Mapa estratégico

A roadmap, Section 2.2.1.2, has been selected as the form of environmental repre- sentation, due to the customisation the resultant node network can provide [119]. As the safety of the pedestrians is paramount, the Voronoi diagram (VD), Fig. 2.5a, is chosen for the basic roadmap structure. The VD guarantees a roadmap of connected nodes that provide a maximum distance safety margin between objects, Fig. 3.9a, described by the vertex point (VP) in the centre of an Apollonius circle between 3 neighbouring objects and centre point of the minimum distance (MD) between each of the neighbouring ellipses. Connecting these describing points will generate an approximated VD of vectors, Fig. 3.9b, allowing the search algorithm to plan paths that keep furthest away from any ellipses and potential collisions. Unfortunately, the VD does not generate very direct paths when moving between objects, and so the visibility graph (VG), Fig. 2.5b, is also used. The VG generates a roadmap of connected nodes that ensures the most direct path can be taken when moving

Fig. 3.8 The model of the AMR. The robot’s spatial representation is modelled using a circle with centre (Rx, Ry) and radii Rr. Its trajectory, Rθ, is orientated relative to the x-axis, along which its velocity, Rv, is directed. The FoV the AMR uses to detect objects within has a distance range, Rf ov, beginning from the centre of the AMR with a angular range, F oVθ, equally spaced either side of its front. The buffer zone, BZ, surrounds the AMR as a safety precaution which stops the AMR moving if a pedestrian enters it.

3.3 The Considerate Path Planner 89

(a) (b) (c) (d)

Fig. 3.9 Visualisation of the construction of the Euclidean vector roadmap. The blue circle is the Apollonius circle, which has its centre at the Voronoi vertex point. The orange lines are the MD between neighbouring ellipses. (a) The green nodes of the VP and MDs are the connectivity graph, from which the roadmap is formed. (b) The yellow lines are the approximated VD vectors. (c) The purple lines are the shortcuts that occur between MDs, in order to create a more natural flow. (d) The complete VD-VG roadmap of all interconnected nodes.

between objects, allowing more efficient paths to be planned. These connections assist in aligning the search more with the direction of the crowd. Shortcuts via MD connections are made that can jump between ellipses when planing paths, Fig. 3.9c, not confining paths that detour along the VD connections. Therefore, for the CPP to plan paths that are both safe and efficient a hybrid roadmap between the VD and VG (VD-VG) has been developed, Fig. 3.9d.

3.3.2.1 Modifying the Visibility Graph for Closed Curve Conic Sections A traditional VG roadmap is formed between multiple 2D polygons, by evaluating if there is a direct line of sight between each polygon’s corners and any other polygon’s corners. Unfortunately, as a circle or ellipse is a closed curve conic section, there are no corner nodes to be connected between each shape. However, there are 4 distinct tangent points on each shape that can be connected to one another, Fig. 3.10a. These provide the most direct connections between objects, either moving alongside (non- intersecting) or crossing between (intersecting) them. However, these connections not only result in paths that are too close to the pedestrians, the tangent points that create the nodes are unique for each conic sections pair, and therefore each node is not guaranteed to connect to another.

To ensure nodes can connect to one another, an MD connection is made between two neighbouring ellipses. Two nodes close to both ellipses are created along this MD, Fig. 3.10b, which is also a describing point of the VD. By observing the resul- tant node network, Fig. 3.10c, connections can be made that represent similar VG connections, as seen in Fig. 3.10a, when travelling between ellipses. The result is a large interconnected network that requires a large search in order to plan any path. However, this approximated VG (aVG) can now have its node network reduced and be combined with the VD.

(a)

(b) (c)

Fig. 3.10 A demonstration of modifying a visibility graph of points to use with ellipses. (a) Demonstration of up to four points connecting one ellipse to another, which are tangential to both. Those connections directly visible from one ellipse to another are comparable to traditional visibility graph connections. Multiple ellipses are unlikely to share the same points with other ellipses, making a connected node network not possible. (b) Proposed node network for the visibility graph of ellipses, formed from the unobstructed connections that form the minimum distance between ellipses. (c) A standard visibility graph can be generated from the proposed node network in (b). The network produces many connections that are similar to those seen in (a).

3.3 The Considerate Path Planner 91

3.3.2.2 Hybrid Voronoi Diagram-Visibility Graph Roadmap (VD-VG)

The aVG will have a lot of path connections, Fig. 3.10c, and provides a lot of potential path diversity, which is beneficial in path planning [154] in order to evaluate the potentials of different path possibilities, Section 2.4.2.1. Comparatively the VD, Fig. 3.11a, provides more limited but very specific path connections that ensure safe paths, by connecting vertex points with their associated MD connections. The aVG uses the points near the ends of the MDs, whilst the VD uses the MD’s mid- points. If the aVG used these mid-point connections instead, Fig. 3.11b, it would still create connections that passed between all neighbouring ellipses, but with less path diversity and a larger clearance of the ellipses. However, by only using the MD mid-points the aVG can be fused with the VD to create created the VD-VG roadmap hybrid, Fig. 3.11c, which would possess the safety of the VD and path efficiency of the VG.

For whatever path finding search algorithm to be used, Section 2.2.2.2, it would be able to plan a route that could mix both elements of the VD-VG and change between safety and path efficiently at different moments along the overall path. Figs. 3.11d to 3.11f highlight an example path through the VD element of the VD- VG, from a start point at the bottom-left to three goals. Figs. 3.11g to 3.11i highlight the VG element of the VD-VG for the same scenario. The effectiveness of the VD- VG will be evaluated in Section 6.3 as an overall roadmap, as well as how effective the VD and the VG elements are when used as separate roadmaps.