b and c are the direction cosines of the
unit vector [??] = [a, b, c] of the normal to the plane (1), which coincides with the vector [??].
The control scheme for the SEHAPF using FSMPWM and
unit vector is depicted in Section 3.
The
unit vector [[??].sub.3] is aligned with the Earth axis of rotation and is positive northward, whereas [[??].sub.1] is aligned with the vernal axis, which corresponds to the intersection of the Earth equatorial plane with the ecliptic plane.
Denoting its angle 0 and the
unit vector along its axis n, one may express the associated quaternion as
Similarly, we set [e.sub.2] to an arbitrary
unit vector with an orthogonal projection on the subspace < [e.sub.-1], [e.sub.0], [e.sub.1] > lying on the line with directing vector [r.sub.1] and the angle between this vector and [r.sub.1] equal [[alpha].sub.1], and select the coefficient [c.sub.2] in such a way that for [r.sub.2] = [r.sub.1]-[c.sub.2][e.sub.2] scalar products ([r.sub.2], -[e.sub.2]), ([r.sub.2], -[e.sub.1]), ([r.sub.2], [e.sub.0]), ([r.sub.2], [e.sub.-1]) are equal or, equivalently, angles between [r.sub.2] and the vectors [e.sub.-1], [e.sub.0], -[e.sub.1], -[e.sub.2] are equal.
Using the mass-metric, if N denotes the
unit vector field normal to [T.sub.x]S in [T.sub.x][F.sub.n](E), the projection of [[nabla].sup.S.sub.v]u of any vector field u on [T.sub.x]S is
The gravity force is given by [[??].sub.g] = - m.g[??] where [??] is a
unit vector along the global fixed axis z in the inertial reference frame (x, y, z).
Both these energy phenomena are described from the same methodological positions within the framework of a general mathematical model in which an essential role is played by the ZS
unit vector (a vector with a unit norm)
propagates along the direction given by the
unit vector N.