An observability analysis of the six-degree-of-freedom attitude and position determination problem using line-of-sight observations is shown. This analysis involves decompositions of the associated error covariance matrix, derived from maximum likelihood, for a number of cases ranging from one vector observation to three or more vector observations. The covariance matrix is shown to be singular when one or two vector observations are used, leading to an unobservable system. For the one-vector case, the observable quantities involve a combination of both attitude and position information that cannot be decoupled. For the two-vector case, the covariance matrix has rank four, but only one axis of attitude and one axis of position is fully observable, with the other two observable quantities involving coupled attitude/position information. When three or more vector observations are present, the covariance matrix has full rank, except for some special cases that are derived. This observability analysis is useful for the design and analysis of estimators using line-of-sight vector observations. [References: 16]
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