Many applications of robots in collaboration with humans require the robot to follow the person autonomously. Depending on the tasks and their context, this type of tracking can be a complex problem. The paper proposes and evaluates a principle of control of autonomous robots for applications of services to people, with the capacity of prediction and adaptation for the problem of following people without the use of cameras (high level of privacy) and with a low computational cost. A robot can easily have a wide set of sensors for different variables, one of the classic sensors in a mobile robot is the distance sensor. Some of these sensors are capable of collecting a large amount of information sufficient to precisely define the positions of objects (and therefore people) around the robot, providing objective and quantitative data that can be very useful for a wide range of tasks, in particular, to perform autonomous tasks of following people. This paper uses the estimated distance from a person to a service robot to predict the behavior of a person, and thus improve performance in autonomous person following tasks. For this, we use an adaptive fuzzy neural network (AFNN) which includes a fuzzy neural network based on Takagi-Sugeno fuzzy inference, and an adaptive learning algorithm to update the membership functions and the rule base. The validity of the proposal is verified both by simulation and on a real prototype. The average RMSE of prediction over the 50 laboratory tests with different people acting as target object was 7.33.