A mathematical model of the evolution of the human–machine interface based on entropy, trust, and digital twins

The article considers the proposed formalization of the human–machine interface as a dynamic system that dynamically transitions from a state of empirical inconsistency to a more orderly and predictable interaction in the “human – robot – environment” loop. A mathematical model is proposed that describes the process of human-machine interface interaction through the integral characteristics of entropy, trust, average connectivity and operator participation weight. This approach allows us to move from an intuitive description to a formalized analysis of the evolution of the system. The model is built in the form of an operator structure that combines the projection of the physical state into the space of digital twins, the generation and evaluation of scenarios, the formation of the interaction structure and the selection of control actions taking into account safety constraints. Special attention is paid to the role of environmental uncertainties that determine the level of entropy and significantly affect the speed and nature of the system’s transition to a consistent regime. It is shown that under conditions when the system accumulates information faster than the level of random disturbances increases, there is a gradual decrease in entropy and a strengthening of connections between its components. This naturally leads to an increase in trust between man and machine. In turn, trust determines to what extent the system can take on part of the control functions, reducing the load on the operator. For a generalized assessment of efficiency, an integral index is additionally introduced, which allows quantitatively comparing different modes of operation. Practical verification of the model was performed in the Gazebo environment using ensemble forecasting and an adaptive interaction matrix. The results obtained coincide with the theoretical conclusions and demonstrate stable dynamics of the main characteristics of the system. In particular, compared with the basic approaches, a decrease in entropy by 25–40 %, an increase in trust by more than two times, as well as a significant decrease in the load on the operator are observed. Thus, the obtained results confirm that the transition to structured interaction occurs primarily in the space of digital twins and is provided by a closed loop of prediction, coordination and decision–making. The proposed approach can be used as a basis for the development of adaptive human–machine interfaces in robotic systems operating under uncertainty.