When we are to calculate the attitude in a satellite, we encounter a similar issue. There is no equipment that directly measures attitude. Instead, what we are required to do is to use measurements from a varied set of devices and use these collectively to ‘estimate’ the attitude.

Multiple sensors could be employed to get measured vector quantities in the body frame of satellite (essentially a set of 3 orthogonal coordinate axes imagined to be fixed to the satellite’s body) . For instance, a sun sensor gives a sun vector and a magnetometer gives the Earth’s magnetic field vector. These quantities are compared with their calculated values from scientific models in the reference frame (represents the required orientation of the coordinate axes of the satellite at a point in space) and the satellite’s attitude is ‘estimated’. <br \>

If you are given two randomly oriented pairs of 3D orthogonal coordinate axes and asked to align them ,what would you do? Is aligning one pair of axes enough? No! The other two pairs could still be rotated in their plane about this axis. So, what if you align another pair? That’s it! They are perfectly matched now. So, intuitively, we could say that to align two 3D frames we would need a minimum of two pairs of vectors in these frames. The same holds true for the body and reference frames of a satellite. <br \>

Mathematically too, it turns out that the rotation matrix (which represents this process of alignment) has nine elements with six constraints imposed on them to qualify as a valid rotation matrix. The other three constraints to uniquely determine this matrix are to come from the vector pairs compared in both frames. Each of these pairs, you might think, provides three additional constraints. After all, we are to match three coordinates of both the vectors in the two frames. Remember, however, that these two vectors depict the same physical quantity in two different frames. So if two of their coordinates are matched, so is their third. This is because their magnitude (and we know this value from what quantity they represent) has to be the same in both frames. In fact, the vector pairs are all normalised to make things less complicated. <br \>