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The Complexity of the Two Dimensional CurvatureConstrained ShortestPath Problem
 IN PROC. THIRD INTERNATIONAL WORKSHOP ON THE ALGORITHMIC FOUNDATIONS OF ROBOTICS
, 1998
"... The motion planning problems for nonholonomic carlike robots have been extensively studied in the literature. The curvatureconstrained shortestpath problem is to plan a path (from an initial configuration to a final configuration, where a configuration is defined by a location and an orienta ..."
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Cited by 33 (4 self)
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The motion planning problems for nonholonomic carlike robots have been extensively studied in the literature. The curvatureconstrained shortestpath problem is to plan a path (from an initial configuration to a final configuration, where a configuration is defined by a location and an orientation) in the presence of obstacles, such that the path is a shortest among all paths with a prescribed curvature bound. The curvatureconstrained shortestpath problem can also be seen as finding a shortest path for a point carlike robot moving forward at constant speed with a radius of curvature upper bounded by some constant. Previously, there is no known hardness result for the 2D curvature constrained shortestpath problem. This paper shows that the above problem in two dimensions is NPhard, when the obstacles are polygons with a total of N vertices and the vertex positions are given within O(N²) bits of precision. Our reduction is computed by a family of polynomialsize ...
Minimum wheelrotation paths for differentialdrive mobile robots,”
 Int. J. Robotics Research,
, 2009
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Timeoptimal Paths for a Dubins airplane
, 2007
"... We consider finding a timeoptimal trajectory for an airplane from some starting point and orientation to some final point and orientation. Our model extends the Dubins car [15] to have altitude, which leads to Dubins airplane. We assume that the system has independent bounded control over the alti ..."
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Cited by 19 (2 self)
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We consider finding a timeoptimal trajectory for an airplane from some starting point and orientation to some final point and orientation. Our model extends the Dubins car [15] to have altitude, which leads to Dubins airplane. We assume that the system has independent bounded control over the altitude velocity as well as the turning rate in the plane. Through the use of the Pontryagin Maximum Principle, we characterize the timeoptimal trajectories for the system. They are composed of turns with minimum radius, straight line segments, and pieces of planar elastica. One motivation for determining these elementary pieces is for use as motion primitives in modern planning and control algorithms that consider obstacles.
Nonuniform Discretization For Kinodynamic Motion Planning And Its Applications
 SIAM J. COMPUT
, 2000
"... The first main result of this paper is a novel nonuniform discretization approximation method for the kinodynamic motionplanning problem. The kinodynamic motionplanning problem is to compute a collisionfree, timeoptimal trajectory for a robot whose accelerations and velocities are bounded. Previ ..."
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Cited by 8 (0 self)
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The first main result of this paper is a novel nonuniform discretization approximation method for the kinodynamic motionplanning problem. The kinodynamic motionplanning problem is to compute a collisionfree, timeoptimal trajectory for a robot whose accelerations and velocities are bounded. Previous approximation methods are all based on a uniform discretization in the time space. On the contrary, our method employs a nonuniform discretization in the configuration space (thus also a nonuniform one in the time space). Compared to the previously best algorithm of Donald and Xavier, the running time of our algorithm reduces in terms of 1/#, roughly from O((1/#) 6d1 ) to O((1/#) 4d2 ), in computing a trajectory in a ddimensional configuration space, such that the time length of the trajectory is within a factor of (1 + #) of the optimal. More importantly, our algorithm is able to take advantage of the obstacle distribution and is expected to perform much better than the analyt...
Reachability By Paths of Bounded Curvature in Convex Polygons
, 2000
"... Let B be a point robot moving in the plane, whose path is constrained to forward motions with a curvature at most 1, and let P be a convex polygon with n vertices. Given a starting conguration (a location and a direction of travel) for B inside P , we characterize the region of all points of P that ..."
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Cited by 6 (0 self)
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Let B be a point robot moving in the plane, whose path is constrained to forward motions with a curvature at most 1, and let P be a convex polygon with n vertices. Given a starting conguration (a location and a direction of travel) for B inside P , we characterize the region of all points of P that can be reached by B, and show that it has linear complexity.
Velocity, acceleration and deceleration bounds for a timeoptimal planner of a wheeled mobile robot
"... SUMMARY Three singlevalued upper boundary functions for velocity, acceleration and deceleration of a wheeled mobile robot (WMR) are defined as closed mathematical forms over its entire spatial path. The limits deal with mechanical, kinematic and dynamic characteristics of the robot and with task a ..."
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Cited by 3 (3 self)
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SUMMARY Three singlevalued upper boundary functions for velocity, acceleration and deceleration of a wheeled mobile robot (WMR) are defined as closed mathematical forms over its entire spatial path. The limits deal with mechanical, kinematic and dynamic characteristics of the robot and with task and operating matters. These boundary functions can be computed making use of any robot model, as complex as is needed, since it works offline. All studies are particularised for the robot RAM. A kinematic and a complete dynamic model of this WMR is built, with special attention on the study of wheelground contact efforts. For this purpose an empiricalanalytical model of rubber wheel rolling is developed.
MultiVehicle Path Planning in Dynamically Changing Environments
, 2009
"... In this paper, we propose a path planning method for nonholonomic multivehicle system in presence of moving obstacles. The objective is to find multiple fixed length paths for multiple vehicles with the following properties: (i) bounded curvature (ii) obstacle avoidant (iii) collision free. Our app ..."
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Cited by 3 (2 self)
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In this paper, we propose a path planning method for nonholonomic multivehicle system in presence of moving obstacles. The objective is to find multiple fixed length paths for multiple vehicles with the following properties: (i) bounded curvature (ii) obstacle avoidant (iii) collision free. Our approach is based on polygonal approximation of a continuous curve. Using this idea, we formulate an arbitrarily fine relaxation of the path planning problem as a nonconvex feasibility optimization problem. Then, we propound a nonsmooth dynamical systems approach to find feasible solutions of this optimization problem. It is shown that the trajectories of the nonsmooth dynamical system always converge to some equilibria that correspond to the set of feasible solutions of the relaxed problem. The proposed framework can handle more complex mission scenarios for multivehicle systems such as rendezvous and area coverage.
Finding CurvatureConstrained Paths that Avoid Polygonal Obstacles (Extended Abstract)
, 2007
"... We describe an algorithm to find a unitcurvature path between specified configurations in an arbitrary polygonal domain. Whenever such a path exists, the algorithm returns an explicit description of one such path in time that is polynomial in n (the number of features of the domain), m (the precisi ..."
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Cited by 2 (1 self)
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We describe an algorithm to find a unitcurvature path between specified configurations in an arbitrary polygonal domain. Whenever such a path exists, the algorithm returns an explicit description of one such path in time that is polynomial in n (the number of features of the domain), m (the precision of the input) and k (the number of segments on the simplest obstaclefree Dubins path connecting the specified configurations). Our algorithm is based on a new normal form for unitcurvature paths and a dynamic path filtering argument that exploits a separation bound for distinct paths in this normal form. The best result known for the feasibility of boundedcurvature motion in the presence of arbitrary polygonal obstacles involves a reduction to the firstorder theory of the reals. It just determines if a feasible path exists (it does not return a path) and requires exponential time and space.
The cost of bounded curvature
 Computing Geometry
, 2013
"... We study the motionplanning problem for a carlike robot whose turning radius is bounded from below by one and which is allowed to move in the forward direction only (Dubins car). For two robot configurations σ, σ′, let `(σ, σ′) be the shortest boundedcurvature path from σ to σ′. For d> 0, let ..."
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We study the motionplanning problem for a carlike robot whose turning radius is bounded from below by one and which is allowed to move in the forward direction only (Dubins car). For two robot configurations σ, σ′, let `(σ, σ′) be the shortest boundedcurvature path from σ to σ′. For d> 0, let `(d) be the supremum of `(σ, σ′), over all pairs (σ, σ′) that are at Euclidean distance d. We study the function dub(d) = `(d) − d, which expresses the difference between the boundedcurvature path length and the Euclidean distance of its endpoints. We show that dub(d) decreases monotonically from dub(0) = 7pi/3 to dub(d∗) = 2pi, and is constant for d> d∗. Here d ∗ ≈ 1.5874. We describe pairs of configurations that exhibit the worstcase of dub(d) for every distance d. 1