Invariants for Homology Classes with Application to Optimal Search and Planning Problem in Robotics
We consider planning problems on a punctured Euclidean spaces, , where is a collection of obstacles. Such spaces are of frequent occurrence as configuration spaces of robots, where represent either physical obstacles that the robots need to avoid (e.g., walls, other robots, etc.) or illegal states (e.g., all legs off-the-ground). As state-planning is translated to path-planning on a configuration space, we collate equivalent plannings via topologically-equivalent paths. This prompts finding or exploring the different homology classes in such environments and finding representative optimal trajectories in each such class.
In this paper we start by considering the problem of finding a complete set of easily computable homology class invariants for -cycles in . We achieve this by finding explicit generators of the de Rham cohomology group of this punctured Euclidean space, and using their integrals to define cocycles. The action of those dual cocycles on -cycles gives the desired complete set of invariants. We illustrate the computation through examples.
We further show that, due to the integral approach, this complete set of invariants is well-suited for efficient search-based planning of optimal robot trajectories with topological constraints. Finally we extend this approach to computation of invariants in spaces derived from by collapsing subspace, thereby permitting application to a wider class of non-Euclidean ambient spaces.
1.1 Motivation: Robot Path Planning with Topological Constraints
In numerous robotics applications, it is important to distinguish between configuration space paths in different topological classes, as a means of categorizing continuous families of plans. This motivation — connected components of paths relative to endpoints — leads to classifying up to homotopy. Examples motivating a classification of homotopy classes of paths include: (1) group exploration of an environment [Bourgault02informationbased], in which an efficient strategy involves allocating one agent per homotopy class; (2) visibility, especially in the tracking of uncertain agents in an environment with dynamic obstacles [occlusion:yan:06]; and (3) multi-agent coordination, in which (Pareto-) optimal planning coincides with homotopy classification [GL:2006].
Although homotopy is a natural topological equivalence relation for paths, the computational bottlenecks involved, especially in higher dimensional configuration spaces, present severe challenges in solving practical problems in robot path planning. Thus we resort to its computationally-simpler cousin — homology (Figure 1). We assume a basic familiarity with first-year algebraic topology, as in [Hatcher:AlgTop] for homology and [bott1982differential] for differential forms and de Rham cohomology.
The methods we employ, following [planning:AURO:12], construct an explicit differential -form, the integration of which along trajectoriesgives complete homology class invariants. Such -forms are elements of the de Rham cohomology group of the configuration space, . To deal with the obstacles, we replace with topologically equivalent codimesion-2 skeleta (e.g., Figure 2) and then compute the degrees (or linking numbers) of closed loops with the skeleta.
1.2 Contributions of this Paper
We generalize the path-planning problem to higher homology classes and linking numbers of arbitrary submanifolds (not merely -dimensional curves representing trajectories). In particular, we will consider -dimensional closed manifolds as generalization of -dimensional curves that constituted the trajectories. Obstacles will be represented by codimension closed manifolds (which, in many cases will be deformation retracts of the original obstacles).
Degree and linking numbers are closely related to homology [Hatcher:AlgTop, dold1995lectures]. We will in fact show that the proposed integration along trajectories give homology class invariants for closed loops (something that was claimed in [planning:AURO:12], but not proved rigorously).
The primary aim of this paper is two-fold:
To find certain differential -forms in the Euclidean space punctured by obstacles, and show that integration of the forms along -dimensional closed manifolds give a complete set of invariants for homology classes of the manifolds in the punctured space (i.e. the value of the integral over two closed manifolds are equal if and only if the manifolds are homologous),
To adapt and extend the tools used in [planning:AURO:12] for robot path planning with topological reasoning to arbitrary dimensional Euclidean configuration spaces punctured by obstacles.
1.3 Overview and Organization of this Paper
The main concept behind the treatment in this paper is to exploit the pairing , which evaluates -cocycles over -cycles. Given a cycle , and a large enough set of cocycles, , one can hope that the set of values will provide some information about the homology class of , that is the value of . In fact choosing the coefficients in , and with some assumptions on , we will show that it is sufficient to choose the elements of such that their cohomology classes generate .
However, the challenge lies in explicitly finding the cochains, , that will serve our purpose and are easy to evaluate on cycles. Due to De Rham’s theorem, the cocycles, , can be represented by some -form, , so that the evaluation of the cocycle over a cycle is, precisely, integration of the form over the cycle. In order to find this form, we exploit the difference map . The codomain of this map is the -dimensional Euclidean space with the origin removed, and is much simpler and well-studied. Thus, if is a differential -form in , a simple pull-back via gives the form . Upon integration of over some -cycle, , one may hope to obtain the desired -form, . Considering the space as a fiber bundle over with as the fibers, one may be tempted to integrate over the fibers. However, the nature of (its topology, dimensionality) can be quite arbitrary in general.
Thus we begin by constructing a suitable skeleton with which to replace , so that the spaces and are identical as far as their homology groups are concerned. However, in that construction, we will ensure that is a collection (disjoint union) of codimension- manifolds, thus simplifying the problem.
Throughout this paper we consider homology and cohomology with coefficients in the field . As a consequence, all the homology and cohomology groups are freely and finitely generated. Also, for simplicity, we will throughout consider to avoid the special treatment of the (co)homology groups. All topological spaces referred to in this paper are assumed to be Hausdorff.
2 On Building Obstacle Equivalents
As preparation for the technical details involving linking numbers, we consider the replacement of our obstacles with their -dimensional representatives. This is trivial for contractible obstacles in the plane (point representatives) and in -dimensional space (cf. the skeletons of [planning:AURO:12]). The intuition is that replacing obstacles by their homotopy equivalents leaves the homology classes of trajectories in the complement unchanged (Figure 2); however, we have dimension constraints, and there exist simple obstacles that do not have a -dimensional deformation retract (e.g. for the case, a hollow torus does not have a dimensional homotopy equivalent). We therefore turn to -dimensional equivalents faithful to homology in the desired dimension (Figure LABEL:fig:obstacle-eqvs-genrators).
In the proposition and related corollaries that follow, we represent the ambient configuration space (without obstacles) by , an obstacle by , and the -dimensional equivalent of the obstacle with which we replace for computational simplicity.
Let be a compact, locally contractible subspace of . Let be a compact, locally contractible subspace of , such that the inclusion induces an isomorphism . Then the inclusion map induces an isomorphism .
Consider the following diagram.