In this paper a novel, geometrically interpreted adaptive control is proposed for controlling the motion of a convoy of automated vehicles that behaves like a virtual train. It is supposed that the first vehicle plays leading role while the other members have local controllers trained for keeping a safe tracking distance between themselves and the preceding car/lorry. These local controllers are assumed to be almost linear for accelerations within a limited range but for higher values they have saturation at a maximum possible value modeling the limited resources of the available drives. To improve energy efficiency no dissipative term (viscous friction) is built in the interaction between the members of the train. Furthermore, the connection between the vehicles acts like an elastic spring with the distinction that no "reaction forces" exist in this model: the (n + 1)th member's motion is influenced by that of the nth member but the members do not affect the motion of the preceding ones. It is supposed that the leading vehicle can freely move, and the aim is to guarantee the possible smoothest motion for the last element of the train that may be followed by various participants of the road communication whose driving comfort has to be kept in mind. (The relative motion of the other elements of the convoy is regarded as some "interior affair" of no significance from the point of view of the public participants of the communication.) It is also assumed that each car has a sensor by the use of which it can measure the distance between its and the preceding car's center point, and that each element can communicate with the leading vehicle to exchange information on the position, velocity, and acceleration of the individual members in realtime. It is shown via computer simulation that keeping safe distance between the members without dissipation can be achieved by relatively "stiff" connection that may result in some "vibration" in the motion of the last vehicle. It is also shown that the adaptive control successfully eliminates this vibration that otherwise cannot simply be estimated due to the saturation in the driving force of the members that appears in the form of a cascade structure of embedded nonlinear functions in its equation of motion.