Fractional potential field in path planning for mobile robot obstacle ...

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Example: z ∗ i = Vi xρ(x)dx We call the tessellation defined by (3) a Centroidal Voronoi Tessellation if and only if Vi ρ(x)dx for i = 1, · · · , k. (4) zi = z ∗ i for i = 1, · · · , k. So, the points zi that serves as the generators for the Voronoi regions Vi are themselves the mass centroids of those regions, as shown in Fig. 1(b). 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 (a) The Voronoi regions constructed by 100 randomly selected points in a square Ω = (−1, 1) 2 . The density function is given by ρ(x, y) = e −8(x2 +y 2 ) . The dots are the Voronoi generators and the circles are the mass centroids of the corresponding Voronoi regions. The generators and the mass centroids do not coincide. 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 (b) The centroidal Voronoi tessellations constructed by 100 points in the same Ω and same density function ρ as in 1(a). These points server both as the generators and the mass centroids of the corresponding regions. More of them aggregate to area where ρ(x, y) is bigger. Fig. 1. The illustration of Voronoi diagram and Centroidal Voronoi Tessellations. Centroidal Voronoi Tessellation has broad applications in many fields. It is the solution to optimal placement of resources, but in general, CVT can only be approximately constructed. For algorithms to implement CVT, refer to [14]. In [12], an example is given to show how CVT can be used to predict the cell divisions. It is shown that, after the cell division process, the new cells’ shapes are very closely approximated10by- 15 Centroidal Voronoi Tessellations corresponding to the increased number of generators. This is an example to show how CVT can be applied in a dynamically evolving environment.
Collision Avoidance Now we consider the safety issue of the mobile robot. It is highly likely that the robot will encounter moving obstacles that travel through the polluted area, for example, the moving cargo vessels. In order to avoid collision between the robot and the moving obstacle effectively, the relative position and velocity between robot and obstacle should be considered. Here we introduce a potential field method from reference 3 that is used for dynamic obstacle avoidance. The repulsive potential generated by the obstacle is given by: Urep(p, v) = ⎧ ⎪⎨ ⎪⎩ 0 : if ρs(p, p obs) − ρm ≥ ρ0 or vRO ≤ 0 η(Urep) : if 0 < ρs(p, p obs) − ρm < ρm and vRO > 0 undefined : if vRO > 0 and ρs(p, p obs) < ρm The positive force between the robot and the obstacle is defined as the negative gradient of the repulsive potential with respect to robot position and velocity. 11 - 15 (3)
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Page 7 and 8: Feedback control of a diffusion pro
Page 9: The objectives are: • Control the
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