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We study motion planning algorithms for collision free control of multiple objects in the presence of moving obstacles. We compute the topological complexity of algorithms solving this problem. We apply topological tools and use information…

Optimization and Control · Mathematics 2007-05-23 Michael Farber , Mark Grant , Sergey Yuzvinsky

This is a continuation of our recent paper in which we developed the theory of sequential parametrized motion planning. A sequential parametrized motion planning algorithm produced a motion of the system which is required to visit a…

Robotics · Computer Science 2022-12-05 Michael Farber , Amit Kumar Paul

We give new lower bounds for the (higher) topological complexity of a space, in terms of the Lusternik-Schnirelmann category of a certain auxiliary space. We also give new lower bounds for the rational topological complexity of a space, and…

Algebraic Topology · Mathematics 2016-01-20 Mark Grant , Gregory Lupton , John Oprea

The topological complexity TC(X) is a homotopy invariant which reflects the complexity of the problem of constructing a motion planning algorithm in the space X, viewed as configuration space of a mechanical system. In this paper we…

Algebraic Topology · Mathematics 2008-06-26 Michael Farber , Mark Grant

This paper presents motion planning algorithms for underactuated systems evolving on rigid rotation and displacement groups. Motion planning is transcribed into (low-dimensional) combinatorial selection and inverse-kinematics problems. We…

Optimization and Control · Mathematics 2007-05-23 Sonia Martinez , Jorge Cortes , Francesco Bullo

Motion planning algorithms often leverage topological information about the environment to improve planner performance. However, these methods often focus only on the environment's connectivity while ignoring other properties such as…

Robotics · Computer Science 2020-03-05 Diane Uwacu , Regina Rex , Bonnie Wang , Shawna Thomas , Nancy M. Amato

We survey results on the topological complexity of classical configuration spaces of distinct ordered points in orientable surfaces and related spaces, including certain orbit configuration spaces and Eilenberg-Mac Lane spaces associated to…

Algebraic Topology · Mathematics 2019-08-27 Daniel C. Cohen

The presence of task constraints imposes a significant challenge to motion planning. Despite all recent advancements, existing algorithms are still computationally expensive for most planning problems. In this paper, we present Constrained…

Robotics · Computer Science 2020-08-11 Ahmed H. Qureshi , Jiangeng Dong , Austin Choe , Michael C. Yip

We determine explicit formulas for geodesics (in the Euclidean metric) in the configuration space of ordered pairs (x,x') of points in R^n which satisfy d(x,x')>=epsilon. We interpret this as two or three (depending on the parity of n)…

Differential Geometry · Mathematics 2020-07-07 Donald M Davis

This paper proposes a novel online motion planning approach to robot navigation based on nonlinear model predictive control. Common approaches rely on pure Euclidean optimization parameters. In robot navigation, however, state spaces often…

Robotics · Computer Science 2022-01-06 Christoph Rösmann , Artemi Makarow , Torsten Bertram

Motion planning techniques for quadrotors have advanced significantly over the past decade. Most successful planners have two stages: a front-end that determines a path that incorporates geometric (or kinematic or input) constraints and…

Robotics · Computer Science 2024-03-11 Yifei Simon Shao , Yuwei Wu , Laura Jarin-Lipschitz , Pratik Chaudhari , Vijay Kumar

In terms of Rudyak's generalization of Farber's topological complexity of the path motion planning problem in robotics, we give a complete description of the topological instabilities in any sequential motion planning algorithm for a system…

Algebraic Topology · Mathematics 2014-01-13 Jesus Gonzalez , Mark Grant

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…

Algebraic Topology · Mathematics 2022-09-15 Seyed Abolfazl Aghili , Hanieh Mirebrahimi , Ameneh Babaee

Motion planning in the presence of multiple dynamic obstacles is an important research problem from the perspective of autonomous vehicles as well as space-constrained multi-robot work environment. In this paper, we address the motion…

Systems and Control · Electrical Eng. & Systems 2019-12-30 Trishant Roy , Anindya Harchowdhury , Leena Vachhani

Motion planning is a difficult problem in robot control. The complexity of the problem is directly related to the dimension of the robot's configuration space. While in many theoretical calculations and practical applications the…

Robotics · Computer Science 2020-05-26 Felix Wiebe , Shivesh Kumar , Daniel Harnack , Malte Langosz , Hendrik Wöhrle , Frank Kirchner

Manifolds occur naturally as configuration spaces of robotic systems. They provide global descriptions of local coordinate systems that are common tools in expressing positions of robots. The purpose of this survey is threefold. Firstly, we…

Geometric Topology · Mathematics 2024-02-13 Stephan Mescher

We compute the higher topological complexity of ordered configuration spaces of orientable surfaces, thus extending Cohen-Farber's description of the ordinary topological complexity of those spaces.

Algebraic Topology · Mathematics 2016-07-27 Jesús González , Bárbara Gutiérrez

Topological complexity for spaces was introduced by M. Farber as a minimal number of continuity domains for motion planning algorithms. It turns out that this notion can be extended to the case of not necessarily commutative C*-algebras.…

Operator Algebras · Mathematics 2017-04-03 Vladimir Manuilov

For a pair of spaces $X$ and $Y$ such that $Y \subseteq X$, we define the relative topological complexity of the pair $(X,Y)$ as a new variant of relative topological complexity. Intuitively, this corresponds to counting the smallest number…

Algebraic Topology · Mathematics 2017-10-18 Robert Short

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…

Algebraic Topology · Mathematics 2025-10-13 Gopal Chandra Dutta , Amit Kumar Paul , Subhankar Sau