The pseudo measurement method is a main approach to equality-constrained state estimation due to its simplicity. It is, however, not popular due to possible numerical problems and increased computational complexity. The work presented here further develops the pseudo measurement method. To avoid numerical problems resulting from singular measurement noise when a matrix inverse is used, the Moore-Penrose (MP) inverse is used instead. Also, to reduce the computational load without performance loss and to simplify the analysis of this type of estimation problem, two sequential forms are obtained. They differ only in the processing order of the physical measurement and the pseudo measurement (i.e., the equality constraint). Although form 1 is the same as some existing results, form 2 is new. This motivates the discussion of processing order for this type of estimation problem, especially in the extension to the nonlinear case. It is found that under certain conditions, the use of the pseudo measurement for filtering is redundant. This differs in effect from update by the physically error-free measurement. However, if there exists model mismatch, update by the pseudo measurement is necessary and helpful. Supporting numerical examples are provided.