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The problem of controlling the wheeled robot was considered. In the robot model used, the current curvature of the trajectory of the objective point which is related by simple algebraic expressions with the angle of rotation of the front wheels was taken as the control parameter. Boundedness of the angle of rotation of the robot front wheels imposes bilateral constraints on the control. The control constraints influence strongly the transients of the robot entering the desired trajectory. Additionally, the nature of the transients depends on the initial conditions. The aim of the paper was to construct the attraction domain guaranteeing the given rate of the transients. The quadratic Lyapunov function was used to approximate this domain. The problem of determining the quadratic Lyapunov functions was reduced to the standard mathematical verification for solvability of the system of linear matrix inequalities. For the case where the aim of control is to drive the robot objective point to the straight segment of the trajectory, the results of calculations of the stability domains were presented for various formulations of the problem.