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In this paper we study the topological invariant ${sf {TC}}(X)$ reflecting the complexity of algorithms for autonomous robot motion. Here, $X$ stands for the configuration space of a system and ${sf {TC}}(X)$ is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in $X$. We focus on the case when the space $X$ is aspherical; then the number ${sf TC}(X)$ depends only on the fundamental group $pi=pi_1(X)$ and we denote it ${sf TC}(pi)$. We prove that ${sf TC}(pi)$ can be characterised as the smallest integer $k$ such that the canonical $pitimespi$-equivariant map of classifying spaces $$E(pitimespi) to E_{mathcal D}(pitimespi)$$ can be equivariantly deformed into the $k$-dimensional skeleton of $E_{mathcal D}(pitimespi)$. The symbol $E(pitimespi)$ denotes the classifying space for free actions and $E_{mathcal D}(pitimespi)$ denotes the classifying space for actions with isotropy in a certain family $mathcal D$ of subgroups of $pitimespi$. Using this result we show how one can estimate ${sf TC}(pi)$ in terms of the equivariant Bredon cohomology theory. We prove that ${sf TC}(pi) le max{3, {rm cd}_{mathcal D}(pitimespi)},$ where ${rm cd}_{mathcal D}(pitimespi)$ denotes the cohomological dimension of $pitimespi$ with respect to the family of subgroups $mathcal D$. We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion free hyperbolic groups as well as all torsion free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher are exactly the classes having Bredon cohomology extensions with respect to the family $mathcal D$.
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We prove equivaria
A defining feature of sampling-based motion planning is the reliance on an implicit representation of the state space, which is enabled by a set of probing samples. Traditionally, these samples are drawn either probabilistically or deterministically to uniformly cover the state space. Yet, the motion of many robotic systems is often restricted to small regions of the state space, due to, for example, differential constraints or collision-avoidance constraints. To accelerate the planning process, it is thus desirable to devise non-uniform sampling strategies that favor sampling in those regions where an optimal solution might lie. This paper proposes a methodology for non-uniform sampling, whereby a sampling distribution is learned from demonstrations, and then used to bias sampling. The sampling distribution is computed through a conditional variational autoencoder, allowing sample generation from the latent space conditioned on the specific planning problem. This methodology is general, can be used in combination with any sampling-based planner, and can effectively exploit the underlying structure of a planning problem while maintaining the theoretical guarantees of sampling-based approaches. Specifically, on several planning problems, the proposed methodology is shown to effectively learn representations for the relevant regions of the state space, resulting in an order of magnitude improvement in terms of success rate and convergence to the optimal cost.
Motion planning for multi-jointed robots is challenging. Due to the inherent complexity of the problem, most existing works decompose motion planning as easier subproblems. However, because of the inconsistent performance metrics, only sub-optimal solution can be found by decomposition based approaches. This paper presents an optimal control based approach to address the path planning and trajectory planning subproblems simultaneously. Unlike similar works which either ignore robot dynamics or require long computation time, an efficient numerical method for trajectory optimization is presented in this paper for motion planning involving complicated robot dynamics. The efficiency and effectiveness of the proposed approach is shown by numerical results. Experimental results are used to show the feasibility of the presented planning algorithm.
We define the orbit category for transitive topological groupoids and their equivariant CW-complexes. By using these constructions we define equivariant Bredon homology and cohomology for actions of transitive topological groupoids. We show how these theories can be obtained by looking at the action of a single isotropy group on a fiber of the anchor map, extending equivariant results for compact group actions. We also show how this extension from a single isotropy group to the entire groupoid action can be applied to the structure of principal bundles and classifying spaces.
Anytime sampling-based methods are an attractive technique for solving kino-dynamic motion planning problems. These algorithms scale well to higher dimensions and can efficiently handle state and control constraints. However, an intelligent exploration strategy is required to accelerate their convergence and avoid redundant computations. Using ideas from reachability analysis, this work defines a Time-Informed Set, that focuses the search for time-optimal kino-dynamic planning after an initial solution is found. Such a Time-Informed Set (TIS) includes all trajectories that can potentially improve the current best solution and hence exploration outside this set is redundant. Benchmarking experiments show that an exploration strategy based on the TIS can accelerate the convergence of sampling-based kino-dynamic motion planners.
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