Triad Algorithm

Triad is one of the simplest deterministic attitude determination algorithm. Triad begins by considering two measurement vectors, such as the direction to the sun and the direction of the Earth's magnetic field. We denote the actual vectors by s and m, respectively. The measured components of the vectors, with respect to the body frame (A reference frame that is attached to the body of the satellite) , are denoted [math]s_b[/math] and [math]m_b[/math], respectively. The known components of the vectors in the inertial frame are [math]s_i[/math] and [math]m_i[/math]. Ideally, the rotation matrix, or attitude matrix, R, satisfies and . Unfortunately, since the problem is overdetermined (because the Rotation Matrix has three independent parameters and we have four constraints by these two equations), it is in general not possible to find such an R. The most simple deterministic attitude determination algorithm is based on neglecting one piece of this information. However, this approach doesn't simply amount to throwing away a component of one of the measured vectors. The algorithm is known as the Triad algorithm, since it is based on constructing two triads of orthonormal unit vectors by making use of the vector information that we have.

The Algorithm

The algorithm considers one vector measurement to be more accurate than the other. For example, suppose we could consider the sun vector to have more accurate components than the magnetic field vector. We consider the accurate vector to be our first vector of the triad. Thus, and
The second vector we chose is perpendicular to both measured vectors:

Note that we are in effect assuming that the measurement of the magnetic field vector is less accurate than the measurement of the sun vector. If the magnetic field vector is more accurate, suitable changes should be made in the triad.
Finally the third vector of the triad is

Now, we construct two matrices by putting the t vector components into the columns of two 3 x 3 matrices. The two matrices are

Note that

Thus,

With this, we have found an appropriate Rotation Matrix.
Notice that in general, s and m may not be perpendicular. So if we directly used [math]t_1 = s[/math] and [math]t_2 = m[/math], the matrix we get by this method may not be orthogonal. When we construct orthogonal vectors from these, we are in effect reducing a constraint of the equation and making the equations solvable.

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