Quaternion Data Fusion

Attitude quaternion [1] is the attitude parameterization of choice for spacecraft attitude estimation for several reasons: 1) it is free of singularities, 2) the attitude matrix is quadratic in the quaternion components, and 3) the kinematics equations is bilinear and an analytic solution exists for the propagation. However, the components of the attitude quaternion are not independent of each other and the norm of the attitude quaternion must be unity. This unity-norm constraint leads to problems for data fusion involving quaternions. The objective of data fusion is to find the optimal estimate from data of various sources. Reference [2] addresses the problem of fusing or averaging a set of quaternions. The fused or averaged quaternion is defined as the optimal solution to a constrained minimization problem subject to one equality constraint (the quaternion constraint). The method of Lagrange multipliers is used to convert the constrained minimization problem to an unconstrained minimization problem. The Lagrange multiplier is the maximum eigenvalue of a 4×4 symmetric matrix composed from the quaternions and weights [3]. The optimal average quaternion is the eigenvector corresponding to the maximum eigenvalue [3]. Reference [3] addresses a more general data fusion problem in which the state vector of interest consists of one quaternion and a set of unconstrained parameters, for example, gyro biases. The data fusion problem is also formulated as a constrained minimization problem subject to one equality constraint. The Lagrange multiplier is now the maximum eigenvalue of an 8× 8 asymmetric matrix or the maximum root of the 8-degree secular equation [3]. Given the Lagrange multiplier, the optimal state estimate is obtained by solving a linear system of equations [3]. This paper addresses an even more general data fusion problem in which the state vector includes two or more quaternions as well as a set of unconstrained parameters. Such a problem appears in formation flying involving multiple vehicles in which two or more