Analysis of the Minimum Required Coefficient of Sliding Friction at Brachistochronic Motion of a Nonholonomic Mechanical System

The paper analyzes the problem of brachistochronic motion of a nonholonomic mechanical system, using an example of a simple car model. The system moves between two default positions at an unaltered value of the mechanical energy during motion. Differential equations of motion, containing the reaction of nonholonomic constraints and control forces, are obtained on the basis of general theorems of dynamics. Here, this is more appropriate than some other methods of analytical mechanics applied to nonholonomic systems, where the provision of a subsequent physical interpretation of the multipliers of constraints is required to solve this problem. By the appropriate choice of the parameters of state as simple a task of optimal control as possible is obtained in this case, which is solved by the application of the Pontryagin maximum principle. Numerical solution of the two-point boundary value problem is obtained by the method of shooting. Based on the thus acquired brachistochronic motion, the active control forces are determined as well as the reaction of constraints. Using the Coulomb laws of friction sliding, the minimum value of the coefficient of friction is determined to avoid car skidding at the points of contact with the ground.


INTRODUCTION
As is well known the classical brachistochrone problem was proposed by Johann Bernoulli in 1696 for the case of a particle moving in a vertical plane under the influence of its own gravity in a homogeneous field of gravity.Much later, the generalization of the classical brachistochrone problem was carried out within the calculus of variations [1] and [2].A detailed review of literature related to the problems of brachistochronic motion can be found in [3] and [7].The problems considered in the present paper involve a review of references on the Bernoulli's case of the classical brachistochrone extended to the system of rigid bodies.Paper [4] considers the Bernoulli's brachistochrone problem extended to the multibody system in the form of a closed kinematic chain without external constraints.Paper [5] considers a special case of the multibody system whose motion is limited by external constraints.
The brachistochronic motion of a multibody system in a stationary field of potential forces, and Coulomb friction, for a special case, was solved in [6] where it has been shown that there is a complete analogy between the brachistochronic motion of a mechanical system with 2-DOF, whose metric tensor is constant, and the brachistochronic motion of a particle acted on by the forces of viscous and Coulomb friction.
Considerations of the brachistochronic motion of a nonholonomic rheonomic mechanical system acted upon not only by control forces but also by both potential and non-potential forces can be found in [7].The results obtained in [6] were extended in [8], where the brachistochronic motion of a mechanical system with unilateral constraints was considered.This paper, using the example of a nonholonomic mechanical system with limited reactions of constraints, presents the procedure of creating the differential equations of motion where both reactions of nonholonomic constraints and control forces figure, based on the general theorems of dynamics [9].Applying the general theorems of dynamics, one obtains, on one hand, more suitable relations in determining the reactions of constraints and control forces, while, on the other hand, physical interpretation of the multipliers of constraints is unnecessary as in the case of applying some other equations of analytical mechanics that refer to nonholonomic mechanical systems [10].This paper also provides the procedure for solving brachistochronic motion of a nonholonomic mechanical system in a plane at the steady value of mechanical energy during motion, when initial and end positions are specified [14].
The formulated brachistochrone problem, with a corresponding choice of the quantities of state, was solved, in this case, as the simplest problem of optimal control by applying the Pontryagin maximum principle [11] and [12].Simultaneously, control forces were also determined so as to realize the brachistochronic motion.The Coulomb laws of sliding friction were the basis for specifying the minimum required value of the coefficient of sliding friction, so that the mechanical system moves in accordance with the nonholonomic bilateral constraints.Thus determined value of the coefficient of sliding friction prevents the occurrence of the system slipping during the entire motion.
The numerical procedure for solving the two-point boundary value problem is based on the shooting method [13].Also, estimates are provided for the intervals of parameters' values being determined.

DESCRIPTION OF A NONHOLONOMIC SYSTEM MODEL
In order to generate differential equations of motion of a nonholonomic mechanical system, using the example of a simplified vehicle model (see Fig. where moments of inertia are known 1 J and 2 J around the main central axes of inertia perpendicular to the plane Axy , respectively, having in mind that is 1 2 J J  .The mass of rear axle and the mass of wheels are disregarded.The vehicle configuration relative to the system Oξη is defined by a set of Lagrangian coordinates   q ,q ,q ,q , where φ  is the angle between the axis Oξ and axis Ax , while 4  q θ  is the angle between the axis Ay and the vehicle front axle axis. Further analysis refers to the case when point A cannot move in the direction of the front axle axis, while point B of the vehicle cannot move in the direction of the rear axle axis (lateral slipping of the front and rear axle is prevented).Due to the imposition of constraints to the motion, there occur horizontal reactions sin cos respectively.Such vehicle motion is limited by two ideal independent nonholonomic constraints The consequence of the imposed constraints to the vehicle motion is that velocityV  of the point B has the direction of the axis Ax , so that relations (1) can be expressed in the following form cos sin tan where During motion, the vehicle is acted on by the control force L , proportionate to the relative angle of the front axle rotation around the vertical axis, where 3 3 L k θ  .Now, the differential equations of vehicle motion, based on the theory of change in momentum and moment of momentum for a movable axis that passes through point B and is perpendicular to the plane of motion [9], have the form where The differential equation of vehicle front axle rotation around the axis that passes through point A, and is perpendicular to the plane of motion, has the form Solving the system of equations ( 4) and ( 5), the reactions of nonholonomic constraints are obtained, as well as the control force and control moment, so as to realize the brachistochronic motion, in function of introduced quantitities of state and corresponding derivatives Also, during vehicle brachistochronic motion the law of the conservation of mechanical energy holds

BRACHISTOCHRONIC MOTION AS THE PROBLEM OF OPTIMAL CONTROL
In this section, using the example of a simplified model of a vehicle with constrained motion, we will formulate the problem of brachistochronic motion as the problem of optimal control.The equations of state that describe the motion of the considered system in state space can be defined in the form where controls 1 u and 2 u represent the vehicle point B velocity and angular velocity vehicle, respectively.
Relative angle of the front axle rotation θ can be given on the basis of (2).
The quantities of state ,

B B
ξ η and φ were determined at the initiation time moment while quantities of state , ξ η and φ at the vehicle final time moment The brachistochrone problem of vehicle motion, described by differential equations (8), consists in determining the controls 1 u and 2 u , as well as their corresponding quantities of state , ξ η and φ , so that the vehicle starting from the initial state ( 9) moves into the final state (10), with unchanged value of mechanical energy (7), in a minimum time.This can be expressed in the form of condition, so that the functional 0 ,  has a minimum value.In order to solve the problem of optimal control, formulated using the Pontryagin maximum principle [11] and [12], we will create Pontryagin's function in the form as follows where 0 const.0 , and φ λ are coordinates of conjugate vector, where it can be taken that 0 1 λ   , while μ is a multiplier corresponding to (7).Based on Pontryagin's function ( 12), the conjugate system of differential equations is If controls belong to an open set, as in this case, the conditions for determining optimal control can be expressed in the form 2 0, 0, , =1,2,3.
Applying the Theorem 1 [11], it follows directly that the value of Pontryagin's function on the optimal trajectory equals zero, for 0 , having in mind, based on (13), that const. .
The application of shooting method requires specified intervals of the values of parameters to be determined.The global estimates for the interval of value of the conjugate vector coordinate ξ λ can be given on the basis of ( 13), ( 14) and ( 16) Shooting consists in determining the unknown coordinates of conjugate vector ξ λ and η λ , having in mind ( 16) and (17), as well as a minimum required time f t , so that the vehicle starting from initial state (9) moves into the final state (10).
The two-point boundary value problem was solved for the following values of parameters   The estimates for the intervals of values of the conjugate vector coordinates ξ λ and η λ can be given on the basis of ( 16), ( 18) and (19)    Based on above considerations, it can be inferred (see Fig. 5.) that a minimum required value of the coefficient of sliding friction, between stationary surface and vehicle wheels, is * 0.16 μ  .

CONCLUSIONS
In this paper, using the example of a simplified vehicle model (see Fig. 1.) with constrained motion (1), we have presented the procedure of generating differential equations of motion ( 4) and ( 5), based on the general theorems of dynamics [9].The formulated brachistochrone problem, with a corresponding choice of the equations of state (8), was solved, in this case, as the simplest problem of optimal control.Applying the Pontryagin maximum principle [11] and [12], the problem of optimal control was solved (see Fig. 2. and Fig. 3.), with a given estimate of the interval of conjugate vector coordinates (18).
Thereafter, the reactions of constraints were determined, as well as the control forces for realizing 204 ▪ VOL.42, No 3, 2014 FME Transactions the brachistochronic motion, as shown in Fig. 4. The performed numerical procedure for solving the twopoint boundary value problem is based on the shooting method [13].
Applying the Coulomb laws of sliding friction, minimum required values for the coefficient of sliding friction were determined, so as to prevent slipping of both vehicle rear and front axle, as presented in Fig. 5.
Authors consider that results obtained in this paper can be extended to the case when the coefficients of sliding friction are below minimum required values.In that case, as well as in the case when control forces are constrained, the problem of optimal control becomes considerably more complex, which will be the subject of future studies.
Now, based on (14) and (15), we obtain the value of a multiplier μ , as well as the control functions in the following form accordance with(11), the time of the vehicle brachistochronic motion, as well as the conjugate vector coordinates, for the given values of the system parameters (19), are 6

Figure 3
Figure 3. Optimal controls   1 u t and   2 .u t

Figure 5
Figure 5. Diagrams Axy coincides with the plane Oξη .The axis of the movable coordinate system Ax is defined by the direction of a normal to the axis of the vehicle rear axle and point C, where C Ax  .Point C represents the center of mass of the vehicle body, while point A is the center of mass of the front axle, and 1 CA l  are given.Unit vectors of the movable coordinate system axes are i , j,   and k,  respectively.A simplified vehicle model consists of a vehicle body of mass 1 M , and a front axle of mass 2 M , the axis Ax , as well as by the drag, proportinate to the first degree of the velocity of point C , with the coefficient of proportionality 2