Most motion planners for a vehicle often assume no-slipping and no-skidding in the context of nonholonomic motion planning. However, there actually exist slipping and skidding in practical situations and it is important to know the force between the tire and the contact plane considering slipping and skidding in the planning problem. In this paper, a Coulomb friction model of the tire friction force of vehicle is extended to nonholonomic constraints. In the modeling, the longitudinal and lateral tire friction forces are modeled corresponding to no slipping and no skidding constraints, respectively. A two directional model based on the maximum dissipation principle is also developed. Some typical movements of vehicles with the proposed friction force model are illustrated by simulations.