coefficients. The latter type includes two kinds; namely, a proportional feedback loop, and a proportional-integral feedback loop. The various schemes are examined through simulations and their performance is compared. It is shown that all schemes are adequate for extracting the angular velocity at an accuracy suitable for control loop damping.