Related papers: Topological complexity of Milnor fibration
We study certain topological problems that are inspired by applications to autonomous robot manipulation. Consider a continuous map $f\colon X\to Y$, where $f$ can be a kinematic map from the configuration space $X$ to the working space $Y$…
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…
We study a generalized motion planning problem involving multiple autonomous robots navigating in a $d$-dimensional Euclidean space in the presence of a set of obstacles whose positions are unknown a priori. Each robot is required to visit…
It has been observed that the very important motion planning problem of robotics mathematically speaking boils down to the problem of finding a section to the path-space fibration, raising the notion of topological complexity, as introduced…
In this paper we present new results about the topology of the Milnor fibrations of analytic function-germs with a special attention to the topology of the fibers. In particular, we provide a short review on the existence of the Milnor…
The paper surveys topological problems relevant to the motion planning problem of robotics and includes some new results and constructions. First we analyse the notion of topological complexity of configuration spaces which is responsible…
We study the topological complexity of work maps with respect to some subspaces of the configuration space and a workspace considered as the target set of the motion of robots. The motivation is to optimize and reduce the number of motion…
There are several topological spaces associated to a complex hyperplane arrangement: the complement and its boundary manifold, as well as the Milnor fiber and its own boundary. All these spaces are related in various ways, primarily by a…
In this paper we combine a survey of the most important topological properties of kinematic maps that appear in robotics, with the exposition of some basic results regarding the topological complexity of a map. In particular, we discuss…
Milnor's fibration theorem is about the geometry and topology of real and complex analytic maps near their critical points, a ubiquitous theme in mathematics. As such, after 50 years, this has become a whole area of research on its own,…
In this paper we study a notion of topological complexity for the motion planning problem. The topological complexity is a number which measures discontinuity of the process of motion planning in the configuration space X. More precisely,…
Robotic manipulation in complex, constrained spaces is vital for widespread applications but challenging, particularly when navigating narrow passages with elongated objects. Existing planning methods often fail in these low-clearance…
In this paper, we deal with the robot motion planning problem in multi-valued function theory. We first enrich the multi-homotopy studies by introducing a multi-homotopy lifting property and a multi-fibration. Then we compute both a…
Despite the attention that the problem of path planning for tethered robots has garnered in the past few decades, the approaches proposed to solve it typically rely on a discrete representation of the configuration space and do not exploit…
Topological complexity $\TC{B}$ of a space $B$ is introduced by M. Farber to measure how much complex the space is, which is first considered on a configuration space of a motion planning of a robot arm. We also consider a stronger version…
We design a motion planning algorithm to coordinate the movements of two robots along a figure eight track, in such a way that no collisions occur. We use a topological approach to robot motion planning that relates instabilities in motion…
We study an elementary problem of the topological robotics: collective motion of a set of $n$ distinct particles which one has to move from an initial configuration to a final configuration, with the requirement that no collisions occur in…
In this paper we introduce and study a new concept of parametrised topological complexity, a topological invariant motivated by the motion planning problem of robotics. In the parametrised setting, a motion planning algorithm has high…
Consider a fibration \[p:E\to S^{p-1}\] with fiber $F$. We have the following natural question: Under what conditions does this fibration admit a cross-section? Our purpose is to discuss this problem when the fibration $p$ is the Milnor…
Planning for multi-robot teams in complex environments is a challenging problem, especially when these teams must coordinate to accomplish a common objective. In general, optimal solutions to these planning problems are computationally…