Antecedentes de la conciliación en Colombia

4.3.1. Structure of the Hybrid Graph

Recall that the goal of our algorithm was to use the faster low-dimensional planning, except for areas of the environment where high-dimensional planning is necessary to ensure the feasibility of the resulting path and the desired cost sub-optimality bound. We want our hybrid state-space to capture this property—namely, we want GAD to consist largely of low-dimensional states, except for the areas where high-dimensional planning needs to be performed, represented by areas of high-dimensional states inGAD. To ensure path feasibil- ity in the high-dimensional regions ofGAD, we have to use high-dimensional transitions. In the low-dimensional areas we can use simpler low-dimensional transitions. However, recall that the transitions we have in THD andTLD connect two states of the same dimensional- ity, which do not allow us to transition from the low-dimensional to the high-dimensional regions. Therefore, we have to construct a transition set TAD that allows for transitions between states of different dimensionality.

4.3.2. Construction of the Hybrid Graph

Our algorithm iteratively constructs GAD, beginning with the low-dimensional state-space SLD_{and introducing a set of high-dimensional regionsRin it. We first explain how the high-}

dimensional regions are being introduced intoGADand connected with the low-dimensional regions. The algorithm that decides when and where to introduce these regions will be explained later.

Once a high-dimensional regionr is introduced, the following changes are made toGAD. If a low-dimensional state X_iLD falls inside a new high-dimensional region r ∈R, we replace it with its high-dimensional projection states inλ−1(X_iLD). Thus,GAD contains both low-

dimensional and high-dimensional states. Notice that if a high-dimensional stateXHD is in SAD, then its low-dimensional projectionλ(XHD) is not in SAD, and also ifXHD 6∈SAD, then λ(XHD) ∈ SAD. Thus, for every state XHD in the original high-dimensional state- space, eitherXHD∈SADorλ(XHD)∈SAD (but not both). Adding new high-dimensional regions or increasing the sizes of existing regions requires the reconstruction of SAD and TAD, and thus, will produce a new instance ofGAD = (SAD, TAD).

Next we define the transition set TAD for the hybrid graph GAD as follows.

Definition 4.4 Transitions in GAD: For any state Xi ∈SAD:

• If Xi is high-dimensional (Xi ∈ SHD), then for all high-dimensional transitions

(Xi, X_jHD) ∈ THD, if X_jHD ∈ SAD then (Xi, X_jHD) ∈ TAD. If X_jHD 6∈ SAD,

then (Xi, λ(XjHD))∈TAD. That is, for high-dimensional states we allow only high-

dimensional transitions to other high-dimensional states if they fall inside SAD, or their low-dimensional projections (Fig. 5 lower left).

• If Xi is low-dimensional (Xi ∈ SLD), then for all low-dimensional transitions

(Xi, XjLD)∈TLD, ifXjLD ∈SAD then(Xi, XjLD)∈TAD and for all high-dimensional

transitions (X, X_jHD)∈THD, where X ∈λ−1(Xi), if XjHD ∈SAD then(Xi, XjHD)∈

TAD. That is, for low-dimensional states we allow low-dimensional transitions if they lead to another low-dimensional state inSAD(Fig. 5 upper left), and high-dimensional transitions from their high-dimensional projections if they lead to a high-dimensional state inSAD (Fig. 5 right).

Notice, that the above definition of TAD _{allows for transitions between states of different}

dimensionality. Figure 5 illustrates the set of transitions in the adaptive graph in the case of 3D (x, y, θ) path planning.

4.3.3. Mapping Hybrid Solutions to the High-Dimensional State-Space

Once we have computed a path through our hybrid graph GAD_{, which can contain low-}

dimensional states and transitions, we need to be able to project it to the high-dimensional state-space in order to ensure that it is feasible and satisfies the desired solution cost sub- optimality bound. Therefore, we define a tunnel τ of radius w around a hybrid path πAD

as follows:

Definition 4.5 A tunnelτ of widthwaround a hybrid pathπADis a sub-graphτ = (Sτ, Tτ)

τ ⊆GHD such that Sτ ⊆SHD Tτ ⊆THD ∀XHD ∈SHD, XHD ∈Sτ iff ∃Xi ∈πAD s.t. dist(λ(XHD), Xi)≤w if Xi ∈SLD or dist(λ(XHD), λ(Xi))≤w if Xi ∈SHD ∀EHD= (Xi, Xj)∈THD, EHD ∈Tτ iff Xi ∈τ and Xj ∈τ

where dist is some pre-defined distance metric in SLD.

In other words,τ is a sub-graph ofGHD, and thus consists only of high-dimensional states and transitions. Moreover, τ contains all high-dimensional states XHD if they fall within distance w of some state Xi ∈ πAD. We include in τ all transitions (Xj, Xk) from THD

such that both Xj and Xk are in τ. It is important to note that the above definition of τ

for tunnel width w = 0 becomes equivalent to the sub-graph produced by projecting the hybrid path πAD to the high-dimensional state-space SHD through the projection function

λ−1. This λ−1 projection method can be used when no distance metric is available in the low-dimensional state-space. The above definition, however, allows for more flexibility when mapping hybrid paths into the high-dimensional state-spaceSHD_{. To produce a high-}

(a)Initial 2D/3D path (b) Tunnel around path (c) Tracking in tunnel (d) Add HD region at point of failure

(e)2D/3D path (f) Tunnel around path (g)Tracking in tunnel (h) Add HD region at point of failure

(i)2D/3D path (j)Tunnel around path (k)Tracking in tunnel (l)Final trajectory

Figure 6: Example of the iterative process of Planning with Adaptive Dimensionality on simple map in the context of 3D (x,y,heading) path planning for a non-holonomic vehicle. Start: top left; goal: bottom right; light gray circles: 3D regions; darker gray outer circles: borders between 2D and 3D regions consisting of 2D states which have valid 3D transitions going into the 3D areas; white: 2D regions; black bars: obstacles.

dimensional path from a hybrid path πAD, we construct a tunnel τ around πAD; then we

perform a graph search from start to goal in τ, which is a small sub-graph of the original high-dimensional state-space. The search, if successful, produces a fully high-dimensional pathπHD corresponding to our hybrid pathπAD.