On the existence and synthesis of curvature-bounded paths inside nonuniform rectangular channels
01 Jun 2010-pp 5382-5387
TL;DR: In this article, the authors present a numerical algorithm for determining the existence of a curvature-bounded path contained within a rectangular channel, where the rectangular cells comprising the channel are assumed to be of arbitrary, non-uniform dimensions and the bounds on curvature are allowed to be different for different cells.
Abstract: Motion planners for autonomous mobile vehicles that are based on rectangular cell decompositions are often required to construct kinematically feasible path - typically curvature-bounded paths - traversing rectangular channels. In this paper, we present a numerical algorithm for determining the existence of a curvature-bounded path contained within a rectangular channel. The rectangular cells comprising the channel are assumed to be of arbitrary, non-uniform dimensions and the bounds on curvature are allowed to be different for different cells. The proposed algorithm is based on the explicit construction of the cone of feasible directions for a bounded-curvature path at the cell exit edge, given the entry point for each cell in the channel. Based on this analysis, we devise a path construction scheme that retains the convenience of cell-by-cell path synthesis but eliminates the guesswork involved in choosing terminal conditions within each cell.
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TL;DR: The proposed iterative algorithm is suitable for real-time implementations, where hard bounds on the available computation time are imposed, and where the original H-cost optimization algorithm may not have sufficient time to converge to a solution at all.
Abstract: Motion planning for mobile vehicles involves the solution of two disparate subproblems: the satisfaction of high-level logical task specifications and the design of low-level vehicle control laws. A hierarchical solution of these two subproblems is efficient, but it may not ensure compatibility between the high-level planner and the constraints that are imposed by the vehicle dynamics. To guarantee such compatibility, we propose a motion-planning framework that is based on a special interaction between these two levels of planning. In particular, we solve a special shortest path problem on a graph at a higher level of planning, and we use a lower level planner to determine the costs of the paths in that graph. The overall approach hinges on two novel ingredients: a graph-search algorithm that operates on sequences of vertices and a lower level planner that ensures consistency between the two levels of hierarchy by providing meaningful costs for the edge transitions of a higher level planner using dynamically feasible, collision-free trajectories.
62 citations
TL;DR: A class of Hamilton–Jacobi equations is derived which models such motions of a vehicle which travels under certain curvature constraints; it unifies two well-known vehicular models, the Dubins’ and Reeds–Shepp’s cars, and gives further generalizations.
Abstract: We propose a PDE approach for computing time-optimal trajectories of a vehicle which travels under certain curvature constraints. We derive a class of Hamilton---Jacobi equations which models such motions; it unifies two well-known vehicular models, the Dubins' and Reeds---Shepp's cars, and gives further generalizations. Numerical methods (finite difference for the Reeds---Shepp's car and semi-Lagrangian for the Dubins' car) are investigated for two-dimensional domains and surfaces.
44 citations
Cites background from "On the existence and synthesis of c..."
...This is a problem posed in [42], where it was solved using purely geometrical arguments....
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15 Jul 2015
TL;DR: A graph-search algorithm that operates on sequences of vertices and a lower level planner that ensures consistency between the two levels of hierarchy by providing meaningful costs for the edge transitions of a higher level planner using dynamically feasible, collision-free trajectories are proposed.
Abstract: New requirements of autonomous mobile vehicles necessitate hierarchical motion-planning techniques that not only find a plan to satisfy high-level specifications, but also guarantee that this plan is suitable for execution under vehicle dynamical constraints. In this context, the H-cost motion-planning technique has been reported in the recent literature. We propose an incremental motion-planning algorithm based on this technique. The proposed algorithm retains the benefits of the original technique, while significantly reducing the associated computational time. In particular, the proposed iterative algorithm presents during intermediate iterations feasible solutions, with the guarantee that the algorithm eventually converges to an optimal solution. The costs of solutions at intermediate iterations are almost always nonincreasing. Therefore, the proposed algorithm is suitable for real-time implementations, where hard bounds on the available computation time are imposed, and where the original H-cost optimization algorithm may not have sufficient time to converge to a solution at all. We illustrate the proposed algorithm with numerical simulation examples.
37 citations
TL;DR: A fast numerical algorithm is presented to determine whether a narrow planar passage can be traversed by a curve that satisfies prespecified upper bounds on its curvature, and it is demonstrated that the proposed algorithm can affirm traversability in cases where the most recent result in the literature fails.
Abstract: We consider the geometric problem of deciding whether a narrow planar passage can be traversed by a curve that satisfies prespecified upper bounds on its curvature. This problem is of importance for path- and motion-planning of autonomous mobile robots, particularly when vehicle dynamical constraints are considered during planning. For a special case of narrow passages, namely, rectangular channels, we present a fast numerical algorithm to determine if a given channel may be traversed via curvature- bounded paths. We demonstrate that the proposed algorithm can affirm traversability in cases where the most recent result in the literature fails.
18 citations
Cites background or methods from "On the existence and synthesis of c..."
...This paper presents detailed algorithms for the solution of the CBTA problem, a complete proof of Theorem 1, and applications of the proposed technique, neither of which appeared in [16]....
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...Preliminary versions of the discussion and results presented in this paper were previously presented in [16]....
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Dissertation•
10 Nov 2011
17 citations
Cites methods from "On the existence and synthesis of c..."
...This can be achieved by appropriately concatenating different Dubins’ paths using geometric arguments [29, 8, 9, 64] or by employing a discrete path planner (high level planning) that yields a set of way-points to be locally connected by means of Dubins-like paths (low level planning) [25, 111, 28, 155, 65]....
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References
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TL;DR: The decidability of the reachability question: “Given a source placement and a target placement, is there a curvature-constrained path from source to target avoiding obstacles?” is shown.
Abstract: We consider the motion planning problem for a point constrained to move along a path with radius of curvature at least one. The point moves in a two-dimensional universe with polygonal obstacles. We show the decidability of the reachability question: “Given a source placement (position and direction pair) and a target placement, is there a curvature-constrained path from source to target avoiding obstacles?” The decision procedure has time and space complexity 2
o(poly(n, m))
wheren is the number of corners andm is the number of bits required to specify the position of corners.
43 citations
01 Dec 2007
TL;DR: This work presents an efficient multiresolution cell decomposition scheme based on the Haar wavelet transform and demonstrates an algorithm to extract the adjacency and transition cost relations of the cells directly from the wavelet Transform coefficients.
Abstract: Path planning techniques based on hierarchical multiresolution cell decompositions are suitable for online implementation due to their simplicity and speed of implementation. We present an efficient multiresolution cell decomposition scheme based on the Haar wavelet transform. The decomposition approximates the environment using high resolution close to the agent and coarse resolution elsewhere. We demonstrate an algorithm to extract the adjacency and transition cost relations of the cells directly from the wavelet transform coefficients.
24 citations
10 Jun 2009
TL;DR: A new algorithm is presented to compute the shortest path on a graph when the node transition costs depend on the prior history of the path to the current node.
Abstract: A new algorithm is presented to compute the shortest path on a graph when the node transition costs depend on the prior history of the path to the current node. The algorithm is applied to solve path planning problems with curvature constraints.
20 citations
"On the existence and synthesis of c..." refers background in this paper
...In [2], we address the first issue by proposing a geometric path planning algorithm based on rectangular cell decomposition which can incorporate any information on the kinematic and dynamic limitations of the vehicle in the highlevel planning itself....
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TL;DR: A minimalbounded curvature path is located in O(n4)time whose length, except in special situations involving arcs of length greater than π , is at most twice the length of a shortest convex and simple path of bounded curvature.
Abstract: . In this paper we study the collision-free path planning problem for a point robot, whose path is of bounded curvature(i.e., constrained to have curvature at most 1), moving in the plane inside an n -sided simple polygon P . Given two points sand tinside Pand two directions of travel, one at sand one at t , the problem is to compute a convex and simple path of bounded curvature inside Pfrom sto tconsisting of straight-line segments and circular arcs such that (i) the radius of each circular arc is at least 1, (ii) each segment on the path is the tangent between the two consecutive circular arcs on the path, (iii) the given initial direction at sis tangent to the path at sand (iv) the given final direction at tis tangent to the path at t . We propose an O(n
4
)time algorithm for this problem. Using the notion of complete visibility, Pis pruned to another polygon P'such that any convex and simple path from sto tinside Palso remains inside P' . Then our algorithm constructs the locus of center of a circle of unit radius translating along the boundary of P'and, using this locus, the algorithm constructs a convex and simple path of bounded curvature. Our algorithm is based on the relationship between the Euclidean shortest path, link paths and paths of bounded curvature, and uses several properties derived here on convex and simple paths of bounded curvature. We also show that the path computed by our algorithm can be transformed in O(n)time to a minimalconvex and simple path of bounded curvature. Using this transformation and two necessary conditions proposed here for the shortest convex and simple path of bounded curvature, a minimalbounded curvature path is located in O(n
4
)time whose length, except in special situations involving arcs of length greater than π , is at most twice the length of a shortest convex and simple path of bounded curvature.
15 citations
"On the existence and synthesis of c..." refers background or methods in this paper
...[10]–[12] is to appropriately choose a finite set of configurations (position and orientation) in the environment, and connecting these configurations by Dubins paths, and then perform a graph search using the length of the Dubins paths as edge costs....
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...References [10]–[12] present algorithms for efficiently finding approximations to shortest curvature-bounded paths in polygonal environments....
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...It may be argued that the given rectangular channel may be treated as a special case of a polygonal channel, and the geometric techniques presented in [11] or [12] could be applied....
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01 Dec 2007
TL;DR: This paper presents a meta-modelling framework for estimating the modeled model of the EMMARM, and describes the architecture, software, and user interfaces used in this system.
Abstract: (c) 2007 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works. Digital Object Identifier : 10.1109/CDC.2007.4434146
12 citations
Additional excerpts
...Since our ultimate objective is to use this trajectory generation scheme in conjunction with multi-resolution cell decompositions [3], [4] we assume channels comprised of rectangular cells of arbitrary dimensions....
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