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Reachability of uncertain linear systems using zonotopes
 IN HYBRID SYSTEMS : COMPUTATION AND CONTROL, LNCS 3414
, 2005
"... We present a method for the computation of reachable sets of uncertain linear systems. The main innovation of the method consists in the use of zonotopes for reachable set representation. Zonotopes are special polytopes with several interesting properties: they can be encoded efficiently, they are ..."
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Cited by 97 (14 self)
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We present a method for the computation of reachable sets of uncertain linear systems. The main innovation of the method consists in the use of zonotopes for reachable set representation. Zonotopes are special polytopes with several interesting properties: they can be encoded efficiently, they are closed under linear transformations and Minkowski sum. The resulting method has been used to treat several examples and has shown great performances for high dimensional systems. An extension of the method for the verification of piecewise linear hybrid systems is proposed.
Zonotope/Hyperplane Intersection for Hybrid Systems Reachability Analysis
 HSCC’08, to appear
, 2008
"... In this paper, we are concerned with the problem of computing the reachable sets of hybrid systems with (possibly high dimensional) linear continuous dynamics and guards defined by switching hyperplanes. For the reachability analysis of the continuous dynamics, we use an efficient approximation algo ..."
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Cited by 24 (1 self)
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In this paper, we are concerned with the problem of computing the reachable sets of hybrid systems with (possibly high dimensional) linear continuous dynamics and guards defined by switching hyperplanes. For the reachability analysis of the continuous dynamics, we use an efficient approximation algorithm based on zonotopes. In order to use this technique for the analysis of hybrid systems, we must also deal with the discrete transitions in a satisfactory (i.e. scalable and accurate) way. For that purpose, we need to approximate the intersection of the continuous reachable sets with the guards enabling the discrete transitions. The main contribution of this paper is a novel algorithm for computing efficiently a tight overapproximation of the intersection of (possibly highorder) zonotopes with a hyperplane. We show the accuracy and the scalability of our approach by considering two examples of reachability analysis of hybrid systems.
Computing Reachable Sets of Hybrid Systems Using a Combination of Zonotopes and Polytopes
, 2009
"... The computation of reachable sets for hybrid systems with linear continuous dynamics is addressed. Zonotopes are used for the representation of reachable sets, resulting in an algorithm with low computational complexity with respect to the dimension of the considered system. However, zonotopes have ..."
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Cited by 15 (3 self)
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The computation of reachable sets for hybrid systems with linear continuous dynamics is addressed. Zonotopes are used for the representation of reachable sets, resulting in an algorithm with low computational complexity with respect to the dimension of the considered system. However, zonotopes have drawbacks when being intersected with transition guards which determine the discrete behavior of the hybrid system. For this reason, in the proposed approach, reachable sets are represented by polytopes within guard sets as an intermediate step in order to enclose them by zonotopes afterwards. Different methods for the conservative conversion from zonotopes to polytopes and vice versa are proposed and numerically evaluated.
Collision detection for rigid bodies: A state of the art review
 In GraphiCon
, 2005
"... Virtual reality applications refer to the use of computers to simulate a physical environment in such a way that humans can readily visualize, explore, and interact with “objects ” in this environment. The design of virtual scenes requires realistic physically based simulation algorithms and in part ..."
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Cited by 8 (0 self)
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Virtual reality applications refer to the use of computers to simulate a physical environment in such a way that humans can readily visualize, explore, and interact with “objects ” in this environment. The design of virtual scenes requires realistic physically based simulation algorithms and in particular efficient collision detection routines. Collision detection prevent penetrations between objects. This paper presents an overview of the collision detection procedures for rigid bodies. It divides those procedures in two families that form a general hybrid collision detection algorithm. The first family is used in the first phase known as broad phase that roughly detects collision. The second family is used in the second one known as narrow phase that exactly detects if collision occurs.
An Accurate Join for Zonotopes, Preserving Affine Input/Output Relations
"... Zonotopes are a convenient abstract domain for the precise analysis of programs with numerical variables. Compared to the domain of convex polyhedra, it is less expensive and may easily handle nonlinear assignments. However, the classical join operator of this abstract domain does not always preser ..."
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Cited by 2 (1 self)
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Zonotopes are a convenient abstract domain for the precise analysis of programs with numerical variables. Compared to the domain of convex polyhedra, it is less expensive and may easily handle nonlinear assignments. However, the classical join operator of this abstract domain does not always preserve linear invariants, unlike the convex hull. We present a global join operator that preserves some affine relations. We end up by showing some experiments conducted on the constrained Taylor1+ domain of Apron. Keywords: Abstract interpretation, zonotopes, affine relations 1
On the Quasivector Space of Zonotopes in the Plane
"... In this work we consider centred zonogons represented implicitly as Minkowski sums of centred segments. Using this representation the order relation inclusion (containment) of zonogons has been studied. ..."
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Cited by 1 (1 self)
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In this work we consider centred zonogons represented implicitly as Minkowski sums of centred segments. Using this representation the order relation inclusion (containment) of zonogons has been studied.
Efficient Bounding Volume Hierarchies for Hair Simulation
"... We study the practicability and performance of different classes of bounding volume hierarchies for self collision detection in complex human hair styles with authentic geometrical properties. In particular, we focus on topology based hierarchies which allow for fast update schemes as well as conven ..."
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Cited by 1 (0 self)
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We study the practicability and performance of different classes of bounding volume hierarchies for self collision detection in complex human hair styles with authentic geometrical properties. In particular, we focus on topology based hierarchies which allow for fast update schemes as well as conventional spatially adapted hierarchies. In this context, we introduce a parameterized kDOP approach based on Grassmanian spaces which allows for a systematic generation of normal sets for arbitrary k. This enables us to empirically determine optimal values of k for collision detection based on DOP hierarchies.