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ISSN 1560-3547, Regular and Chaotic Dynamics, 2012, Vol. 17, No. 2, pp. 191­198. c Pleiades Publishing, Ltd., 2012.

Two Non-holonomic Integrable Problems Tracing Back to Chaplygin
Alexey V. Borisov* and Ivan S. Mamaev**
Institute of Computer Science, Udmurt State University ul. Universitetskaya 1, Izhevsk, 426034 Russia
Received August 14, 2011; accepted November 29, 2011

Abstract--The paper considers two new integrable systems which go back to Chaplygin. The systems consist of a spherical shell that rolls on a plane; within the shell there is a ball or Lagrange's gyroscope. All necessary first integrals and an invariant measure are found. The solutions are shown to be expressed in terms of quadratures. MSC2010 numbers: 76M23, 34A05 DOI: 10.1134/S1560354712020074 Keywords: non-holonomic constraint, integrability, invariant measure, gyroscope, quadrature, coupled rigid bo dies

INTRODUCTION Recent advances in the design of controlled devices that use one or several balls for their propulsion (see, e.g., [3, 4, 7­10, 12­14]) has recently evoked an increasing interest in various models (in particular, non-holonomic ones) for rolling motion of spherical shells, including the case where some of the shells contain intricate mechanisms inside. The classical problem of rolling motion of a dynamically non-symmetric balanced Chaplygin ball has been sufficiently well investigated [11]. A newer version of it, namely the rolling of a Chaplygin's ball over a sphere, has recently been discussed in a number of papers [1, 5, 6]. Here we consider two new integrable systems that trace back to Chaplygin. His paper [2] is essentially concerned with generalized conditions for the existence of integrals linear in velocities for mechanical systems that consist of several spheres. These conditions are even now far from being completely appreciated. Chaplygin discusses in detail two problems. The first of them deals with a system that consists of a spherical, geometrically and dynamically symmetric shell with a homogeneous ball rolling inside; the shell itself rolls without slipping on a horizontal plane (Fig. 1). Chaplygin established the integrability of this system by expressing the solutions in terms of quadratures. Although his method for obtaining quadratures is quite natural (and also applies to the problem of rolling of a body of revolution on a plane), we believe that in solving this problem Chaplygin committed a few inaccuracies which resulted in enormously complicated and hardly verifiable formulas. Besides, Chaplygin missed one of the additional first integrals. The complexity of the results seems to discourage Chaplygin himself. Indeed, for each newly obtained analytical solution he always pursued clarification of its dynamical and geometrical aspects, but not in that case! Here we present a new approach to this system, a complete set of first integrals and reduce the problem to quadratures. The second problem is concerned with rolling of a shell (with a spherical pendulum inside) on a plane. For this problem the equations of motion were integrated by Chaplygin. We show that a more general system which consists of a Lagrange's gyroscope placed inside a rolling sphere is also integrable. The equations can be integrated using some generalized versions of the Chaplygin vector integrals and two additional linear integrals.
* **

E-mail: borisov@rcd.ru E-mail: mamaev@rcd.ru

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1. THE CHAPLYGIN SYSTEM 1.1. Equations of Motion Consider a system that consists of two bodies: 1) a spherical shell rolling on a horizontal plane inside which there is a spherical cavity such that its center coincides with the shell's center and 2) a ball placed into the spherical cavity of the shell (Fig. 1). Both bodies are assumed to be balanced (i. e. the center of mass coincides with the geometrical center) and dynamically symmetric (i. e. the tensor of inertia is spherical). We denote their moments of inertia by I and i respectively.

Fig. 1

Choose a fixed reference frame Oxy z with origin in a horizontal plane through the center of the shell and with the Oz -axis directed vertically downwards. Let us write the equations for balance of the angular and linear momentum of the bodies. For the shell the balance of the angular momentum relative to its center Gs and the balance of the linear momentum read I = Ro k â No + Ri n â Ni , where and V are the angular the mass of the shell, Ro and Ri the force of gravity, No and Ni with the plane Qo and the ball relative to its center Gb are M V = No + Ni + Mg k, (1.1) velocity of the shell and the linear velocity of its center Gs , M is are its inner and outer radii, k = (0, 0, 1) is the unit vector along are the reaction forces acting on the shell at the points of contact Qi , g is the acceleration of gravity. Similar equations for the ball mv = -Ni + mg k, (1.2)

i = Rb n â (-Ni ),

here and v are the ball's angular velocity and the linear velocity of its center Gb , m is the mass of the ball, and Rb is its radius. We shall assume that there is no slipping at the points of contact Qo and Qi , i. e. the velocity of the point of contact Qo is zero: V + â Ro k = 0, V + â Ri n = v + â Rb n. (1.3) (1.4) and at the point Qi the velocities of the contacting elements of the shell and the ball coincide: These equations express the (non-holonomic) constraints imposed on the system. Using them, we eliminate the reaction forces from (1.1) and (1.2); this allows derivation of a closed system of equations governing the evolution of the vectors , and n. According to the definition of the vector n (see Fig. 1), we have v - V = (Ri - Rb )n, and (1.4) gives (Ri - Rb )n = (Ri - Rb ) â n. We express No from the second equation in the system (1.1) and V from (1.3), whence we find Ro k â No = Ro k â (-Ni - Mg k - MRo â k).
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This allows us to represent the equations for and in the form
2 I + MRo k â ( â k) = J = a â Ni ,

i = -Rb n â Ni ,
-

(1.5)

2 2 where J = diag(I + MRo ,I + MRo ,I ) and a = Ri n - Ro k = Qo Qi is the vector connecting the points of contact, and the reaction force Ni can be found from the second equation (1.2) and the constraint (1.4):

Ni = -mv + mg k,

v = â a - â Rb n.

(1.6)

Differentiating v and simplifying, we ultimately obtain: The equations of motion for the shel l and the bal l rol ling without slipping on a plane can be written as J + ma â ( â a) - mRb a â ( â n) = -ma â (Ri - Rb ) â n + mg Ri n â k, 2 i + mRb n â ( â n) - mRb n â â a = mRb n â (Ri - Rb ) â n - mg Rb n â k, (Ri - Rb )n = (Ri - Rb ) â n, (1.7) where a = Ri n - Ro k, k = (0, 0, 1). The trajectory of the shel l's center rs (t) (point of contact) is given by rs = V = Ro k â . 1.2. Invariant Measure, First Integrals, Gyroscopic Function Equations (1.7) admit an invariant measure d d dn whose density on the level surface of the geometrical integral n2 = 1 is as follows: = 1 2 ,
2 2 mRb I + MRo + me (Ri - Ro )2 2Ri (1 - n2 ), + (1 - n3 )+ 3 2 m e Ro Ro i 2 2 I + MRo + me (Ri - Ro )2 2Ri MRi (1 - n2 ), + (1 - n3 )+ 2 (n3 ) = 3 2 m e Ro Ro I

(1.8)

1 (n3 ) =

(1.9)

where, as in celestial mechanics, we refer to me =

im 2 i+mRb

as the reduced mass of the ball.

Remark. Concatenating the angular velocities into a single vector w = (, ), we can write the first six equations of the system (1.7) as Gw = b(w, n), where G is a 6 â 6frm[o]­matrix whose components depend on n and b is a six-dimensional vector. It turns out that det G = 1 2 , where is a constant. The equations of motion (1.7) admit obvious first integrals: the geometric integral n2 = 1, 1 energy E = (M V 2 + I 2 + mv 2 + i 2 ) - mg (Ri - Rb )(k, n), 2 where V and v are expressed from (1.3) and (1.4) as follows: V = Ro â k ,
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(1.10)

v = â a - â Rb n.
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Moreover, Eqs. (1.7) also admit the vector integral K = J + Ri i + mRo v â k Rb (1.11)

which is linear in the angular velocities and . (The simplest way to find this integral is to use (1.5) and (1.6).) In addition to these three integrals, there are two more integrals linear in the angular velocities: MRo Ri F1 = (J,Ri n - Ro k) - IRb ( , n) - MRo Ri (v â k, n)+ (i , k ), mRb F2 = (Ri - Rb , n) 2 . (1.12)

Remark. The existence of the vector integral K was discovered by S. A. Chaplygin [2] during his study of general mechanical systems that admit vector integrals linear in velocities. In this context, Chaplygin treated this system as a comparatively trivial but ultimately edifying example. For this specific system Chaplygin found the general integrals (1.10), (1.11) and the integral F1 (but did not indicate the integral F2 and measure). His attempts of integrating the equations by means of various substitutions led to a collection of bulky and unwieldy expressions (which most likely are fallacious). Thus, the system of nine equations (1.7) admits seven first integrals and an invariant measure and, therefore, is integrable (by the Euler ­ Jacobi theorem). Moreover, on a fixed level of the first integrals M, = {(, , n) | n2 = 1, K = ,F1 = 1 ,F2 = 2 } the energy integral can be represented as E= 1 3 i 1 n2 2 - mg (Ri - Rb )n3 + U (n3 )+ 2 2 1 - n3 I +i mR i=i I +i
2 Ri 2 Rb 2 2 Ri o R2 b 2 Ri 2 Rb

2 + 1

2 2 2 o

,

+(m + M )R

-1
2 o

,

+(M + m)R

B0 B1 C0 2 . U (n3 ) = A0 2 + A1 1 3 + A2 2 + 1 2 + 3 2 + 1 3 2 2 2 2 Here Ak are quadratic polynomials in n3 , Bk are cubic polynomials, and C0 are polynomials of degree four. The coefficients of the polynomials depend in a complicated fashion on the parameters of the system I , i, M ,..., and for this reason we do not write them out here (they can be easily found using any system of analytic computations, for example, Maple, Mathematica, etc.). For a fixed value of energy E= 1 2I +i
2 Ri 2 Rb

2 + 1

2 2 2 o

+h

+(m + M )R

the derivative n3 can be found from the equation n2 = 3 2(1 - n2 ) 3 i1 (n3 ) h + mg (Ri - Ro )n3 - U (n3 ) , (1.13)

where the gyroscopic function of the system is contained within the brackets.
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1.3. A spherical shell with a pendulum There is an interesting limiting case of this Chaplygin system which is equivalent to the problem of rolling of a shell with a spherical pendulum fixed at its center. To illustrate this, consider a formal problem: an ball rolls (without slipping) over the outer side of a spherical surface which is fixed at the center of the shell (Fig. 2). Since the vector from the point of contact Qi to the center of the ball Gb is now pointed in the opposite direction, we must replace Rb by -Rb in Eqs. (1.7). Then we let Ri go to zero and thereby obtain a system for which the functions (1.10) and (1.11) remain integrals of motion and the second of the functions (1.12) simplifies to ( , n) = const. On the zero level of this integral ( , n) = 0 we get a system that is equivalent to a ball with a spherical pendulum.

Fig. 2

It is interesting to note that the first of the integrals (1.12) becomes degenerate and naturally "reincarnates" into the integral 3 = const. Here we do not analyze this problem in detail because in the next section we consider a more general system that includes it as a particular case. Remark. dealt with absolutely gyroscopic Chaplygin himself also considered the problem of a shell with a spherical pendulum. He the configuration depicted in Fig. 1, but assumed the inner surface of the shell to be smooth. In this case he explicitly found all necessary integrals and the corresponding function. 2. A SPHERICAL SHELL WITH LAGRANGE'S TOP 2.1. Equations of Motion We now consider in greater detail a more general system which cannot be obtained as a particular or limiting case of the Chaplygin system, namely a spherical shell rolling without slipping on a horizontal plane with an axisymmetric top at the center of the shell (Fig. 3). Similar problems arise in the design of control strategies for ballbots. One of the most intriguing pro jects here is a ballbot for interplanetary missions (e.g. a Mars rover, [14]). As in the previous case, we associate a fixed reference frame Oxy z with a horizontal plane Oxy passing through the center of the ball. Let Gs be the center of mass of the shell, Gt the center of mass of the top and let Rt = |Gs Gt | denote the distance between them. Assuming that in the top's principal axes frame its tensor of inertia is i = diag(i, i, i + j ), we can represent the kinetic energy of the system as 1 1 T = (M V 2 + I 2 )+ (mv 2 + i 2 + j ( , n)2 ), 2 2
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Fig. 3

Here V and are the linear velocity of the center of the shell and the shell's angular velocity, M and I stand for its mass and moment of inertia, v and are the linear velocity of the center of mass of the top and its angular velocity, and finally m is the mass of the top. The equations for balance of the angular momentum relative to the point Gs and balance of the shell's linear momentum read T
.

= I = Ro k â No ,

T V

.

= M V = No + Nt + Mg k,

where No , Nt are the reaction forces on the shell applied at the point of contact Qo and the point where the top Gs is attached. Similar equations for the top (relative to its center of mass Gt ) are T
·

= (i + j ( , n)n)· = Rt n â Nt ,

T v

·

= mv = mg k - Nt .

The no-slip condition at the point of contact Qo implies that V = Ro k â , whereas the velocity of the center of mass of the top is v = V + Rt â n = Ro k â + Rt â n. The evolution of the vector n obtained from the equation Rt n = v - V reads n = â n. Using these relations and the fact that there is an integral ( , n) = const, we can get rid of the reaction forces No , Nt (as this was done for the Chaplygin system) and thus obtain the following result:

The equations of motion for a spherical shel l with an axisymmetric top fixed at its geometrical center can be represented as
2 J + mRo k â ( â k) - mRo Rt k â ( â n) = mRo Rt k â ( â n), 2 i - mRt n â ( â n) - mRo Rt n â ( â k)

n = â n.

2 = -j ( , n)n - mRt n â ( â n)+ mg Rt n â k,

(2.1)

2 2 Here J = diag(I + MRo ,I + MRo ,I ) and k = (0, 0, 1). The trajectory of the point of contact of the shel l with the plane is then also given by (1.8).

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2.2. Invariant Measure, First Integrals and Gyroscopic Function To save space, we put
2 I = I +(M + m)Ro ,

Js = diag(I, I, I ), .

=

2 2 j I - mRt (I + mRo ) 2 2 iI + mRt (I + mRo )

The density of the invariant measure d d dn from Eqs. (2.1) is form (n3 ) =
2 (i + mRt )I 2 2 R2 - n3 . m 2 Rt o

As in the previous case, this system admits seven first integrals: geometrical integral energy Chaplygin's vector integral linear integrals where V = Ro â k , On a common level of the first integrals M, = (, , n) | n2 = 1, K = ,F1 = 1 ,F2 = the energy of the system can be written as
22 m2 Ro Rt (n3 ) 2 n3 1 + U(n3 ) - mg Rt n3 , E = (, J-1 )+ s 2 1 - n2 2I 3 2

n 2 = 1, 1 1 E = (M V 2 + I 2 )+ (mv 2 + i 2 + j ( , n)2 ) - mg Rt (n, k), 2 2 K = J + mRo v â k, F1 = 3 + ( , n)n3 , F2 = ( , n ) , v = Ro â k + Rt â n.

U =

1 2(1 - n2 ) 3

i+

2 2 mRt (I + MRo )

I

2 - 2(i + j )n3 1 2 +(i + j )(1 + n2 )2 . 1 32

Thus, on a level surface of the energy integral 1 E = (, J-1 )+ h s 2 the evolution n3 is governed by the equation n2 = 3 2I (1 - n2 ) 3 h + mg Rt n3 - U(n3 ) , 22 m2 Ro Rt (n3 )

where the gyroscopic function of the system is contained within the brackets. 3. DISCUSSION In conclusion we highlight several problems which can be solved using the results of this paper. 1. Note that the property of integrability is preserved when a ball rolls on the outer surface of a shell (Fig. 4). This can seriously facilitate a complete analysis of stability (stabilization) of rotation and rolling of the ball on the top of the shell. Moreover, the shell itself can be allowed to roll. (It is interesting to note that similar arrangements are often used in circus performances where maintaining equilibrium is the actor's concern.) 2. The systems described in this paper are ubiquitous in control applications (an extensive list of references can be found in [9]). Taking advantage of their integrability, one can develop a set of elementary stable solutions which can be used for solving various control problems.
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Fig. 4

ACKNOWLEDGMENTS This research was supported by the Grant of the Government of the Russian Federation for state support of scientific research conducted under supervision of leading scientists in Russian educational institutions of higher professional education (contract no. 11.G34.31.0039) and the Federal target programme "Scientific and Scientific-Pedagogical Personnel of Innovative Russia", measure 1.5 "Topology and Mechanics" (pro ject code 14.740.11.0876). The work was supported by the Grant of the President of the Russian Federation for the Leading Scientific Schools of the Russian Federation (NSh-2519.2012.1). REFERENCES
1. Borisov, A. V., Kilin, A. A., and Mamaev, I. S., Generalized Chaplygin's Transformation and Explicit Integration of a System with a Spherical Support, Regul. Chaotic Dyn., 2012, vol. 8, no. 2, pp. 170­190. 2. Chaplygin, S. A., On Some Generalization of the Area Theorem with Applications to the Problem of Rolling Balls, Regul. Chaotic Dyn., 2012, vol. 8, no. 2, pp. 199­217 [Russian original: Mat. Sb., 1897, Vol. 20; reprinted in: Col lected Works: Vol. 1, Moscow­Leningrad: Gostekhizdat, 1948, pp. 26­56]. 3. Alves, J. and Dias, J., Design and Control of a Spherical Mobile Robot, J. Systems and Control Engineering, 2003, vol. 217, pp. 457­467. 4. Bhattacharya, S. and Agrawal, S. K., Design, Experiments and Motion Planning of a Spherical Rolling Robot, in Proc. of the IEEE Internat. Conf. on Robotics and Automation (San Francisco, CA, April 2000), IEEE, 2000, pp. 1207­1212. 5. Borisov, A. V., Kilin, A. A., and Mamaev, I. S., Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere, Regul. Chaotic Dyn., 2011, vol. 16, no. 5, pp. 465­483. 6. Borisov, A. V., Fedorov, Yu. N., and Mamaev, I. S., Chaplygin Ball over a Fixed Sphere: An Explicit Integration, Regul. Chaotic Dyn., 2008, vol. 13, no. 6, pp. 557­571. 7. Camicia, C., Conticelli, F., and Bicchi, A., Nonholonimic Kinematics and Dynamics of the Sphericle, in Proc. of the 2000 IEEE/RSJ Internat. Conf. on Intel ligent Robots and Systems (Takamatsu, Japan, Oct. 31 ­ Nov. 5 2000), IEEE, 2000, pp. 805­810. 8. Chung, W., Nonholonomic Manipulators, Springer Tracts in Advanced Robotics, vol. 13, Berlin: Springer, 2004. 9. Crossley, V. A., A Literature Review on the Design of Spherical Rol ling Robots, Pittsburgh, PA, 2006. 10. Goncharenko, I., Svinin, M., and Hoso e, S., Dynamic Mo del, Haptic Solution, and Human-inspired Motion Planning for Rolling-based Manipulation, J. of Computing and Information Science in Engineering, 2009, vol. 9, no. 1, 011004, 10 pp. 11. Kilin, A. A., The Dynamics of Chaplygin Ball: The Qualitative and Computer Analysis, Regul. Chaotic Dyn., 2001, vol. 6, no. 3, pp. 291­306. 12. Michaud, F. and Caron, S., Roball, the Rolling Robot, Autonomous Robots, 2002, vol. 12, pp. 211­222. 13. Mukherjee, R., Minor, M. A., and Pukrushpan, J. T., Motion Planning for a Spherical Mobile Robot: Revisiting the Classical Ball-plate Problem, J. Dyn. Syst. Meas. Control, 2002, vol. 124, pp. 502­511. 14. Wilson, J. L., Mazzoleni, A. P., DeJarnette, F. R., Antol, J., Ha jos, G. A., and Strickland, C. V., Design, Analysis, and Testing of Mars Tumbleweed Rover Concepts, J. of Spacecraft and Rockets, 2008, vol. 45, no. 2, pp. 370­382.

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