A Predictive Attitude Determination Algorithm

In this paper, a new and efficient algorithm is developed for attitude determination from vector observations. The new algorithm, called the Predictive Attitude Determination (PAD) algorithm, is derived from a general nonlinear predictive filter approach. Traditional deterministic algorithms are shown to be suboptimal for anisotropic measurement errors. The major advantage of the PAD algorithm is that it can be easily be applied to the case where anisotropic measurement errors exist. Also, an analytical expression is derived for the steady-state attitude error covariance, which is shown to be equivalent to the optimal covariance derived from maximum likelihood techniques. Simulation studies indicate that the new algorithm is able to accurately determine the attitude of a spacecraft, even for radically anisotropic measurement errors. Introduction Attitude determination refers to the identification of a proper orthogonal rotation matrix so that the measured observations in the body frame equal the reference frame observations mapped by that matrix into the body frame. If all the measured and reference vectors are error free, then the rotation (attitude) matrix is the same for all sets of observations. However, if measurements errors exist, such as noise, then a least-squares type method must be used. For this case, the most common method for determining the attitude matrix uses a loss function first posed by Wahba [1]. This problem involves finding an orthogonal rotation matrix which minimizes the weighted sum of the squares of the observation residuals. Since its origination in 1965, there have been many algorithms developed which minimize Wahba’s loss function. The first practical method was given by Davenport’s q-method [2], which solves for the quaternion representation of the rotation matrix. However, this method requires an eigenvalue/eigenvector decomposition of a dimension 4 matrix, which may be computationally intense. A more efficient method was proposed by Shuster [3], called the QUEST algorithm, which simplifies the q-method approach by solving for the components of a Gibbs vector, and uses the fact that any meromorphic function of a dimension 3 matrix can be represented as a quadratic in that matrix. Other methods solve for the attitude matrix directly (e.g., see Refs. [4-5]). In particular, the FOAM algorithm [5] has been shown to be comparable to QUEST in computational speed, and has also been shown to be more robust in some cases. Still other methods which address Wahba’s problem can be found in Ref. [6-8]. In Wahba’s problem each vector residual is weighted by a scalar number to reflect the relative importance of each sensor. Shuster [9] has shown that Wahba’s problem is equivalent to a maximum likelihood estimation problem, where the scalar weight is equal to the scalar inverse variance of the measurement error process. Shuster [10] has further shown that a scalar variance is a good approximation of the actual measurement errors, except in the case where the measurement