1,635 research outputs found
Optimal Sampling-Based Motion Planning under Differential Constraints: the Drift Case with Linear Affine Dynamics
In this paper we provide a thorough, rigorous theoretical framework to assess
optimality guarantees of sampling-based algorithms for drift control systems:
systems that, loosely speaking, can not stop instantaneously due to momentum.
We exploit this framework to design and analyze a sampling-based algorithm (the
Differential Fast Marching Tree algorithm) that is asymptotically optimal, that
is, it is guaranteed to converge, as the number of samples increases, to an
optimal solution. In addition, our approach allows us to provide concrete
bounds on the rate of this convergence. The focus of this paper is on mixed
time/control energy cost functions and on linear affine dynamical systems,
which encompass a range of models of interest to applications (e.g.,
double-integrators) and represent a necessary step to design, via successive
linearization, sampling-based and provably-correct algorithms for non-linear
drift control systems. Our analysis relies on an original perturbation analysis
for two-point boundary value problems, which could be of independent interest
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
Optimal Sampling-Based Motion Planning under Differential Constraints: the Driftless Case
Motion planning under differential constraints is a classic problem in
robotics. To date, the state of the art is represented by sampling-based
techniques, with the Rapidly-exploring Random Tree algorithm as a leading
example. Yet, the problem is still open in many aspects, including guarantees
on the quality of the obtained solution. In this paper we provide a thorough
theoretical framework to assess optimality guarantees of sampling-based
algorithms for planning under differential constraints. We exploit this
framework to design and analyze two novel sampling-based algorithms that are
guaranteed to converge, as the number of samples increases, to an optimal
solution (namely, the Differential Probabilistic RoadMap algorithm and the
Differential Fast Marching Tree algorithm). Our focus is on driftless
control-affine dynamical models, which accurately model a large class of
robotic systems. In this paper we use the notion of convergence in probability
(as opposed to convergence almost surely): the extra mathematical flexibility
of this approach yields convergence rate bounds - a first in the field of
optimal sampling-based motion planning under differential constraints.
Numerical experiments corroborating our theoretical results are presented and
discussed
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