ESOQ: A Closed-Form Solution to the Wahba Problem
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ESOQ: A Closed-Form Solution to the Wahba Problem' Daniele Mortarf Abstract A closed-form solution to the problem of optimal spacecraft attitude estimation based on vector observation, known as the Wahba problem, is presented. The algorithm first provides the closed-form expressions of a 4 X 4 matrix eigenvalues and then computes the eigenvector associated with the greatest of them (representing the optimal quatemion) using two different methods. The first method uses a vector cross-product in a four-dimensional space, while the second uses an equivalent technique requiring a 3 X 3 nonsingular matrix inversion. The resulting "EStimator of the Optimal Quaternion" (ESOQ) algorithm does not present any singularities and allows an easy identification of the approaching of the unresolvable condition of quasi-parallel observed vectors. Numerical accuracy tests, showing the average and the variance of the maximum attitude errors, are presented. Speed numerical tests, which demonstrate ESOQ as the fastest optimal attitude estimation algorithm to date, validate ESOQ as the most suitable algorithm when a fast and optimal attitude determination is required.
Introduction The spacecraft attitude estimation, that is, the evaluation of a Spacecraft Body System (SBS) orientation with respect to another reference system, such as the Inertial Reference System (IRS), is an important task of spacecraft navigation, dynamics, and control problems. This task, which has to be executed in the fastest and most precise manner, is often accomplished using the information of n ~ 2 unit vectors S i s observed by the attitude sensors in the SBS, and the same directions evaluated by proper codes in the IRS, which are the Vi unit vectors. In the SBS, the unit vector Vi is represented by the unit vector TVi' where T is the true attitude matrix indicating the spacecraft orientation. Let f3i be the precision of the ith sensor. This means that the angle between the true and the observed direction is smaller than f3 i- The sensor relative precisions I Extracted
from Paper AAS-96-173 presented at the Sixth Annual AIAAJAAS Space Flight Mechanics Meeting, Austin, Texas, Feb. 11-15, 1996. 2Assistant Professor, Aerospace Department, Universita' degli Studi "La Sapienza" di Roma, Via Salaria 851,00138 Rome (Italy). E-mail: [email protected]. Member AAS. 195
Daniele Mortari
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a, are derived from the f3i as follows
1
(1)
As is easily observed, the relative precisions a, are positive, less than unity, of a greater magnitude when the sensors are more accurate, and satisfy the relative condition !,iai = 1. The optimal attitude estimation problem, which is known as the Wahba problem, means to estimate the spacecraft attitude by satisfying an optimality criterion and using the n ~ 2 unit vector pairs [SiVi] together with the n sensor relative precisions a.. Up to now the optimality criterion, originally introduced by Wahba [1] in 1965 and then completed by including the weights a, as well as the 1/2 multiplier, defines as optimal the attit
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