An extended Gibbs-Appell (G-A) formulation is presented for the derivation of motion equations in variable-mass systems subject to holonomic and nonholonomic constraints. The formulation incorporates time-varying mass into the classical G-A framework, thereby enabling a rigorous treatment of dynamic systems in which mass distribution changes during operation. By employing quasi-velocities, the motion equations were expressed in a simplified form, eliminating the necessity of Lagrange multipliers. The methodology was demonstrated through the dynamic modeling of a mobile robot sprayer for precision agriculture, where the mass of the liquid tank decreased during spraying. In this application, wheeled motion constraints and joint mechanics were explicitly captured, allowing accurate representation of navigation and spraying dynamics. Numerical simulations were conducted in MATLAB, where a proportional-integral-derivative (PID) control algorithm was implemented to follow a prescribed circular trajectory. The results indicate a mean tracking error of 0.2346 m and a mean orientation error of 0.0039 rad, confirming the robustness of the proposed framework. Beyond agricultural robotics, the extended G-A formulation establishes a versatile foundation for the analysis of constrained variable-mass systems in aerospace engineering, robotic mobility, and other domains where dynamic mass variation significantly influences system performance.
Two-wheeled self-balancing vehicles inherently exhibit nonlinear, unstable, and strongly coupled dynamic characteristics, and their analysis and control remain of substantial relevance to military, industrial, and intelligent transportation applications. To address these challenges, a Bluetooth-enabled self-balancing vehicle system was designed with enhanced sensing, estimation, and hierarchical control capabilities. An improved Kalman filter (KF) algorithm was developed to overcome the limitations of conventional sensor fusion approaches. In the proposed method, gyroscope and accelerometer measurements were adaptively fused, enabling higher accuracy in attitude estimation while suppressing cumulative drift and transient disturbances. On this basis, a hierarchical proportional–integral–derivative (PID) control strategy was formulated to enhance responsiveness, stability, and tunability. Optimal attitude angles and reference velocities were processed within this framework to generate pulse-width modulation (PWM) signals for motor actuation. In parallel, a Bluetooth module was integrated to receive real-time commands from a mobile application, enabling precise execution of forward motion, reverse motion, and differential steering maneuvers. Experimental validation demonstrated that the system maintained stable posture, resisted external perturbations, responded rapidly to mobile control inputs, and executed commanded trajectories with high accuracy. The overall performance indicates that the proposed design provides a reliable and scalable platform for self-balancing vehicle research and offers potential applicability in human-robot interaction, intelligent mobility, and adaptive control studies.
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