Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://ics.org.ru/upload/iblock/db4/RCD739.pdf Äàòà èçìåíåíèÿ: Thu Dec 3 15:28:47 2015 Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 23:55:42 2016 Êîäèðîâêà:

ISSN 1560-3547, Regular and Chaotic Dynamics, 2015, Vol. 20, No. 6, pp. 739­751. c Pleiades Publishing, Ltd., 2015.

Qualitative Analysis of the Dynamics of a Wheeled Vehicle
Alexey V. Borisov* , Ivan S. Mamaev** , Alexander A. Kilin*** , and Ivan A. Bizyaev****
Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
Received August 25, 2015; accepted October 29, 2015

Abstract--This paper is concerned with the problem of the motion of a wheeled vehicle on a plane in the case where one of the wheel pairs is fixed. In addition, the motion of a wheeled vehicle on a plane in the case of two free wheel pairs is considered. A metho d for obtaining equations of motion for the vehicle with an arbitrary geometry is presented. Possible kinds of motion of the vehicle with a fixed wheel pair are determined. MSC2010 numbers: 70F25, 37J60 DOI: 10.1134/S156035471506009X Keywords: nonholonomic constraint, system dynamics, wheeled vehicle, Chaplygin system

INTRODUCTION In this paper we consider the problem of the motion of a wheeled vehicle on a plane. By the wheeled vehicle we mean two wheel pairs which are attached to the platform and which in the general case can rotate in the horizontal plane independently of the platform. Among the classical works on wheeled vehicles we mention those of Bottema [3] and Stuckler [10, ¨ 11]. In particular, in [3] the tra jectories of the vehicle with a fixed rear axis are analyzed and remarkable motions are presented. In [10, 11], two models of an automobile with a fixed rear axis are considered. In the first model the front axis freely rotates about the normal to the plane and in the second model, due to the trapezoid mechanism (Jeantaud's mechanism) the wheels are always parallel to each other, but not normal to the axis of the axle. In [10, 11] the focus is on the analysis of reactions and the application of the Hamel equations. Later, the studies carried out in [10, 11] were continued in [8], in which no essentially new results were added. A stability analysis of the simplest design consisting of two two-wheeled vehicles, called rol lerracer, was investigated by Rocard in his well-known book [16]. By the way, he revealed the asymptotic instability of rectilinear motion, which is typical of nonholonomic systems. We note that the asymptotic instability and the resulting absence of a smooth invariant measure lead, as a rule, to complex, almost dissipative behavior of nonholonomic systems, which is characterized by the presence of strange attractors. For a rattleback such dynamics are examined in detail in [2] (for other examples of nonholonomic systems related to the rolling motion of rigid bodies see, for example, [18, 19]). Modern studies on the dynamics of wheeled vehicles are mainly initiated by robotic developments and provide, as a rule, insufficiently detailed dynamical analyses. The Bottema system [3] was rediscovered in [12], and various particular cases (examples) of a three-wheeled vehicle are considered in [6, 9, 13]. As a rule, the authors of these papers restrict themselves to a derivation of equations of motion in general form and, in view of their complexity, consider only the behavior of the system in some particular cases, without investigating all possible tra jectories of the wheeled vehicles.
* ** *** ****

E-mail: E-mail: E-mail: E-mail:

borisov@rcd.ru mamaev@rcd.ru aka@rcd.ru bizaev 90@mail.ru

739


740

BORISOV et al.

The most insightful and interesting analysis of the dynamics of an articulated n-trailer vehicle (nonholonomic multilinked vehicle) is presented in [4]. We note that the same system has been considered recently in [7], where the equations of motion are not presented in explicit form, but some experimental data are presented. By the way, discussions of the kinematics of wheeled systems can be found in [5, 21] and in an extensive literature on control (see, e.g., references in [4]). In this paper, two particular cases of wheeled vehicles are examined in detail. A symmetric vehicle with one fixed axle is considered. The equivalence of this system to the three-wheeled vehicle is shown, a complete classification of possible types of motion is presented and their dynamics are analyzed. In addition, a symmetric vehicle with unfixed axles is discussed. It is shown that in this case the reduced system may exhibit limit cycles. 1. THE CHAPLYGIN SYSTEMS Let the system be described by generalized coordinates q = (q1 ,... ,qn ), x = (x1 ,...,xm ) such that: 1. the nonintegrable constraints imposed on it are represented as
n

xi =
=1

ai (q ) , q

i = 1, ...,m;

(1.1)

2. the Lagrangian function of the free system, i.e., when the constraints (1.1) are ignored, does not explicitly depend on the coordinates xi , but only on the velocities xi L = L( q , q ). x, Let us denote the Lagrangian function after substituting the constraints into (1.2) as L(q, q ) = L( q , q ) x,
x=A(q ) q

(1.2)

,

A(q ) = ai (q ) .

(1.3)

The following simple assumption holds. Assumption 1. The equations of motion of the system with Lagrangian (1.2) with the constraints (1.1) taken into account are represented as d L dt q - L = q
n n

S q ,
=1

xi =
=1 m

ai (q ) , q ai ai - q q

i = 1,...,m, = 1,... ,n, (1.4) L xi .
x=Aq

S = -S =
i=1

Proof. Write the equations of motion of the system (1.2) with undetermined multipliers corresponding to the constraints d L dt xi fj = xj -


=
j

j

fj , xj

d L dt q

-

L = q

j
j

fj , qj (1.5)

aj (q ) , q

i, j = 1, ... ,m,

, = 1, ... ,n.

This gives i = d L . dt xi
Vol. 20 No. 6 2015

REGULAR AND CHAOTIC DYNAMICS


QUALITATIVE ANALYSIS OF THE DYNAMICS OF A WHEELED VEHICLE

741

Substituting this relation into the equation for q in (1.5), using the relations L = q L L ai + , xi q L = q L L ai q + , xi q q

i

i,

and simplifying yields Eqs. (1.4). 2. WHEELED VEHICLE 2.1. General Formalism As an example of the Chaplygin system we consider the model of a vehicle with a rotating axle (or axles). We assume that there is a rigid platform to which two moving frames with fixed wheel pairs are fastened and that the frames can rotate about the vertical axles attached to the platform (see Fig. 1).

Fig. 1

Let O1 , O2 denote the points of attachment of the wheel pairs to the platform (which do not necessarily coincide with the centers of the wheel pairs) and , 1 , 2 the angles of rotation of the entire system and the wheel pairs, respectively (see Fig. 1). Also, let (x, y ) be the coordinates of the point O fixed on the platform and let ij be the angles of rotation of the wheels about their own axes. Constraints. The constraint equations, which imply the absence of slipping at the points of contact of the wheels with the supporting surface (i.e., the velocities of the wheels at the points of contact with the support are zero), have the form v + Jz ri +( + i )Jz rij + hij ni = 0, i , j = 1, 2 , (2.1) ni = cos(i + ), sin(i + ) , v = (x, y ), ri = (±bi cos , ±bi sin ), di cos(i + ) - sin(i + ) 0 -1 , , Jz = rik = (-1)k+1 aik 10 sin(i + ) cos(i + )

where bi , aij , and di are some constants defining the configuration of the system, h is the radius of the wheels, and the vectors ri , rij , ni are shown in Fig. 1. Remark 1. The product Jz r is equivalent to the pro jection of the vector product ez â r onto the plane x, y .
REGULAR AND CHAOTIC DYNAMICS Vol. 20 No. 6 2015


742

BORISOV et al.

The system (2.1) contains eight linear equations in nine unknowns x, y, , i , ij . It can be shown that (if simultaneously 1 + 2 and 2 + 2 ) only six of them are linearly independent, and moreover, the constraints can be written in the form (1.1) as follows:
-(b1 + b2 )cos 2 - d1 cos(1 - 2 ) - a11 sin(1 - 2 ) ( + 1 )+ d2 ( + 2 ) , 11 = h sin(1 - 2 ) -(b1 + b2 )cos 2 - d1 cos(1 - 2 ) - a12 sin(1 - 2 ) ( + 1 )+ d2 ( + 2 ) 12 = , h sin(1 - 2 ) -(b1 + b2 )cos 2 - d1 ( + 1 )+ d2 cos(1 - 2 ) - a21 sin(1 - 2 ) ( + 2 ) 21 = , h sin(1 - 2 ) -(b1 + b2 )cos 2 - d1 ( + 1 )+ d2 cos(1 - 2 ) - a22 sin(1 - 2 ) ( + 2 ) 22 = , h sin(1 - 2 ) b1 cos 1 cos( + 2 )+ b2 cos 2 cos( + 1 ) + d1 cos( + 2 )( + 1 ) - d2 cos( + 1 )( + 2 ) x= , sin(1 - 2 ) b1 cos 1 sin( + 2 )+ b2 cos 2 sin( + 1 ) + d1 sin( + 2 )( + 1 ) - d2 sin( + 1 )( + 2 ) y= . sin(1 - 2 ) (2.2)

Kinetic energy. The kinetic energy of this system with the constraints ignored can be written by summing the kinetic energies of individual parts so that we finally find 1 1 T = Mv 2 + I0 2 + 2 2 +M (vy Rx - vx Ry )
(0) I0 2 1 r1

1 I1 ( + 1 )2 + 2 (1) + m1 (vy Rx
(1)

1 1 2 2 2 2 I2 ( + 2 )2 + (I11 11 + I12 12 + I21 21 + I22 22 )+ 2 2 (1) (2) (2) - vx Ry )( + 1 )+ m2 (vy Rx - vx Ry )( + 2 )+
(2)

+m1 (r1 , R I0 = +m +m
2 2 r2

)( + 1 )+ m2 (r2 , R
(0) I1

)( + 2 ), I2 = I2 + m21 r ,
(0) 2 21 2 + m22 r22 ,

,

I1 =

R= where M, m1 , m2 , m
ij are (0) I0 is

m

2 2 + m11 r11 + m12 r12 , (0) + m r + m r 0R 11 22

(2.3) the masses of the entire system, of the frames with wheels and of the

M

the moment of inertia of the platform (without frames) relative to wheels, respectively; (0) (0) point O; I1 , I2 are the moments of inertia of the frames without wheels relative to the axes O1 , O2 ; Iij are the moments of inertia of the wheels; R(0) is the vector from point O to the center of mass of the platform; and R(i) are the vectors from points Oi to the centers of mass of the frames with wheels. We shall not present equations of motion for the most general case, but consider some of the most frequently used and illustrative examples. 2.2. Symmetric Vehicle with a Fixed Axle Consider the motion of a vehicle sub ject to the following additional conditions: · the points of attachment of the wheel pairs O1 , O2 coincide with the centers of mass of these pairs and with the middles of the axles supporting the wheels: d 1 = d 2 = 0, a11 = a12 = a1 , a21 = a22 = a2 ;

· one axle (which we call the rear axle) is attached at a fixed angle: 2 = 0 = const, 1 = ;

REGULAR AND CHAOTIC DYNAMICS

Vol. 20

No. 6

2015


QUALITATIVE ANALYSIS OF THE DYNAMICS OF A WHEELED VEHICLE

743

Fig. 2. Scheme of a symmetric vehicle and the notation used. In the case where the rear axle is fixed 2 = 0 = const, 1 =

· the point O on the straight line O1 O2 is chosen on the perpendicular to this straight line passing through the center of mass of the system C , so that: R = (- sin , cos ), |CO| = ;

· the wheels have the same mass mw and the same moment of inertia: Iij = mw h2 , Equations of motion and first integrals. equations in this case can be represented as x= i , j = 1, 2 . Let us denote 1 = . Then the constraint

(b1 cos 1 cos( + 0 )+ b2 cos 0 cos( + )), sin( - 0 ) (b1 cos sin( + 0 )+ b2 cos 0 sin( + )), y= sin( - 0 ) (b1 + b2 )cos 0 ± a1 ( + ), h11/12 = - sin( - 0 ) (b1 + b2 )cos ± a2 . h21/22 = - sin( - 0 )

(2.4)

Substituting the constraints (2.4) into the kinetic energy (2.3), we find 1 1 T = I1 ( + )2 + g ()2 , 2 2 ( - 0 ) J0 sin2 ( - 0 )+ J1 cos2 + J2 cos2 0 + J3 cos 0 cos sin(0 - ) , J0 = I0 + I2 +2mw a2 - Mb1 b2 , 2 J3 = 2M (b1 + b2 ), J2 = (b1 + b2 ) Mb2 +2mw (b1 + b2 ) . 1 g 2 . 0

g () = sin-

2

I1 = I1 +2mw a2 , 1

(2.5)

J1 = (b1 + b2 ) Mb1 +2mw (b1 + b2 ) ,

For the matrix defining the nonholonomic terms in (1.4) we obtain 0 S=

1 g - 2 Using Assumption 1, we obtain the equations of motion of the system in the following form: 1 1 g () ¨ . = - = ¨ 2 g ()
REGULAR AND CHAOTIC DYNAMICS Vol. 20 No. 6 2015

(2.6)


744

BORISOV et al.

This system has the obvious integrals of motion 1 (2.7) E = g ( ) 2 . 2 The singularity for = 0 = /2 in the constraint equations (2.4) and in Eq. (2.6) is inessential: it appears due to the fact that for = 0 the angular velocity of the axis O1 O2 vanishes, i.e., = 0, which follows from the initial constraint equations (2.1). This singularity is eliminated by choosing a new independent variable C = + , v = , sin( - 0 )

and in the variables v , , the equations of motion do not contain singularities. Qualitative analysis for 0 = 0. Let us make a detailed qualitative analysis of the vehicle dynamics, when the rear axle is fixed at a right angle to the straight line O1 O2 . Reduced system. In the space of variables z = (v , , ), mod 2, the tra jectory is determined by the intersection of the level surfaces of the first integrals (2.7), which are represented as + v sin = c, A=
2 (1 + A sin2 - B sin cos )v = 2E ,

I0 + I2 + Mb2 +2mw a2 - (M +2mw )(b1 + b2 )2 J0 - J1 2 2 = , J1 + J2 (M +4mw )(b1 + b2 )2 2M J3 , = B= J1 + J2 (M +4mw )(b1 + b2 )

(2.8)

where c, E are the constants of the first integrals. Assuming that the mass of the wheel is much smaller than that of the entire system, i.e., M , we can find the domains of definition of the constants A and B . We transform the mw equality for the constant A from (2.8): A= b2 I0 + I2 2 + - 1. 2 M (b1 + b2 ) (b1 + b2 )2 (2.9)

The value of the second term in (2.9) belongs to the interval (0, 1). Then from (2.9) we obtain the following inequality: A I0 + I2 - 1, M (b1 + b2 )2 (2.10)

since (I0 + I2 )/M (b1 + b2 ) 0 A -1. M , the equality for the constant B becomes Assuming that mw 2 , B= (b1 + b2 )

(2.11)

since (b1 + b2 ) > 0, the sign of the constant B is determined by the sign of , and since the value of is not limited and can be both positive and negative, B (+, -). We note that for = 0 and hence for B = 0 we obtain the equations of motion and the quadratures (2.6), (2.4) for the case of a symmetric vehicle with a fixed wheel pair. Equations (2.6) (more precisely, the equations for (, , v )) describe the so-called reduced system, i.e., the dynamics of relative positions of parts of the vehicle. From the known solutions of the reduced system, one can determine the position and orientation of the vehicle on the plane by using quadratures (i.e., one can perform the so-called reconstruction of the dynamics): = v sin , x = v b1 cos cos + b2 cos( + ) , y = v b1 cos sin + b2 sin( + ) . (2.12)

It is interesting that the length of the front half-axle a1 appears neither in the equations of motion for the variables z nor in the constraint equations (2.12). This implies, in particular, that
REGULAR AND CHAOTIC DYNAMICS Vol. 20 No. 6 2015


QUALITATIVE ANALYSIS OF THE DYNAMICS OF A WHEELED VEHICLE

745

a)

b)

Fig. 3. Pro jection of the tra jectories corresponding to different values of E on the development of the cylinder (, ) for A = 1, c = 1 (a) B = 0, (b) B = 1.6

the "three-wheeler" is equivalent to the usual (wheel) vehicle with some specially chosen constants A and B . Thus, Eqs. (2.8) define the two-parameter family of tra jectories (parameterized by c and E ), which entirely fill the three-dimensional phase space of the reduced system defined by the coordinates z (and which, due to the uniqueness of the solutions, apparently do not intersect each other). To describe possible motions of the vehicle, we proceed by analogy with Hamiltonian systems with one degree of freedom as follows. We fix the value of the linear integral c and pro ject the resulting one-parameter family of tra jectories onto the cylinder (, mod 2 ). The pro jections of the tra jectories on this cylinder are defined by (2.13) (1 + A sin2 - B sin cos )( - )2 = 2E sin2 , and, as is seen from Fig. 3, intersect each other at the points = 0, , = c. It is evident that these intersections are only the singularities of the pro jection and in the three-dimensional space (v , , ) the tra jectories do not intersect each other. This is easily checked by pro jecting them onto the cylinder (v , mod 2 ). One of the differences of the above system from a standard Hamiltonian system with one degree of freedom is that the level lines of the integral (2.13) do not intersect the straight line = 0 at a right angle. Hence, on the tra jectory in a neighborhood of the point of intersection = the following equality holds: = k ( - )+ O | - |2 , k = const (tangent of the inclination angle of the curve). The time dependence of the angle, (t), for large t on this tra jectory is defined by = + ekt + O(e
2kt

),

t -, for k > 0, t +, for k < 0.

(2.14)

From this we notice that on the straight line = that divide the level lines of the integral (2.13) from (2.14) that for k > 0 these fixed points are Thus, the reduced system (2.8) has four types T1 -- fixed points on the axis = 0;

0 there are (degenerate) fixed points of the system into a pair of asymptotic tra jectories; it follows unstable, while for k < 0 they are stable. of solutions (see Fig. 3):

T2 -- asymptotic trajectories tending to the axis = 0; T3 -- periodic trajectories = c, E = 0; T4 -- periodic trajectories that do not intersect the axis = 0. On the bifurcation Absolute each of the plane of first integrals the type of the reduced system is defined by the following diagram: dynamics. We now describe the dynamics of the entire vehicle in absolute space for above-mentioned types of tra jectories.
Vol. 20 No. 6 2015

REGULAR AND CHAOTIC DYNAMICS


746

BORISOV et al.

Fig. 4. Bifurcation diagram of the reduced system. The type of tra jectory in each region is denoted by the corresponding letters Tk , k = 1 .. . 4. The tra jectories of type T3 lie only on the axis E = 0.

(T1 ) In this case the moving axis is fixed, i.e., = = const, and according to (2.12) = 0 + ct, b1 cos cos ct + b2 cos(ct + ) b1 cos sin ct + b2 sin(ct + ) , y=- . x= sin sin Thus, the vehicle rolls so that the center of mass moves in a circle of radius R : R= b2 cos2 +2b1 b2 cos2 + b2 1 2 . sin2 (2.15)

(2.16) from the unstable fixed u = 0, = . Hence, the (2.15), as t -, to an

(T2 ) In the reduced system the tra jectory emanates (as t -) s point = 0, = , and exponentially tends to the stable point vehicle on the plane changes the type of motion described by analogous type of motion (see Fig. 5).

a)

b)

Fig. 5. Typical tra jectories for the second type of solutions T2 for the system parameters E = 1, A = 2, and c = 0.8. a) Pro jection of the tra jectory on the development of the cylinder (, ). b) Tra jectory of the center of mass.

(T3 ) In this case = c, = 0, x = y = 0, i.e., the vehicle stands still and its moving axle rotates uniformly. (T4 ) The reduced system executes periodic motion whose period is defined by the quadrature
2

T=
0

d c- 2E 2 1+ A sin - B sin cos

.

REGULAR AND CHAOTIC DYNAMICS

Vol. 20

No. 6

2015


QUALITATIVE ANALYSIS OF THE DYNAMICS OF A WHEELED VEHICLE

747

The vehicle executes quasi-periodic two-frequency motion in the bounded region on the plane (see Fig. 6).

Fig. 6. Typical tra jectories for the fourth type of solutions T4 for the system parameters E = 2.3, A = 2, and c = 0.8. (a) Pro jection of the tra jectory on the development of the cylinder (, ). (b) Tra jectory of the center of mass.

2.3. Symmetric Vehicle with Moving Axles Consider the motion of the vehicle sub ject to the following additional conditions: · the points of attachment of the wheel pairs O1 , O2 coincide with the centers of mass of these pairs and with the middles of the axles supporting the wheels: d 1 = d 2 = 0, a11 = a12 = a1 , a21 = a22 = a2 ; · the point O on the straight line O1 O2 is chosen on the perpendicular to this straight line passing through the center of mass of the system C , so that: R = (- sin , cos ), |CO| = ;

· the wheels have the same mass mw and the same moment of inertia: Iij = mw h2 , Equations of motion and first integrals. represented as x= (b1 cos 1 cos( sin(2 - 1 ) (b1 cos 1 sin( y= sin(2 - 1 ) (b1 h11/12 = - sin(2 - 1 ) (b1 h21/22 = - sin(2 - 1 ) i , j = 1, 2 . The constraint equations in this case are

+ 2 )+ b2 cos 2 cos( + 1 )), + 2 )+ b2 cos 2 sin( + 1 )), (2.17) + b2 )cos 2 ± a1 ( + 1 ), + b2 )cos 1 ± a2 ( + 2 ).

Substituting the constraints (2.17) into the kinetic energy (2.3), we find 1 T = I1 ( + 1 )2 + 2 g (1 ,2 ) = sin2 (2 - 1 ) J0 sin2 (2 - 1 )+ Ii = Ii +2mw a2 , i 1 1 I2 ( + 2 )2 + g (1 ,2 )2 , 2 2 J1 cos2 1 + J2 cos2 2 + J3 cos 1 cos 2 sin(2 - 1 ) J3 = 2M (b1 + b2 ), (2.18)
REGULAR AND CHAOTIC DYNAMICS Vol. 20 No. 6 2015

J0 = I0 + I1 + I2 - Mb1 b2 ,

J1 = (b1 + b2 ) Mb1 +2mw (b1 + b2 ) ,

J2 = (b1 + b2 ) Mb2 +2mw (b1 + b2 ) .


748

BORISOV et al.

For the matrix defining the nonholonomic terms in (1.4) we obtain 0 1 S= - 2 1 - 2 g 1 g 2 1 g 2 1 0 0 1 g 2 2 0 0 .

Using Assumption 1, we obtain the equations of motion of the system in the following form: ¨ ¨ 1 = 2 = - = ¨ g (1 ,2 ) g (1 ,2 ) 1 g (1 ,2 ) = . 1 + 2 g (1 ,2 ) 1 2 2 2 g (1 ,2 ) C2 = + 2 , , sin(1 - 2 ) (2.20) (2.19)

This system has the obvious integrals of motion C1 = + 1 , F= G(1 ,2 )

G(1 ,2 ) = 1 + cos(1 + 2 )cos(1 - 2 )+ A sin2 -B sin(1 - 2 )(cos(1 + 2 )+cos(1 - 2 )) - C sin(1 J3 J1 2J0 , B= , C= A= J1 + J2 J1 + J2 J1

(1 - 2 )- - 2 )sin(1 + 2 ) - J2 . + J2

Thus, for the system (2.19) to be integrable by the Euler ­ Jacobi theorem (see [17] for details), an invariant measure is required. As in the case of a vehicle with a fixed axle, the singularity for 1 = 2 = /2 in the constraint equations (2.17) and Eqs. (2.20) is inessential; it appears due to the fact that for 1 = 2 the angular velocity of the axis O1 O2 vanishes, i.e., = 0, which follows from the initial constraint equations (2.1). This singularity is eliminated by choosing, instead of , a new independent variable: v = Making the change of variables 1 + 2 = , 1 - 2 = , where [-, ), [0, 2 ) are the angle variables on the torus, and performing a reduction to the level set of first integrals C1 = c1 , C2 = c2 , F = f, we obtain equations of motion of the reduced system 2f sin , = c1 - c2 , = c1 + c2 - (2.22) G where G = G( + , - ). 2 2 Numerical investigations show that the system (2.22) has no smooth invariant measure, since it contains limit cycles. An example of the phase portrait of the system (2.22) containing limit cycles is given in Fig. 7. In this figure, the heavy solid line corresponds to a stable cycle and the dashed line indicates an unstable cycle. The solid line in Fig. 7 indicates a tra jectory that starts from an unstable cycle (as t -) and then tends to a stable cycle (as t +). Another example of a nonholonomic system which reduces to analysis of the flow on a (two-dimensional) torus without a smooth invariant measure is given in [20]. We note that the right-hand sides of the system (2.22) contain isolated singular points at = 0, = and = - , = 0, which are denoted in Fig. 7 by A and B , respectively. An example of the tra jectory passing near the point = 0, = is given in Fig. 8a, and its enlarged fragment near a singular point is shown in Fig. 8b. The dependence (t) corresponding to this tra jectory passing in a neighborhood of the point = 0, = is shown in Fig. 8c, from which it is evident that when the tra jectory passes near the singular point, an abrupt jump of occurs.
REGULAR AND CHAOTIC DYNAMICS Vol. 20 No. 6 2015

. sin(1 - 2 )

(2.21)


QUALITATIVE ANALYSIS OF THE DYNAMICS OF A WHEELED VEHICLE

749

Fig. 7. Tra jectories of the system (2.22) for fixed values of the first integrals and system parameters c1 = 38, c2 = 2, f = -87, A = 0.5, B = 0.8, C = 0.3.

Fig. 8. (a) and (b) tra jectory of the system (2.22) passing in a neighborhood of the singular point = 0, = , (c) dependence (t); all figures have been plotted for fixed values of the first integrals and system parameters c1 = 38, c2 = 2, f = -87, A = 0.5, B = 0.8, C = 0.3.

The above singularity of the equations of motion has an interesting physical interpretation. Indeed, expressing from (2.20), we obtain = f sin G Hence, when the tra jectory passes near singular points, undergoes a jump and changes sign, that is, the vehicle abruptly reverses the direction of rotation. This behavior is typical of systems with impacts. In this connection it would be interesting to examine in more detail the motion of the vehicle near singular points. To conclude this section, we describe the qualitative behavior of the vehicle in the fixed coordinate system in the presence of limit cycles in the system (2.22). We note that the tra jectory of the vehicle in absolute space is recovered from the known solutions 1 (t), 2 (t), and (t) by the quadratures (2.17). In the general case, to the limit cycles of the reduced system there correspond quasi-periodic tra jectories in absolute space. In the case considered here, the arbitrary tra jectory of the vehicle starts from an unstable quasi-periodic solution (as t -) and then tends to a stable quasi-periodic solution (as t +). A typical tra jectory of the vehicle for this case is shown in Fig. 9. The question of boundedness of the vehicle's tra jectories depending on its parameters and values of the first integrals remains open (for the Chaplygin ball this problem was investigated in [1]).
REGULAR AND CHAOTIC DYNAMICS Vol. 20 No. 6 2015

.


750

BORISOV et al.

Fig. 9. Tra jectory of point O in the fixed coordinate system for fixed values of the first integrals and system parameters c1 = 38, c2 = 2, f = -79, A = 0.5, B = 0.6, C = 0.5, b1 = 2, b2 = 1.

ACKNOWLEDGMENTS This work was supported by the Russian Scientific Foundation (pro ject 14-50-00005) and was carried out at the Steklov Mathematical Institute of the Russian Academy of Sciences. REFERENCES
1. Borisov, A. V., Kilin, A. A., and Mamaev, I. S., The Problem of Drift and Recurrence for the Rolling Chaplygin Ball, Regul. Chaotic Dyn., 2013, vol. 18, no. 6, pp. 832­859. 2. Borisov, A. V., Jalnine, A. Yu., Kuznetsov, S. P., Sataev, I. R., and Sedova, J. V., Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Mo del of the Rattleback, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 512­532. 3. Bottema, O., Die Bewegung eines einfachen Wagenmo dells, Z. Angew. Math. Mech., 1964, vol. 44, no. 12, pp. 585­593. 4. Bravo-Do ddoli, A. and Garc´ ia-Naranjo, L. C., The Dynamics of an Articulated n-Trailer Vehicle, Regul. Chaotic Dyn., 2015, vol. 20, no. 5, pp. 497­517. 5. Campion, G., Bastin, G., and d'Andr´ ea-Novel, B., Structural Properties and Classification of Kinematic and Dynamic Mo dels of Wheeled Mobile Robots, IEEE Trans. Robot. Autom., 1996, vol. 12, no. 1, pp. 47­62. 6. Giergiel, J. and Zylski, W., Description of Motion of a Mobile Robot by Maggie's Equations, J. Theor. Appl. Mech., 2005, vol. 43, no. 3, pp. 511­521. 7. Khala ji, A. K. and Moosavian, S. A. A., Dynamic Modeling and Tracking Control of a Car with n Trailers, Multibody Syst. Dyn., 2015, 15 pp. 8. Lobas, L. G., On a Nonholonomic System Serving as a Mo del of a Four-Wheeled Vehicle, Z. Angew. Math. Mech., 1979, vol. 59, no. 2, pp. 85­91. 9. Staicu, S., Dynamics Equations of a Mobile Robot Provided with Caster Wheel, Nonlinear Dynam., 2009, vol. 58, no. 1, pp. 237­248. ¨ 10. Stuckler, B., Uber die Differentialgleichungen fur die Bewegung eines idealisierten Kraftwagens, Arch. ¨ ¨ Appl. Mech., 1952, vol. 20, no. 5, pp. 337­356. ¨ 11. Stuckler, B., Uber die Berechnung der an rollenden Fahrzeugen wirkenden Haftreibungen, Arch. Appl. ¨ Mech., 1955, vol. 23, no. 4, pp. 279­287. 12. Martynenko, Yu. G., The Theory of the Generalized Magnus Effect for Non-Holonomic Mechanical Systems, J. Appl. Math. Mech., 2004, vol. 68, no. 6, pp. 847­855; see also: Prikl. Mat. Mekh., 2004, vol. 68, no. 6, pp. 948­957. 13. Martynenko, Yu. G., Motion Control of Mobile Wheeled Robots, J. Math. Sci. (N. Y.), 2007, vol. 147, no. 2, pp. 6569­6606; see also: Fundam. Prikl. Mat., 2005, vol. 11, no. 8, pp. 29­80.
REGULAR AND CHAOTIC DYNAMICS Vol. 20 No. 6 2015


QUALITATIVE ANALYSIS OF THE DYNAMICS OF A WHEELED VEHICLE

751

14. Markeev, A. P., Theoretical Mechanics, Izhevsk: R&C Dynamics, Institute of Computer Science, 2007 (Russian). 15. Neimark, Ju. I. and Fufaev, N. A., Dynamics of Nonholonomic Systems, Trans. Math. Monogr., vol. 33, Providence, R.I.: AMS, 1972. 16. Ro card, Y., L'instabilit´ en m´ anique: Automobiles, avions, ponts suspendus, Paris: Masson, 1954. e ec 17. Bolsinov A. V., Borisov A. V., Mamaev I. S. Hamiltonization of non-holonomic systems in the neighborho o d of invariant manifolds //Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 443­464. 18. Borisov, A. V., Kilin, A. A., and Mamaev, I. S. How to Control Chaplygin's Sphere Using Rotors, Regul. Chaotic Dyn., 2012, vol. 17, nos. 3­4, pp. 258­272. 19. Borisov, A. V. and Mamaev, I. S., Rolling of a Non-homogeneous Ball over a Sphere without Slipping and Twisting, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 153­159. 20. Bolsinov, A. V., Borisov, A. V., and Mamaev, I. S., Rolling of a Ball without Spinning on a Plane: The Absence of an Invariant Measure in a System with a Complete Set of Integrals, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 571­579. 21. Altafini, C., Some Properties of the General n-trailer, Int. J. Control, 2001, vol. 74, no. 4, pp. 409­424.

REGULAR AND CHAOTIC DYNAMICS

Vol. 20

No. 6

2015