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
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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.
39 citations
Cites background from "On the existence and synthesis of c..."
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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
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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.
15 citations
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References
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2,613 citations
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01 May 1998
TL;DR: Guidelines in nonholonomic motion planning for mobile robots and collision detection algorithms for motion planning are presented.
Abstract: Guidelines in nonholonomic motion planning for mobile robots.- Geometry of nonholonomic systems.- Optimal trajectories for nonholonomic mobile robots.- Feedback control of a nonholonomic car-like robot.- Probabilistic path planning.- Collision detection algorithms for motion planning.
802 citations
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01 Oct 1992
TL;DR: In this article, nonholonomic kinematics and the role of elliptic functions in constructive controllability, R.W. Murray and S.J. Sussmann planning smooth paths for mobile robots, P. Jacobs and J.P. Laumond motion planning for non-holonomic dynamic systems, M. Reyhanoglu et al a differential geometric approach to motion planning, G.G. Lafferriere and H.
Abstract: Nonholonomic kinematics and the role of elliptic functions in constructive controllability, R.W. Brockett and L. Dai steering nonholonomic control systems using sinusoids, R.M. Murray and S. Shakar Sastry smooth time-periodic feedback solutions for nonholonomic motion planning, L. Gurvits and Zexiang Li lie bracket extensions and averaging - the single-bracket case, H.J. Sussmann and Wensheng Liu singularities and topological aspects in nonholonomic motion planning, J.-P. Laumond motion planning for nonholonomic dynamic systems, M. Reyhanoglu et al a differential geometric approach to motion planning, G. Lafferriere and H.J. Sussmann planning smooth paths for mobile robots, P. Jacobs and J. Canny nonholonomic control and gauge theory, R. Montgomery optimal nonholonomic motion planning for a falling cat, C. Fernandes et al nonholonomic behaviour in free-floating space manipulators and its utilization, E.G. Papadopoulos.
359 citations
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08 May 1994
TL;DR: This work calculates the partition of the configuration space R/sup 2//spl times/S/sup 1/ of a car-like robot, only moving forwards, with respect to the type of the length optimal paths, and gives new optimality conditions on these paths.
Abstract: We calculate the partition of the configuration space R/sup 2//spl times/S/sup 1/ of a car-like robot, only moving forwards, with respect to the type of the length optimal paths. This kind of robot is subject to kinematic constraints on its path curvature and its orientation. Starting from the results on shortest paths, we give new optimality conditions on these paths, and compute the partition for any horizontal plane of the configuration space. >
171 citations
"On the existence and synthesis of c..." refers methods in this paper
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12 May 1992
TL;DR: Given two oriented points in the plane, the authors determine and compute the shortest paths of bounded curvature joining them and propose a solution based on the minimum principle of Pontryagin.
Abstract: Given two oriented points in the plane, the authors determine and compute the shortest paths of bounded curvature joining them. This problem has been solved by L.E. Dubins (1957) in the no-cusp case, and by J.A. Reeds and L.A. Shepp (1990) with cusps. A solution based on the minimum principle of Pontryagin is proposed. The approach simplifies the proofs and makes clear the global or local nature of the results. The no-cusp case and the more difficult case with cusps are discussed. >
130 citations
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