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ISSN 1560-3547, Regular and Chaotic Dynamics, 2015, Vol. 20, No. 6, pp. 752­766. c Pleiades Publishing, Ltd., 2015.

On the Hadamard ­ Hamel Problem and the Dynamics of Wheeled Vehicles
Alexey V. Borisov1 * , Alexander A. Kilin2 ** , and Ivan S. Mamaev3
1 Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, 141700 Russia

***

Udmurt State University ul. Universitetskaya 1, Izhevsk, 426034 Russia
3

2

Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
Received October 12, 2015; accepted November 9, 2015

Abstract--In this paper, we develop the results obtained by J. Hadamard and G. Hamel concerning the possibility of substituting nonholonomic constraints into the Lagrangian of the system without changing the form of the equations of motion. We formulate the conditions for correctness of such a substitution for a particular case of nonholonomic systems in the simplest and universal form. These conditions are presented in terms of both generalized velocities and quasi-velocities. We also discuss the derivation and reduction of the equations of motion of an arbitrary wheeled vehicle. In particular, we prove the equivalence (up to additional quadratures) of problems of an arbitrary wheeled vehicle and an analogous vehicle whose wheels have been replaced with skates. As examples, we consider the problems of a one-wheeled vehicle and a wheeled vehicle with two rotating wheel pairs. MSC2010 numbers: 37J60, 37N05 DOI: 10.1134/S1560354715060106 Keywords: nonholonomic constraint, wheeled vehicle, reduction, equations of motion

INTRODUCTION 1 The problem of the possibility of writing the equations of motion of nonholonomic systems in Lagrangian form goes back to Hertz, who pointed out the inapplicability of the Hamilton principle for nonholonomic systems (for a further discussion of this problem see Hamel [4] and, for example, [29]). The impossibility of Lagrangian (Hamiltonian) description in the general case implies the incorrectness of substituting nonholonomic constraints into the Lagrangian when equations of motion are set up. In particular, the well-known error of E. Lindelof, who used the ¨ Lagrange equations to solve the problem of a body of revolution rolling on a plane, and those of others1) prompted S. A. Chaplygin[26] to advance considerably in the study of nonholonomic mechanics and to introduce the Chaplygin systems[25], which are a special kind of nonholonomic systems. The analysis of these systems provided a deeper understanding of the problem of the possibility of substituting nonholonomic constraints into a Lagrangian and of writing the equations of motion of nonholonomic systems in Lagrangian form (more precisely, the problem of the incorrectness of such an operation in the general case). Moreover, for such systems S.A.Chaplygin developed the reducing multiplier theory (which nowadays is the basis of Hamiltonization theory for nonholonomic systems [30]).
* ** *** 1)

E-mail: borisov@rcd.ru E-mail: aka@rcd.ru E-mail: mamaev@rcd.ru Analogous errors were made by C. Neumann and H. Schouten.

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Before the publication of S. A. Chaplygin's work the issue of the correctness of substituting constraints into a Lagrangian had been discussed by A. Vierkandt [10], who gave an explicit example showing the incorrectness of such a substitution in the general case. We note that in the general case such a substitution leads to incorrect equations of motion which depend on the method of eliminating constraints [9]. We consider in more detail two particularly complete, interesting and now classical works in this direction [3, 4]. The goal of this paper is to pursue a further development of these works. The first of them, [3] (1895), was written by J. Hadamard at the time when he was a student of P. Appel, and was his only work concerned with nonholonomic mechanics. This work is also known for the fact that it introduced the model of pure rolling (called by J. Koiller more than 100 years later rubber rol ling [14]). This work was republished in [11] as an appendix to a small book by Appel, where the well-known problem of the motion of a circular disk on a horizontal plane was analyzed. In [3] J. Hadamard examined in detail the problem of the correctness of substituting nonholonomic constraints into the Lagrangian. The main results obtained by J. Hadamard can be formulated as follows. Let a system be described by configuration variables y = (y1 , . . . , ym ), z = (z1 , . . . , zn ), and suppose that m nonholonomic constraints y = g (y , z , z ) are imposed on it, where g are arbitrary functions linear in the velocities z . Further, assume that this system of constraints is completely nonholonomic2) (see, e.g., [5, 8]). Then from these constraints one can form m - n(n - 1)/2 "substitutable" combinations which can be substituted into the Lagrangian before writing the equations of motion. J. Hadamard did not simply point to the existence of such combinations, but presented a constructive algorithm for forming these combinations. In the same work he stated that all m constraints can be substituted into the Lagrangian if and only if these constraints are integrable. We note that the result obtained by J. Hadamard is associated not only with the specificity of the constraints and does not depend on the Lagrangian of the system. That is, it is a sufficient condition for the substitution of constraints (more precisely, their combinations) into the Lagrangian. The second work, which is closely related to our research, is the main and most extensive work of G. Hamel on nonholonomic mechanics [4] (1904), where his well-known equations of motion were obtained. By the way, it formed the content of his thesis and was partially included in his wellknown textbook on theoretical mechanics [13]. Part of this work is also devoted to the possibility of using a "prohibited form"3) of the kinetic energy to write the equations of nonholonomic systems in Lagrangian form. In particular, G. Hamel notes that the conditions obtained by J. Hadamard for the substitution of constraints are sufficient and generalizes the result of J. Hadamard to the case of writing the equations of motion in quasi-velocities. As part of the quasi-velocities he uses the constraints imposed on the system (or their linear combinations). Such an approach simplifies the general analysis and allows taking into account the dependence on the Lagrangian of the system. However, when specific systems are considered, the possibility of constructive formation of "substitutable" combinations of constraints is lost. It can only be determined which of the constraints chosen as quasi-velocities can be substituted into the Lagrangian. The calculations involved are rather laborious and, in fact, reduce to writing the equations of motion in the chosen (and fixed) quasi-velocities. 2 In this paper we develop the results of the works [3, 4] on the possibility of substituting nonholonomic constraints into the Lagrangian of the system without changing the form of the equations of motion. In contrast to the above-mentioned works we divide the constraints immediately into two groups, one of which we check for the possibility of substituting into the Lagrangian. We assume that the first group of constraints does not depend on the coordinates y and has the form y = g (z , z ).
2)

( 0. 1)

3)

A not completely nonholonomic system of constraints can always be reduced to a completely nonholonomic system. To do this, it is necessary to decouple the holonomic part of constraints from the system (according to the Frobenius theorem) and to restrict the remaining nonholonomic constraints to them. By the prohibited form G. Hamel, following C. Neumann, means restriction of the kinetic energy to nonholonomic constraints. Vol. 20 No. 6 2015

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The second group of constraints has an arbitrary form and and does not depend on the coordinates y and velocities y . Moreover, we assume that the Lagrangian of the system is also independent of the coordinates y . We call the constraints (0.1) whose substitution into the Lagrangian is correct quasi-holonomic. We note that in the case of absence of the second group of constraints the system under consideration becomes a Chaplygin system [25]. Thus, the quasi-holonomy of the constraints (0.1) is equivalent to the possibility of writing the Chaplygin equations in Lagrangian form in the presence of additional nonholonomic constraints. Also, the problem of quasi-holonomy of constraints is closely related to the problems of reduction and Hamiltonization of nonholonomic systems and has been studied most completely for Chaplygin systems [6, 38­40]. The analysis of constraints of a more general form (depending on y ) leads to a considerable complication of the quasi-holonomy conditions [3]. Moreover, in this case the equations for the variables z do not decouple from the complete system. And, although formally they have a Lagrangian form, they depend on the coordinates y , the dynamics of which is determined by the constraint equations. Thus, in the variables (y , z ) the system is not Lagrangian. The restrictions imposed by us on the constraints and the Lagrangian allow us to formulate the quasi-holonomy conditions in the simplest and universal form. In particular, they take into account the form of the Lagrangian and can be written in terms of both generalized velocities and quasi-velocities. These conditions do not require laborious calculations and reduce to verifying the linear dependence of some vectors. 3 The resulting conditions for the quasi-holonomy of constraints allow one to establish the equivalence of various problems ranging from simple ones, considered in mechanics courses separately [12, 13] to complex nonholonomic multibody systems. Of special note is the convenience of the proposed approach for applications to the dynamics of wheeled vehicles, since it is in these problems that a part of the constraints can be decoupled and represented in the Chaplygin form (0.1). In particular, this approach makes it possible to set up the equations of motion for wheeled systems in the simplest and most transparent form, to verify them, and to perform a qualitative and topological analysis. Among the classical works on the dynamics of wheeled vehicles we mention those of B. Stuck¨ ler [15, 16] and O. Bottema [18], in which various models of an automobile with a fixed rear axle were investigated, and the well-known book of Y. Rocard [17] concerned with the dynamics of the simplest two-linked wheeled vehicle (called later rol ler-racer, see, e.g., [12]). Modern studies on the dynamics of wheeled vehicles were prompted mainly by robotic developments [19, 20, 22, 23, 27]. As a rule, the authors of these papers restrict themselves to a derivation of the equations of motion in general form and, in view of their complexity, consider only the behavior of the system in some particular cases, without performing a complete study of the system's dynamics. Among the most insightful and interesting papers published recently we mention the work on the dynamics of an articulated n-trailer vehicle (nonholonomic multilink vehicle) [21]. We note that the kinematics of wheeled vehicles has been studied much more deeply. This is due to the solution of practical control problems arising in the case of autonomous motion of automobiles and robotized wheeled vehicles. A discussion of the kinematics of wheeled vehicles can be found in [24, 36] and in an extensive literature on control theory. 4 In this paper we discuss the derivation and reduction of the equations of motion of an arbitrary wheeled vehicle. In particular, using the property of quasi-holonomy of constraints, we prove the equivalence (up to additional quadratures) of the problems of an arbitrary wheeled vehicle and an analogous vehicle whose wheels have been replaced with skates. Thus, we justify the replacement of the wheels with the skates4) not only from the viewpoint of the kinematics (the imposed constraints), but also from the viewpoint of the system's dynamics. The use of this analogy allows the equations of motion of wheeled vehicles to be represented in a transparent form convenient for verification and further research.
4)

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We note that in this paper we perform direct calculations by using, as an example, a vehicle with two rotating wheel pairs. However, all theorems and statements that are proved in this paper are general in character and hold for arbitrary wheeled vehicles. 5 As mentioned above, the dynamics of wheeled vehicles still remain poorly understood, although this area of research is of great practical importance for the investigation of the dynamics and control of mobile robots. To advance in this area, it is apparently necessary to use the general concept of the hierarchy of dynamics of nonholonomic systems, introduced in [31]. From this point of view the existence of various tensor invariants (conservation laws) significantly influences the system's dynamics. In particular, nonholonomic systems can exhibit behavior typical of both Hamiltonian and dissipative systems, moreover, their behavior can be regular or chaotic [41­43]. As already mentioned above, many problems in the dynamics of wheeled vehicles are equivalent to each other, and their equivalence is not always obvious due to the complexity of equations and due to the choice of various local coordinates. Accordingly, a classification is needed of problems of the dynamics of wheeled vehicles, not only from the viewpoint of tensor invariants, but also from the viewpoint of reducibility to some "basic" systems. Some of such systems, which exhibit regular dynamics and may be studied by methods of qualitative analysis, are discussed in [32, 33]. Using the approach presented in this paper, one can also consider other "basic" problems in the dynamics of wheeled vehicles. Of special note is the paper [27], in which the methods of geometrical mechanics are applied to the above-mentioned roller-racer problem, which is one of the basic and regular problems. In our point of view, both in the dynamics of wheeled vehicles and in nonholonomic mechanics in general this approach is not fruitful. The authors of this approach, in their fascination for imaginary geometric (pseudo-)clearness and linguistic constructions, often stray from specific dynamical problems and also ignore the real geometry and topology associated with the dynamical problem under study (a culmination of the geometric approach, where any dynamical meaning is lost altogether, is reached in the review [44])5) . Meanwhile, as shown in many of our studies [30, 34], concrete nonholonomic problems have many important aspects which may well be studied using modern methods of qualitative and topological analysis, Poisson geometry etc. Many new results for analysis of nonintegrable (in particular, nonholonomic) systems can now be obtained via modern efficient computer calculations. In future publications, we are planning to consider a number of model problems concerning the dynamics of wheeled vehicles exhibiting regular and chaotic properties. 1. EQUATIONS OF MOTION Let us consider the problem of a vehicle rolling on a horizontal plane without slipping. Assume that the vehicle consists of a rigid platform to which two moving frames with rigidly fixed wheel pairs are attached (see Fig. 1a). We shall call such frames links (see Fig. 1b), and by a wheel pair we shall mean a pair of identical coaxial wheels which can rotate independently of one another. The links in this vehicle model can rotate about the vertical axes passing through their points of attachment to the platform (see Fig. 1a). These points may or may not coincide with the geometric center of the wheel pairs. Moreover, we shall assume that the vehicle rolls without slipping of the wheels relative to the plane. Let us choose two coordinate systems: a space-fixed (inertial) coordinate system OX Y and a moving coordinate system oxy attached to the platform (see Fig. 1a). Let us specify the position of the vehicle by the coordinates (x, y ) of the origin of the moving system oxy , its orientation by the angle of rotation of the moving axes relative to the fixed ones, and let 1 , 2 denote the angles of rotation of the links relative to the platform and k1 , k2 , k = 1, 2 the angles of rotation of each of the wheels relative to their own axes. Thus, the vector of the generalized coordinates has the form q = (x, y , , 1 , 2 , 11 , 12 , 21 , 22 ), and the configuration space of the system is the product of the group of motions of the plane and the six-dimensional torus E (2) â T6 .
5)

Our attitude to geometrization is close to the critical remarks of V. I. Arnold concerning the "bourbakization" of mathematics. Unfortunately, this process has been going on; it has affected the most interesting applied branches of mechanics. Physical intuition and the basic dynamical principles play here a key role. In the general case, for mechanics problems any meaningful mathematical equation should in principle be experimentally observable (starting with the Kepler laws and the Newtonian law of gravitation). Vol. 20 No. 6 2015

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Fig. 1. Schematic of the general design of the vehicle (a) and of one of the links with a wheel pair (b)

To describe the dynamics of the system under consideration, we introduce the following notation for the geometrical and dynamical characteristics of the platform: rk is the radius vector of the point of attachment of the k -th link, Rf is the radius vector of the center of mass of the platform, and mf and If are the mass and the moment of inertia of the platform relative to the vertical axis passing through point O. We also introduce the following notation for the geometrical and dynamical characteristics of each of the links: Ok is the point of attachment of the link to the platform, k is the radius vector from the point of attachment to the center of mass of the wheel pair, hk is the radius of the wheels, lk is half the distance between the wheels, nk and k are, respectively, the normal unit vector and the tangent unit vector to the wheels' plane (both lying in the horizontal plane), ik is the axial moment of inertia of each wheel, mk and jk are the total mass of the link and its total mass of inertia relative to the vertical axis passing through the point of attachment Ok , and Rk is the radius vector from the point of attachment to the center of mass of the link. We note that the vectors nk , k , k and Rk depend on the angle of rotation of the link k . Referred to the moving coordinate system, they have the form nk = (cos k , sin k ), k = Qk 0 , k k = (- sin k , cos k ), cos k - sin k 0 , R k = Qk R k , Qk = sin k cos k

0 where Rk and 0 are the constant radius vectors of the center of mass of the link and of the center k of mass of the wheel pair, referred to the axes of the coordinate system Ok nk k attached to this link. To abbreviate some of our forthcoming formulae, we define the vector of the vertical ez (perpendicular to the plane OX Y ) and the operation of the vector product of the two-dimensional vectors by each other and by the vector of the vertical as follows: 0 -1 a. a â b = (a1 b2 - a2 b1 )ez , a â ez = 10

The vectors nk and k are related by nk â k = ez .

Here and in the sequel (unless otherwise specified) we shal l refer al l vectors to the moving coordinate oxy .
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The total kinetic energy of the system can be represented as
2

T = Tf +
k =1

Tk ,

1 1 ( 1. 1) mf v 2 + mf (Rf â v , ez ) + If 2 , 2 2 1 1 1 2 2 Tk = mk vk + mk k (Rk â vk , ez ) + jk k + ik 2 1 + 2 2 , k = 1, 2, k k 2 2 2 where v = (v1 , v2 ) and = are the velocity and the angular velocity of the platform (the moving coordinate system) and the velocities of the points of attachment vk , and the angular velocities k of each link are expressed in terms of them as follows: Tf = vk = v + ez â rk , k = + k , k = 1, 2 . ( 1. 2) The conditions of the absence of slipping of the wheels relative to the plane are represented in the form of eight constraint equations vk + k ez â (k + lk nk ) + hk k1 k = 0, ( 1. 3) vk + k ez â (k - lk nk ) + hk k2 k = 0, k = 1, 2. Performing a scalar multiplication of these equations by nk , one can show that only six of the eight constraints are independent and can be represented as fk = (vk , nk ) + k (ez â k , nk ) = 0, g = (vk , k ) + k (ez â (k + lk nk ), k ) + hk k1 = 0, = (vk , k ) + k (ez â (k - lk nk ), k ) + hk k2 = 0, k = 1, 2. g
k1

( 1. 4)

k2

Using the equations of motion of nonholonomic systems in quasi-velocities (see, e.g., [5]), we obtain for this case T v T T j T j
·

T + ez â = v + ez , v â T v

2


k =1 2

k

gk 1 gk 2 fk + µk 1 + µk 2 , v v v
k

·

=
k =1

gk 1 gk 2 fk + µk 1 + µk 2 ,

·

2

( 1. 5)

=
k =1 · 1

gk 1 gk 2 fk + µk 1 + µk 2 , k j j j T j
·

gj 1 , = µj 1 j 1

=µ
2

j2

gj 2 , j 2

j = 1, 2 ,

where k , µk1 and µk2 are the undetermined multipliers, which are found from the common solution of Eqs. (1.5) and the time derivatives of the constraints (1.4). Equations (1.5) along with the kinematic relations x = v1 cos - v2 sin , y = v1 sin + v2 cos ( 1. 6) form a closed system of equations describing the motion of the vehicle. 2. QUASI-HOLONOMIC CONSTRAINTS Consider a system described by the generalized coordinates z = (z1 , . . . , zn ) and y = (y1 , . . . , ym ). Let two groups of constraints yi = gi (z , z ), i = 1, . . . , m, ( 2. 1) f (z , z ) = 0, = 1, . . . , k (k < n), ( 2. 2)
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be imposed on this system and suppose that the Lagrangian of the system has the form L = L( z , z , y ) . Assume that all constraints (2.1) and (2.2) are independent. We shall call the constraints (2.1) quasi-holonomic if after restriction to them the equations of motion can be represented as d dt


L z

-

L = z

k


=1



f , z

( 2. 3)

where ( ) denotes the restriction to the constraints (2.1) and are the undetermined Lagrange multipliers corresponding to the constraints (2.2). In other words, the quasi-holonomic constraints can be eliminated by substituting the velocities y into the Lagrangian of the system. In this case the equations of motion for the variables z preserve the form of the D'Alembert ­ Lagrange equations of genus 2 only with the constraints (2.2). We note that in the case of absence of the second group of constraints (2.2) the constraints (2.1) are quasi-holonomic with an arbitrary Lagrangian if and only if they are integrable (holonomic) [3]. Therefore, the existence of the second group of constraints is a key condition for quasi-holonomy of nonintegrable constraints. Further, we define the conditions for quasi-holonomy of the constraints (2.1). Theorem 1. The constraints (2.1) are quasi-holonomic if and only if the restriction of the vector
m

N=
i=1

L yi



d dt

gi z

-

gi z , = 1, . . . , k .

( 2. 4)

to the constraints (2.2) lies in the linear span of the vectors

f z

Proof. For ease of calculations we introduce the Lagrangian derivative of an arbitrary function u(z , z ) with respect to the coordinates z in the form [u]z = d dt u z - u . z ( 2. 5)

Then the equations of the initial system can be represented as
k

[ L] z =
=1





f + z

m

µ
i=1

i

gi , z

d dt

L yi

= -µi ,

i = 1, . . . , m ,

( 2. 6)

where and µi are the undetermined Lagrange multipliers defined from the common solution of equations (2.6) and the time derivatives of the constraints (2.1) and (2.2). Considering L = L(z , z , g (z , z )) as a complex function, it is easy to show that the following relation holds
m

[L ]z =



[ L] z +
i=1

L yi

·

gi z



m

+
i=1

L yi



[ gi ] z .

( 2. 7)

Substituting (2.6) into (2.7), we obtain the equations of motion for the variables z in the form
k

[L ]z = N +
=1







f . z

( 2. 8)

Let us now represent the vector N in the form of the sum
k

N=
=1



f + N, z
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where are the coefficients of decomposition of the restriction of N to the constraints (2.2) with respect to the vectors f and N is some (arbitrary) vector satisfying the condition z N Substituting (2.9) into (2.8) yields [ L ] z =
=1 f =0 k

= 0. f + N, z ( 2. 10)



where = from (2.10) constraints The necessi

+ and ( 2. 2) ty of

are the new undetermined (2.3) after the elimination of are identical. Thus, we have s this condition is obvious and

multipliers. It is obvious that the equations resulting the undetermined multipliers and restriction to the hown that the condition of the theorem is sufficient. follows from the comparison of (2.3) and (2.8).

Remark 1. It is easy to verify that for holonomic constraints the vector N is equal to zero and the conditions of Theorem 1 hold. Thus, the verification of the quasi-holonomy criterion does not require that the condition of complete nonholonomy of the system of constraints (2.1) is fulfilled. This is convenient in cases where the constraints are reduced to a completely nonholonomic form by making a complicated change of variables. In many examples it is convenient to use, instead of the generalized velocities z , the quasi velocities w = (w1 , . . . , wn ) related to z by the linear relations
n

zi =
j =1

aj i (z )wj ,

i = 1 . . . n.

( 2. 11)

In this case the equations of motion and the constraints are rewritten as d dt L wj
n

- aj (L) =
s,l=1

L + c ws wl
l sj

k =1

f + wj

m

µ
i=1

i

gi , wj

j = 1 . . . n, ( 2. 12)

d dt and

L yi

= -µi ,

i = 1...m i = 1 . . . m, = 1 . . . k, ( 2. 13) ( 2. 14)

yi = gi (z , w) = gi (z , z (w)), f (z , w) = f (z , z (w)) = 0,
n z

where L(z , w, y ) is the Lagrangian function expressed in terms of the quasi-velocities with the help of (2.11), aj = transformation (2.11), and
i=1 cl j s

aj i (z )

i

is the basis of the vector fields corresponding to the

are the coefficients of decomposition of their commutators
n

[as , aj ] =
l =1

cl j al , s

s, j = 1 . . . n.

In this c not change holonomic. of the form

ase, as above, the constraints (2.13) the substitution of which into the Lagrangian does the form of the equations of motion (2.12) for quasi-velocities w will be called quasiIt is easy to generalize the condition for quasi-holonomy of the constraints to systems (2.12).

Theorem 2. The constraints (2.13) are quasi-holonomic if and only if the restriction of the vector N with the components m n d gi gi L Nj = - aj (gi ) - cl j ws s yi dt wj wl
i=1 s,l=1

to the constraints (2.14) lies in the linear span of the vectors
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The proof of Theorem 2 is similar to that of Theorem 1, but in this case it is necessary to replace z with w and the Lagrangian derivative (2.5) with the generalizing operator [ . ]w whose components act on the arbitrary function u(z , w) as follows d ([u]w)j = dt u wj
n

- aj (u) -
s,l=1

cl j ws s

u . wl

( 2. 15)

3. EQUIVALENCE OF SKATES AND WHEELS We show that the equations of motion of the vehicle (1.5) after restriction to the constraints gk1 , gk2 coincide with the equations of motion of the sleigh of a similar design in which the wheel pairs have been replaced with skates located at the centers of mass of the pairs perpendicularly to their axes. That is, the evolution of the variables v , , 1 , 2 is governed by the Lagrange equations of genus 2 in the quasi-coordinates T v T T k with the constraint equations fk = (vk , nk ) + k (ez â k , nk ) = 0, and the kinetic energy T = Tf +
k =1 2 s Tk , ·

+ ez â
·

T = v

2

k
k =1 2

fk , v
k

+
·

T ez , v â v T = 0, k

=
k =1

fk ,

( 3. 1)

-

k = 1, 2 ,

k = 1, 2 ,

( 3. 2)

1s2 1 2 s s Tk = ms vk + ms (Rk â vk , ez )k + jk k , k 2k 2

( 3. 3)

s where Tf is the kinetic energy of the platform (1.1), Tk is the kinetic energy of the k-th link with s is its mass, j s is its moment of inertia relative to the vertical axis passing through the a skate, mk k s point of attachment to the platform, and Rk is the radius vector from the point of attachment to its center of mass. To prove the above equivalence of the equation, we first show that the constraints gk1 , gk2 , k = 1 . . . 2 are quasi-holonomic. For ease of calculations we make the change of variables

k =

k1 + k2 , 2

k =

k1 - k2 . 2

( 3. 4)

In the new variables the constraint equations (1.4) (more precisely, their linear combination) b e c om e fk (q ) = (vk , nk ) + k (ez â k , nk ) = 0, ( 3. 5) 1 ( 3. 6) k = gk (q ) = - (vk , k ) + k (ez â k , k ), hk lk k = gk (q ) = - ( + k ), ( 3. 7) hk where q = (x, y , , 1 , 2 , 1 , 2 , 1 , 2 ) is the vector of the generalized coordinates. The kinetic energy of the link (1.1) takes the form Tk = 1 1 2 2 2 k mk vk + mk k (Rk â vk , ez ) + jk k + ik k + ik 2 . 2 2
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Remark. Obviously, the constraints k = sk (q ) are holonomic and after integration can be represented as hk k + lk ( + k ) = const. ( 3. 9) These constraints can be eliminated by direct substitution of k , expressed from (3.9), into the kinetic energy (3.8). After their elimination the remaining equations of motion for the multilink wheeled vehicle of an arbitrary design coincide with the equations of motion of the same wheeled vehicle whose wheel pair has been replaced with one wheel located at the center of mass of the wheel pair. In this case the connection between the dynamical characteristics of the initial link and the link with one wheel is defined by the relations i w = 2i k , k
w jk = j k + 2 2 lk ik , h2 k

mw = mk , k

w Rk = Rk .

( 3. 10)

We divide the configuration variables q into two groups y = (1 , 2 , 1 , 2 ), z = (x, y , , 1 , 2 ) and define the vector of quasi-velocities w = (v1 , v2 , , 1 , 2 ). By direct calculations it is easy to show that 1 fk , = ( + k ) ( 3. 11) [ g k ] w = 0, [ g k ] w hk w fk =0 where the operator [ . ]w is defined by (2.15). By Theorem 2 it follows from (3.11) that the constraints (3.6) and (3.7) are quasi-holonomic. After restriction to the quasi-holonomic constraints we obtain a system of equations (3.1) with
2

the constraints (3.2) and with the kinetic energy T = Tf +
k =1

Tk , where Tk is the restriction of

the energy (3.8) to the constraints (3.6), (3.7). Below we show that these equations coincide with s the same equations with kinetic energy (3.3), i.e., Tk coincides with Tk up to terms that do not influence the equations of motion. For this, we use the following proposition. Prop osition 1. Let L(q , w) be the Lagrangian of some system with generalized coordinates q and quasi-velocities w, and let fk (q , w) = 0, k = 1, . . . , n be the nonholonomic constraints imposed on the system. Then the equations of motion with the Lagrangian
n

L=L+
k,m=1

akm (q )fk fm ,

where akm (q ) are arbitrary (smooth) functions of the coordinates q , coincide with the equations of motion of the initial system. It is easy to prove the proposition via explicit calculations. We note that Tk can be represented as
Tk =

1 2

mk +

2i k h2 k

2 vk + k 2 k

mk Rk + ik 2 - 2 fk . hk

2i k k â vk , e h2 k

z

1 + 2

2i k 2 jk + 2 2 + lk hk k

( 3. 12)

Comparing (3.3) with (3.12) and using the above proposition, we arrive at the following conclusion. After restriction to the quasi-holonomic constraints the equations of motion of the wheeled vehicle of an arbitrary design coincide with those of the sleigh of a similar design whose wheel pairs have been replaced with skates located at the centers of mass of these pairs perpendicularly to their axes. In this case the relation between the mass-geometric characteristics of the links with the wheel pairs and skates is given by ms = mk + k 2i k , h2 k
s ms R k = mk R k + k

2i k k , h2 k
2015

s jk = j k +

2i k 2 2 ( + lk ). h2 k k

( 3. 13)

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As shown above, the wheel pair is equivalent to both a single wheel (see the remark on page 759) and to a skate. Hence, a vertically fastened single wheel is equivalent to a skate. That is, the equations of motion of the vehicle (with wheels) and the sleigh (with skates) coincide up to quadratures describing the rotation of the wheels. Moreover, from the comparison of (3.10) and (3.13) it follows that the difference of the parameters of the skate from those of the wheel lies only in an increase of its mass. Thus, the following statement holds. After eliminating quasi-holonomic constraints the equations of motion of the vehicle of an arbitrary design with vertical ly fastened wheels coincide with the equations of motion of the sleigh of a similar design whose wheels have been replaced with skates. The principal moments of inertia of the skates and the wheels relative to the vertical coincide, and the mass the skate ms is related to the mass-geometric characteristics of the wheel by ms = mw + i . h2

4. EXAMPLES 4.1. The Problem of a One-wheeled Vehicle As a simple example we consider the problem of a one-wheeled vehicle, i.e., a system consisting of a platform with a single wheel rigidly attached to it. Choose a moving coordinate system so that its axes are parallel to the vectors n and and the origin of coordinates is at the point of contact of the wheel with the plane (see Fig. 2). The kinetic energy and the nonholonomic constraints of such a system have the form 1 1 1 T = m v 2 + m (R â v , ez ) + j 2 + i 2 2 2 and f = ( v , n ) = 0, 1 = - (v , ). h ( 4. 2)
2

( 4. 1)

After eliminating the angular velocity of rotation of the wheel (restriction to the quasi-holonomic constraint) and adding the i term 2h2 f 2 to the kinetic energy, the kinetic energy takes the form T= 1 2 m+ i h2
2 2 vn + v + m (Rn v - R vn ) +

12 j , ( 4. 3) 2

Fig. 2. Schematic of the design of a one-wheeled vehicle

where vn , v , Rn and R are the pro jections of the vectors v and R onto the axes of the moving coordinate system and is the angular velocity of the one-wheeled vehicle. In the chosen notation the constraint remaining after elimination of the angular velocity of rotation of the wheel, , has the form vn = 0. ( 4. 4)

As we can see, the system with the kinetic energy (4.3) and the constraint (4.4) describes the motion of the Chaplygin sleigh (the mass of the skate of which has been increased by the quantity i as compared to the the mass of the wheel). h2 Remark 2. Let us write the constraint equations (4.2) in the form x = h si n y = -h cos . If now, following J. Hadamard, we calculate the combination of constraints which can be substituted into the Lagrangian, we obtain the second constraint (4.2), which, as we have shown, is quasiholonomic.
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4.2. A Vehicle with Two Symmetric Links Equations of motion. Consider the problem of motion of a vehicle with two symmetric links. By a symmetric link we shall mean a link whose center of mass coincides with the center of mass of the wheel pair and with the point of attachment of the link to the platform. In the chosen notation this means that in all formulas it is necessary to set k = 0, Rk = 0. Moreover, by choosing the coordinate system oxy (see Fig. 3) we reduce the parameters describing the geometry of the vehicle to the form r1 = (0, b1 ), where Rc Also, we shown ab design of is the radius vector of assume that all wheels ove, the vehicle of this such a sleigh is presente r2 = (0, -b2 ), R c = ( , 0) , the center of mass of the whole vehicle (including the wheels). are identical, that is, i1 = i2 = i, j1 = j2 = j , h1 = h2 = h. As design is equivalent to the sleigh with two rotating skates. The d i n F i g. 3.

Fig. 3. Design of a sleigh with two rotating skates

The kinetic energy of the system considered has the form T= 1 2 2 1 2 2 1 m(v1 + v2 ) + m v2 + I 2 + j (1 + 2 ) + j (1 + 2 ), 2 2 2 ( 4. 5)

where m and I are, respectively, the total mass (including the skates) and the total moment of inertia of the sleigh relative to the vertical axis passing through point O. Let us express the vectors nk and k in terms of the angles of rotation of the wheels k = (- sin k , cos k ), Then the constraint equations fk become f1 = (v1 - b1 ) cos 1 + v2 sin 1 = 0, f2 = (v1 + b2 ) cos 2 + v2 sin 2 = 0. The equations of motion (3.1) in the case at hand are
2

nk = (cos k , sin k ).

( 4. 6)

¨ v + ez â R c + ez â v - 2 R c =
2 k =1

k n k , ( 4. 7)

¨ 2 = c
k =1

k (k , Rc - rk ), k = 1, 2 , 2 = c
2 I - m R c - 2j , m

¨ ¨ k + = 0,

where k are found from the common solution of equations (4.7) and the derivatives of the constraints (4.6) fk = 0.
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Equations (4.7) define the phase flow in the seven-dimensional space M7 = {(v1 ,v2 , ,1 ,2 ,1 ,2 )}, and the constraints (4.6) are its integrals of motion. The sought-for equations of motion are a restriction of the flow (4.7) to the five-dimensional invariant manifold in M7 , which is given by (4.6). Reduction to the level set of first integrals. It follows from the third equation of (4.7) that these equations admit, along with the energy integral (4.5), two first integrals of motion T = k + = Ck , k = 1, 2. ( 4. 8) k The physical meaning of these integrals is that the vertical components of the absolute angular velocities of rotation of the skates of the (wheel pairs) remain constant during the whole time of motion. Thus, the five-dimensional invariant manifold (4.6) is foliated by two-dimensional submanifolds given by the integrals (4.5) and (4.8). We perform a partial reduction of Eqs. (4.7) by fixing the level set of the integrals (4.8). It is more convenient to do this before explicitly eliminating the undetermined multipliers k . This leads us to consider the flow in the five-dimensional phase space M 5 = R3 â T2 = {(v1 , v2 , , 1 , 2 )} given by 2 ¨ v + ez â R c + ez â v - 2 R c = k nk,
2 k =1

( 4. 9) k = 1, 2 ,

2 c

¨ =
k =1

(k k , Rc - rk ) ,

k = Ck - ,

where k are calculated as above. After restricting Eqs. (4.9) to the constraints (4.6), we obtain a flow on the three-dimensional invariant manifold in M5 . This manifold is foliated by two-dimensional invariant submanifolds (isoenergetic levels) given by the value of the energy integral, which, up to constant terms, can be written as 2 1 1 E = m v + ez â Rc + m2 2 . ( 4. 10) c 2 2 CONCLUSION To conclude, we consider in greater detail an important special feature of the nonholonomic systems, namely, that at some points of configuration space the constraint equations become linearly dependent. For example, for the system discussed in Section 4.2, under the conditions cos 1 = 0, two constraints (4.6) reduce to one equation6 The undeterm infinity under defined at the Thus, from
)

cos 2 = 0

( 4. 11)

v2 = 0. ined multipliers 1 and 2 , which correspond to the constraint reactions, tend to the conditions (4.11). Hence, the vector field defining the system's dynamics is not points of the phase space defined by (4.11). the geometrical point of view, in the phase space of the system (4.7) M5 = {(v1 , v2 , , 1 , 2 , 1 , 2 ) | f1 = 0, f2 = 0} 0 Eqs. (4.11) describe a four-dimensional (unconnected) submanifold M4 M5 on which the flow s 0 is not defined. Moreover, the two-dimensional invariant integral submanifolds of the system, which are defined by the energy integral and the linear integrals (4.8), are divided by the submanifold M4 s into several parts. In this connection we formulate some open problems concerning the description of the system's dynamics.
6)

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1. Give a topological description of two-dimensional integral submanifolds and study the behavior of the tra jectories on them. We note that in this case it is important to properly choose suitable local coordinates, since otherwise the pro jection may have singularities of Whitney umbrella type [2]. 2. Analyze the behavior of the tra jectories near the points of degeneration of the constraints and describe the resulting dynamical effects. Preliminary numerical studies show that in the neighborhood of these points the vehicle abruptly changes the direction of rotation. We note that the system of nonholonomic constraints may have another singularity -- the inconstancy of the growth vector of the distribution of the constraints [7, 8, 28], usually referred to as one of the obstacles to control. It is unknown whether the dynamical behavior of a nonholonomic system has any singularities near points where the growth vector is changed. We also note that, like systems with friction, nonholonomic systems may exhibit paradoxes that were first found by P. Painlev´ [45]. Most of them are very poorly explored. Only one of the e examples of such paradoxes is investigated in detail in [37] for a system combining the nonholonomic two-wheeled vehicle and the action of dry friction. ACKNOWLEDGMENTS The work of A. V. Borisov was carried out within the framework of the state assignment for institutions of higher education. The work of A. A. Kilin was supported by the RFBR grant no. 15-38-20879 mol a ved. The work of I. S. Mamaev was supported by the RFBR grant no. 13-01-12462-ofi m. REFERENCES
1. Chaplygin, S. A., On a Ball's Rolling on a Horizontal Plane, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 131­148; see also: Math. Sb., 1903, vol. 24, no. 1, pp. 139­168. 2. Zakalyukin, I. V., Dynamics of a Beam with Two Sleights via Systems of Implicit Differential Equations, Trudy MAI, 2011, no. 42, 25 pp. 3. Hadamard, J., Sur les mouvements de roulement, M´moires de la Soci´t´ des sciences physiques et e ee naturel les de Bordeaux, 4e s´rie, 1895, vol. 5, pp. 397­417. e 4. Hamel, G., Die Lagrange-Eulerschen Gleichungen der Mechanik, Z. Math. u. Phys., 1904, vol. 50, pp. 1­ 57. 5. Borisov, A. V. and Mamaev, I. S., Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems, Regul. Chaotic Dyn., 2008, vol. 13, no. 5, pp. 443­490. 6. Borisov, A. V. and Mamaev, I. S., Symmetries and Reduction in Nonholonomic Mechanics, Regul. Chaotic Dyn., 2015, vol. 20, no. 5, pp. 553­604. 7. Jean, F., The Car with N Trailers: Characterization of the Singular Configurations, ESAIM Control Optim. Calc. Var., 1996, vol. 1, pp. 241­266. 8. Agrachev, A. A. and Sachkov, Yu. L., Control Theory from the Geometric Viewpoint, Encyclopaedia Math. Sci., vol. 87, Berlin: Springer, 2004. 9. Kozlov V. V. On the realization of constraints in dynamics, J. Appl. Math. Mech., 1992, vol. 56, no. 4, pp. 594­600; see also: Prikl. Mat. Mekh., 1992, vol. 56, no. 4, pp. 692­698. ¨ 10. Vierkandt, A., Uber gleitende und rollende Bewegung, Monatsh. Math. Phys., 1892, vol. 3, no. 1, pp. 31­ 38, 97­116. ´ 11. Appell, P., Les mouvements de roulement en dynamique, Evreux: H´rissey, 1899. e 12. Bloch, A., Nonholonomic Mechanics and Control, New York: Springer, 2003. 13. Hamel, G., Theoretische Mechanik: Eine einheitliche Einfuhrung in die gesamte Mechanik, 2nd ed., ¨ Berlin: Springer, 1978. 14. Ehlers, K. M. and Koiller, J., Rubber Rolling: Geometry and Dynamics of 2 - 3 - 5 Distributions, in Proc. IUTAM Symposium 2006 on Hamiltonian Dynamics, Vortex Structures, Turbulence (Moscow, Russia, 25­30 August 2006), pp. 469­480. ¨ 15. Stuckler, B., Uber die Differentialgleichungen fur die Bewegung eines idealisierten Kraftwagens, Arch. ¨ ¨ Appl. Mech., 1952, vol. 20, no. 5, pp. 337­356. ¨ 16. Stuckler, B., Uber die Berechnung der an rollenden Fahrzeugen wirkenden Haftreibungen, Arch. Appl. ¨ Mech., 1955, vol. 23, no. 4, pp. 279­287. 17. Rocard, Y., L'instabilit´ en m´canique: Automobiles, avions, ponts suspendus, Paris: Masson, 1954. e e 18. Bottema, O., Die Bewegung eines einfachen Wagenmodells, Z. Angew. Math. Mech., 1964, vol. 44, no. 12, pp. 585­593. 19. Staicu, S., Dynamics Equations of a Mobile Robot Provided with Caster Wheel, Nonlinear Dynam., 2009, vol. 58, no. 1, pp. 237­248.
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