where

' Ay g = f - de = yk has been used. Subsequently, consider

2 yyint = 2 y( dxint + g) = 2 dyxint + 2 gy > (15)

such that (14) and (15) inserted into (13) gives

(1 + d2 )x2nt + 2(dg - dy - x)xint + (x2 + y2 + g2 - 2gy - r2) = 0, (16)

which is a standard, analytically-solvable second order equation. Then, denote a = 1 + d2 b = 2(dg - dy - x)

c 4 x2 + y2 + g2 - 2 gy - r% from which the solution of (16) becomes

, , . „ , -b Wb2 - 4ac , r A „ , -b ^b2 - 4ac TT

where if Ax >0, then xint =-, and if Ax <0, then xint =-. Having

2a 2a calculated xint, yint is easily obtained from (12). Note that when Ay = 0, yint = yk (= yk+x). Case 2: Ax = 0

If Ax = 0, only equation (10) is valid, which means that yint = y ±vr2 - (xint - x)2, (18)

where x^t = xk (= xk+x). If Ay >0, then yint = y + ,Jr2 - (xint - x)2 , and if Ay <0, then yint = y -^r2 - (xint - x)2 . When Ax = 0, Ay = 0 is not an option.

4.1.2 Lookahead-based steering

Here, the steering assignment is separated into two parts

where

is the path-tangential angle, while

is a velocity-path relative angle which ensures that the velocity is directed toward a point on the path that is located a lookahead distance A >0 ahead of the direct projection of p(t) onto the path (Papoulias 1991), see Fig. 5.

As can be immediately noticed, this lookahead-based steering scheme is less computationally intensive than the enclosure-based approach. It is also valid for all cross-track errors, whereas the enclosure-based strategy requires r > |e(t)| . Furthermore, Fig. 5 shows that e2 +A2 = r2, (22)

which means that the enclosure-based approach corresponds to a lookahead-based scheme with a time-varying A(t) = r2 - e(t)2 , varying between 0 (when |e(t)| = r ) and r (when |e(t)| = 0). Only lookahead-based steering will be considered in the following.

If a path is made up of n straight-line segments connected by n+1 waypoints, a strategy must be employed to purposefully switch between these segments as they are traversed. In (Fossen 2002), it is suggested to associate a so-called circle of acceptance with each waypoint, with radius Rk+1 > 0 for waypoint k +1, such that the corresponding switching criterion becomes

i.e., to switch when p(t) has entered the waypoint-enclosing circle. Note that for the enclosure-based approach, such a switching criterion entails the additional (conservative) requirement r > Rk+x .

A perhaps more suitable switching criterion solely involves the along-track distance s(t), such that if the total along-track distance between waypoints pk and pk+x is denoted s2+j, a switch is made when

which is similar to (23), but has the advantage that p(t) does not need to enter the waypoint-enclosing circle for a switch to occur, i.e., no restrictions are put on the cross-track error. Thus, if no intrinsic value is associated with visiting the waypoints, and their only purpose is to implicitly define a piecewise linear path, there is no reason to apply the circle-of-acceptance switching criterion (23).

Denote the center of a circle with radius rc >0 as pc = [xc, yc ]T e R2. Subsequently, consider a path-fixed reference frame with origin at the direct projection of p(t) onto the circular

where

and Ae{-1,l} with 1 = -1 corresponding to anti-clockwise motion and 1 = 1 to clockwise motion. Hence, xp becomes time-varying for circular (curved) motion, as opposed to the constant xp associated with straight lines (20). Also, note that (26) is undefined for p(t) = pc, i.e., when the kinematic vehicle is located at the circle center. In this case, any projection of p(t) onto the circular path is valid, but in practice this problem can be alleviated by, e.g., purposefully choosing %c(t) based on the motion of p(t).

Since the path-following control objective for circles is identical to (8), lookahead-based steering can be employed, implemented by using (19) with (25) instead of (20), and

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