IP t

a,max

where Ua,max >0 specifies the maximum approach speed toward the target, and Ap >0 affects the transient interceptor-target rendezvous behavior.

Note that CB guidance becomes equal to PP guidance for a stationary target, i.e., the basic difference between the two guidance schemes is whether the target velocity is used as a kinematic feedforward or not.

Returning to the example on motion camouflage, it seems that two main strategies are in use; camouflage against an object close by and camouflage against an object at infinity. The first strategy clearly corresponds to LOS guidance, while the second strategy equals CB guidance since it entails a non-rotating predator-prey line of sight.

4. Guidance laws for path scenarios

In this section, guidance laws for different path scenarios are considered, including path following, path tracking, and path maneuvering. Specifically, the guidance laws are composed of speed and steering laws, which can be combined in various ways to achieve different motion control objectives. The speed is denoted U(t) = |v(t)| = ^x(t)2 + y(t)2 > 0, while the steering is denoted x(t) = atan2 (y(t), x(t)) e S = [-n,n], where atan2 (y, x) is the four-quadrant version of arctan (y / x) e (-n /2,n/ 2j .

Path following is ensured by proper assignments to %(t) as long as U(t) > 0 since the scenario only involves a spatial constraint, while the spatio-temporal path-tracking and path-maneuvering scenarios both require explicit speed laws in addition to the steering laws. The following material is adapted from (Breivik & Fossen 2004a), (Breivik & Fossen 2005b), and (Breivik et al. 2008).

4.1 Steering laws for straight lines

Consider a straight-line path implicitly defined by two waypoints through which it passes.

Denote these waypoints as pk = [xk, yk ]T e R2 and pk+t = [xk+x, yk+t ]T e R2, respectively.

Also, consider a path-fixed reference frame with origin in p k , whose x-axis has been rotated a positive angle ak = atan2 (yk+1 - yk, xk+1 - xk )e S relative to the x-axis of the stationary reference frame. Hence, the coordinates of the kinematic vehicle in the path-fixed reference frame can be computed by e(t) = RK)T(p(t) - Pk),

where

sinak cosak and e(t) = [s(t),e(t)]T e R2 consists of the along-track distance s(t) and the cross-track error e(t), see Fig. 5. For path-following purposes, only the cross-track error is relevant since

Fig. 5. The main variables associated with steering laws for straight-line paths e(t) = 0 means that the vehicle has converged to the straight line. Expanding (5), the cross-

track error can be explicitly stated by

Fig. 5. The main variables associated with steering laws for straight-line paths e(t) = 0 means that the vehicle has converged to the straight line. Expanding (5), the cross-

track error can be explicitly stated by

In the following, two steering laws that ensure stabilization of e(t) to the origin will be presented. The first method is used in ship motion control systems (Fossen 2002), and will be referred to as enclosure-based steering. The second method is called lookahead-based steering, and has links to the classical guidance principles from the missile literature. The two steering methods essentially operate by the same principle, but as will be made clear, the lookahead-based scheme has several advantages over the enclosure-based approach.

4.1.1 Enclosure-based steering

Imagine a circle with radius r >0 enclosing p(t). If the circle radius is chosen sufficiently large, the circle will intersect the straight line at two points. The enclosure-based strategy for driving e(t) to zero is then to direct the velocity toward the intersection point that corresponds to the desired direction of travel, which is implicitly defined by the sequence in which the waypoints are ordered. Such a solution involves directly assigning e(t) = -(x(t) - xk)sinak + (y(t) - y k)cos¬ęk/ and the associated control objective for straight-line path following becomes

where pint(t) = [xtat(t), ytat(t)]T e R2 represents the intersection point of interest. In order to calculate pint(t) (two unknowns), the following two equations must be solved

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