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DIFFERENTIALLY CONSTRAINED MOTION PLANNING WITH STATE LATTICE MOTION PRIMITIVES
, 2012
"... Robot motion planning with differential constraints has received a great deal of attention in the last few decades, yet it still remains a challenging problem. Among a number of reasons, three stand out. First, the differential constraints that most physical robots exhibit render the coupling betwee ..."
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Robot motion planning with differential constraints has received a great deal of attention in the last few decades, yet it still remains a challenging problem. Among a number of reasons, three stand out. First, the differential constraints that most physical robots exhibit render the coupling between the control and state spaces quite complicated. Second, it is commonly accepted that robots must be able to operate in environments that are partially or entirely unknown; classical motion planning techniques that assume known structure of the world frequently encounter difficulties when applied in this setting. Third, such robots are typically expected to operate with speed that is commensurate with that of humans. This poses stringent limitations on available runtime and often hard real-time requirements on the motion planner. The impressive advances in computing capacity in recent years have been unable, by themselves, to meet the computational challenge of this problem. New algorithmic approaches to tackle its difficulties continue to be developed to this day. The approach advocated in this thesis is based on encapsulating some of the complexity of satisfying the differential constraints in pre-computed controls that serve as motion primitives, elementary motions that are combined to form the solution trajectory for the system. The contribution of this work is in developing a general approach to constructing such motion primitives,
A geometric method for determining intersection relations between a movable convex object and a set of planar polygons
- In: Robotics, IEEE Transactions on
, 2004
"... Abstract—In this paper, we investigate how to topologically and geometrically characterize the intersection relations between a movable convex polygon and a set of possibly overlapping polygons fixed in the plane. More specifically, a subset is called an intersection relation if there exists a pl ..."
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Abstract—In this paper, we investigate how to topologically and geometrically characterize the intersection relations between a movable convex polygon and a set of possibly overlapping polygons fixed in the plane. More specifically, a subset is called an intersection relation if there exists a placement of that intersects, and only intersects, . The objective of this paper is to design an efficient algorithm that finds a finite and discrete representation of all of the intersection relations between and . Past related research only focuses on the complexity of the free space of the configuration space between and and how to move or place an object in this free space. However, there are many applications that require the knowledge of not only the free space, but also the intersection relations. Examples are presented to demonstrate the rich applications of the formulated problem on intersection relations. Index Terms—Configuration space, critical curves and points, geometric and algebraic structure, intersection relation. I.
Pipes, Cigars, and Kreplach: The Union of Minkowski Sums in Three Dimensions
- In ACM Symposium on Computational Geometry
, 1999
"... Let\Omega be a set of pairwise-disjoint polyhedral obstacles in R 3 with a total of n vertices, and let B be a ball in R 3 . We show that the combinatorial complexity of the free configuration space F of B amid\Omega\Gamma i.e., (the closure of) the set of all placements of B at which B does ..."
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Let\Omega be a set of pairwise-disjoint polyhedral obstacles in R 3 with a total of n vertices, and let B be a ball in R 3 . We show that the combinatorial complexity of the free configuration space F of B amid\Omega\Gamma i.e., (the closure of) the set of all placements of B at which B does not intersect any obstacle, is O(n 2+" ), for any " ? 0; the constant of proportionality depends on ". This upper bound almost matches the known quadratic lower bound on the maximum possible complexity of F . The special case in which\Omega is a set of lines, for which F is a the complement of the union of n congruent cylinders, is studied separately. We also present several extensions of this result, including a randomized algorithm for computing the boundary of F whose expected running time is O(n 2+" ), for any " ? 0. 0 A preliminary version of this paper appeared in Proc. 15th Annual ACM Symposium on Computational Geometry, 1999, pp. 143--153. Work by P.A. was supported by Army R...
Universidad de Buenos Aires Facultad de Ciencias Exactas y Naturales
, 2013
"... Sistema de navegación monocular para robots móviles en ambientes interiores/exteriores ..."
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Sistema de navegación monocular para robots móviles en ambientes interiores/exteriores
Pianos are not Flat: Rigid Motion Planning in Three Dimensions
"... Consider a robot R that is either a line segment or the Minkowski sum of a line segment and a 3-ball, and a set S of polyhedral obstacles with a total of n vertices in R 3. We design near-optimal exact algorithms for planning the motion of R among S when R is allowed to translate and rotate. Specifi ..."
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Consider a robot R that is either a line segment or the Minkowski sum of a line segment and a 3-ball, and a set S of polyhedral obstacles with a total of n vertices in R 3. We design near-optimal exact algorithms for planning the motion of R among S when R is allowed to translate and rotate. Specifically, we can preprocess S in time O(n 4+ε) for any ε> 0 into a data structure that given two placements α and β of R, can decide in time O(log n) whether a collision-free rigid motion of R between α and β exists and if so, output such a motion in time asymptotically proportional to its complexity. Furthermore, we can find in time O(n 4+ε) for any ε> 0 the largest placement of a similar (translated, rotated and scaled) copy of R that does not intersect S. A number of additional stronger results are provided. Our line segment motion planning algorithm improves the result of Ke and O’Rourke by two orders of magnitude and almost matches their lower bound, thus settling a classical motion planning problem first considered by Schwartz and Sharir in 1984. This implies a number of natural directions for future work concerning rigid motion planning in three dimensions.
Motion Planning of a Ball Amid Segments in Three Dimensions
"... Let S be a set of n pairwise disjoint segments in R 3 , and let B be a ball of radius 1. The free configuration space F of B amid S is the set of all placements of B at which (the interior of) B does not intersect any segment of S. We show that the combinatorial complexity of F is O(n 5=2+" ..."
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Let S be a set of n pairwise disjoint segments in R 3 , and let B be a ball of radius 1. The free configuration space F of B amid S is the set of all placements of B at which (the interior of) B does not intersect any segment of S. We show that the combinatorial complexity of F is O(n 5=2+" ), for any " ? 0, with the constant of proportionality depending on ". This is the first subcubic bound on the complexity of the free configuration space even when S is a set of lines in R 3 . We also present a randomized algorithm that can compute the boundary of the free configuration space in O(n 5=2+" ) expected time. 1 Introduction Problem statement. Let S be a collection of n pairwise disjoint segments in R 3 , and let B be a ball of radius 1. We regard S as a set of obstacles and consider the motion-planning problem in which B is allowed to move (translate) freely in R 3 without intersecting any segment of S. The free configuration space F of B with respect to S is the set of ...
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"... Abstract—In this paper we investigate how to topologically and geometrically characterize the intersection relations between a movable convex polygon A and a set Ξ of possibly overlapping polygons fixed in the plane. More specifically, a subset Φ⊂Ξ is called an intersection relation if there exists ..."
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Abstract—In this paper we investigate how to topologically and geometrically characterize the intersection relations between a movable convex polygon A and a set Ξ of possibly overlapping polygons fixed in the plane. More specifically, a subset Φ⊂Ξ is called an intersection relation if there exists a placement of A that intersects and only intersects Φ. The objective of this paper is to design an efficient algorithm that finds a finite and discrete representation of all the intersection relations between A and Ξ. Past related research only focuses on the complexity of the free space of the configuration space between A and Ξ and how to move or place an object in this free space. However, there are many applications that require the knowledge of not only the free space but also the intersection relations. Examples are presented to demonstrate the rich applications of the formulated problem on intersection relations. Index Terms—Configuration space, intersection relation, geometric and algebraic structure, critical curves and points. I.
