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Curvatureconstrained shortest paths in a convex polygon (Extended Abstract)
 SIAM JOURNAL ON COMPUTING
, 2002
"... Let B be a point robot moving in the plane, whose path is constrained to have curvature at most1, and let P be a convex polygon with n vertices. We study the collisionfree, optimal pathplanning problem for B moving between two configurations inside P( a configuration specifies both a location and ..."
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Let B be a point robot moving in the plane, whose path is constrained to have curvature at most1, and let P be a convex polygon with n vertices. We study the collisionfree, optimal pathplanning problem for B moving between two configurations inside P( a configuration specifies both a location and a direction of travel). We present an O(n²log n) time algorithm for determining whether a collisionfree path exists for Bbetween two given configurations. If such a path exists, the algorithm returns a shortest one. We provide a detailed classification of curvatureconstrained shortest paths inside a convex polygon and prove several properties of them, which are interesting in their own right. Some of the properties are quite general and shed some light on curvatureconstrained shortest paths amid obstacles.
Polytime Algorithm for the Shortest Path in a Homotopy Class amidst SemiAlgebraic Obstacles in the Plane
, 1998
"... Given a set of semialgebraic obstacles in the plane and two points in the same connected component of the complement, the problem is to construct the shortest path between these points in a given homotopy class. This path is unique and has some canonical form. We use the representation of homoto ..."
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Cited by 18 (1 self)
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Given a set of semialgebraic obstacles in the plane and two points in the same connected component of the complement, the problem is to construct the shortest path between these points in a given homotopy class. This path is unique and has some canonical form. We use the representation of homotopy classes in a way that is as general as the classical one. It consists in representing generators of a free group which describes the classes of homotopy by disjoint cuts [GS97] homeomorphic to rays. We show that given such a system of generators and a word representing a homotopy class, one can contruct the shortest path of this class in time polynomial in the size of the word and in the size of the representation of the obstacles and the cuts. The homotopy class may also be represented by a path, then the polynomial complexity will depend on the size of the representation of this path. As a technical notion we introduce one particular system of cuts, which we call an extremity ba...
Implementing a generalpurpose edge router
 Proceedings of Graph Drawing 97, LNCS 1353
, 1997
"... Abstract. Although routing is a wellstudied problem in various contexts, there remain unsolved problems in routing edges for graph layouts. In contrast with techniques from other domains such as VLSI CAD and robotics, where physical constraints play a major role, aesthetics play the more important ..."
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Cited by 17 (2 self)
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Abstract. Although routing is a wellstudied problem in various contexts, there remain unsolved problems in routing edges for graph layouts. In contrast with techniques from other domains such as VLSI CAD and robotics, where physical constraints play a major role, aesthetics play the more important role in graph layout. For graphs, we seek paths that are easy to follow and add meaning to the layout. We describe a collection of aesthetic attributes applicable to drawing edges in graphs, and present a general approach for routing individual edges subject to these principles. We also give implementation details and survey di culties that arise in an implementation. 1
On the Boundary Complexity of the Union of Fat Triangles
, 2000
"... A triangle is said to be fat if its smallest angle is at least > 0. A connected component of the complement of the union of a family of triangles is called hole. It is shown that any family of n fat triangles in the plane determines at most O n log 2 holes. This improves on some earlier ..."
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Cited by 16 (2 self)
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A triangle is said to be fat if its smallest angle is at least > 0. A connected component of the complement of the union of a family of triangles is called hole. It is shown that any family of n fat triangles in the plane determines at most O n log 2 holes. This improves on some earlier bounds of Efrat, Rote, Sharir, Matousek et al. Solving a problem of Agarwal and Bern, we also give a general upper bound for the number of holes determined by n triangles in the plane with given angles. As a corollary, we obtain improved upper bounds for the boundary complexity of the union of fat polygons in the plane, which, in turn, leads to better upper bounds for the running times of some known algorithms for motion planning, for finding a separator line for a set of segments, etc.
State of the Union (of Geometric Objects)
 CONTEMPORARY MATHEMATICS
"... Let C be a set of geometric objects in R d. The combinatorial complexity of the union of C is the total number of faces of all dimensions on its boundary. We survey the known upper bounds on the complexity of the union of n geometric objects satisfying various natural conditions. These bounds play ..."
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Cited by 11 (7 self)
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Let C be a set of geometric objects in R d. The combinatorial complexity of the union of C is the total number of faces of all dimensions on its boundary. We survey the known upper bounds on the complexity of the union of n geometric objects satisfying various natural conditions. These bounds play a central role in the analysis of many geometric algorithms, and the techniques used to attain these bounds are interesting in their own right.
Physically Realistic Motion Synthesis in Animation
, 1993
"... Motionsynthesis problems arise in the creation of physically realistic animations involving autonomous characters. Typically characters are required to perform goal tasks, subject to physical law and other constraints on their motion. Witkin and Kass (1988) dubbed this class of problems "Space ..."
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Cited by 11 (6 self)
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Motionsynthesis problems arise in the creation of physically realistic animations involving autonomous characters. Typically characters are required to perform goal tasks, subject to physical law and other constraints on their motion. Witkin and Kass (1988) dubbed this class of problems "Spacetime Constraints" (SC), and presented results for specific problems involving an articulated figure. Their approach was based on a procedure for the local optimization of an initial approximate trajectory supplied by the user. Unfortunately, SC problems are typically multimodal and discontinuous, and the number of decision alternatives available at each time step can be exponential in the number of degrees of freedom in the system. Thus, constructing even coarse trajectories for subsequent optimization can be difficult. We present an algorithm that constructs such trajectories de novo, without directive input from the user. Rather than use a timeseries representation, which might be appropriate...
Extending the pathplanning horizon
, 2005
"... ii SINCE typical mobile robotic vehicles have mobility sensors (such as LADAR or stereo) that can only acquire data up to a few tens of meters, a navigation system has no knowledge about the world beyond this sensing horizon. As a result, path planners that rely only on this knowledge to compute pat ..."
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Cited by 10 (1 self)
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ii SINCE typical mobile robotic vehicles have mobility sensors (such as LADAR or stereo) that can only acquire data up to a few tens of meters, a navigation system has no knowledge about the world beyond this sensing horizon. As a result, path planners that rely only on this knowledge to compute paths are unable to anticipate obstacles sufficiently early and has no choice than to plan inefficient paths that trace obstacle boundaries. To alleviate this problem, We present an opportunistic navigation and view planning strategy that incorporates lookahead sensing of possible obstacle configurations. This planning strategy is based on a “whatif ” analysis of hypothetical future configurations of the environment. Candidate vantage positions are evaluated based on their ability of observing anticipated obstacles. These vantage positions identified by this forwardsimulation framework are used by the planner as intermediate waypoints. The validity of the strategy is supported by results from simulations as well as field experiments with a real robotic platform. These results also show that opportunistically significant reduction in path length can be achieved by using this framework.
Piecewise Linear Paths Among Convex Obstacles
 Discrete and Computational Geometry
, 1993
"... Let B be a set of n arbitrary (possibly intersecting) convex obstacles in R d . It is shown that any two points which can be connected by a path avoiding the obstacles can also be connected by a path consisting of O(n (d\Gamma1)bd=2+1c ) segments. The bound cannot be improved below \Omega\Gamma ..."
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Cited by 8 (0 self)
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Let B be a set of n arbitrary (possibly intersecting) convex obstacles in R d . It is shown that any two points which can be connected by a path avoiding the obstacles can also be connected by a path consisting of O(n (d\Gamma1)bd=2+1c ) segments. The bound cannot be improved below \Omega\Gamma n d ); thus in R 3 , the answer is between n 3 and n 4 . For open disjoint convex obstacles, a \Theta(n) bound is proved. By a wellknown reduction, the general case result also upper bounds the complexity for a translational motion of an arbitrary convex robot among convex obstacles. In the planar case, asymptotically tight bounds and efficient algorithms are given. 1 Introduction The results presented in this paper are motivated by the following problem. Consider n disjoint convex obstacles of an arbitrary shape in the plane or in R 3 , and a convex robot R. Suppose that a position q of R can be reached from a position p by a translational motion of R avoiding the obstacles. I...