A companion robot is capable of performing a variety of activities and thus supporting the elderly and people with disabilities. It should be able to overcome obstacles on its own, respond to what is happening around it in real time, and communicate with its surroundings. It is particularly important to pay attention to these issues, as a companion robot is likely to become a participant in traffic. The research aims to develop a mathematical model that considers the use of two navigation solutions in the companion robot. Thanks to this, it will be possible to use the obtained mathematical relationships to compare various types of navigation and make a rational choice, enabling the implementation of the assumed activities in a specific external environment. What is new in this paper is the analysis of several navigation methods and the presentation of research conducted in real time using an actual robot.