Underground mining machines, such as wheel-tyre drilling rigs, are articulated and equipped with booms that project far beyond the undercarriage. Such a structure makes these machines prone to losing stability. Hence, it is necessary to analyse the distribution of masses and geometry as well as their broadly understood stability during the entire design process, taking into account many factors resulting from the manner and conditions of their operation. However, there are no appropriate computational models that would enable analytical tests to be carried out for machines with this kind of construction. This article is concerned with the author’s computational model, which allows the stability of single- and twin-boom drilling rigs to be quickly assessed. The model makes it possible to perform analyses without having to solve differential equations that are present in dynamic models or using specialist software based on CAD and CAE tools. The developed model allows determination of the pressure of wheels and jacks as a function of many important parameters and variables. Additionally, the distances of the centre of gravity from the tipping edge are calculated. The developed computational model was verified by comparing the obtained results with the results of the full dynamic model, the results of model tests carried out in the CAD/CAE program, and the results of empirical tests of wheel and jack pressures on the ground for the selected drilling rig. The model was subjected to verification and validation, which proved that it was fully correct and useful. The model was used to prepare a practical and user-friendly calculation sheet. Apart from the numerical values, the calculation sheet contains a graphical representation of the machine, the location of the centre of gravity, the tipping edges, as well as graphs of the wheel and jack pressures. Next, analytical tests of the stability of the selected drilling rig were carried out. The obtained calculation results are consistent with the results of empirical research. The computational model and the spreadsheet provide handy tools used during the design process by one of the Polish company’s producing drilling rigs.