and g(Xp, Up, Xr , Ur ) = - sin Xp sin Xr cos Ur - cos Xp sin Up sin Ur +

and the elevation angle u(up,Xr,ur) = arcsin(sinup cosXr cosur + cosup sinur). (62)

Through the use of trigonometric addition formulas, it can be shown that (59) is equivalent to (19) in the 2D case, i.e., when up = ur = 0 .

5.1 Path parameterizations

Applicable (arc-length) parameterizations of straight lines and helices are now given.

5.1.1 Parameterization of straight lines

A spatial straight line can be parameterized by me R as xp(m) = xf +m cos a cos ¡5 (63)

where pf = [xf, yf, zf ]T e R3 represents a fixed point on the path (for which m is defined relative to), and a e S represents the azimuth angle of the path, while fie S represents the elevation angle of the path (both corresponding to the direction of increasing m).

5.1.2 Parameterization of helices

A helix can be parameterized by me R as

where pc = [xc, yc,zc ]T e R3 represents the origin of the helix center (for which w is defined relative to), rc >0 represents the radius of the horizontally-projected circle of the helix, and X e {-1,1} decides in which direction this horizontally-projected circle is traced; X = -1 for anti-clockwise motion and X = 1 for clockwise motion. Here, an increase in w corresponds to movement in the negative direction of the z-axis of the stationary frame.

6. Conclusions

This work has given an overview of guidance laws applicable to motion control of AUVs in 2D and 3D. Specifically, considered scenarios have included target tracking, where only instantaneous information about the target motion is available, as well as path scenarios, where spatial information is available apriori. For target-tracking purposes, classical guidance laws from the missile literature were reviewed, in particular line of sight, pure pursuit, and constant bearing. For the path scenarios, enclosure-based and lookahead-based guidance laws were presented. Relations between the guidance laws have been discussed, as well as interpretations toward saturated control.

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