Lower bound on the curvature radius

Given a cost function t, our goal is to insure the feasibility of any trajectory C for an AUV v before computing the distance function u. Mathematically speaking, we want V(x1,x2) e Q2,Rmta(C) > r(v) knowing that C = argmin|C |p(x1,x2). For this purpose we will express a formal link between the cost function t and the lower bound Rmin(C) for any curve C minimizing the metric p between two configurations.

Using the differential geometry framework, it is shown in (Caselles et al., 1997) that the Euler-Lagrange equation associated with the functional minimization (14) is:

where N is the normal unit vector to a curve C.

From equation (16), it is deduced in (Cohen & Kimmel, 1997) that the curvature magnitude k is bounded along any curve C minimizing p. The lower bound Rmin is then:

The conclusion is that to increase the lower bound on the curvature radius Rmin(C) of an optimal trajectory C, two choices are possible:

• smoothing the cost function t to decrease sup, {|Vx| |}

• adding an offset to the cost function to increase the numerator inf,T without affecting the denominator.

The following illustrations depict some trajectories computed using the FM* algorithm after smoothing the cost map (figure 3) and after smoothing the cost map and adding an offset (figure 4).

Fig. 3. Influence of smoothing the cost function. a) A binary 100x100 cost function t, t(C-free) = 1 and T(C-obstacles) = 11 and the related optimal trajectory Ca, Rmin(Ca) = 332 (in arbitrary units). b) t after smoothing using a 11x11 average filter, Rmin(Cb) = 1216. c) t after smoothing using a 21x21 average filter, Rmin(Cc) = 1377.

Fig. 3. Influence of smoothing the cost function. a) A binary 100x100 cost function t, t(C-free) = 1 and T(C-obstacles) = 11 and the related optimal trajectory Ca, Rmin(Ca) = 332 (in arbitrary units). b) t after smoothing using a 11x11 average filter, Rmin(Cb) = 1216. c) t after smoothing using a 21x21 average filter, Rmin(Cc) = 1377.

Fig. 4. Influence of both smoothing and adding an offset. The original cost function t is similar to the one in figure 3.a. a) Offset = 5, average filter 7x7, Rmin(Ca) = 1977 (in arbitrary units). b) Offset = 5, average filter 15x15, Rmin(Cb) = 2787.
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