The article discusses the theoretical issues of developing an ultrasonic drill for drilling soil on celestial bodies as part of a robotic platform and formulates the tasks of its control. The specifics of using the drill require a wide temperature range during operation, low energy consumption, an efficient operating cycle and the requirement to ensure the preservation of the structure and composition of the soil when exposed to the tool. One of the promising areas of development is devices with vibration decoupling from the ground by ultrasound due to the transformation of ultrasonic vibrations into repeated impacts of sound frequency using a specially introduced “free” body between the ultrasonic and impact parts of the drill (bit). This allows continuous drilling regardless of the condition of the impact bit, including jamming. The purpose of the work is to select the most effective drill design and compile a mathematical model that provides simulation modeling. Materials and methods. Technical solutions of both domestic and foreign authors were considered, and a prototype was selected. Theoretical works were analyzed and it was found that the materials presented in them do not contain detailed mathematical models necessary for simulation modeling at the stage of design and optimization of the structure. In the theoretical part of the article, mathematical models are proposed, obtained on the basis of equivalent parameters found from a distributed description of the drill design elements. Methods of mathematical physics, classical mechanics and similarity theory were used. The results are that two alternative mathematical models are developed. One of the models has a continuous description with a significantly nonlinear rigidity of the “wall” upon impact, while the impact motion is considered as a type of self-oscillation. The second model is based on the theory of impact. Conclusion. A parametric study was carried out and the dependences of the oscillation frequency of a free body on the size of the impact gap, body mass and amplitude of oscillations of the concentrator, etc. were obtained. The considered models can be used for simulation modeling when developing a drill design. The tasks of controlling a robotic drilling platform are formulated from the position of ensuring the operability of the drill when drilling various soils.