# Modeling Movement Stability of Machine-Tractor Units Based on Modular Type Tractor
## 2 Theoretical Premises  
It is convenient to consider the agricultural MTU model functioning based on a modular tractor in its reactions to input control and disturbing influences. The indicated responses of the unit in the first case determine the controllability and stability of its movement in the second. In this case, the turning angle of the energy module front wheels (\(\alpha\)) is taken as the control action. Concerning the main assumptions and simplifications made in the work, it is assumed that the quantity and nature of the disturbing influences are different for each specific MTU and depend entirely on its design and functional purpose. To solve the problem, let us imagine that an MTU based on a modular tractor performs an even translational movement at a \(V_{o}\) velocity together with the XOY plane (Figure 2). In the process of performing work, the energy and technological modules of the tractor, under the influence of random disturbances, deviate from their original positions and receive additional velocities. At the same time, EM turns in the XOY plane through an angle \(\varphi\). The mass center of the entire tractor (point S), located on the axis passing through the vertical hinge of the TM, moves in the transverse direction by an amount determined by the \(X_{S}\) coordinate. The technological module performs an independent turning movement, the measure of which is the angle \(\psi\). Thus, an MTU based on a modular tractor with a free vertical hinge TM and front-steered wheels in the horizontal plane has three degrees of freedom, corresponding to three generalized coordinates: \(\varphi\) and \(\psi\) angles, and the \(X_{S}\) abscissa. The number of external forces acting on the MTU under consideration during its plane-parallel motion includes rolling resistance forces of the front (\(P_{fa}\)) and rear (\(P_{fb}\)) EM wheels and TM wheels (\(P_{fc}\)); gross traction forces of three tractor axles \(P_{ka}\), \(P_{kb}\), and \(P_{fc}\); lateral forces \(P_{sa}\), \(P_{sb}\), and \(P_{sc}\) leading to the emergence of yaw angles (\(\delta_{a}\), \(\delta_{b}\), and \(\delta_{c}\)) of the front and rear axles of the energy module and the bridge of the technological module, respectively; the components  
Figure 2: Diagram of forces acting on a modular tractor in a horizontal plane. of the main vector of forces (\(R_{nx},\ R_{ny}\)) reduced to the EM mass center, as well as the central moment of forces \(M_{P}\), acting from the front and side agricultural machines (diagram A, Figure 3); components of the main vector of forces (\(R_{mx},\ R_{my}\)) reduced to the TM mass center and the central moment of forces \(M_{C}\), acting from the rear and side machines (diagram B, Figure 3). As noted above, the joining of the technological module to the energy one in the horizontal plane can be either locked or hinged. Both design solutions provide at least three options for controlling the direction of the MTU operating movement based on this modular tractor (Table 1). From the point of view of technical feasibility, option 1 is the simplest since eliminating the angular mobility in the horizontal plane of the technological module relative to the energy one (option 1a) does not present any particular difficulty. From the standpoint of mathematical description, option 1b, due to the presence of an additional degree of freedom in the form of the TM turning angle (\(\psi\)), is the most complex. But since this is the most general case of a modular tractor diagram, from the MTU motion equations based on it, through appropriate transformations, mathematical models of units that implement all other methods of controlling a modular tractor listed in Table 1 can be obtained. In order to simplify the calculations for building the mathematical model, it was assumed that at small angles of rotation of the front wheels of the energy module \(\mathrm{tg}(\alpha)=\alpha\). This assumption led to a slight discrepancy in the units of measurement in the final formulas of the system of Equation (1).