This paper deals with two problems related with autonomous mobile robots. The first problem is to determine the control strategy so that the robot describes the shortest trajectory linking an arbitrary position inside a working area and a right-line path to be followed. This problem is solved by considering an equivalent minimum-time nonlinear control problem. The control switching functions which minimize the Hamiltonian function of the minimum-time control problem has been determined in a general form. The second problem analyzed consists in designing a nonlinear observer for the dynamic estimation of the variables required to implement the shortest-trajectory control strategy. A Luenberger-like nonlinear observer was designed. It was found that the observer estimates fastly enough the state variables of the mobile robot so that path tracking can be efficiently implemented.