which means that the kinematic vehicle speeds up to catch the target when located behind it, and speeds down to wait when located in front of it. Hence, this approach entails a synchronization-law extension of the path-following scenario, where no corners are cut.
The path-maneuvering scenario involves the use of knowledge about vehicle maneuverability constraints to design purposeful speed and steering laws that allow for feasible path negotiation. Since this work only deals with kinematic considerations, such deliberations are outside of its scope. However, relevant work in this vein include (Sheridan 1966), (Yoshimoto et al. 2000), (Skjetne et al. 2004), (B0rhaug et al. 2006), (Subbotin et al. 2006), (Gomes et al. 2006), and (Sharp 2007). Much work still remains to be done on this topic, which represents a rich source of interesting and challenging problems.
4.7 Steering laws as saturated control laws
Rewriting (21) as
it can be seen that the lookahead-based steering law is equivalent to a saturated proportional control law, effectively mapping e e R into Xr(e) /2,n/2^ .
As can be inferred from the geometry of Fig. 5, a small lookahead distance implies aggressive steering, which intuitively is confirmed by a correspondingly large proportional gain in the saturated control interpretation. This interpretation also suggests the possibility of introducing, e.g., integral action into the steering law, such that
where kt >0 represents the integral gain. Note that such integral action is not necessary in a purely kinematic setting, but can be particularly useful for underactuated AUVs that can only steer by attitude information, enabling them to follow straight-line paths while under the influence of constant ocean currents even without having access to velocity information. Thus, considering horizontal path following along straight lines, the desired yaw angle can be computed by
with xr(e) as in (44). In practice, to avoid overshoot and windup effects, care must be taken when using integral action in the steering law. Specifically, the integral term should only be used when a steady-state off-track condition has been detected.
For those AUVs that do have access to velocity information, temporal integration can be replaced by spatial integration in order to minimize overshoot and windup problems (Davidson et al. 2002), employing
where for straight-line paths i0Se(-)d- = J>(T)^dfdT (47)
which means that integration only occurs when the velocity has a component along the path. Also, derivative action can be added to the steering law in order to obtain a damped transient response toward the path.
In this section, guidance laws for 3D motion control scenarios are considered. For spatial target-tracking purposes, the guidance principles of Section 3 remain equally valid, and the velocity assignment (2) is directly applicable for 3D target tracking. However, the steering laws of Section 4 need to be extended. Specifically, in what follows, lookahead-based steering will be put into a spatial framework for regularly parameterized paths, adapted from (Breivik & Fossen 2005b). Note that the path-tracking speed law (42) need not be modified, and can be directly applied to 3D scenarios.
Now, represent the kinematic vehicle by its spatial position p(i) = [x(i), y(t), z(t)]T e R3 and velocity v(t) = p(t) e R3, stated relative to some stationary reference frame. Also, the speed is represented by U(t) = |v(i)| = ^x(t)2 + y(t)2 + Z(t)2 > 0, while the steering is characterized by the two angular variables x(t) = atan2 (y(f), x(t)) e S (the azimuth angle) and u(t) = atan2 (-Z(t),^x(t)2 + y(i)2) e S (the elevation angle). Path following is then ensured by proper assignments to %(t) and u(t) as long as U(t) > 0 .
Then, consider a spatial 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 R3. Thus, the path can be expressed by the set
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