Model of Carriage Movement on Plane with Dry Friction Forces

hekkoki@gmail.com Abstract— Wheel slip model is an important aspect of vehicle driving stabil-ity and safety control. However, in most of the carriage movement models friction forces and wheel slip effect are being neglected. This paper raises the problem of wheel slip in dangerous driving and autonomous vehicles under critical driving modes, then tasks the modeling of movement of an individual vehicle is considered. These are two tasks of nonholonomic mechanics: 1) the movement of a wheelset without wheel slip (two disks freely mounted on an axis) along an inclined plane in the field of gravity, and 2) the movement of a flat wheel model, which, under certain assumptions, can be a four-wheeled carriage. In these tasks, in addition to the standard dynamics (continuous motion without wheel slip), critical situations can also be observed associated with wheel sleep and separation of wheels from


Dangerous driving
The steady increase in motorization has made personal vehicles widely available. The number of cars in the streets is constantly increasing. Due to the fact that the accident rate of road transport causes enormous physical and financial damage to society in general and citizens in particular, the problem of transport safety is one of the priority tasks for any government [1]. This problem is especially acute in large cities where saturated complex transport networks are overloaded and population density is up to maximum limit [2]. There are many reasons why traffic accidents occur. Among them are natural disasters, destruction of the roadway, slippery road surface, failure of traffic lights, ignoring road signs and markings, inattention of road users to the situation on the road and to other road users, state of alcohol or drug affect, strokes and sudden health problems of drivers, as well as an unbalanced mental state of mind or a state of panic. Since the road transport network is a complex system, in order to maintain normal functioning, it requires constant involvement and attention from all its participants [3]. Therefore, any neglect of this complex system in favor of driver's personal interests can cause fatal consequences.
More and more the situation of the constant use of mobile phones while driving is used as an example of such attitude. However, another, no less frequently occurring side of the problem is aggressive driving or driving in a state of panic. Panic and aggressiveness from the psychological point of view lead to the fact when a person should not feel like a member of a group, feel fear for his own life, and begins to act logically, but not in the laws and not according to the rules of groups, but in his own individual manner, violating the coherence of the whole system [4]. In this state, the driver can make decisions that are unpredictable for other participants, so they may either not have time to react, or these actions will provoke emergency or panic actions, which will lead to a dangerous situation or to a collision. This phenomenon has various names: dangerous driving, aggressive driving, careless driving and reckless driving. Laws of many countries provide for various fines and other penalties for such actions. The Government of the Russian Federation identifies the following types of dangerous driving [5]: • Multiple and sudden lane changes • Lane change with small gaps (headways) • Tailgating (decrease of headways) • Decreasing lateral interval • Emergency breaking and rubber-necking • Obstructing overtaking Nevertheless, despite the apparent clarity of the classification of dangerous driving, there are many questions from the law point of view. As noted on the official website of the Road Traffic Organization of the Russian Federation, "the line between creating a threat is very thin, subjective, and uncertain" [6]. Therefore, in most cases it becomes possible to qualify dangerous driving only after accident occurred. And the punishment for dangerous driving, if it is proved to be the cause of the accident, is additional to the main consequences for the immediate actual accident.
The attempts to formulate the qualitative characteristics of such type of dangerous driving as "Multiple and sudden lane changes" by analyzing the vehicle's trajectory along GPS coordinates were made by our research team in the article [7].

Autonomous transport
The high injury risk of road transport caused by the instability of human psychology can be eliminated by increasing the level of autonomy of the car. Most specialists believe that the exclusion of the human factor from the transport system will allow reducing accident rate and optimizing the efficiency of transport networks.
Self-driving cars, or autonomous vehicles (AVs), connected and autonomous vehicles (CAVs), driverless cars, robo-cars or robotic cars, development of which begin history back in the 20s of the last century, have become an obvious trend of our time, and plans for the development and implementation of autonomous vehicles have become part of the national programs of many countries, automotive manufacturing companies, as well as institutes and research centers. SAE International has established 6 levels of autonomy for self-driving cars (See Fig.1). These levels from zero to fifth characterize the degree of control of artificial intelligence over the vehicle and the degree of human involvement in the process of driving. Level zero is a car without any automation; ABS and ordinary cruise control do not count. Level 1 includes cars with steering and braking control, adaptive cruise control. Level 2 cars can control taxiing and braking in certain areas, such as highways, but the driver must still be vigilant and be ready to take control at any time. Level 3 allows the driver not to watch the headway all the time. Level 4 is already a full autopilot, but it only works in certain conditions and in a marked area, it even allows the driver to take a nap. Level 5 is a complete autopilot without any restrictions. No steering wheel is required at this level [8]. The implementation of the autonomous vehicle system includes the following hardware elements: cameras and optical instruments, system-on-chip (SoC), radar, a communication system with other vehicles and infrastructure (V2V/V2I), active remote sensing system LiDAR (Light Detection and Ranging), converters of digital signal to mechanical actions, ultrasonic sensors, GPS, odometric sensors, etc., as well as software elements such as: high-resolution maps, methods of simultaneous localization and map building (SLAM, simultaneous localization and mapping), algorithms for data analysis and processing, recognition of obstacles and objects, prediction of trajectories and actions of moving objects, decision-making algorithms, such as planning routes, making maneuvers, accelerating, braking, taxiing, an operating system for driving a car, a supervisory control system for identifying and eliminating errors, as well as a system for interacting with an external data collection center and cloud services [9].
The development of robotics, machine vision, artificial intelligence, increasing the speed of data processing made it possible to implement autonomous vehicle. But the new concept of transport systems without the active participation of the person who delegated the right to drive to autonomous systems, together with solving a number of traditional issues, brings an even greater number of new problems. These include problems of liability in the case of damage, loss of the ability to drive a vehicle independently, reliability and vulnerability of software, loss of privacy, the danger of using autonomous cars for terrorist activities, the ethical issue of minimizing casualties when making decisions in critical situations etc. [10].
Another important aspect of driving in extreme situations is the study of the mechanical characteristics of the adhesion of the wheels of a moving car to the roadway. The fact is that when performing a speed maneuver while decreasing the distance between the cars, the driver makes decisions based on his experience, may not take into account all conditions, such as abrasion of treads, defects in the road surface, various degrees of moisture in the roadway, etc. This problem is relevant both for those cases when a person is driving a car, and for improving the control algorithms of autonomous vehicles.

2
Description of the Wheelset Model

Description of the wheelset system without wheel slip
Consider a wheelset on an inclined plane, shown in Figure 2. This design consists of two homogeneous disks, each with mass т and radius b, freely mounted on the axis O1O2, which is a uniform axle of mass М and length 2l. Thus, disks (wheels) can freely and independently rotate around the axis О1О2 without friction.

Fig. 2. Wheelset on surface
The described system moves under the influence of gravity along a rough inclined plane (the angle of inclination to the horizontal is а) so that at the points of contact of the wheels with the plane (points Р1, Р2, P1, P2, see Fig. 2), and no wheel slip (neither lateral nor longitudinal) occurs.

Basic kinematic variables
Let Сbe the center of mass of the system (it is clear that it is in the middle of the segment О1О2), and С -is the coordinate system rigidly connected with the axis О1О2, moreover С is directed along the axis О1О2, Сdirected upward along the normally inclined plane, and Сis perpendicular to С and С and forms the right coordinate system with them.
The movement of the wheelset occurs under the influence of gravity and reaction forces of the plane that occur at the contact points Р1 and Р2. These forces are shown in fig. 2: N1, N2normal reaction forces, and the values of their projections onto the axis С should be non-negative (N1⩾0, N2⩾0), F1, F2lateral, and F1, F2longitudinal forces adhesion (static friction forces). In the future, we will assume that separation does not occur and F1, N2 are substantially positive values. In addition, let the friction coefficient of the reference plane be so large that there is no wheel slip at points Р1, Р2, thus VP1 = VP2 = 0.
Let 1, 2 be the proper angles of rotation of the wheels relative to the axis О1О2,  is the angle of rotation of the axis О1О2 in an inclined plane, counted counterclockwise from the line of the largest slope (if viewed from the positive end of the axis С).
Thus, the angular velocity of the axis О1О2 (and hence the of system С) in the projections on the axis , ,  is: The angular velocity of each of the wheels, in projections on the same axis: Let's find the speed of point О1, considering it to belong to the wheel 1: Hence, since 1 = 0, we obtain: On the other hand, since the point О1 belongs to the axis О1О2, we obtain: or, in projections on the axis , ,  we obtain: Similar logic for the point О2 leads to the relations: Thus, no wheel slip condition leads to the following conclusions: 1. The speed of point С has a nonzero projection only onto the longitudinal axis Сh, we will denote it hereafter by , which is u x = 0, u h = u, u z = 0 (1.5) 2. Two relations are satisfied:

The equation of motion of wheelset without wheel slip
The equations of motion of the center of mass of the system in the axes xhz, taking into account (1.1) and (1.4), we obtain: Where 0 = + 2 is the aggregate mass of the system.

Analysis of the system of equations (1.6)-(1.10)
Let the initial conditions be: { (0), j(0), j(0), ẏ1(0), ẏ2(0)} such that equalities (1.5) and (1.6) are satisfied at t=0 (the moment of start of the wheelset movement in the plane, so that the kinematic coupling conditions are not violated). It is required to determine the solution of system (1.6)-(1.10), which satisfies the continuity condition: > 0, = 1,2. In a more accurate statement, if the friction coefficient f of the plane with wheels is specified, it is also necessary to satisfy Coulomb's inequalities in the process of motion: This kind of problem was considered for the Chapylygin sleigh on an inclined plane with friction in work [11].
As shown in [12], the solution has the form: Let x be the coordinate of the center of mass C along the line of the largest slope (down the inclined plane), ybe the coordinate perpendicular to it, then we have:
Integration of the last equalities leads to the conclusion that the center of mass of motion along the cycloid perpendicular to the line of the largest slope, that is, the wheelset does not roll down.

Critical situations
Example: For example, for 0 = 0, 0 = 0 we have < /(2 ), that is, a rollover can happen for sufficiently large a: > /(2 ). If inequality (1.15) is violated, then there will definitely come a point in time that a normal reaction is zero, for example, 1 . So, let 1 ( 1 ) = 0 and further movement is possible only with separation of wheel 1 (while maintaining the contact of wheel 2). Consider the beginning of such a separation, that is, the resulting rotation of the wheelset around the axis h. We will determine such a rotation by the angle x, count it counterclockwise, if we look from the positive part of the axis h (see Fig. 2). Considering the further movement, we assume that ≈ 0,̇≈ 0,̈≠ 0. Then we have: It is clear that choosing 0 small enough, we can achieve large ( 1 ), and formula (1.13) then gives us ⃛ < 0, that is, a paradoxical situation arises when the separation occurs. These considerations lead us to the conclusion that, in cases where there can be a zeroing of the normal reaction, it is necessary to take into account the finite coefficient of friction and consider the process of movement of the wheelset with possible transitions to the situation of wheel slippage, which, of course, complicates the solution of the problem.

3
Description of the Carriage Model Figure 3 shows a flat wheel model of two wheelsets 1 2 , 1 2 , connected at their centers of mass by a rigid axle AB. The joints in connection points A and B are assumed to be ideal and cylindrical, and the rolling of wheels 1 , 2 , 1 , 2 along the supporting plane occurs without slipping, i.e. a nonholonomic connection is imposed.

Model description
In this model, we assume (same as in [1][2][3][4]), that this coupling is realized only by the transverse forces and , applied to 1 2 and 1 2 , respectively (see Fig. 3). The consideration of longitudinal forces was considered, for example, in paper [7].
The presence of a non-holonomic connection gives the right to assume that the velocities of the centers of mass of the wheelsets, denoted as and , are always directed perpendicular to the corresponding axes, i.e. ⊥ 1 2 , ⊥ 1 2 (see Fig.3).

Description of symbols
Further, the following notation is introduced: 1 , 2the deflection angles of the wheelsets 1 , 2 , 1 , 2 from the longitudinal axis of the flat wheel model, and 1 is counted clockwise, and, 2 is counterclockwise; = , = , C is the center of mass of the entire system, which we assume to be unchanged, that = , = ; is the angle of rotation of the axle AB relative to the invariably oriented axis , counted counterclockwise; P is the instantaneous velocity center of the axle AB, which, due to nonholonomic constraints, is the intersection of lines 1 2 and 1 2 ; , , are the masses of axle AB, of wheelsets 1 , 2 , 1 , 2 ; is the moment of inertia of the axle AB together with the concentrated masses at its ends relative to point C; , are the moments of inertia of the wheel pairs, respectively

Equations of motion
For the three angles , 1 , 2 we can write three equations of dynamics that do not contain unknown forces and reaction forces in the hinges A and B:  .3) are the equations of kinetic moments of wheel sets relative to their own centers of mass. The following notation is introduced in these equations: = + + is the total mass of the system; is the kinetic moment of the system relative to the point P, is the speed of the point P (as geometric relative to the motionless coordinate system), is the velocity of the center of mass C; , control moments in joints A and B (if there are any). Let D be the projection of the point P onto the axle AB (see Fig 3). Denote = 0 . Then we have, in accordance with Koenig's theorem: After several operations we obtain:

Case of the equation with fixed angles
We control the movement of the wheelset, that and so that 1 = 2 = 0 (the same deviations of the wheelsets, but in different directions). Then from (2.2) (2.6)

Fixed wheelset
Consider a special case: a fixed wheelset 1 , 2 , that 1 = 0, then the energy integral (2.6) has the form of: .
It is easy to show that all these integrals are taken in elementary functions. Thus, in this case, the equations of motion of the system are integrated in time functions for any parameter values and initial conditions. The resulting equation is the main equation for the movement of a controlled wheelset.

Conclusion
In this paper we've found equation for the movement of a wheelset without wheel slip, where two wheels freely mounted on an axis and moving along an inclined plane in the field of gravity, another equation of movement of a wheel model, which, under certain assumptions, can be a four-wheeled carriage. In addition to dynamics of continuous motion without wheel slip, critical situations were observed associated with wheel slip and separation of the carriage wheels from the plane. The related parameters were found and considered.