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The subject of study is the physical processes of translational and angular motion of a two-wheeled experimental sample. The goal is to develop physical, mathematical, and graphic models of the translational and angular motions of a two-wheeled experimental sample as an object of automatic control. The objectives: to form physical models of a two-wheeled experimental sample; to develop a nonlinear mathematical description of the processes of translational and angular sample`s motions using the Lagrange approach; to obtain a linearized mathematical sample`s description as an object of automatic control in the state space and frequency domain; to generate graphic models in the form of structural diagrams in the time and frequency domains; to analyze the functional properties of an object of automatic control: stability, controllability, observability, structural and signal diagnosability concerning violations of the functional properties of electric drives and sensors of the angular position of the body and wheels. The methods of the study: the Lagrange method, Taylor series, state-space method, Laplace transformations, Lyapunov, Kalman criteria, and diagnosability criterion. The results: physical models of a two-wheeled experimental sample have been obtained in the form of a kinematic diagram of the mechanical part and the electric circuit of an electric drive; mathematical descriptions of translational and angular motions have been developed in nonlinear and linearized forms; structural diagrams have been developed; functional characteristics of a two-wheeled experimental model as an object of automatic control have been analyzed to solve problems of control algorithms synthesis. Conclusions. The scientific novelty lies in obtaining new models that describe the translational and angular motion of a two-wheeled experimental model as an object of automatic control. The obtained models differ from the known ones by considering the dynamic properties of sensors and electric drives, as well as the relationship of movements.