Considered is the quazi-optimal control design problem for planar motion of a wheeled robot. The control objective is to place the reference point in the prespecied trajectory and to stabilize the motion of the reference point along this trajectory. The trajectory is a line. In the mathematical model under consideration, the current curvature of the trajectory of the reference point is taken as control. The control is subject to two-sided constraints due to limitations on the steering angle of the front wheels. For the control law proposed, the attraction domain in the space distance to the trajectoryangle deviation is constracted. For all initial points from the interior region of this domain to assure xed time stability. Numerical examples are given to illustrate the theoretical results.