A complete and careful foundation is presented for maximum-likelihood attitude estimation and the calculation of measurement sensitivity matrices with the intent of revealing heretofore undisclosed pitfalls associated with unconstrained quaternion estimation. Efficient formulas are developed for computing the measurement sensitivity matrix for any attitude representation for which an efficient formula for the inverse kinematic equation is known. In particular, it is shown that the measurement sensitivity matrix for the quaternion is ambiguous and may take on a wide range of values. Hence, estimates of a quaternion which do not take correct account of the norm constraint will be physically meaningless. It is shown also that within Maximum Likelihood Estimation the form of the Wahba cost function for attitude estimation is incorrect when the attitude constraint is relaxed. A simple physical example is presented for quaternion estimation from noise-free vector measurements which fails when the norm constraint on the quaternion is relaxed. Part I of this work provides the basis for more detailed investigations of unconstrained attitude estimation in Part II [1].