Estrategias de Marketing Mix.

5.3

Avoiding Homotopic Path Repetition

In the same way as other roadmaps, the VD-VG forms multiple connections that converge on the same nodes. As mentioned in Section 5.2.1, the search along a vector will automatically terminate if another frontier is propagating along another vector that shares the same end node, and will arrive first. However, the paths converted from the frontier, once the search terminates, can belong to the same homotopy class by moving: between the same ellipses, in the same direction around an ellipse, and when moving through open space.

Moving Through the Same Gap Between Two Ellipses

All VD-VG nodes are either VPs, with 3 ellipses in common or MDs with 2 ellipses in common, which can either be inside or outside an ellipse, Fig. 5.9a. If any of the frontier paths have at least 2 ellipses in common, and point in the same direction with the same internal/external ellipse component, Fig. 5.9b, then only the most desirable is chosen, as they belong to the same homotopy class, Figs. 5.9c and 5.9d.

Moving Around the Same Ellipse

To achieve considerate movement, one of the main principles set out in the aims & objectives, Section 1.3.2, is to avoid interfering with a pedestrians’ trajectories. A potential path can either collide with; pass on either side of; or move away from, an agent’s uncertainty ellipse. As ellipse orientations represent agent movement vectors, the orientations of the frontier paths are compared with them. Comparisons between paths are made. Fig. 5.10, in order to determine if two paths are competing to move in similar directions:

a1 Only the most desirable path is chosen if one of the following three criteria are

met:

a1.1 Both paths collide with the ellipse, Fig. 5.10a.

• As a collision will likely occur for both paths, increasing the number of paths is not desirable.

a1.2 Both paths pass by the ellipse on the same side, Fig. 5.10c.

• This is particularly effective if the CPP searches an environment where all agents appear only to one side of the AMR, as it will prevent multiple paths repeating on the same side of the crowd.

(a)

(b) (c) (d)

Fig. 5.9 Visualisation of how homotopic path repetition is reduced. (a) Example of the VD-VG roadmap for 4 ellipses, with the area of interest outlined with the orange square. The search will propagate along the roadmap beginning from the bottom- left. (b) There are 6 homotopy classes that are formed by the different vector orientations, highlighted in various shades of green. (c) If the search terminates here the 3 vectors share two homotopy classes; 2 are chosen (green markers), 1 ignored (red marker). The middle vector is more desirable than the lower vector, which are homotopic, however the middle is removed due to the upper vector and so the lower can remain. (d) If the search terminates here the 5 vectors belong to 3 homotopy classes; 3 are chosen (green arrowheads) and 2 ignored (red arrowheads). The search also terminated at the lower VP (red square marker) earlier on, as the other nodes connected to it would be reached by alternative vectors earlier, Algorithm 2.

5.3 Avoiding Homotopic Path Repetition 147

(a) (b)

(c) (d)

(e)

Fig. 5.10 Visualisation of how similar paths are avoided. A comparison is made between the semi-major axis ends of the ellipse, e0 and ev, and the position of the potential paths, p1 and p2, as well as their orientation to the front or back of the

robot (blue circle). (a) Choose the best path, as p1 and p2 both collide with the

ellipse. (b) Keep both paths, as p1collides with and p2passes the ellipse. (c) Choose

the best path, as p1 and p2 both pass on same side of the ellipse. (d) Keep both

paths, as p1 and p2 pass on different sides of the ellipse. (e) Choose one path, as p1

a1.3 Both paths are moving away from the ellipse, Fig. 5.10e.

• A special condition where the orientation of the ellipse is irrelevant, and neither of the paths will interfere with the agent.

a2 Both paths are kept if one of the following two criteria are met:

a2.1 One path collides with the ellipse, whilst the other passes by, Fig. 5.10b.

• Both paths contribute differently, even if one has a much higher col- lision potential.

a2.2 Both paths pass by the ellipse on opposite sides, Fig. 5.10d.

• Both paths will circumnavigate the agents via opposite routes, help- ing to explore the environment more.

The procedure to determine which frontier paths to use involves a simple comparison between which side of two frontier path vectors the ellipse’s S-Ma axis limits, e0and ev, lie (+ ⇒ left, − ⇒ right): If paths pass on the same side (the symbols are the same) Then keep both paths Else choose the most desirable path only. However, before this comparison can be made the orientation of the paths w.r.t. the ellipse must be checked: If e0 or ev is behind a path vector (the perpendicular line at its start) Then use the other vector to find the correct symbolinstead, e.g. Fig. 5.10c.

Moving into Open Space

A special condition arises if both paths move away from an ellipse, whereby the orientation of the agent is irrelevant. If both paths remain behind the ellipse, Thenonly the most desirable is chosen. No significant divergence is possible under this condition, as there is no potential for the AMR to begin interacting with the agent. This prevents multiple paths being created that would only ever move into open space, where the agent will not be.