INVESTIGATION OF ROAD TRAIN DIVERGENT STABILITY LOSS WHEN MOVING ALONG A PROGRAM TRAJECTORY

The object of research is transport driven multilink wheel systems. In the development of the realized possibility of controlled movement of a semi-trailer truck along a program trajectory, the possibility of constructing a bifurcation set by a velocity parameter is considered. The velocity value is calculated for each discrete value of the calculated real trajectory. The trajectory can be specified in an explicit, implicit, parametric form or by the law of variation of the curvature radius. The study of this parameter is one of the most problematic places for analyzing the stability of the movement of a road train. Changes in this parameter at certain values, called bifurcation, lead to changes in the qualitative structure of the solutions of the system of differential equations and, as a result, divergent stability of the road train. For such an investigation of the phenomenon, the method of continuation by parameter and the first Lyapunov method are applied. During the research, many bifurcation velocity values are obtained. This is due to the fact that the proposed approach has a number of features, in particular, an iteration is performed for all control parameters of the program trajectory, and for each such value, the velocity has been iterated until its bifurcation value is reached. At each iteration, the roots of the characteristic equation are checked for the presence of at least one root with a positive real part, which corresponds to the bifurcation value of the parameter of the velocity of a road train according to Lyapunov. Due to this, it is possible to obtain this set by an exclusively analytical method using computer calculations, without resorting to the use of graphic-analytical methods. Obtaining these bifurcation sets can practically be used both to limit the velocity of a road train and to warn of its excess. Compared with similar known methods, this provides such advantages as a significant acceleration of the construction of this set and, as a result, its use in real time.


Introduction
From the point of view of the safety of the movement of a semitrailer truck, special attention should be paid to the analysis of the conditions under which an abrupt change in the orientation of the semitrailer is possible (internal and exter nal). This is the socalled fold bifurcation, which corresponds to the divergent loss of stability of the circular stationary mode with a variation of the longitudinal velocity. The set of parameters at which such abrupt transitions of stationary states occur is a critical set or a bifurcation set. Among them, for the system under consideration, it is possible to distin guish both control parameters (movement velocity, steering wheel angle of rotation) and characteristic parameters (linear dimensions of the system components, their masses, etc.). This research addresses the task of building a bifurcation set of velocities for a synthesized [1] controlled motion along a given programmed trajectory containing a set of coordinates of the tractor and trailer, the angle of rotation of the steering wheels and the folding angle.
Obtaining set of these parameters is relevant because it allows to organize the management of the road train using feedback. Planned studies can provide additional control over the maximum allowable velocity of movement in or der to avoid bifurcation effects, thereby contributing to increased traffic safety.

The object of research and its technological audit
The object of research is transport driven multilink wheel systems. A flat bicycle model of the movement of the semitrailer truck [2] is considered. The articulated crew diagram is shown in Fig. 1, where: v -longitudinal component of the center of mass of the tractor; θ -rotation angle of the controlled module; a, b -distance from the center of mass of the tractor to the centers of the front (steered) axis and the rear axle of the tractor; c -distance from the center of mass of the tractor to the point of coupling with the second link; d 1 -distance from the center of mass of the second link to the hitch point with the tractor; Y i -removal forces on the axes, drag coefficients on the axes (k 1 ; k 2 ; k 3 ); TECHNOLOGY AUDIT AND PRODUCTION RESERVES -№ 6/2(44), 2018 ISSN 2226-3780 m -tractor mass; u -transverse projection of the velocity vector of the center of mass of the tractor; ω -angular velocity of the tractor relative to the vertical axis; m 2 -mass of the second link; v 1 , u 1 -longitudinal and transverse projections of the velocity vector of the center of mass of the semitrailer; j -folding angle (the angle between the longitudinal axis of the tractor and the semitrailer).
The subject of research is mathematical models of the dy namics of controlled motion of multilink wheeled transport systems with the possibility of divergent loss of stability.
One of the most problematic places in the movement of road trains is the possibility of divergent loss of stability when the maximum permissible velocities are exceeded. The construction of such a set of velocities when moving along a program trajectory is implemented in this research with the aim of improving traffic safety.

The aim and objectives of research
The aim of research is realization of the possibility of analytically constructing a bifurcation set of velocities when moving along a controlled program trajectory using the first Lyapunov method for calculating bifurcation values together with the method of continuation by parameter.
To achieve this aim, the following objectives are: 1. Numerical integration of a system of differential equa tions with a synthesized control law.
2. Construction of a bifurcation set of velocities for a discrete set of control values using the first Lyapunov method and the method of continuation over a parameter by iterating over velocity.
3. Visualization of the set of critical control parame ters (bifurcation sets) for various program trajectories and modeling (visualization) of the system motion.

Research of existing solutions of the problem
Among the studies of the problem of loss of diver gent stability by road train, let's note the work on real bifurcations of twolink systems with rolling [3]. The is sues of qualitative analysis of nonlinear models of wheeled vehicles with the involvement of elements of the theory of bifurcations are studied in [4]. In [5], an approach is implemented that allows one to obtain a more com plete representation of the bifurcation set of the model of a semitrailer truck in an analytical form. It should be noted also the work on the determination and analysis of the stability of circular stationary driving modes of the model of a semitrailer truck [6,7]. The analysis of the movement stability of a semitrailer truck model from the point of view of determining its maneuverability is also considered in [8,9]. Mathematical modeling of a semi trailer truck with a controlled semitrailer and analysis of its stability is performed in [10]. The studies mentioned above, as a rule, are aimed at the further development of the grapho analytical approach to analyzing the set of sta tionary modes of the nonlinear model of a two link trailer using the ideas of bifurcation analysis. However, the problems of divergent loss of stability of a train when moving along a program trajec tory are not considered in these studies and are proposed for the first time in this paper.
Some studies [11,12] are purely experimental in nature and are useful in creating a specialized robotic installation tractorsemitrailer, which rep resents a certain rarity. In connection with the proposed further experiments with this installa tion and its improvement, attention is drawn to specialized studies on unstable driving conditions at high velocities [13]. And also on the features of the synthesis of safe control for AHV (Articulated Heavy Vehicle) systems [14].
Of particular interest are studies of the stability of the yaw of a tractorsemitrailer system in constant driving conditions, analyzed using the theory of bifurcations [15].
Note that recently the theory of bifurcations and the corresponding analytical methods have been successfully applied to the study and control of some engineering systems, such as associated satellite systems, jet engine compressors, longitudinal flight dynamics and system power [16,17]. The technologies of bifurcation analy sis described in these studies are also of considerable interest.
In addition to the divergence loss of stability considered in this research (fold bifurcation), the Hopf bifurcation (limit cycle) is also studied in papers [18,19], which also occurs during the road train movement.
An analysis of these publications leads to the conclu sion in favor of the proposed research methods that allow for effective computer implementation.

Methods of research
In this research, an exclusively analytical approach is used using the first Lyapunov method [20] and the parameter continuation method [21,22], while practically identical results are obtained for identical models.
To solve these problems, let's the methods of the dy namics of a system of connected bodies, the mathematical apparatus of the theory of stability, the theory of bifurca tions and controls, symbolic transformations and numerical methods, and heuristic search algorithms.
In these expressions (the system of differential equa tions of motion): -  (chp,lambda)).
The application determines the moment of the appea rance of the root with a nonnegative real part by varying the velocity value. Fig. 2 is general view of the application with the pre bifurcation state of the road train (all the roots of the characteristic equation with a negative real part). Divergent instability is realized at a velocity of v = = 11.5 m/s -obtained in [5], based on the numerical analytical method of continuation with two parameters.
Using the developed application, it is possible to set an arbitrarily small calculation error; a more accurate value of 11.5009 m/s is obtained.
The bifurcation velocity value is obtained by the con tinuation method with respect to the parameter with a step of 0.0001 (this is the required accuracy), in Fig. 3 -loss of divergent stability -the appearance of roots with a po sitive real part.
To build a set of bifurcation values, iteration is per formed over all control parameters of the program tra jectory, and for each such value, the velocity is iterated until its bifurcation value is reached. To velocity up such a process, initial (threshold) values of velocities are em pirically determined. At each iteration, the roots of the characteristic equation are checked for the presence of at least one root with a positive real part, which will correspond to the bifurcation value of the parameter of the velocity of movement of the train.
Thus, Fig. 4 shows the generated bifurcation sets of velocities for various program trajectories. In this case, the bifurcation set of velocities in m/s is displayed in red, the control in degrees is green.
In this case, the parabola is set: The construction of a bifurcation set makes it pos sible to determine the maximum allowable velocity of mo vement along a program trajectory without divergent loss of stability; this is the minimum velocity value in the bi furcation set. At subcritical velocity, the motion is stable, the program and real trajectories (red and blue) coincide (Fig. 5).
When it is exceeded, bifurcation phenomena are ob served (Fig. 6, the real trajectory is shown in blue). TECHNOLOGY AUDIT AND PRODUCTION RESERVES -№ 6/2(44), 2018 ISSN 2226-3780  For the initial model, the divergent loss of stability for this maneuver is realized at a velocity of 11.3 m/s -its minimum value in the bifurcation set. The lower the velocity, the larger part of the clothoid can be passed without divergent buckling.
In addition to obtaining bifurcation velocities, the ap plication allows to vary the control parameter (at a constant velocity or simultaneously). It is also possible to vary the characteristic parameters, which, obviously, is important at the design stage of twolink systems, constructing ap propriate bifurcation sets for them.

SWOT analysis of research results
Strengths. The strengths of the proposed method in clude the fact that the basis of the mathematical model of the controlled motion of semitrailer truck is based on the classical principles of the mechanics of a system of solids (taking into account the presence of nonholonomic bonds). The task of synthesizing softwarecontrolled motion is solved on the basis of strictly grounded approaches of the theory of automatic control and control of dynamic systems. The correctness of the obtained results is veri fied on the basis of an independent numerical simulation of the system. The implementation of the construction of a bifurca tion set of velocities by an exclusively analytical method allows to significantly velocity up the process of such construction and makes it possible to apply this approach in real time, informing the driver of the road train about the inadmissibility of velocity.
Weaknesses. The weak sides should be attributed to the insufficient accumulation of experimental empirical material, although the experiments carried out with a spe cially developed robotic installation (tractorsemitrailer) show good agreement with the theory. The developed in stallation at the moment has a number of design flaws (limited control, wheel slip), which requires its technical improvement.
Opportunities. Opportunities for further research are the technical improvement of the experimental setup de veloped and the further study of driver's maneuvers with the construction of bifurcation sets for them.
Threats. When introducing the results of this research, it is necessary to develop a technical device with func tions that help improve the safety and efficiency of driving a vehicle by obtaining values from a bifurcation set of velocities in real time. For each alert, the device can play a beep or display information on the display.

Conclusions
1. A numerical integration of a system of differential equations with a synthesized control law in the system of symbolic Maple calculations is performed. Such integra tion is performed on each discrete controlled trajectory in a specially designed application. The results of numerical integration are visualized in the form of data of phase portraits and solutions of the characteristic equation of the system.
2. The bifurcation velocity set is constructed for a dis crete set of control values using the first Lyapunov method and the parameter continuation method by iterating over velocity. The method of continuation by parameter is also applied to the control and the characteristic parameters of the model, which makes it possible to obtain bifurcation sets depending on changes in these parameters. These re sults can be used at the design stage of multilink systems, determining their optimal linear dimensions.
3. Visualization of a set of critical parameters of velo city and control for various software trajectories (typical driver maneuvers) and modeling of the system movement in the Unity 3D environment and using the developed robotic installation are implemented. This visualization allows to explore the movement of a road train at the time of bifurcation and develop recommendations for the management of a road train to get out of this situation. In the absence of slippage and surface irregularities, the experimental results of the installation correspond to the theoretical ones.