Related papers: Linear motion planning with controlled collisions …

In this paper we generalize the discrete r-homotopy to the discrete (s, r)-homotopy. Then by this notion, we introduce the discrete motion planning for robots which can move discreetly. Moreover, in this case the number of motion planning,…

In this series of papers, we present a motion planning framework for planning comfortable and customizable motion of nonholonomic mobile robots such as intelligent wheelchairs and autonomous cars. In Part I, we presented the mathematical…

We determine the Lusternik-Schnirelmann category of the projective product spaces introduced by D. Davis. We also obtained an upper bound for the topological complexity of these spaces, which improves the estimate given by J. Gonz\'alez, M.…

Computing the Jones polynomial of general link diagrams is known to be $\#$P-hard, while restricting the computation to braid closures on fixed number of strands allows for a polynomial time algorithm. We investigate polynomial time…

Let $\mathfrak{G}=\mathfrak{S}_{q} \overleftrightarrow{\times} \mathfrak{S}_q$ be the $\mathbb{Z}/2$-extension of the product of two symmetric groups $\mathfrak{S}_{q} \times \mathfrak{S}_q$. In this paper, we compute the…

This paper reports a novel result: with proper robot models on matrix Lie groups, one can formulate the kinodynamic motion planning problem for rigid body systems as \emph{exact} polynomial optimization problems that can be relaxed as…

The LS-category of a topological space is a numerical homotopy invariant, introduced originally in a course on the global calculus of variations by Lyusternik and Schnirelmann, to estimate the number of critical points of a smooth function.…

Using a recent result of Bartels and Lueck (arXiv:0901.0442) we deduce that the Farrell-Jones Fibered Isomorphism conjecture in L-theory is true for any group which contains a finite index strongly poly-free normal subgroup, in particular,…

The real torus manifolds are a generalization of small covers, and the Dold manifolds of real torus type are a class of non-trivial fibre bundles over the projective product spaces with real torus manifolds as fibres. In this paper, first,…

For each braided category $\mathcal{C}$ we show that, under mild hypotheses, there is an associated category of "half braided algebras" and their bimodules internal to $\mathcal{C}$ which is not only monoidal but even braided and balanced.…

For non-abelian simple objects in a unitary modular category, the density of their braid group representations, the #P-hard evaluation of their associated link invariants, and the BQP-completeness of their anyonic quantum computing models…

A topological theory initiated recently by the author uses methods of algebraic topology to estimate numerically the character of instabilities arising in motion planning algorithms. The present paper studies random motion planning…

We prove the compactness of weighted Sobolev trace operators in outward cuspidal domains by using composition operators on Sobolev spaces. This result allows us to formulate the non-linear Steklov problem in outward cuspidal domains in a…

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…

In the study of the relation between the mapping class group M of a surface and the theory of finite-type invariants of homology 3-spheres, three subgroups of the mapping class group play a large role. They are the Torelli group, the…

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…

We consider the path planning problem for a 2-link robot amidst polygonal obstacles. Our robot is parametrizable by the lengths $\ell_1, \ell_2>0$ of its two links, the thickness $\tau \ge 0$ of the links, and an angle $\kappa$ that…

Periodic solutions of the planar $N$-body problem determine braids through the trajectory of $N$ bodies. Braid types can be used to classify periodic solutions. According to the Nielsen-Thurston classification of surface automorphisms,…

In this paper, we introduce higher symmetric simplicial complexity $SC_n^{\Sigma}(K)$ of a simplicial complex $K$ and higher symmetric combinatorial complexity $CC_n^{\Sigma}(P)$ of a finite poset $P$. These are simplicial and combinatorial…

In 2010, M. Sakai and the first author showed that the topological complexity of a space $X$ coincides with the fibrewise unpointed L-S category of a pointed fibrewise space $\operatorname{pr}_{1} : X \times X \to X$ with the diagonal map…