A super-fast optimal attitude matrix in attitude determination

The fast optimal attitude matrix (FOAM) for minimizing Wahba's loss function solves the optimal attitude matrix directly and efficiently. This method is more intuitive compared with the method of calculating attitude quaternion. A method named super-FOAM (SFOAM) is proposed with a more precise initial iteration value and a faster iterative algorithm. A condition number estimation method of third-order matrices is presented for cases without exact singular value. When the matrix condition number is not particularly large, the root-seeking formula for the quartic equation is directly used to obtain the maximum characteristic root. Otherwise, under large condition numbers, a high-accuracy estimation algorithm for the maximum eigenvalue is given to achieve a high-accuracy initial value set, which is beneficial to iteration reduction and convergence speed improvement. Finally, some comparative simulations show that the SFOAM method has the same accuracy as the singular value decomposition (Matlab) method even in the case of a very large matrix condition number, while the SFOAM's computation is only 50%-60% of the traditional FOAM algorithm. This algorithm has good generalization and application values.