https://hal.inria.fr/inria-00073323Génot, FrankFrankGénotBIP - Biped Robot - Inria Grenoble - Rhône-Alpes - Inria - Institut National de Recherche en Informatique et en AutomatiqueBrogliato, BernardBernardBrogliatoNew Results on Painlevé ParadoxesHAL CCSD1998PAINLEVE EXAMPLEHYBRID DYNAMICAL SYSTEMLCPGENERALIZED FRICTION CONESINGULAR ODEIMPACT WITHOUT COLLISION[INFO.INFO-OH] Computer Science [cs]/Other [cs.OH]Inria, Rapport De Recherche2006-05-24 12:32:302022-02-04 03:22:542006-05-31 14:24:27enReportsapplication/pdf1In this note, we deal with dynamics of mechanical system subject to unilateral constraints. In particular, we focus on the integration of such a system that is known to be closely related to so-called Linear Complementarity Problem (LCP). Except in very simple case like codimension 1, frictionless contraints, the problem of well-posedness (existence and uniqueness of solution) to such hybrid dynamic systems (smooth dynamics + LCP + shock dynamics) is a big challenge. We concentrate on the well-known Painlevé example [Painlevé 1895], whose dynamics in a sliding regime may be singular depending on the friction coefficient. A new critical friction coefficient is exhibited below which contact forces remain bounded. Moreover a detailed analysis of the vector field nearby the singularity shows that the eventual divergence of the contact force does not call into question the model well-posedness.