arXiv subjects

Subhrajit Bhattacharya

Publications and source records attributed to Subhrajit Bhattacharya.

22 records · Page 2Linked to original sources

arXiv2016

A Search Algorithm for Simplicial Complexes

We present the `Basic S*' algorithm for computing shortest path through a metric simplicial complex. In particular, given a metric graph, $G$, which is constructed as a discrete representation of an underlying configuration space (a larger "continuous" space/manifold typically of dimension greater than one), we consider the Rips complex, $\mathcal{R}(G)$, associated with it. Such a complex, and hence shortest paths in it, represent the underlying metric space more closely than what the graph does. While discrete graph representations of continuous spaces is convenient for motion planning in configuration spaces of robotic systems, the metric induced in them by the ambient configuration space is significantly different from the metric of the configuration space itself. We remedy this problem using the simplicial complex representation. Our algorithm requires only an abstract graph, $G=(V,E)$, and a cost/length function, $d:E\rightarrow \mathbb{R}_+$, as inputs, and no global information such as an embedding or a global coordinate chart is required. The complexity of the Basic S* algorithm is comparable to that of Dijkstra's search, but, as the results presented in this paper demonstrate, the shortest paths obtained using the proposed algorithm represent/approximate the geodesic paths in the original metric space significantly more closely.

cs.DM↗

arXiv2014

A Classification of Configuration Spaces of Planar Robot Arms with Application to a Continuous Inverse Kinematics Problem

Using results on the topology of moduli space of polygons [Jaggi, 92; Kapovich and Millson, 94], it can be shown that for a planar robot arm with $n$ segments there are some values of the base-length, $z$, at which the configuration space of the constrained arm (arm with its end effector fixed) has two disconnected components, while at other values the constrained configuration space has one connected component. We first review some of these known results. Then the main design problem addressed in this paper is the construction of pairs of continuous inverse kinematics for arbitrary robot arms, with the property that the two inverse kinematics agree when the constrained configuration space has a single connected component, but they give distinct configurations (one in each connected component) when the configuration space of the constrained arm has two components. This design is made possible by a fundamental theoretical contribution in this paper -- a classification of configuration spaces of robot arms such that the type of path that the system (robot arm) takes through certain critical values of the forward kinematics function is completely determined by the class to which the configuration space of the arm belongs. This classification result makes the aforesaid design problem tractable, making it sufficient to design a pair of inverse kinematics for each class of configuration spaces (three of them in total). We discuss the motivation for this work, which comes from a more extensive problem of motion planning for the end effector of a robot arm requiring us to continuously sample one configuration from each connected component of the constrained configuration spaces. We demonstrate the low complexity of the presented algorithm through a Javascript + HTML5 based implementation available at http://hans.math.upenn.edu/~subhrabh/nowiki/robot_arm_JS-HTML5/arm.html

math.DG↗

arXiv2012

Invariants for Homology Classes with Application to Optimal Search and Planning Problem in Robotics

We consider planning problems on a punctured Euclidean spaces, $\mathbb{R}^D - \widetilde{\mathcal{O}}$, where $\widetilde{\mathcal{O}}$ is a collection of obstacles. Such spaces are of frequent occurrence as configuration spaces of robots, where $\widetilde{\mathcal{O}}$ represent either physical obstacles that the robots need to avoid (e.g., walls, other robots, etc.) or illegal states (e.g., all legs off-the-ground). As state-planning is translated to path-planning on a configuration space, we collate equivalent plannings via topologically-equivalent paths. This prompts finding or exploring the different homology classes in such environments and finding representative optimal trajectories in each such class. In this paper we start by considering the problem of finding a complete set of easily computable homology class invariants for $(N-1)$-cycles in $(\mathbb{R}^D - \widetilde{\mathcal{O}})$. We achieve this by finding explicit generators of the $(N-1)^{st}$ de Rham cohomology group of this punctured Euclidean space, and using their integrals to define cocycles. The action of those dual cocycles on $(N-1)$-cycles gives the desired complete set of invariants. We illustrate the computation through examples. We further show that, due to the integral approach, this complete set of invariants is well-suited for efficient search-based planning of optimal robot trajectories with topological constraints. Finally we extend this approach to computation of invariants in spaces derived from $(\mathbb{R}^D - \widetilde{\mathcal{O}})$ by collapsing subspace, thereby permitting application to a wider class of non-Euclidean ambient spaces.

math.AT↗

arXiv2011

A Homotopy-like Class Invariant for Sub-manifolds of Punctured Euclidean Spaces

We consider the $D$-dimensional Euclidean space, $\mathbb{R}^D$, with certain $(D-N)$-dimensional compact, closed and orientable sub-manifolds (which we call \emph{singularity manifolds} and represent by $\widetilde{\mathcal{S}}$) removed from it. We define and investigate the problem of finding a homotopy-like class invariant ($χ$-homotopy) for certain $(N-1)$-dimensional compact, closed and orientable sub-manifolds (which we call \emph{candidate manifolds} and represent by $ω$) of $\mathbb{R}^D \setminus \widetilde{\mathcal{S}}$, with special emphasis on computational aspects of the problem. We determine a differential $(N-1)$-form, $ψ_{\widetilde{\mathcal{S}}}$, such that $χ_{\widetilde{\mathcal{S}}}(ω) = \int_ωψ_{\widetilde{\mathcal{S}}}$ is a class invariant for such candidate manifolds. We show that the formula agrees with formulae from Cauchy integral theorem and Residue theorem of complex analysis (when $D=2,N=2$), Biot-Savart law and Ampere's law of theory of electromagnetism (when $D=3,N=2$), and the Gauss divergence theorem (when $D=3,N=3$), and discover that the underlying equivalence relation suggested by each of these well-known theorems is the $χ$-homotopy of sub-manifolds of these low dimensional punctured Euclidean spaces. We describe numerical techniques for computing $ψ_{\widetilde{\mathcal{S}}}$ and its integral on $ω$, and give numerical validations of the proposed theory for a problem in a 5-dimensional Euclidean space. We also discuss a specific application from \emph{robot path planning problem}, when N=2, and describe a method for computing least cost paths with homotopy class constraints using \emph{graph search techniques}.

math.DG↗