Abstract: This article is devoted to the dynamical analysis of the two-wheeled self-balancing vehicles in the horizontal plane. We obtained a modified dynamical formulation of the system previously. This new formulation is more detailed, in comparison with the commonly used formulations. The most important difference is the appearing of a new nonlinear coupling term in our dynamical equations. This term has significant effects on the dynamical behavior of the system. In this paper the effect of the new term on the free response of the system is shown and paths of the chassis in the horizontal plane are depicted under different initial conditions. The importance of this term and its effects on the paths of the system is shown by different simulations.