Analysis and Validation of a Real‐Time AR‐IoT Training System via Nonlinear Dynamical Modeling

This paper presents a formally validated real‐time AR‐IoT firearm training framework, in which classical control‐theoretic tools are embedded as a system‐level validation layer rather than as novel theoretical contributions. Unlike prior AR‐based training systems that rely primarily on heuristic or empirical design, the proposed framework introduces a mathematically grounded structure that explicitly accounts for user interactions, synchronization delays, and energy constraints encountered in real‐world deployment. The system is modeled using discrete‐time dynamical equations under bounded disturbances, and five theoretical assumptions related to state boundedness, delay tolerance, and energy consumption are formalized. Classical stability and input‐to‐state stability (ISS) concepts are employed to provide an analysis‐based validation of boundedness, synchronization accuracy, and robustness, serving as a formal certification mechanism for the training system. A set of standard analytical results (recalling classical nonlinear stability/ISS tools) is used and supported by numerical experiments under diverse conditions, including sensor noise and delayed inputs. The theoretical framework is further implemented in a practical AR‐IoT firearm training prototype that integrates marker‐based AR overlays, Bluetooth‐enabled microcontrollers, and IPSC‐compliant scoring mechanisms. Real‐time data from laser‐triggered events, motion sensors, and visual feedback are processed to deliver adaptive scoring and trainee guidance. Experimental results demonstrate that the system consistently converges to stable feedback patterns while maintaining low‐latency performance under dynamic conditions. Overall, this work contributes a system‐level integration of mathematical validation and real‐time deployment, offering a scalable and verifiably reliable solution for immersive firearm training in safety‐critical environments.