Path Homotopy Invariants and their Application to Optimal Trajectory Planning
We consider the problem of optimal path planning in different homotopy classes in a given environment.
Though important in robotics applications, path-planning with reasoning about homotopy classes of trajectories has typically focused on subsets of the Euclidean plane in the robotics literature.
The problem of finding optimal trajectories in different homotopy classes in more general configuration spaces (or even characterizing the homotopy classes of such trajectories) can be difficult.
In this paper we propose automated solutions to this problem in several general classes of configuration spaces by constructing presentations of fundamental groups and giving algorithms for solving the word problem in such groups.
We present explicit results that apply to knot and link complements in 3-space, discuss how to extend to cylindrically-deleted coordination spaces of arbitrary dimension, and also present results in the coordination space of robots navigating on an Euclidean plane.
In the context of robot motion planning, one often encounters problems requiring optimal trajectories (paths) in different homotopy classes. For example, consideration of homotopy classes is vital in planning trajectories for robot teams separating/caging and transporting objects using a flexible cable (Bhattacharya et al., 2015), or in planning optimal trajectories for robots that are tethered to a base using a fixed-length flexible cable (Kim et al., 2014), or in human-robot collaborative exploration problems in context of search-and-rescue missions (Govindarajan et al., 2014). This paper addresses the problem of optimal path planning with homotopy class as the optimization constraint.
There is, certainly, a large literature on minimal path-planning in computational geometry (for a brief sampling and overview, see (Mitchell and Sharir, 2004)). The topology of a configuration space plays a key role in solving path-planning problems. To that end, the study of the topological invariants of robot configuration spaces and the properties of motion planning problems in such configuration spaces is not new (Farber, 2001; Costa and Farber, 2010; Farber et al., 2006; Cohen and Pruidze, 2007). However, most of this body of research has primarily focused on deduction of topological invariants of configuration/coordination spaces, addressed existence questions, have studied properties of the motion planning problem itself, but do not explicitly implement algorithms for finding optimal paths in configuration spaces of robots. Classification of paths based on homotopy has also been studied in the robotics literature in two-dimensional planar workspaces using geometric methods (Grigoriev and Slissenko, 1998; Hershberger and Snoeyink, 1991), probabilistic road-map construction (Schmitzberger et al., 2002) techniques and triangulation-based path planning (Demyen and Buro, 2006). While in a planar configuration space such methods can be used for determining whether or not two trajectories belong to the same homotopy class, efficient planning for least cost trajectories with homotopy class constraints is difficult using such representations even in -dimensions. To that end in prior work we developed a graph search-based method for computing shortest paths in different homology classes in , and higher dimensional Euclidean spaces with obstacles (Bhattacharya et al., 2012, 2013), and in different homotopy classes in planar configuration spaces with obstacles (Bhattacharya et al., 2015).
Of course, since the problem of computing shortest paths (even for a 3-d simply-connected polygonal domain) is NP-hard (Canny and Reif, 1987), we must restrict attention to subclasses of spaces, even when using homotopy path constraints. In particular, we construct a discrete graph representation of a configuration space and thus reduce the problem to shortest path computation on the graph. This representation, along with path homotopy invariants of the configuration space, allows us to compute shortest paths in different homotopy classes using graph search-based path planning algorithms.
This paper focuses on computation of optimal paths in different homotopy classes in two interesting and completely different types of configuration spaces: (1) knot and link complements in 3-d; and (2) cylindrically-deleted coordination spaces (Ghrist and LaValle, 2006). Compared to our prior conference publication (Bhattacharya and Ghrist, 2015), in this paper we have (i) provided a rigorous Proposition on the presentation of the fundamental group of knot/link complements in -d and hence the justification behind the proposed homotopy invariants in such spaces, (ii) have provided an explicit presentation of the fundamental group of cylindrically deleted coordination space for robots navigating on a plane, and (iii) hence have demonstrated through simulation the problem of optimal path computation in different homotopy classes for three robots navigating on a simple planar domain.
2 Configuration Spaces with Free Fundamental Groups
In this section we introduce a few fundamental definitions and concepts. Each definition is accompanied by a reference to a standard text on the subject, where the reader can find more details on these topics.
Homotopy Classes of Paths: Consider oriented/directed curves in a topological space, . Two curves connecting the same start and end points, , are called homotopic (or belonging to the same homotopy class) if one can be continuously deformed into the other without intersecting any obstacle (i.e.. there exists a continuous map such that , and (Hatcher, 2001)). A set of all homotopically equivalent paths (i.e., homotopic paths) constitute a homotopy class. We denote the homotopy class of a path a .
Fundamental Group: The fundamental group or the first homotopy group of a topological space, , is the set of all homotopy classes of oriented closed loops (paths with ) in with a group structure imposed on the set as follows: i. The identity element is the class of all loops that can be contracted/homotoped to the point ; ii. The inverse of a homotopy class, , is the homotopy class of loops constituting of the same loops as in , but with reversed orientation, and is denoted as or ; iii. The composition (group operation) of two classes and is the class of loops that are obtained by concatenating a curve in with a curve in (i.e., it is the class of the loop ).
Free Group and Free Product of Groups: A free group over a set of letters/symbols is the group whose elements consists of all words constructed out of the letters and their formal inverses, with identity element being the empty word, and the group operation being word concatenation (with any letter juxtaposed with its inverse reducing to the identity) (Scott and Scott, 1964). Given two groups, and , the free product of the groups is the group of words that can be constructed with all the elements of the groups as the letters. The free product is thus written as .
2.2 Motivation: Homotopy Invariant in
We are interested in constructing computable homotopy invariants for trajectories in a configuration space that are amenable to graph search-based path planning. To that end there is a very simple construction for configuration spaces of the form (Euclidean plane punctured by obstacles) (Grigoriev and Slissenko, 1998; Hershberger and Snoeyink, 1991; Tovar et al., 2008; Bhattacharya et al., 2015; Kim et al., 2014): We start by placing representative points, , inside the connected component of the obstacles, . We then construct non-intersecting rays, , emanating from the representative points (this is always possible, for example, by choosing the rays to be parallel to each other). Now, given a curve in , we construct a word by tracing the curve, and every time we cross a ray from its right to left, we insert the letter “” into the word, and every time we cross it from left to right, we insert a letter “” into the word, with consecutive and canceling each other. The word thus constructed is written as . For example, in Figure 1(a), . This word, called the reduced word for the trajectory , is a complete homotopy invariant for trajectories connecting the same set of points. That is, , with are homotopic if and only if .
2.3 Words as Homotopy Invariants in Spaces with Free Fundamental Groups
In a more general setting the aforesaid construction can be generalized as follows:
Given a -dimensional manifold (possibly with boundary), , suppose are -dimensional orientable sub-manifolds (not necessarily smooth and possibly with boundaries) such that . Then, for any curve, (connecting fixed start and end points, ), which is in general position (transverse) w.r.t. the ’s, one can construct a word by tracing the curve and inserting into the word a letter, or , whenever the curve intersects with a positive or negative orientation respectively.
The proposition below is a direct consequence of a simple version of the Van Kampen’s Theorem (of which several different generalizations are available in the literature).
Words constructed as described in Construction 1 are complete homotopy invariants for curves in joining the given start and end points if the following conditions hold:
is path connected and simply-connected, and,
Proof. Consider the spaces and . Due to the aforesaid properties of the ’s the set constitutes an open cover of , is closed under intersection, the pairwise intersections are simply-connected (and hence path connected), and so are .
The proof, when is a closed loop (i.e. ), then follows directly from the Seifert-van Kampen theorem (Hatcher, 2001; Crowell, 1959) by observing that , the free product of copies of , each generated due to the restriction of the curve to .
When and are curves (not necessarily closed) joining points and , they are in the same homotopy class iff is null-homotopic — that is, (where by “” we mean word concatenation).
The Construction 1 gives a presentation (Epstein, 1992) of the fundamental group of (which, in this case, is a free group due to the Van Kampen’s theorem) as the group generated by the set of letters , and is written as . In our earlier construction with the rays, was the configuration space, and were the support of the rays in the configuration space. It is easy to check that the conditions in the above proposition are satisfied with these choices. However such choices of rays is not the only possible construction of the ’s satisfying the conditions of Proposition 1. Figure 1(b) shows a different choice of the ’s that satisfy all the conditions.
2.4 Simple Extension to with Unlinked Unknotted Obstacles
The construction described in Section 2.2 can be easily extended to the -dimensional Euclidean space punctured by a finite number of un-knotted and un-linked toroidal (possibly of multi-genus) obstacles. Instead of “rays”, in this case the ’s are -dimensional sub-manifolds that satisfy the conditions in Proposition 1, with a letter, (or ), being inserted in every time the curve, , crosses/intersects a . This is illustrated in Figures 2(a) and 2(b).
However, a little investigation makes it obvious that such -dimensional sub-manifolds cannot always be constructed when the obstacle are knotted or linked (Figure 2(c)). One can indeed construct surfaces (e.g. Seifert surfaces) satisfying some of the properties, but not all.
2.5 Application to Graph Search-based Path Planning
Using the homotopy invariants described in the previous sub-section, we describe a graph construction for use in search-based path planning for computing optimal (in the graph) trajectories in different homotopy classes.
Suppose is a discrete graph representation of a configuration space, . That is, , is a set of points (vertices) sampled from , and for any two neighboring points (where neighbors are typically determined by a distance threshold), there exists an edge .
We first fix the set of sub-manifolds as described earlier. Now, given a discrete graph representation of the configuration space, , we construct an -augmented graph, , which is essentially a lift of into the universal covering space of (Hatcher, 2001). The construction of such augmented graphs has been described in our prior work (Bhattacharya et al., 2015; Kim et al., 2014; Govindarajan et al., 2014), and the explicit construction of can be described as follows:
Vertices in are tuples of the form , where and is a word made out of letters and .
For every edge and every vertex , there exists an edge , where denotes the directed curve that constitutes the edge .
The length/cost of an edge in is same as its projection in : .
The item ’1’ is just a qualitative description of the vertices in . Item ’2’ describes one particular vertex in , and using that, item ’3’ describes an incremental construction of the entire graph . The topology of can be described as a lift of into the universal covering space, , of , and is illustrated in Figure 3 for a uniform cylindrically discretized space with a single disk-shaped obstacle.
Such an incremental construction is well-suited for use in graph search algorithms such as Dijkstra’s or A* (Cormen et al., 2001), in which one initiates an open set using the start vertex (in item 2), and then gradually expands vertices, generating only the neighbors at every expansion (the recipe for which is given by item ’3’). Executing a search (Dijkstra’s/A*) in from to vertices of the form (where ’’ denotes any word), and projecting it back to , gives us optimal trajectories in that belong to different homotopy classes. Figure 4(a) shows such optimal trajectories in the graph, connecting a given start and goal vertex, where was constructed by an uniform hexagonal discretization of the planar configuration space. One can then employ a simple curve shortening algorithm (Kim et al., 2014) to obtain ones more optimal than the ones restricted to (Figure 4(b)). Similarly, shortest trajectories connecting and can be obtained in -dimensional configuration spaces with free fundamental group (e.g., Figure 4(d), showing paths connecting two fixed points in with two un-linked toroidal obstacles).
3 Knot and Link Complements
As described earlier, when the obstacle set in consists of knots and links, it is in general not possible to find the sub-manifolds as required by Proposition 1. However, thankfully we have more generalized versions of the Van Kampen theorem at our disposal that lets us extend the proposed methodology to such spaces. We first illustrate the generalization in using knot/link diagrams.
3.1 Dehn Presentation of Fundamental Group of Knot/Link Complements
For simplicity we consider knots and links in as obstacles. We assume that the knots/links are described by polygons, all of which together constitute . The thickened obstacles (the knots/links with the tubular neighborhoods) will be referred to as . We consider a knot/link diagram (Lickorish, 1997) of the obstacles: Given a projection map, , the knot/link diagram is the projection of the knot/link, , along with additional information about the -ordering at the self-intersections of . We assume that in this diagram the self-intersections are all transverse (which can always be achieved through infinitesimal perturbations) and that the diagram divides the plane into simply-connected regions (say counts of them) each bounded by segments of the projected obstacles, and one unbounded exterior region. The boundary (the boundary of the closure) of each of the bounded regions is itself a polygon, (Figure 5(a)). Clearly (the preimage of in the original obstacle) will be a discontinuous polygon, with discontinuities at the preimages of the self-intersection points on the knot diagram. But these discontinuities can be removed simply by “connecting” the preimages at each self-intersection point, resulting into a spatial polygon, with the property that . A simple triangulation can then be employed to construct a surface, , in , such that its boundary is and is the simply-connected region bounded by (this can be achieved by first triangulating the planar region, , and then lifting the triangulation to ) – see Figure 5(a).
The ’s thus constructed satisfy properties (2) and (3) of Proposition 1, but not property (1), nor do they satisfy the property . The consequence of this is that near the regions where the ’s intersect, there can be closed loops in which are null-homotopic, but words constructed simply by tracing the loop and inserting letters corresponding to intersections with the ’s, as we did earlier, may not be the empty word (identity element). An example is illustrated in Figure 5(b)). Due to our construction, such intersection of the ’s happen only along lines passing through the pre-image of the self-intersections in the knot diagram, for each of which we end up getting a null-homotopic closed loop with non-empty word.
The Dehn presentation (Weinbaum, 1971) uses surfaces as constructed to describe the fundamental group of knot/link complements. We consider the free group, . In general, for every self-intersection in the knot/link diagram, there are four adjacent surfaces, and in the order as shown in Figure 6 (when the self-intersection is adjacent to the unbounded region in the knot diagram, there are only three). Correspondingly, there is a closed null-homotopic loop, , that has a word . Thus we have such words (assuming there are counts of self-intersections) that represent null-homotopic loops. These words are called relations and we call the set the relation set. It can be easily noted that inverses and cyclic permutations of each also corresponds to null-homotopic loops. We thus define the symmetricized relation set, , as the set containing all the words in , all their inverses, and all cyclic permutation of each of those.
Let the normal subgroup of generated by be (normal closure of in ). It is easy to observe that a closed loop, , in , has a word that is an element of iff it is null-homotopic. Due to a more general version of the Van Kampen’s theorem (Hatcher, 2001), the fundamental group of is the quotient group, — the group in which, under the quotient map, elements of are mapped to the identity element. This is summarized and generalized in the following proposition.
Given a -dimensional manifold (possibly with boundary), , suppose are -dimensional orientable sub-manifolds (not necessarily smooth and possibly with boundaries) such that and
a) The intersections of and , if non-empty, are transverse (hence -dimensional),
is path connected and simply-connected, and,
For a loop in , as before, one can construct a word as described in Construction 1. Let the set of all the letters constituting such words be .
For every -dimensional intersection of the , one can construct a loop in the tubular neighborhood of in that links with . Let the set of words corresponding to such loops be .
Then the fundamental group of is isomorphic to .
3.2 The Word Problem and Dehn Algorithm
Due to the discussion above, two trajectories, , connecting and in the knot/link complement, , belong to the same homotopy class iff the word belongs to . This problem in group theory is known as the word problem (Epstein, 1992), and there are various algorithms, each suitable for specific types of groups, for solving the word problem. We, in particular, will focus on a very simple algorithm due to Max Dehn (Lyndon and Schupp, 2001; Greendlinger and Greendlinger, 1986), which is applicable to a wide class of groups and their presentations.
Dehn’s metric algorithm: Given a presentation of a group, , we construct the symmetricized relation set as described earlier. Given a cyclically reduced word, , made up of letters (and their inverses) from , one checks for every element if and share a common sub-words that is of length greater than ( being the length of ). If they do (say, , with being a sub-word appearing in , and ), we replace the sub-word with the shorter equivalent that one obtains by setting to the identity element (i.e., replace by in ). This process is repeated, and the algorithm terminates when no more such sub-words are found. The final word at which the algorithm terminates indicates if the initial word, , is in (whether it maps to the identity element in ).
This algorithm can be used in conjunction with search in as before for finding optimal trajectories in different homotopy classes, with two vertices being the same iff reduces to the empty word upon applying the Dehn’s metric algorithm.
3.3 Guarantees of Dehn Algorithm
It’s well known (Greendlinger and Greendlinger, 1986) that if Dehn algorithm terminates at the empty word, then . However, the converse is not necessarily true. One can derive several sufficient (and often highly restrictive) conditions on the presentation under which the converse holds (Lyndon and Schupp, 2001). If, for a given presentation of a group the converse holds, we say that Dehn algorithm is complete for that presentation (or that the presentation is complete with respect to the Dehn algorithm, or that the word problem is solvable using the specific presentation and Dehn algorithm).
Due to the result of (Weinbaum, 1971), the Dehn presentation of the fundamental groups of the complement of a tame, alternating, prime knot is complete with respect to the Dehn algorithm. It is also known (Epstein, 1992) that automatic groups (including hyperbolic groups) have presentations that are complete with respect to Dehn algorithm.
4 Cylindrically-deleted Configuration Spaces
The previous results are limited to 3-dimensional spaces: one suspects that higher dimensions are more difficult. However, there are some classes of spaces for which optimal path-planning with homotopy constraints is still computable via a Dehn algorithm, independent of dimension. The following class of examples is inspired by robot coordination problems, in which individual agents with predetermined motion paths have to coordinate their motions so as to avoid collision.
Consider a collection of graphs , each embedded in a common workspace (usually or ) with intersections permitted. In the simplest case, each will be homeomorphic to a closed interval, but more general graphs are permitted, such as roadmap approximations to a configuration space. On each , a robot with some particular fixed size/shape is free to move. Such motion may be Euclidean (by translation/rotation); more general motions are possible, so long as the region occupied by the robot in the workspace is purely a function of location on . A point in the product space determines the locations of the robots in the common workspace. Certain configurations are illegal, due to collisions. For example, if the robots are point-like, and each is identical, then the configuration space of points on is the cross product minus the pairwise diagonal . If the robots are given finite extent, then this system has a configuration space obtained by the graph product minus an -neighborhood of the pairwise diagonal. However, more general types of collisions can be defined, say, if the robots are irregularly shaped and the graphs are all different. In this most general case, the natural analogue of a configuration space is the coordination spaces of (Ghrist and LaValle, 2006).
The coordination space of this system is defined to be the space of all configurations in for which there are no collisions – the geometric robots have no overlaps in the workspace. Under the assumption that collisions between robots are pairwise-defined, the coordination space is cylindrically deleted and of the form
for some (open, “collision”) sets where . In what follows, we assume that the are sufficiently tame (e.g., semialgebraic) so as to avoid issues of non-finitely-generated . Given (internal, path-) metrics on each , the coordination space inherits a locally-Euclidean metric on products of edges in the graphs. Such are complete path-spaces and thus the problem of geodesics is well-posed. Their fundamental groups can be (highly) nontrivial, depending on the obstacle set . However, finding optimal paths subject to homotopy classes is still computable. To that end one can construct the subspaces of co-dimension , and the relation set , and use them to design complete homotopy invariants as before. We do not discuss the explicit construction of the ’s for cylindrically-deleted coordination spaces in this paper, but provide the following theorem on solvability of the word problem in such spaces.
Any compact cylindrically-deleted coordination space admits a Dehn algorithm for .
Proof. Any such is realized as a Hausdorff limit of cubcial complexes which were shown in (Ghrist and LaValle, 2006, Thm 4.4) to be nonpositively-curved and to stabilize in by tameness. All nonpositively-curved piecewise-Euclidean cube complexes have fundamental groups which are, by a famous result of Niblo-Reeves (Niblo and Reeves, 1998), biautomatic. Biautomatic groups all admit a Dehn algorithm (specifically, there is a quadratic isoperimetric inequality) (Epstein, 1992).
It is worth noting that -shortest paths are perhaps not the most natural optimization for coordination spaces. It would be interesting to consider other (, ) pointwise norms.
In the next section we consider point robots navigating on a plane. The configuration space of each robot is the Euclidean plane. For such a configuration space for the individual robots, the result of Theorem 1 may not hold, since the fundamental group of the coordination space may not be biautomatic. Nevertheless, we can construct a presentation of the fundamental group, which we do, and still apply Dehn metric algorithm to it, although the Dehn algorithm may not be complete for the proposed presentation.
4.1 Presentation of the Fundamental Group of a Cylindrically Deleted Coordination Space for Point Robots Navigating on a Plane
We consider point robots navigating on a plane. Thus, in this case , the configuration space of the robot, is the Euclidean plane coordinatized by . A collision set , is a -dimensional hyperplane embedded in the joint configuration space of the robots and .
The joint configuration space of robots (with the collision sets included) is . The “cylindrical” obstacles in this configuration space created due to are thus
which are co-dimension subspaces (hyperplanes) embedded in the dimensional joint configuration space. Thus the coordination space is .
Design of co-dimension manifolds, :
As before, we are interested in constructing a set of -dimensional sub-manifolds, , in such that removing all but one of these sub-manifolds will give us a space with fundamental group isomorphic to . This would let us apply the generalized Van Kampen theorem as before by allowing us to construct words based on transverse intersection of paths with the surfaces, (which would be of co-dimension in ). We outline a general construction, and then show how that can be specialized for .
Consider a single -dimensional obstacle , which is a hyperplane of co-dimension in . Thus the homotopy group of is isomorphic to and is generated by a -dimensional loop that links with in this space. Our first construction corresponding to the obstacle is thus the following half-space:
This a -dimensional (co-dimension in ) half-hyperplane with at its boundary. However, for another pair of indices, , the obstacle in general intersects in a -dimensional half-hyperplane, and the space intersects in a -dimensional half-hyperplane. In particular,
is of co-dimension in , and,
is of co-dimension in (a half-hyperplane) and is non-empty. This is schematically shown in Figure 7(a).
is path connected.
Sketch of proof. Suppose . If , but , then whichever path, , is chosen to connect and , there will be a for which . However, if at this point, if , then the path will be intersecting or . Clearly, this can be prevented by simply altering the and in a small neighborhood of , without altering any other coordinate and hence the alteration itself not leading to intersection with any other . Thus an arbitrary pair of points in an be connected using a path that does not intersect any of the ’s.
Any loop in linked only with can be homotoped into any other loop linked only with , through a sequence of loops linked to ,
without intersecting .
not without intersecting if either or , otherwise without intersecting .
Sketch of proof. The proof is based on being able to construct, or an obstruction to constructing, a homotopy satisfying certain properties between small loops (in a tubular neighborhood of ) lying in a plane transverse to and linking to it.
Case I: Distinct and :
We first consider the case when and are all distinct. Consider a -dimensional affine plane transverse to described by (where and refers to the set of parameters describing the plane), and coordinatized by and (so that intersects the plane at its origin).
The intersection of this plane with is, in general, empty except for carefully chosen values for the parameters (in particular, for parameters such that ), when they intersect over the entire . Likewise, the intersection of this plane with is, in general, empty except when the parameters are such that ), when, again, they intersect at the entire .
Given two such affine planes, and , for two different sets of parameters, one can easily choose a path, , through the parameter space avoiding simultaneously at any . This gives a homotopy for a loop in around the origin (linking with) to a loop in around the origin, without intersecting .
Similarly, it is possible to choose the path such that and does not happen simultaneously.
Case II: and not all distinct:
Next consider the case when are not distinct. Choose to be the non-distinct indices such that either , or . Then the possible obstacles are , and , with the indices of any pair of obstacle not all distinct.
Consider the plane , on which and . This intersects obstacle at points where and . This gives . Thus, on the plane, the coordinates of the intersection point are . Once again, it is possible to choose the path from to such that and are not simultaneously zero for any . This argument also holds for the other two pairs of obstacles.
Again, using the chosen coordinates on , intersects at the ray , and intersects at the ray . Because the rays point in opposite directions along the axis, given the parameters and , it is always possible to find the path (in particular, ) such that the rays of intersection with and on does not pass through the origin (if and are of opposite signs, and if for some , this can be achieved by choosing ). Similar argument holds for intersection of with and .
However, intersects at the ray , and at the ray . These rays point in the same direction. Clearly, given the parameters and , if and are of opposite sign, it is not possible to find the path such that neither of these rays of intersection do not pass through the origin in .
The consequence of the above Lemma is that the obstruction to fundamental group of being are the intersections of with and , for every such that . This leads us to split by the the hyperplane at which it is intersected by or , for every . It can however be noted that for , and are subsets of the same -dimensional hyperplane. Thus we have the following splitting for only due to its intersection with and :