where e(t) = [s(t), e(t), h(t)]T e R3 represents the along-track, cross-track, and vertical-track errors relative to pp (m), decomposed in the path-fixed reference frame. The path-following control objective is identical to (32), and e(t) can be reduced to zero by assigning an appropriate steering law to the velocity of p(t) as well as a purposeful collaborative behavior to pp (m) . Specifically, the steering law involves
which is equivalent to (21) with A > 0, used to shape the convergence behavior toward the xz-plane of the path-fixed frame, and ur(h) = arctan
used to shape the convergence behavior toward the xy-plane of the path-fixed frame, see Fig. 8. Also, pp(m) moves collaboratively toward the direct projection of p(t) onto the x-axis of the path-fixed reference frame by
where y>0 and |pp(m)| = (m)2 + yp(m)2 + zp(m)2 . In sum, four angular variables (50),
(51), (56), and (57) are used to specify the 3D steering law required for path-following purposes. Fortunately, these variables can be compactly represented by the azimuth angle
X(Xp,Up, Xr,Ur) = atan2 ( f (Xp,Up, XrU I g(Xp ,Up, Xr,Ur)) , (^
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