Chapoutot, Alexandre

dit Sandretto, Julien Alexandre

Garion, Christophe

Rauh, Andreas

Schön, Steffen

Thirioux, Xavier

Ziat, Ghiles

English

Institutionelles Repositorium der Leibniz Universität Hannover

A constraint programming approach forpolytopic simulation of ordinarydi↵erential equations - a collisiondetection applicationJulien Alexandre dit Sandretto1, AlexandreChapoutot1,Christophe Garion2, Xavier Thirioux2 and GhilesZiat21ENSTA Paris, Institut Polytechnique de Paris,828 boulevard des Maréchaux, France{alexandre,chapoutot}@ensta.fr2ISAE-SUPAERO, Université de Toulouse, France{garion,thirioux,ziat}@isae-supaero.frKeywords: Constraint programming, Abstract domains, ODEsIntroductionStarting from a set of possible initial points, the solution of an ODEcan be represented by a reachable tube describing the evolution of thesystem from this initial set. Abstract domains can be used to enclosethe tube: boxes (cartesian product of intervals), zonotopes, ellipsoids,and nonconvex sets such as Taylor models (see [2] for a review of theseabstract domains). The more accurate an abstract domain is, i.e. thesmallest the di↵erence between the hull of the abstraction and theabstracted set is, the more accurate the enclosure of the reachabletube will be. Polytope enclosure is a promising approach as it is moreprecise than the interval or zonotope abstract domains, but su↵ersfrom the expensiveness of its geometrical computation. Considering apolytope as an intersection of zonotopes and therefore benefiting froma ne arithmetic is a possible solution to overcome the limitations (e.g.zonotope bundles [3], i.e., a set of zonotopes is used and therefore theintersection is not computed; or the intersection is computed whennecessary [1]).Reachable tubes as abstract treesConsidering a tube, a disjunction of predicates, T = (t1 ^ e1) _ (t2 ^e2) · · · _ (tn ^ en) where each ei represents the set of values of solutionfunctions within time frame ti, it is to be understood as the followingproperty: the solution is either in set e1 during the time frame t1, or inset e2 during the time frame t2, etc. Considering initial values given asa polytope P , we decompose P as the intersection of s zonotopes Zi.The reachable tube of the corresponding ODE is therefore describedby the conjunction of s tubes T 1 ^ · · · ^ T i ^ · · · ^ T s, each tube T ibeing obtained by the zonotopic simulation with initial value taken inZi. This conjunction of disjunctions can be e ciently solved with con-straint programming and polytopes as abstract domains. For obstacleavoidance or collision detection, a predicate (or several ones) of theform “and not in” is added to the previous formula.AcknowledgementThis work was supported by the Defense Innovation Agency (AID) ofthe French Ministry of Defense.References[1] Julien Alexandre dit Sandretto and Jian Wan, Reacha-bility analysis of nonlinear ODEs using polytopic based validatedRunge-Kutta, Reachability Problems, Springer, 2018.[2] Matthias Althoff, Goran Frehse, and Antoine Girard,Set Propagation Techniques for Reachability Analysis, Annual Re-view of Control, Robotics, and Autonomous Systems, 4(1), 2021.[3] Matthias Althoff and Bruce H. Krogh, Zonotope bundlesfor the e cient computation of reachable sets, IEEE conference ondecision and control and European control conference, 2011

A constraint programming approach for polytopic simulation of ordinary differential equations - a collision detection application

https://www.repo.uni-hannover.de/bitstream/123456789/13315/1/Sandretto_etal_SWIM22.pdf

A constraint programming approach for polytopic simulation of ordinary differential equations - a collision detection application

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