The path following problem for a wheeled robot with constrained resource moving along a given curvilinear path is studied. With the help of an earlier introduced change of variables, the path following problem is reduced to that of stability of the zero solution, and a control law linearizing the system in the case of the unconstrained control resource is synthesized. For the closedloop system, the problem of finding the best ellipsoidal approximation of the attraction domain of the target path is set. To take into account the control constraint, an approach based on absolute stability theory is used. In the framework of this approach, construction of an approximating ellipse reduces to solving a parameterized system of linear matrix inequalities. The LMI system in the considered case can be solved analytically. Owing to this, construction of the best ellipsoidal approximation is reduced to solv ing a standard constrained optimization problem for a function of two variables. The proposed method is further extended to finding the best ellipsoidal approximation with an additional constraint on the maximum deviation from the target path. The discussion is illustrated by numerical example