On the Power of Manifold Samples in Exploring Configuration Spaces and the Dimensionality of Narrow Passages

We extend our study of Motion Planning via Manifold Samples (MMS), a general algorithmic framework that combines geometric methods for the exact and complete analysis of low-dimensional configuration spaces with sampling-based approaches that are appropriate for higher dimensions. The framework explores the configuration space by taking samples that are entire low-dimensional manifolds of the configuration space capturing its connectivity much better than isolated point samples. The contributions of this paper are as follows: (i) We present a recursive application of MMS in a six-dimensional configuration space, enabling the coordination of two polygonal robots translating and rotating amidst polygonal obstacles. In the adduced experiments for the more demanding test cases MMS clearly outperforms PRM, with over 20-fold speedup in a coordination-tight setting. (ii) A probabilistic completeness proof for the most prevalent case, namely MMS with samples that are affine subspaces. (iii) A closer examination of the test cases reveals that MMS has, in comparison to standard sampling-based algorithms, a significant advantage in scenarios containing high-dimensional narrow passages. This provokes a novel characterization of narrow passages which attempts to capture their dimensionality, an attribute that had been (to a large extent) unattended in previous definitions.Comment: 20 page

Generalizing Informed Sampling for Asymptotically Optimal Sampling-based Kinodynamic Planning via Markov Chain Monte Carlo

Asymptotically-optimal motion planners such as RRT* have been shown to incrementally approximate the shortest path between start and goal states. Once an initial solution is found, their performance can be dramatically improved by restricting subsequent samples to regions of the state space that can potentially improve the current solution. When the motion planning problem lies in a Euclidean space, this region $X_{inf}$, called the informed set, can be sampled directly. However, when planning with differential constraints in non-Euclidean state spaces, no analytic solutions exists to sampling $X_{inf}$ directly. State-of-the-art approaches to sampling $X_{inf}$ in such domains such as Hierarchical Rejection Sampling (HRS) may still be slow in high-dimensional state space. This may cause the planning algorithm to spend most of its time trying to produces samples in $X_{inf}$ rather than explore it. In this paper, we suggest an alternative approach to produce samples in the informed set $X_{inf}$ for a wide range of settings. Our main insight is to recast this problem as one of sampling uniformly within the sub-level-set of an implicit non-convex function. This recasting enables us to apply Monte Carlo sampling methods, used very effectively in the Machine Learning and Optimization communities, to solve our problem. We show for a wide range of scenarios that using our sampler can accelerate the convergence rate to high-quality solutions in high-dimensional problems

Robust Execution of Contact-Rich Motion Plans by Hybrid Force-Velocity Control

In hybrid force-velocity control, the robot can use velocity control in some directions to follow a trajectory, while performing force control in other directions to maintain contacts with the environment regardless of positional errors. We call this way of executing a trajectory hybrid servoing. We propose an algorithm to compute hybrid force-velocity control actions for hybrid servoing. We quantify the robustness of a control action and make trade-offs between different requirements by formulating the control synthesis as optimization problems. Our method can efficiently compute the dimensions, directions and magnitudes of force and velocity controls. We demonstrated by experiments the effectiveness of our method in several contact-rich manipulation tasks. Link to the video: https://youtu.be/KtSNmvwOenM.Comment: Proceedings of IEEE International Conference on Robotics and Automation (ICRA2019

The Reach-Avoid Problem for Constant-Rate Multi-Mode Systems

A constant-rate multi-mode system is a hybrid system that can switch freely among a finite set of modes, and whose dynamics is specified by a finite number of real-valued variables with mode-dependent constant rates. Alur, Wojtczak, and Trivedi have shown that reachability problems for constant-rate multi-mode systems for open and convex safety sets can be solved in polynomial time. In this paper, we study the reachability problem for non-convex state spaces and show that this problem is in general undecidable. We recover decidability by making certain assumptions about the safety set. We present a new algorithm to solve this problem and compare its performance with the popular sampling based algorithm rapidly-exploring random tree (RRT) as implemented in the Open Motion Planning Library (OMPL).Comment: 26 page

A Game-theoretic Formulation of the Homogeneous Self-Reconfiguration Problem

In this paper we formulate the homogeneous two- and three-dimensional self-reconfiguration problem over discrete grids as a constrained potential game. We develop a game-theoretic learning algorithm based on the Metropolis-Hastings algorithm that solves the self-reconfiguration problem in a globally optimal fashion. Both a centralized and a fully distributed algorithm are presented and we show that the only stochastically stable state is the potential function maximizer, i.e. the desired target configuration. These algorithms compute transition probabilities in such a way that even though each agent acts in a self-interested way, the overall collective goal of self-reconfiguration is achieved. Simulation results confirm the feasibility of our approach and show convergence to desired target configurations.Comment: 8 pages, 5 figures, 2 algorithm