Et rc pt pj

in (21), see Fig. 6. Note that the lookahead distance A is no longer defined along the path, but (in general) along the x-axis of the path-fixed frame (i.e., along the path tangential associated with the origin of the path-fixed frame). An along-track distance s(t) can also be computed relative to some fixed point on the circle perimeter if required.

4.4 Steering for regularly parameterized paths

Consider a planar path continuously parameterized by a scalar variable me R, such that the position of a point belonging to the path is represented by pp(m) e R2 . Thus, the path is a one-dimensional manifold that can be expressed by the set

Regularly parameterized paths belong to the subset of P for which |pp(m)| = |dpp(m) / dm|

is non-zero and finite, which means that such paths never degenerate into a point nor have corners. These paths include both straight lines (zero curvature) and circles (constant curvature). However, most are paths with varying curvature. For such paths, it is not trivial to calculate the cross-track error e(t) required in (21).

Although it is possible to calculate the exact projection of p(t) onto the path by applying the so-called Serret-Frenet equations, such an approach suffers from a kinematic singularity associated with the osculating circle of the instantaneous projection point (Samson 1992). For every point along a curved path, there exists an associated tangent circle with radius r(m) = 1 / c(m), where c(m) is the curvature at the path point. This circle is known as the osculating circle, and if at any time p(t) is located at the origin of the osculating circle, the projected point on the path will have to move infinitely fast, which is not possible. This kinematic singularity effect necessitates a different approach to obtain the cross-track error required for steering purposes. The solution considered here seems to first have been suggested in (Aicardi et al. 1995), then refined and put into a differential-geometric framework in (Lapierre et al. 2003), and finally extended into the form presented below in (Breivik & Fossen 2004b).

Thus, consider an arbitrary path point pp(m) . Subsequently, consider a path-fixed reference frame with origin at pp (m), whose x-axis has been rotated a positive angle (relative to the x-axis of the stationary reference frame)

such that

where e(t) = [s(t), e(t)]T e R2 represents the along-track and cross-track errors relative to pp(m), decomposed in the path-fixed reference frame by

Fig. 7. The main variables associated with steering for regularly parameterized paths In contrast to (8), the path-following control objective now becomes

and in order to reduce e(t) to zero, p(t) and pp(m) can collaborate with each other. Specifically, pp(m) can contribute by moving toward the direct projection of p(t) onto the x-axis of the path-fixed reference frame by assigning

where xr(e) is given by (21), Y>0, and |pp(m)| = ^xp(m)2 + yp(m)2 . As can be seen, the first element of the numerator represents a kinematic feedforward of the projected speed of p(t) onto the path tangential, while the second element represents a linear feedback term whose purpose is to reduce the along-track error to zero. Hence, the path-constrained attractor pp(m) tracks the motion of p(t), which steers by the location of pp(m) through the cross-

track error of (30) by employing (19) with (29) and (21) for U(t) > 0 . Such an approach suffers from no kinematic singularities, and ensures that e(t) is reduced to zero for regularly parameterized paths. To avoid initial transients in e(t), the initial along-track error s(0) can be minimized offline.

4.4.1 Relations to classical guidance laws

Drawing a connection to the classical guidance principles of the missile literature, lookahead-based steering can be interpreted as pure pursuit of the lookahead point. Convergence to pp(m) is thus achieved as p(t) in vain chases a carrot located a distance A

further ahead along the path tangential. However, in (Papoulias 1992), the lookahead point is suggested to be placed further ahead along the path instead of along the path tangential, which leads to a steady-state offset in the cross-track error for curved paths. In this case, the velocity of p(t) cannot be aligned with the velocity of pp(w) for zero cross-track error. This distinction is vital for curved paths, but not for straight-line paths, where the path tangential is always directed along the path. Thus, in general, the pursued carrot must be located along the path tangential and not along the path itself. Nevertheless, the along-path approach has been widely reported in the literature, see, e.g., (Ollero & Heredia 1995), (Rankin et al. 1997), and (CastaƱo et al. 2005).

4.4.2 Off-path traversing of curved paths

In some applications, it can be desirable to perform off-path traversing of regularly parameterized paths. Specifically, off-path traversing of curved paths requires the use of two virtual points to avoid kinematic singularities. This concept was originally suggested in (Breivik et al. 2006), and used for formation control of ships in (Breivik et al. 2008).

4.4.3 Path parameterizations

Although the recently-presented guidance method also can be applied for both straight lines and circles, the analytic, path-specific approaches presented previously are often preferable since they do not require numerical integrations such as (33). However, for completeness, applicable (arc-length) parameterizations of straight lines and circles are given in the following.

Parameterization of straight lines

A planar straight line can be parameterized by me R as xp(w) = xf + w cos a (34)

where pf = [xf, yf ]T e R2 represents a fixed point on the path (for which w is defined relative to), and a e S represents the orientation of the path relative to the x-axis of the stationary reference frame (corresponding to the direction of increasing w ).

Parameterization of circles

A planar circle can be parameterized by we R as xp(w) = xc + rc cos W j (36)

where pc = [xc, yc ]T e R2 represents the circle center, rc >0 represents the circle radius, and 1e{-1,l} decides in which direction pp(w) traces the circumference; 1 = -1 for anticlockwise motion and 1 = 1 for clockwise motion.

4.5 Speed law for path tracking

As previously stated, the control objective of a path-tracking scenario is to track a target that is constrained to move along a path. Denoting the path-parameterization variable associated with the path-traversing target by mt(t) e R , the control objective is identical to (1) with pt(t) = pp(mt(t)). Here, mt(t) can be updated by

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