552 research outputs found
Probabilistic completeness of RRT for geometric and kinodynamic planning with forward propagation
The Rapidly-exploring Random Tree (RRT) algorithm has been one of the most
prevalent and popular motion-planning techniques for two decades now.
Surprisingly, in spite of its centrality, there has been an active debate under
which conditions RRT is probabilistically complete. We provide two new proofs
of probabilistic completeness (PC) of RRT with a reduced set of assumptions.
The first one for the purely geometric setting, where we only require that the
solution path has a certain clearance from the obstacles. For the kinodynamic
case with forward propagation of random controls and duration, we only consider
in addition mild Lipschitz-continuity conditions. These proofs fill a gap in
the study of RRT itself. They also lay sound foundations for a variety of more
recent and alternative sampling-based methods, whose PC property relies on that
of RRT
Sampling-based optimal kinodynamic planning with motion primitives
This paper proposes a novel sampling-based motion planner, which integrates
in RRT* (Rapidly exploring Random Tree star) a database of pre-computed motion
primitives to alleviate its computational load and allow for motion planning in
a dynamic or partially known environment. The database is built by considering
a set of initial and final state pairs in some grid space, and determining for
each pair an optimal trajectory that is compatible with the system dynamics and
constraints, while minimizing a cost. Nodes are progressively added to the tree
{of feasible trajectories in the RRT* by extracting at random a sample in the
gridded state space and selecting the best obstacle-free motion primitive in
the database that joins it to an existing node. The tree is rewired if some
nodes can be reached from the new sampled state through an obstacle-free motion
primitive with lower cost. The computationally more intensive part of motion
planning is thus moved to the preliminary offline phase of the database
construction at the price of some performance degradation due to gridding. Grid
resolution can be tuned so as to compromise between (sub)optimality and size of
the database. The planner is shown to be asymptotically optimal as the grid
resolution goes to zero and the number of sampled states grows to infinity
Refined Analysis of Asymptotically-Optimal Kinodynamic Planning in the State-Cost Space
We present a novel analysis of AO-RRT: a tree-based planner for motion
planning with kinodynamic constraints, originally described by Hauser and Zhou
(AO-X, 2016). AO-RRT explores the state-cost space and has been shown to
efficiently obtain high-quality solutions in practice without relying on the
availability of a computationally-intensive two-point boundary-value solver.
Our main contribution is an optimality proof for the single-tree version of the
algorithm---a variant that was not analyzed before. Our proof only requires a
mild and easily-verifiable set of assumptions on the problem and system:
Lipschitz-continuity of the cost function and the dynamics. In particular, we
prove that for any system satisfying these assumptions, any trajectory having a
piecewise-constant control function and positive clearance from the obstacles
can be approximated arbitrarily well by a trajectory found by AO-RRT. We also
discuss practical aspects of AO-RRT and present experimental comparisons of
variants of the algorithm
Bidirectional Sampling Based Search Without Two Point Boundary Value Solution
Bidirectional motion planning approaches decrease planning time, on average,
compared to their unidirectional counterparts. In single-query feasible motion
planning, using bidirectional search to find a continuous motion plan requires
an edge connection between the forward and reverse search trees. Such a
tree-tree connection requires solving a two-point Boundary Value Problem (BVP).
However, a two-point BVP solution can be difficult or impossible to calculate
for many systems. We present a novel bidirectional search strategy that does
not require solving the two-point BVP. Instead of connecting the forward and
reverse trees directly, the reverse tree's cost information is used as a
guiding heuristic for the forward search. This enables the forward search to
quickly converge to a feasible solution without solving the two-point BVP. We
propose two new algorithms (GBRRT and GABRRT) that use this strategy and run
multiple software simulations using multiple dynamical systems and real-world
hardware experiments to show that our algorithms perform on-par or better than
existing state-of-the-art methods in quickly finding an initial feasible
solution.Comment: Journal version (Video: https://youtu.be/Rumg66UHfyQ
Robust-RRT: Probabilistically-Complete Motion Planning for Uncertain Nonlinear Systems
Robust motion planning entails computing a global motion plan that is safe
under all possible uncertainty realizations, be it in the system dynamics, the
robot's initial position, or with respect to external disturbances. Current
approaches for robust motion planning either lack theoretical guarantees, or
make restrictive assumptions on the system dynamics and uncertainty
distributions. In this paper, we address these limitations by proposing the
robust rapidly-exploring random-tree (Robust-RRT) algorithm, which integrates
forward reachability analysis directly into sampling-based control trajectory
synthesis. We prove that Robust-RRT is probabilistically complete (PC) for
nonlinear Lipschitz continuous dynamical systems with bounded uncertainty. In
other words, Robust-RRT eventually finds a robust motion plan that is feasible
under all possible uncertainty realizations assuming such a plan exists. Our
analysis applies even to unstable systems that admit only short-horizon
feasible plans; this is because we explicitly consider the time evolution of
reachable sets along control trajectories. Thanks to the explicit consideration
of time dependency in our analysis, PC applies to unstabilizable systems. To
the best of our knowledge, this is the most general PC proof for robust
sampling-based motion planning, in terms of the types of uncertainties and
dynamical systems it can handle. Considering that an exact computation of
reachable sets can be computationally expensive for some dynamical systems, we
incorporate sampling-based reachability analysis into Robust-RRT and
demonstrate our robust planner on nonlinear, underactuated, and hybrid systems.Comment: 16 pages of main text + 5 pages of appendix, 5 figures, submitted to
the 2022 International Symposium on Robotics Researc
Sampling-Based Approximation Algorithms for Reachability Analysis with Provable Guarantees
The successful deployment of many autonomous systems in part hinges on providing rigorous guarantees on their performance and safety through a formal verification method, such as reachability analysis. In this work, we present a simple-to-implement, sampling-based algorithm for reachability
analysis that is provably optimal up to any desired approximation accuracy. Our method achieves computational efficiency by judiciously sampling a finite subset of the state space and generating an approximate reachable set by conducting reachability analysis on this finite set of states. We prove that the reachable set generated by our algorithm approximates the ground-truth
reachable set for any user-specified approximation accuracy. As a corollary to our main method, we introduce an asymptoticallyoptimal, anytime algorithm for reachability analysis. We present simulation results that reaffirm the theoretical properties of our algorithm and demonstrate its effectiveness in real-world inspired scenariosNational Science Foundation (U.S.
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