Attitude determination using vector observations: A fast optimal matrix algorithm

The attitude matrix minimizing Wahba's loss function is computed directly by a method that is competitive with the fastest known algorithm for finding this optimal estimate. The method also provides an estimate of the attitude error covariance matrix. Analysis of the special case of two vector observations identifies those cases for which the TRIAD or algebraic method minimizes Wahba's loss function

Simultaneous quaternion estimation (QUEST) and bias determination

Tests of a new method for the simultaneous estimation of spacecraft attitude and sensor biases, based on a quaternion estimation algorithm minimizing Wahba's loss function are presented. The new method is compared with a conventional batch least-squares differential correction algorithm. The estimates are based on data from strapdown gyros and star trackers, simulated with varying levels of Gaussian noise for both inertially-fixed and Earth-pointing reference attitudes. Both algorithms solve for the spacecraft attitude and the gyro drift rate biases. They converge to the same estimates at the same rate for inertially-fixed attitude, but the new algorithm converges more slowly than the differential correction for Earth-pointing attitude. The slower convergence of the new method for non-zero attitude rates is believed to be due to the use of an inadequate approximation for a partial derivative matrix. The new method requires about twice the computational effort of the differential correction. Improving the approximation for the partial derivative matrix in the new method is expected to improve its convergence at the cost of increased computational effort

Attitude Determination Using Two Vector Measurements

Many spacecraft attitude determination methods use exactly two vector measurements. The two vectors are typically the unit vector to the Sun and the Earth's magnetic field vector for coarse "sun-mag" attitude determination or unit vectors to two stars tracked by two star trackers for fine attitude determination. TRIAD, the earliest published algorithm for determining spacecraft attitude from two vector measurements, has been widely used in both ground-based and onboard attitude determination. Later attitude determination methods have been based on Wahba's optimality criterion for n arbitrarily weighted observations. The solution of Wahba's problem is somewhat difficult in the general case, but there is a simple closed-form solution in the two-observation case. This solution reduces to the TRIAD solution for certain choices of measurement weights. This paper presents and compares these algorithms as well as sub-optimal algorithms proposed by Bar-Itzhack, Harman, and Reynolds. Some new results will be presented, but the paper is primarily a review and tutorial

Minimal parameter solution of the orthogonal matrix differential equation

As demonstrated in this work, all orthogonal matrices solve a first order differential equation. The straightforward solution of this equation requires n sup 2 integrations to obtain the element of the nth order matrix. There are, however, only n(n-1)/2 independent parameters which determine an orthogonal matrix. The questions of choosing them, finding their differential equation and expressing the orthogonal matrix in terms of these parameters are considered. Several possibilities which are based on attitude determination in three dimensions are examined. It is shown that not all 3-D methods have useful extensions to higher dimensions. It is also shown why the rate of change of the matrix elements, which are the elements of the angular rate vector in 3-D, are the elements of a tensor of the second rank (dyadic) in spaces other than three dimensional. It is proven that the 3-D Gibbs vector (or Cayley Parameters) are extendable to other dimensions. An algorithm is developed employing the resulting parameters, which are termed Extended Rodrigues Parameters, and numerical results are presented of the application of the algorithm to a fourth order matrix

HUMBLE PROBLEMS

Harold Morton introduced a talk by saying that when you wind up an old professor, he tends to talk for a microcentury. I will attempt to keep my comments to that canonical time span. Having failed to find some unifying theme for this talk, I decided to just ramble through my career with a focus on the algorithms, spacecraft, and people I've had the privilege and pleasure to work with. The algorithms, and certainly the spacecraft, are not all mine. The people are some of those whose ideas that have most influenced and inspired my career. The organization of the paper is largely chronological, but I do not hesitate to jump forward or backward in time when the material demands it. The coverage is broad but necessarily shallow; the interested reader can find more detail in the reference

Statistical Attitude Determination

All spacecraft require attitude determination at some level of accuracy. This can be a very coarse requirement of tens of degrees, in order to point solar arrays at the sun, or a very fine requirement in the milliarcsecond range, as required by Hubble Space Telescope. A toolbox of attitude determination methods, applicable across this wide range, has been developed over the years. There have been many advances in the thirty years since the publication of Reference, but the fundamentals remain the same. One significant change is that onboard attitude determination has largely superseded ground-based attitude determination, due to the greatly increased power of onboard computers. The availability of relatively inexpensive radiation-hardened microprocessors has led to the development of "smart" sensors, with autonomous star trackers being the first spacecraft application. Another new development is attitude determination using interferometry of radio signals from the Global Positioning System (GPS) constellation. This article reviews both the classic material and these newer developments at approximately the level of, with emphasis on. methods suitable for use onboard a spacecraft. We discuss both "single frame" methods that are based on measurements taken at a single point in time, and sequential methods that use information about spacecraft dynamics to combine the information from a time series of measurements

Kepler Equation solver

Kepler's Equation is solved over the entire range of elliptic motion by a fifth-order refinement of the solution of a cubic equation. This method is not iterative, and requires only four transcendental function evaluations: a square root, a cube root, and two trigonometric functions. The maximum relative error of the algorithm is less than one part in 10(exp 18), exceeding the capability of double-precision computer arithmetic. Roundoff errors in double-precision implementation of the algorithm are addressed, and procedures to avoid them are developed

Lessons Learned

This paper is organized around three themes that have interested me over my career: covariance analysis, constrained estimation, and angular momentum. Both analytical and numerical approaches to covariance analysis are discussed. The section on constraints begins with general considerations arising from the nature of the rotation group before moving to Wahba's Problem and quaternion Kalman filters. The discussion of angular momentum includes its use in attitude determination and control as well as in the detection and diagnosis of spacecraft anomalies. Relevant illustrative examples are included

I Love My Attitude Problem

This viewgraph presentation provides a survey of modern methods for attitude estimation. Two algorithms are discussed: Kalman Filters and Wahba's Problem. Several spacecraft Attitude Determination And Control Subsystem (ADACS) are discussed including: Solar Maximum Mission (SMM), Solar, Anomalous, and Magnetospheric Particle Explorer (SAMPEX), Hubble Space Telescope (HST), Tropical Rainfall Measuring Mission (TRMM), and Wilkinson Microwave Anisotropy Probe (WMAP). Also included is a discussion about Zero Gyro Sunpoint (ZGSP), and the WMAP Attitude Anomaly

Minimal parameter solution of the orthogonal matrix differential equation

As demonstrated in this work, all orthogonal matrices solve a first order differential equation. The straightforward solution of this equation requires n sup 2 integrations to obtain the element of the nth order matrix. There are, however, only n(n-1)/2 independent parameters which determine an orthogonal matrix. The questions of choosing them, finding their differential equation and expressing the orthogonal matrix in terms of these parameters are considered. Several possibilities which are based on attitude determination in three dimensions are examined. It is shown that not all 3-D methods have useful extensions to higher dimensions. It is also shown why the rate of change of the matrix elements, which are the elements of the angular rate vector in 3-D, are the elements of a tensor of the second rank (dyadic) in spaces other than three dimensional. It is proven that the 3-D Gibbs vector (or Cayley Parameters) are extendable to other dimensions. An algorithm is developed employing the resulting parameters, which are termed Extended Rodrigues Parameters, and numerical results are presented of the application of the algorithm to a fourth order matrix