Suppose Say we have a set of N unit vectors <math>v_k</math>, k = 1,2...N. For each vector, weWe have a sensor measurement in the body framefor each vector, <math>v_{kb}</math>, and a mathematical model of the components in the inertial frame, <math>v_{ki}</math>. We want Our aim is to find a rotation matrix <math>R_{bi}</math>, such that<br \>

for each of the N vectors. Obviously It is clear that this set of equations is will be overdetermined if for N > 2,and therefore hence the equation cannot, in general, cannot be satisfied for each k = 1,2..N. ThusHence,we want wish to find a solution for <math>R_{bi} </math> that in some sense way minimizes the overall error for

[[File:Equation24Equation72.png|frame|center]]In this expression, J is the loss function that has to be minimized, k is the counter for theN observations, <math>w_k </math> is weight assigned to the kth measurement, <math>v_{kb} </math> is the matrix consisting of the measuredcomponents in the body frame, and vki <math>v_{ki}</math> is the matrix consisting of components in the inertialframe as determined by using appropriate mathematical models. This loss function is asum of the squared errors for each vector measurement. If the measurements and0andmathematical models are all perfect, then J = 0, since the first equation on this page will be satisfied for all N vectorsand J = 0. If there are any errors or noisy measurements, then J > 0. The smallerwe can make J, the better the approximation of <math>R_{bi}</math>.

In this section we present one method of the several methods to solve this minimization problem known as the Q-method.

[[File:Equation26Equation73.png|frame|center]]Assuming that the normalized vectors are normalized , we see that,[[File:Equation27Equation74.png|frame|center]]

[[File:Equation28Equation75.png|frame|center]]We can see that minimizing in order to minimize J is equivalent , we need to maximizing maximize g(R). This problem can be written in a better and more elegant way by representing rotation matrix R in terms of the quaternion

R can be written in terms of q as<math>R = (q_4^2-q^Tq)I+2qq^T-2q_4[[File:Equation30.png|frame|center]q \times]</math><br \>

[[File:Equation33Equation77.png|frame|center]]Wherein,<br \><math>B = \sum w_k(v_{kb}v_{ki}^T)</math><br \><math>S = B+B^T</math><br \><math>Z = \sum w_k(v_{kb} \times v_{ki})</math><br \><math>\sigma = tr(B)</math> <br \>[[File:Equation34.png|frame|center]]Now we want to maximise g. However, but noting that quaternion elements are not independent, but the constraint [[File:Equation35.png]] <math>\bar{q}^Tq = 1</math> must also be satisfied. So we use the lagrangian multiplier method and differentiate with respect to [[File:Equation36.png]] <math>\bar{q}</math> we get [[File:Equation37<math>K\bar{q} = \lambda \bar{q}</math>.png]] So now, we have reduced the problem has been reduced to an eigenvalue problem for the matrix K. Since, K is a 4x4 matrix, it can have at most 4 different eigenvalues. Substituting [[File:Equation37.png]] <math>K\bar{q} = \lambda \bar{q}</math> in [[File:Equation38.png]]<math>g(\bar{q}) = \bar{q}^TKq</math>, we get