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.
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