The Udwadia-Kalaba formulation is proposed to model the longitudinal dynamics of a road vehicle. To render complex situations such as spinning on a slippery road, an original approach is implemented by the relaxation of constraints in the Udwadia-Kalaba formulation for the rolling of a wheel. In a combined approach of both slip and stiction in the contact section, the constraints equations of pure rolling are associated with stiction. Such constraints are lifted as slip occurs to allow the dynamics of the wheel to take over the normally imposed kinematic constraints. The relaxation of constraints is achieved by the extension of the Udwadia-Kalaba formulation with the introduction of weights on both the minimum norm and the semi-least-squares solutions of the constraints equations. This sets biases on the constraints equations based on the description of weight functions that takes into account a friction conditionality without branching. Which leads to the smooth activation or deactivation of selected constraints equations and associated forces. Without the need to rewrite the equations of motion. Thanks to the extended Udwadia-Kalaba formulation, a new multibody formalism is introduced with the relaxation matrix for systems with intermittent constraints.

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