In this paper we considered the issues of controllability and stability of wheeled tractors on the slopes with the help of mathematical modeling and optimal control. The well-known methods of modeling and research for improving the stability of the tractor do not allow solving the problem of stable motion of the tractor on the slopes, as they do not provide sufficient correction of the moving direction, which depends on the character of external disturbances. The application of modern optimal control methods allows to investigate this problem at the design phase of the machine with using mathematical models. To solve the problem, we created the equations of motion of a wheeled tractor using the Lagrange equations of the second kind. On the basis of the equations of motion we developed models and algorithms for optimal control of a wheeled tractor. The necessary conditions for optimal control of the motion using the Pontryagin maximum principle were investigated. With the help of auxiliary functions of Hamilton-Pontryagin, we have determined the coefficients of stiffness and viscous resistance of wheel tractor tires. The boundary value problem of the maximum principle to determine the transient process motion of the tractor is formulated and on its basis the equations of horizontal and vertical oscillations of the tractor were solved at an uneven distribution of mass between the front and rear driven wheels and the coefficient of adhesion of the wheels and the lateral slip of the tractor in turning were calculated.

The article presents the methods of modeling the horizontal and vertical movement of a small truck Labo. Using the Lagrange equation of the second type, mathematical models of the movement of a small truck Labo on uneven roads developed. The mathematical models developed by the Runge-Kutta-number method solved and the values of the horizontal and vertical oscillations, speeds of movement and the forces acting and acting on each wheel of the automobile found. On the given masses, the parameters of the automobile’s leaf springs and the depreciation part are determined.