On the topological behaviour of motion planners

Let X be a contractible, path connected and compact manifold. On the field of topological robotics, motion planning algorithms (MPA) are some kind of algorithms which need (as an input) a pair of point (A,B) ∈ X×X and produce (as an output) a path in X from A to B without collision. We focus here on the topological study of the setM(X) of that algorithms. We first topologizeM(X) with the open compact topology and show that it is also contractible. Secondly, we equip it with a metric, that induces the same topology than the open-comact one, and for whichM(X) is complete. This leads us to a contravariant functorM : X 7→ M(X) between the category of contractible, path connected and compact manifolds X and that of their associated sets of motion planning algorithmsM(X). This enable us to classify, up to isometry, all motion planning algorithms. Many kind of topological and category theoretic interpretations, but also open questions, will arise for each established result. The awesome one is that the motion planning algorithms of a robot may inherit the topological behaviour of the configuration space on which the robot moves.


Introduction
In Robotics, one means by motion planning algorithm or navigation problem, the process to reach (under some constraints, e.g Piano problem) a fixed target from a given starting point.It is interesting to notice that the first navigation systems emerged from the end of the 60's in the very first International Joint Conferences on Artificial Intelligence (Shakey, Jason, ...).
At the end of the 70's the studies of motion planning popularized the notion of configuration space of a mechanical system; in this space the robot becomes a point.The motion planning for a mechanical system is reduced to pathfinding for a point in the configuration space.
In the early of the 21th century, topology interfers with robotics thanks to the Farber's founder ideas introduced in [2] to measure the complexity of the stability of a robot motion.He defined a motion planning algorithm on a path-connected topological space X to be any continuous section s : , where PX = {γ : [0, 1] −→ X continuous} denoted the set of paths on X.He (see Theorem 1, [2]) showed that such algorithms exist if and only if X is contractible.
Hence, in this paper, all topological spaces X are assumed to be path-connected and contractible, and M(X) will denote the non empty set of associated MPAs.Our approach to study M(X), is homotopical, topological and category theoretical.For this purpose, we view M(X) as a subset of map(X × X, PX) and topologize it with the induced open compact topology.Our first result states that: Theorem 1.1.If X is a path-connected and contractible CW-complex, then M(X) is also contractible.

This leads us to ask the natural following questions
Open Questions 1.2.
(1) The topological behaviour of M(X) resembles to that of X?In particular: (2) Are any two continuous sections of the free path fibration on X homotopic through sections?This corresponds to M(X) being path-connected.(3) If Y is a second path-connected and contractible topological space which is homeomorphic to X, is it true that M(X) and M(Y) are also homeomorphic?
A positive and partial answer for the third question will be given by Theorem 1.5 in terms of compact manifolds.Secondly, we endow X with a metric d X (thanks to some Urysohn Theorems) and define on M(X) a metric M(d X ) which induces on M(X) the same topology that the open compact one.We show that: Hence the first question in the Open Questions 1.2 arise once again, and one would be tempted to ask: Open Questions 1.4.What about the compactness of (M(X), M(d X )), whenever X is a contractible pathconnected and compact manifold?May one define a manifold structure over M(X)?
The completeness of (M(X), M(d X )) means that for a fixed pair of points (departure, target), and from any wider class of closed motion planning algorithms, emerges one (their limit).This can be interpreted by a kind example from Nature, the Ant colony optimization algorithm based on the behaviour of ants seeking for an optimal path nest-food.
The compactness of (M(X), M(d X )) will mean more, that if one fix a pair of points (departure, target), then from any wider class of motion planning algorithms (not necessary closed), emerges one (their limit).Finally, we will prove this main and crucial result which enable us to classify motion planning algorithms up to isometry.The Theorem states that: In the proof of Theorem 1.5, to any isometry ϕ : ) and a well defined functor emerges.Positive answers to both Open Questions (1.2) and (1.4) yields to a functor Here Mfld C-C P-C denotes the set category whose objects are compact and contractible path-connected manifolds, and whose arrows are isometries.
By the way, we prove that: Theorem 1.6.The group (H(I), •) acts continuously on M(X), whenever X is compact and contractible path-connected manifold.
Where I = [0, 1] and where H(I) denotes the set of homeomorphisms α : I → I such that α(0) = 0 and α(1) = 1.One may ask: Open Questions 1.7.What is the nature of this action?What are their orbits, stabilizers?What kind of interpretations on the robot motion one may conclude from this action?...The remaining of the paper is devoted to proofs and is organized as follows: In section 2 we prove Theorem 1.1.In section 3, proofs of both Theorem 1.3, Theorem 1.5 and Theorem 1.6 will be given.The functor M defined in (1) will be discussed in the section 4.

Motion planning algorithms: homotopical view point
Proof (Proof of Theorem 1.1).The evaluation map is a fibration, and M(X) is the fiber of the following fibration between mapping spaces map(X × X, PX) −→ map(X × X, X × X) at the identity in X × X.Since X × X is contractible, then so is the target mapping space, therefore the fiber is weakly equivalent to the source, which is also contractible since PX X * .
In fact Theorem 1.1 is a direct conclusion of a more general one, the following: Lemma 2.1.The space of sections for a fibration E −→ B where E and B are both contractible is also contractible.

Consider the lifting problem
Let Γ f be the space of solutions to this lifting problem; i.e. the space of maps X −→ E making the diagram commute.When f is the identity and A is empty, we get the space of sections of p which we write Γ(B).Notation: When A → X, then Map(X, Y, ) will denote the space of maps which coincide with on A. So if A is basepoint, Map(X, Y, ) is the space of based maps sending x 0 to (x 0 ).Lemma 2.2.The projection Map(X, E, ) −→ Map(X, B, p • ) which sends f to p • f , is a fibration where the fiber over f is Γ f .Proof.The fiber over f is obviously Γ f by definition, and the important claim here is that this is a fibration.This would imply that if f 1 and f 2 are in the same component of Map(X, B, p • ), then Γ f 1 Γ f 2 (the homotopy fiber).can be chosen to be the section space of E −→ B. If B is contractible, then both mapping spaces above are contractible.A fibration between contractible spaces must have contractible fiber (all spaces are of the homotopy type of CW complexes).This achieves the proof of Lemma 2.1 and the answer for our main claim is yes, the section space is contractible.

Motion planning algorithms: topological view point
The main information given by Theorem 1.1 is that, homotopically, M(X) is trivial.We focus now on the case when X is contractible path-connected and compact manifold, and as it will be seen through this section, the topology of M(X) is so rich and enable us to make many interpretations.
We know from Urysohn metrization theorem (see [1]) that X is metrizable.Let d X be such a metric on X, and define on M(X) the metric M(d X ) by for any pair of sections (s, s ) ∈ M(X).
Recall now this classical fact (see [3]): if X is any metric space and Y is a compact Hausdorff space, then the open compact topology on the space X Y of continuous maps f : Y −→ X agrees with the sup-norm metric topology.When applying this to Y = [0, 1] and to X × X, we conclude that the open compact topology of M(X) is the same than that one induced by the metric M(d X ).This is a very interesting fact, which gives us a technical tool to give a sense of the closeness of two motion planning algorithms in term of distances.
Proof (Proof of Theorem 1.3).Consider (s n ) n a Cauchy sequence in (M(X), M(d X )), ε > 0 and an integer N such that for any m > n ≥ N and any (A, B, t) ∈ X × X × [0, 1] we have d(s m (A, B)(t), s n (A, B)(t)) < ε. ( For any fixed pair (A, B) ∈ X×X, (s m (A, B)) m is a Cauchy sequence in PX which is well known to be complete.
Put s(A, B) = lim s m (A, B), fix n in (2) and reach out m to the infinity, then M(d X )(s n , s) < ε for any n ≥ N. The continuity of s and the fact that s is a section of ev are easy to check.Thus s = lim s m in (M(X), δ).
If X = B and A is empty, we get the fibration Map(B, E) −→ Map(B, B).The fiber has constant homotopy type over each component of Map(B, B).Let Map f (B, B) be the component containing the map f and let s be a section of p. Then the homotopy fiber of Map s (B, E) −→ Map id (B, B)