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352
Optimal paths for a car that goes both forwards and backwards
 Pacific Journal of Mathematics
, 1990
"... The path taken by a car with a given minimum turning radius has a lower bound on its radius of curvature at each point, but the path has cusps if the car shifts into or out of reverse gear. What is the shortest such path a car can travel between two points if its starting and ending directions are s ..."
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Cited by 207 (0 self)
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The path taken by a car with a given minimum turning radius has a lower bound on its radius of curvature at each point, but the path has cusps if the car shifts into or out of reverse gear. What is the shortest such path a car can travel between two points if its starting and ending directions are specified? One need consider only paths with at most 2 cusps or reversals. We give a set of paths which is sufficient in the sense that it always contains a shortest path and small in the sense that there are at most 68, but usually many fewer paths in the set for any pair of endpoints and directions. We give these paths by explicit formula. Calculating the length of each of these paths and selecting the (not necessarily unique) path with smallest length yields a simple algorithm for a shortest path in each case. These optimal paths or geodesies may be described as follows: If C is an arc of a circle of the minimal turning radius and S is a line segment, then it is sufficient to consider only certain paths of the form CCSCC where arcs and segments fit smoothly, one or more of the arcs or segments may vanish, and where reversals, or equivalently cusps, between arcs or segments are allowed. This contrasts with the case when cusps are not allowed, where Dubins (1957) has shown that paths of the form CCC and CSC suffice. 1. Introduction. We
RealTime Motion Planning For Agile Autonomous Vehicles
 AIAA JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS
, 2000
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Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to ..."
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Cited by 160 (14 self)
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Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
Guidelines in nonholonomic motion planning for mobile robots
 ROBOT MOTION PLANNNING AND CONTROL
, 1998
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Shortest Paths For The ReedsShepp Car: A Worked Out Example Of The Use Of Geometric Techniques In Nonlinear Optimal Control.
, 1991
"... We illustrate the use of the techniques of modern geometric optimal control theory by studying the shortest paths for a model of a car that can move forwards and backwards. This problem was discussed in recent work by Reeds and Shepp who showed, by special methods, (a) that shortest path motion coul ..."
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Cited by 55 (5 self)
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We illustrate the use of the techniques of modern geometric optimal control theory by studying the shortest paths for a model of a car that can move forwards and backwards. This problem was discussed in recent work by Reeds and Shepp who showed, by special methods, (a) that shortest path motion could always be achieved by means of trajectories of a special kind, namely, concatenations of at most five pieces, each of which is either a straight line or a circle, and (b) that these concatenations can be classified into 48 threeparameter families. We show how these results fit in a much more general framework, and can be discovered and proved by applying in a systematic way the techniques of Optimal Control Theory. It turns out that the "classical" optimal control tools developed in the 1960's, such as the Pontryagin Maximum Principle and theorems on the existence of optimal trajectories, are helpful to go part of the way and get some information on the shortest paths, but do not suffice ...
The Stochastic Motion Roadmap: A sampling framework for planning with Markov motion uncertainty
 in Robotics: Science and Systems III (Proc. RSS 2007
, 2008
"... Abstract — We present a new motion planning framework that explicitly considers uncertainty in robot motion to maximize the probability of avoiding collisions and successfully reaching a goal. In many motion planning applications ranging from maneuvering vehicles over unfamiliar terrain to steering ..."
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Cited by 54 (16 self)
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Abstract — We present a new motion planning framework that explicitly considers uncertainty in robot motion to maximize the probability of avoiding collisions and successfully reaching a goal. In many motion planning applications ranging from maneuvering vehicles over unfamiliar terrain to steering flexible medical needles through human tissue, the response of a robot to commanded actions cannot be precisely predicted. We propose to build a roadmap by sampling collisionfree states in the configuration space and then locally sampling motions at each state to estimate state transition probabilities for each possible action. Given a query specifying initial and goal configurations, we use the roadmap to formulate a Markov Decision Process (MDP), which we solve using Infinite Horizon Dynamic Programming in polynomial time to compute stochastically optimal plans. The Stochastic Motion Roadmap (SMR) thus combines a samplingbased roadmap representation of the configuration space, as in PRM’s, with the wellestablished theory of MDP’s. Generating both states and transition probabilities by sampling is far more flexible than previous Markov motion planning approaches based on problemspecific or gridbased discretizations. We demonstrate the SMR framework by applying it to nonholonomic steerable needles, a new class of medical needles that follow curved paths through soft tissue, and confirm that SMR’s generate motion plans with significantly higher probabilities of success compared to traditional shortestpath plans. I.
Multiple UAV cooperative search under collision avoidance and limited range communication constraints
 In IEEE CDC
, 2003
"... This paper focuses on the problem of cooperatively searching, using a team of unmanned air vehicles (UAVs), an area of interest that contains regions of opportunity and regions of potential hazard. The objective of the UAV team is to visit as many opportunities as possible, while avoiding as many ha ..."
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Cited by 54 (1 self)
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This paper focuses on the problem of cooperatively searching, using a team of unmanned air vehicles (UAVs), an area of interest that contains regions of opportunity and regions of potential hazard. The objective of the UAV team is to visit as many opportunities as possible, while avoiding as many hazards as possible. To enable cooperation, the UAVs are constrained to stay within communication range of one another. Collision avoidance is also required. Algorithms for teamoptimal and individuallyoptimal/teamsuboptimal solutions are developed and their computational complexity compared. Simulation results demonstrating the feasibility of the cooperative search algorithms are presented. 1
On Traveling Salesperson Problems for Dubins' vehicle: stochastic and dynamic environments
 CDC 2005, TO APPEAR
, 2005
"... In this paper we propose some novel planning and routing strategies for Dubins’ vehicle, i.e., for a nonholonomic vehicle moving along paths with bounded curvature, without reversing direction. First, we study a stochastic version of the Traveling Salesperson Problem (TSP): given n targets randomly ..."
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Cited by 46 (18 self)
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In this paper we propose some novel planning and routing strategies for Dubins’ vehicle, i.e., for a nonholonomic vehicle moving along paths with bounded curvature, without reversing direction. First, we study a stochastic version of the Traveling Salesperson Problem (TSP): given n targets randomly sampled from a uniform distribution in a rectangle, what is the shortest Dubins ’ tour through the targets and what is its length? We show that the expected length of such a tour is Ω(n 2/3) and we propose a novel algorithm that generates a tour of length O(n 2/3 log(n) 1/3) with high probability. Second, we study a dynamic version of the TSP (known as “Dynamic Traveling Repairperson Problem” in the Operations Research literature): given a stochastic process that generates targets, is there a policy that allows a Dubins vehicle to stabilize the system, in the sense that the number of unvisited targets does not diverge over time? If such policies exist, what is the minimum expected waiting period between the time a target is generated and the time it is visited? We propose a novel recedinghorizon algorithm whose performance is almost within a constant factor from the optimum.
Motion Planning for Multiple Mobile Manipulators
 Proc. of IEEE Int. Conf. on Robotics and Automation
, 1996
"... We address the motion planning for 'xtureless" materialhandling with multiple manipulators on nonholonomic carts. The mobile manipulators possess the ability to manipulate and transport objects while holding them in a stable grasp. We present a general approach that allows generation of o ..."
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Cited by 34 (6 self)
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We address the motion planning for 'xtureless" materialhandling with multiple manipulators on nonholonomic carts. The mobile manipulators possess the ability to manipulate and transport objects while holding them in a stable grasp. We present a general approach that allows generation of optimal trajectories and actuator inputs for any given maneuver. Constraints such as limitations on the turning radii of the mobile manipulators or bounds on their separation can be easily incorporated into the planning scheme. Numerical solutions for several maneuvers including abrupt turns, parallel parking in cluttered environments and changes in formation are computed. Finally, we present escperimental results with two mobile manipulators.
Smooth motion planning for carlike vehicles
 IEEE Transactions on Robotics and Automation
, 2001
"... Abstract—This paper presents a steering method for a carlike vehicle providing smooth paths subjected to curvature constraints. We show how to integrate this steering method in a global motion planning scheme taking obstacles into account. The main idea of the paper is to consider the car as a 4D ..."
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Cited by 33 (0 self)
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Abstract—This paper presents a steering method for a carlike vehicle providing smooth paths subjected to curvature constraints. We show how to integrate this steering method in a global motion planning scheme taking obstacles into account. The main idea of the paper is to consider the car as a 4D system from a kinematic point of view and as a 3D system from a geometric point of view of collision checking. The resulting planned motions are guaranteed to be collisionfree and between two cusp points. Index Terms—Nonholonomic carlike robot, smooth motion planning. I.