Topological complexity of the work map
We introduce the topological complexity of the work map associated to a robot system. In broad terms, this measures the complexity of any algorithm controlling, not just the motion of the configuration space of the given system, but the task for which the system has been designed. From a purely topological point of view, this is a homotopy invariant of a map which generalizes the classical topological complexity of a space.
The theory of topological complexity was initiated by Michael Farber [far, far2004] and it has become one of most active fields in the area of applied topology during last decade. In broad terms, this theory measures the complexity of any algorithm controlling the motion planing of a given configuration space. One can also regard this invariant as the minimum number of navigational instabilities of such a motion planning.
However, in many situations in robotics, more important than controlling the motion of a given configuration space, it is designing the resulting motion on the corresponding workspace. We briefly support this assertion by looking at two different key examples:
In robotics, see for instance [craig, Chap. 1], a robot manipulator or robot arm consists of multiple rigid segments (sub-arms, links) where successive, neighboring links are connected by joints of different kind (rotational, translational, polar, cylindrical). The base of a robot manipulator is the end of the first link which is fixed to a point through a given joint. The end effector is the device (screw driver, welding device,…) at the end of the last link of the robotic arm, designed to interact with the environment performing the proposed task. Finally, The workspace of a robot arm is defined as the set of points that can be reached by the end effector.
From the topological point of view, the configuration space of a robot arm, i.e., the set of all possible states of such a manipulator, was first modeled in [Gottlieb], see also [Pfalzgraf] or the modern reference [far3] for a complete treatment. Less attention has been given to the topological study of the workspace as this is not an (even diffeomorphic) invariant of the configuration space. For instance, consider two robot manipulators and , each of which consisting of two links of lengths with and both with one degree of freedom. In other words, both and are diffeomorphic to .
However, If we assume to be “planar”, then its workspace is an annulus of radius [far3, §1.2]. On the other hand, if in , the circle generated by the end effector link is “transversal” to the plane containing the circle generated by the based link, then is a torus. This is discussed in detail in Example 2.6.
Nevertheless, both the configuration space and the workspace of a given robot arm are connected by the continuous work map
which assigns to each state of the configuration space the position of the end effector at that state.
This map is a crucial object for implementing algorithms controlling the task performed by the robot manipulator. Indeed, the input of such an algorithm are pairs of points of the workspace, that is, pairs of possible positions of the end effector. The output for such a pair ought to be a curve in the configuration space such that and , where is the space of curves in (that is the space of continuous maps ).
One may argue that a motion planner of the configuration space produces such an algorithm by composing with the work map. However, the efficiency of such an algorithm might not be optimal as, for instance, the work map is not injective in general and therefore, many states of the configuration space may give the same position of the end effector.
The second example to which the above can be applied is the following: according to [Bajd], a multi-robot system consists of two or more robots executing a task requiring collaboration among them. Assume that such a multi-robot system is formed by autonomous mobile robots running in a space without colliding. The configuration space of such a system is the standard -th configuration space
with the subspace topology of the -fold Cartesian product of . On the other hand, the task requiring the collaboration of the robots can be described as a continuous work map
depending on the locations of the robots, and with values on the workspace which is often described as a subspace of some Euclidean space . Hence, an algorithm controlling the task performed by the multi-robot system can be described, as before, in terms of the work map.
These examples motivate the main purpose of this article which is to propose the notion of topological complexity of a (work) map as a generalization of the topological complexity of a given configuration space.
Given a continuous map , the topological complexity of , , is the least integer such that can be covered by open sets on each of which there is a continuous map satisfying .
Here, is the path fibration, .
We first introduce and study in Section 1, for a given pair of maps , the -sectional category of , , a generalization of the Svarc genus or sectional category. Then, the topological complexity of a map can be thought of as . As such, we show in Section 2 that is a an invariant of the homotopy type of and is if and only if is inessential. Moreover, . We also provide upper and lower bounds for the topological complexity of a map, see propositions LABEL:cotas and LABEL:cotasco:
This indicates in particular that the complexity of an algorithm controlling the task performed by a system is in general smaller than the one controlling just the motion of the system. We finish the section with several examples.
In Section 3 we give a characterization of the topological complexity of the rationalization of a given map between simply connected spaces in terms of its Sullivan models. This is highly computable in algebraic terms and becomes a lower bound of the topological complexity of as . As an application, we show that the topological complexity of a formal map always coincides with its cohomological lower bound.
Finally, we would like to stress that our purpose is not being exhaustive in the study of the the topological complexity of a map, but just laying the groundwork for its further development and presenting the general behaviour of this new invariant.
1. -Sectional Category
In what follows, and unless explicitly stated otherwise, a topological space will always be pointed, path-connected, and of the homotopy type of a CW-complex. Continuous maps are assumed to preserve base points.
We recall the definition of the most classical Lusternik-Schnirelmann invariants. The category of a space , , is the least such that can be covered by open sets contractible within . On the other hand, the category of a map , , is the least such that the domain can be covered by open sets on each of which the restriction of is homotopically trivial. Finally, given a map , the sectional category of [s] denoted by , is the least for which can be covered by open sets on each of which there is a local homotopy section of . Here we extend this invariant.
Let be two continuous maps. An open set is -categorical if there is a map such that . We call an -section. The -sectional category of , , is the least for which admits a covering of -categorical open sets.
Obviously, if and , then . Also, observe that .
Recall that is said to be dominated by if there is a (homotopy) commutative diagram,