1,580 research outputs found
Incremental Sampling-based Algorithms for Optimal Motion Planning
During the last decade, incremental sampling-based motion planning
algorithms, such as the Rapidly-exploring Random Trees (RRTs) have been shown
to work well in practice and to possess theoretical guarantees such as
probabilistic completeness. However, no theoretical bounds on the quality of
the solution obtained by these algorithms have been established so far. The
first contribution of this paper is a negative result: it is proven that, under
mild technical conditions, the cost of the best path in the RRT converges
almost surely to a non-optimal value. Second, a new algorithm is considered,
called the Rapidly-exploring Random Graph (RRG), and it is shown that the cost
of the best path in the RRG converges to the optimum almost surely. Third, a
tree version of RRG is introduced, called the RRT algorithm, which
preserves the asymptotic optimality of RRG while maintaining a tree structure
like RRT. The analysis of the new algorithms hinges on novel connections
between sampling-based motion planning algorithms and the theory of random
geometric graphs. In terms of computational complexity, it is shown that the
number of simple operations required by both the RRG and RRT algorithms is
asymptotically within a constant factor of that required by RRT.Comment: 20 pages, 10 figures, this manuscript is submitted to the
International Journal of Robotics Research, a short version is to appear at
the 2010 Robotics: Science and Systems Conference
The Critical Radius in Sampling-based Motion Planning
We develop a new analysis of sampling-based motion planning in Euclidean
space with uniform random sampling, which significantly improves upon the
celebrated result of Karaman and Frazzoli (2011) and subsequent work.
Particularly, we prove the existence of a critical connection radius
proportional to for samples and dimensions:
Below this value the planner is guaranteed to fail (similarly shown by the
aforementioned work, ibid.). More importantly, for larger radius values the
planner is asymptotically (near-)optimal. Furthermore, our analysis yields an
explicit lower bound of on the probability of success. A
practical implication of our work is that asymptotic (near-)optimality is
achieved when each sample is connected to only neighbors. This is
in stark contrast to previous work which requires
connections, that are induced by a radius of order . Our analysis is not restricted to PRM and applies to a
variety of PRM-based planners, including RRG, FMT* and BTT. Continuum
percolation plays an important role in our proofs. Lastly, we develop similar
theory for all the aforementioned planners when constructed with deterministic
samples, which are then sparsified in a randomized fashion. We believe that
this new model, and its analysis, is interesting in its own right
Sampling-based Algorithms for Optimal Motion Planning
During the last decade, sampling-based path planning algorithms, such as
Probabilistic RoadMaps (PRM) and Rapidly-exploring Random Trees (RRT), have
been shown to work well in practice and possess theoretical guarantees such as
probabilistic completeness. However, little effort has been devoted to the
formal analysis of the quality of the solution returned by such algorithms,
e.g., as a function of the number of samples. The purpose of this paper is to
fill this gap, by rigorously analyzing the asymptotic behavior of the cost of
the solution returned by stochastic sampling-based algorithms as the number of
samples increases. A number of negative results are provided, characterizing
existing algorithms, e.g., showing that, under mild technical conditions, the
cost of the solution returned by broadly used sampling-based algorithms
converges almost surely to a non-optimal value. The main contribution of the
paper is the introduction of new algorithms, namely, PRM* and RRT*, which are
provably asymptotically optimal, i.e., such that the cost of the returned
solution converges almost surely to the optimum. Moreover, it is shown that the
computational complexity of the new algorithms is within a constant factor of
that of their probabilistically complete (but not asymptotically optimal)
counterparts. The analysis in this paper hinges on novel connections between
stochastic sampling-based path planning algorithms and the theory of random
geometric graphs.Comment: 76 pages, 26 figures, to appear in International Journal of Robotics
Researc
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
Asymptotically Optimal Sampling-Based Motion Planning Methods
Motion planning is a fundamental problem in autonomous robotics that requires
finding a path to a specified goal that avoids obstacles and takes into account
a robot's limitations and constraints. It is often desirable for this path to
also optimize a cost function, such as path length.
Formal path-quality guarantees for continuously valued search spaces are an
active area of research interest. Recent results have proven that some
sampling-based planning methods probabilistically converge toward the optimal
solution as computational effort approaches infinity. This survey summarizes
the assumptions behind these popular asymptotically optimal techniques and
provides an introduction to the significant ongoing research on this topic.Comment: Posted with permission from the Annual Review of Control, Robotics,
and Autonomous Systems, Volume 4. Copyright 2021 by Annual Reviews,
https://www.annualreviews.org/. 25 pages. 2 figure
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