Prospects for the self-driven vehicles and the existing adaptive cruise-control systems for vehicles put more and more stringent requirements to the longitudinal motion control models. The authors believe that one of the key targets in the contemporary studies is development and research of the motion models suitable for the robot-aided vehicles, showing the difference between the piloted, semi-automated and automated control models. This article is devoted to the theoretical mechanical models of the guided longitudinal motion of vehicles. Such models, precise and robust, are essential for analyzing and construction of complex transportation system models. The article is mainly devoted to the design of the analytical models of the linear movement of the vehicles' control; theoretical and mechanical studies of the longitudinal motion of a single car and a pair of cars are carried out, a paradigm of the constructive mathematics is used. Notably, a traffic flow is introduced as a chain of the vehicle "couples", since their motion is longitudinal. A new approach to the coupled vehicles' movement, taking into account the velocity, acceleration and dynamic parameters of the transport units, is introduced. The article presents a concept of a predetermined (program) motion; the relative vehicles' motion is presented in the deviations from a predetermined motion. Possible relative motion control models are introduced. Ways of keeping the required mode of the coupled movement for different types of control are considered. The asymmetry of the control options for different vehicles is detected and studied.