The target, that is, DCM, is shifted to an attitude quaternion accordingly. A robust framework was proposed by Davenport in 1968, who analytically changed the Wahba problem into an eigenvalue problem. This in fact has a demand on later matrix correction that deduces the optimality of the solution. Such solution is not only time and space consuming, but it will also cause nonorthogonality of the DCM. Brute force calculation of such a system requires a large matrix memory and its sophisticated generalized inverse. For instance, the above loss function can be explicitly given by the augmented rotation equations.

Initial research on this problem mainly focuses on the exact solution rather than its robustness. Here, R is the 3-dimensional direction cosine matrix (DCM) belonging to SO(3). a i is the positive weight of the ith pair with the sum of 1. (1) L R = 1 2 в€‘ i = 1 n a i D i b в€’ R D i r 2 ,in which D i b and D i r are the ith pair of vector observation that was obtained in the body frame and reference frame, respectively. This approach is referred to as the Wahba problem, posed in 1965, which minimizes the following loss function In such applications, the attitude determination from vector observation pairs is usually an effective solution. This leads to different algorithms of attitude determination on various platforms, for example, remote sensing and marine navigation. With the development of modern global navigation satellite systems (GNSS), GNSS-based attitude determination has been a vital part in spacecraft instrumentation and measurement. In aerial applications, strapdown vector sensors are often required to directly give the rotation matrix between the body frame and inertial frame.

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