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ISSN 1560-3547, Regular and Chaotic Dynamics, 2013, Vol. 18, Nos. 1­2, pp. 144­158. c Pleiades Publishing, Ltd., 2013.

How to Control the Chaplygin Ball Using Rotors. II
Alexey V. Borisov* , Alexander A. Kilin** , and Ivan S. Mamaev***
Institute of Computer Science; Laboratory of Nonlinear Analysis and the Design of New Types of Vehicles Udmurt State University, Universitetskaya 1, Izhevsk, 426034 Russia A.A. Blagonravov Mechanical Engineering Research Institute of RAS Bardina str. 4, Moscow, 117334, Russia Institute of Mathematics and Mechanics of the Ural Branch of RAS S. Kovalevskaja str. 16, Ekaterinburg, 620990, Russia
Received December 12, 2012; accepted February 16, 2013

Abstract--In our earlier paper [3] we examined the problem of control of a balanced dynamically nonsymmetric sphere with rotors with no-slip condition at the point of contact. In this paper we investigate the controllability of a ball in the presence of friction. We also study the problem of the existence and stability of singular dissipation-free perio dic solutions for a free ball in the presence of friction forces. The issues of constructive realization of the proposed algorithms are discussed. MSC2010 numbers: 37J60, 37J35, 70E18, 70F25, 70H45 DOI: 10.1134/S1560354713010103 Keywords: non-holonomic constraint, control, dry friction, viscous friction, stability, perio dic solutions

Contents
INTRODUCTION 1 EQUATIONS OF MOTION AND FIRST INTEGRALS 2 THE CONTROL PROBLEM 3 CRITICAL AND DISSIPATION-FREE MOTIONS OF A FREE BALL 3.1 3.2 Critical Perio dic Solutions and their Stability An Invariant Submanifold with Zero Velo city of the Point of Contact 145 145 147 149 151 154 157 158 158

DISCUSSION ACKNOWLEDGMENTS REFERENCES

* ** ***

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

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INTRODUCTION After publication of our first paper [3] we discovered several papers of other authors [7, 17, 19, 20] devoted to similar problems. First of all, we mention the papers [19, 20] in which the problem of controlling the nonholonomic Chaplygin ball using rotors is solved as well. These works actually do not discuss the problem of controlling three rotors (regarding it as too simple) but concentrate on the control by means of two rotors. Such a system is more simple and has a smaller weight (a discussion of engineering issues is contained in the concluding section of this paper), but possesses a deficit of control parameters. In those papers it is shown that it is not possible to dynamically realize any kinematically possible motion of the ball using two rotors. For possible motions, examples are given in [1, 19­21] for controls along the simplest tra jectories allowing the motion by successive steps to any given point. In addition, the problem of optimal control is discussed and the tra jectories of minimal length connecting the given points are numerically constructed. We note that the forms of equations used in [19, 20] are different from those presented in [3] and that additional recalculations are necessary to compare the results. 1. EQUATIONS OF MOTION AND FIRST INTEGRALS Consider the rolling of a dynamically asymmetric balanced ball (the center of mass coincides with the geometric center) on a plane (Fig. 1). We shall assume that the friction force F acts at the point of contact P , which in the case of slipping of the point of contact opposes its velocity Vp : F = -Vp , > 0. (1.1) In addition, we shall assume that three rotors with non-coplanar axes of rotation are attached to the body and that these axes are set in motion by electric motors.

Fig. 1

Choose a body-fixed frame Oex ey ez whose axes are aligned with the principal axes of inertia of the ball (see Fig. 1). We shall parametrize the relative rotation of the axes by the orthogonal matrix Q SO(3), assuming that the coordinates of the moving vectors e1 , e2 , e3 in the fixed axes ex , ey , ez lie on the matrix rows: e1x e1y e1z Q = e2x e2y e2z . e3 x e3 y e3 z Let r = (x, y ) R2 denote the coordinates of the center of mass of the ball C relative to the fixed coordinate system. Thus, the configuration space of the system is R2 â SO(3), and the pair (r , Q) uniquely defines the position and orientation of the ball. Let V be the velocity of the ball's center and its angular velocity. Then the equations of motion expressing the balance principles of linear and angular momentum of the system can be represented as mV = F ,
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(I + K )· = R â F ,
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(1.2)

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where R = -aez is the vector from the center of mass to the point of contact, K is the gyroscopic momentum of the rotors, I is the tensor of inertia of the ball relative to the center of mass in the fixed axes, it is related to the mass moment of inertia tensor I = diag(I1 ,I2 ,I3 ) by I = QT IQ. (1.3)

In order for the velocity of the center to remain parallel to the plane, i.e. Vz 0, we shall assume that in Eqs. (1.2) the force also satisfies the condition Fz 0. These equations must be supplemented with kinematic relations governing the rotation of the moving axes and the motion of the center of mass. They can be represented in matrix and scalar form as follows: 0 z -y Q = Q, = -z 0 x , (1.4) y -x 0 x = Vx , y = Vy .

Note that for an arbitrary three-dimensional vector a the relation a = a â is satisfied. It is also straightforward to show that the evolution of the tensor I is governed by I = [I, ], where [· , · ] is the matrix commutator. The system of equations (1.2) and (1.4) admits a vector integral, which is the angular momentum relative to the point of contact: M = I + K + mV â R = const. (1.6) (1.5)

As we shall show below, the presence of this integral substantially simplifies the control of the ball along the prescribed tra jectory. Remark 1. If the ball experiences spinning friction in addition to sliding friction, the equations of motion (1.2) change only slightly: mV = F , where M s is the spinning friction torque. In this case only two components of the angular momentum vector (1.6) are integrals: Mx = const, My = const. I + K
·

= R â F + M s ez ,

In most of the well-known applications (in particular, in control problems) the gyrostatic momentum vector of rotors is a prescribed function of time in the moving axes attached to the body C e1 e2 e3 (we denote it by k(t)). Then the following relations hold K = QT k(t), K = â K + QT k(t).

The evolution of the energy of the ball is governed by 1 1 E = mV 2 + (, I), 2 2 = (F , Vp ) - (, K ) = (F , Vp ) - (, QT k), E (1.7)

where Vp = V + â R is the velocity of the point of contact. Since the force satisfies the relation (1.1), it follows that if k = const, then E < 0, i.e. the energy strongly decreases (dissipates).
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2. THE CONTROL PROBLEM Consider the problem of controlling the motion of a ball by specifying the rotational velocities of rotors, which, as was shown in the previous paper, is equivalent to finding the vector function k(t), which we shall call control. We are required to choose the control law such that the center of mass moves in a prescribed manner Rc (t)=(x(t),y (t), 0). The velocity of the center of mass V (t) = (x(t), y (t), 0) and its acceleration V (t)=(x(t), y(t), 0) are obviously prescribed functions as ¨ ¨ well. Using the specificity of the system (1.2), more precisely, the fact that K does not appear in the first equation and the fact that there exists an integral M , one can use the following algorithm for finding a control law: 1. using the law of friction we express the friction force F in terms of velocities V , ; 2. from the first equation of the system (1.2) we find x (t), y (t) as prescribed functions of time; 3. assuming z (t) to be an arbitrary function of time, using the integral (1.6) we find the control k(t). Arbitrariness in the choice of the function z (t) can be used for some additional orientation of the ball either in the process of motion or at the endpoint of the tra jectory. We illustrate this with appropriate examples. a. Nonholonomic rolling for M = 0. First of all, we generalize the result of the previous paper for rolling without slipping to the case where the momentum relative to the point of contact is not equal to zero. (For dry friction this corresponds to the condition that the force F lie inside the cone of friction: |F | < mg ). The condition of rolling without slipping and the angular momentum can be represented as V + â R = 0, M = I + mR â ( â R)+ K = const. The first relation for the prescribed tra jectory yields y (t ) x(t) , y = - . a a We shall assume the component z (t) to be a prescribed function of time (in the previous paper it was chosen such that the prescribed tra jectories on a plane and a sphere coincide). Then we obtain x = - k(t) = Q(t) M - I(t)(t) - ma2 ez â ((t) â ez ) , where Q(t), I(t) are found by solving Eqs. (1.4) and (1.5) with the prescribed (t). We note that if M is not parallel to the axis ez , then the center of the ball cannot be at rest when the electric motors are shut off (k = 0). b. Viscous friction. In this case F = -Vp = -(V + â R), (2.1) where is the coefficient of friction, which is a positive constant. From the first equation of the system (1.2) we find x = - 1 a y (t)+ m y (t ) , ¨ y = 1 a x(t)+ m x(t) . ¨

Assuming z (t) to be some given function of time and solving (1.6) for K , we find k(t) = Q(t) M - I(t)(t)+ maV (t) â e
z

,

where the laws of evolution Q(t) and I(t) are found from the solution of differential equations (1.3) and (1.4). These relations make it possible to explicitly obtain the control of the ball along the prescribed tra jectory.
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c. Dry friction.

In this case, for slipping of the point of contact Vp V +âr = -mg , F = -mg |Vp | |V + â r |

where (coefficient of friction) is a positive constant. Substituting this into the first of the equations of the system (1.2), we obtain the following system for the components of velocity Vp x2 (t) - 2 g ¨ y (t ) ¨
2 2 Vpx 2 2 2 ¨ Vpx + x2 (t)V

+ y (t ) - ¨ V
px

2 py 22 2 g Vpy

= 0, = 0.

(2.2)

In the general case, these equations admit only the trivial solution = Vpy = 0, i.e. the velocity of the point of contact in a controllable motion should vanish, and we return to the initial nonholonomic system. The system (2.2) admits nontrivial solutions only under the condition ¨ x2 (t)+ y 2 (t) = 2 g 2 , ¨ (2.3) when the determinant of the corresponding matrix is equal to zero. The solution can be represented in one of the following two forms: V or V
2 px 2 px

= 1 x2 (t), ¨

2 Vpy = 1 x2 (t) - 2 g ¨ 2

2

,

1 > 0, 2 > 0.

= 2 y 2 (t) - 2 g ¨

,

2 ¨ Vpy = 2 y 2 (t),

Due to the condition (2.3) the physical solutions (i.e. those for which the squares of velocity ¨ components are positive) exist only under the condition x2 (t) = 2 g 2 , y (t) = const or y 2 (t) = 2 g 2 , ¨ x(t) = const. This means that the ball can move in a parabola or a straight line, but not along the prescribed tra jectory. Thus, in the case of dry friction with slipping of the point of contact the control using rotors along the prescribed trajectory is impossible. d. Calculation of controls for motors. In the previous paper there was an inaccuracy in the equations determining the voltage on controling electric motors (see the remark on p.262 in [3]) because the torque of the shell was not taken into account. We present a correct derivation in the notation used in this paper. According to the previous paper [3], the kinetic energy of the shell + rotors system is 1 1 T = mV 2 + (, I)+ i 2 2 1 k (Nk , )+ i 2 2 , k
k

(2.4)

k

where i, k and Nk are the moment of inertia, the angle of rotation and the directing vector of the rotation axis of the k -th rotor, respectively (the rotors are assumed to be identical). According to the d'Alembert ­ Lagrange equations of the 2-nd kind, the moment Mk which is developed by the electric motor can be written as T · . Mk = k Using the linear approximation of the moment supplied by the DC electric motor with commuta tor [14] and taking into account the relation Nk = â Nk , we obtain i k +(, Nk ) = cu uk (t) - cv k . ¨ According to [19], the quantities cu , cv are expressed in terms of the starting torque Ms and the rated torque Mr as well as its rated angular velocity r and voltage Ur , which are indicated in the data sheet for the electric motor, using the formulae Ms Ms - Mr , cv = . cu = Ur r
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We make use of the definition of the gyroscopic momentum K=i k Nk = QT k = QT i
k

k n

k

,

where nk are the constant directing vectors of the axes of rotors in the coordinate system attached to the body. For three rotors we denote the constant component matrix of these vectors by N = nkl . Inverting the above relations, we obtain u( t ) = c
-1 u

N

-1

cv k(t)+ k(t) + iNQ, i

u(t) = (u1 (t),u2 (t),u3 (t)).

Remark 2. We note that when deriving the kinetic energy (2.4) in our previous paper [3] we implicitly assumed that both for the shell and for the rotors separately the center of mass coincides with the geometrical center of the ball. The expression (2.4) remains valid in the more general case as well -- when only the general center of mass of the system coincides with the geometrical center. Indeed, in the general case the kinetic energy of the shell and the rotors is 1 1 T0 = m0 V 2 + (, I0 )+ m0 (V â , R0 ), 2 2 1 1 2 Tk = mk V + + k Nk , Ik ( + k Nk ) + mk (V â , Rk ), 2 2 where m0 , mk are the masses of the shell and the rotors, I0 , Ik are the tensors of inertia relative to the center of the ball, and R0 , Rk are the vectors from the geometric center to the centers of mass of the shell and the rotors. Combining these expressions and using the relations m0 R0 + k mk Rk =0, Ik Nk = iNk , we obtain (2.4), where I = I0 + k Ik . 3. CRITICAL AND DISSIPATION-FREE MOTIONS OF A FREE BALL We examine in more detail the problem of free (uncontrollable) motion of a ball with K = 0. Since the equations of motion (1.2), (1.4) do not depend on the coordinates of the center of mass, we shall consider only the system governing the evolution of the quantities V , , Q: mV = F , (I)· = R âF , =
ij k

Q = Q, k ,

I = QT IQ,

(3.1)

which possesses a vector integral of the angular momentum (1.6): M = I + mV âR, and is characterized by dissipation of the energy (1.7): E = (F , Vp ) 0, 1 1 E = mV 2 + (, I). 2 2 (3.3) (3.2)

It is obvious that this system admits the three simplest periodic solutions (found by Gallop [5]) -- permanent rotations about the principal axes of inertia ek , which are directed vertically, i.e. ek ez : V = 0, = (0, 0, 0 ), Qk cos(0 +0 t) Q0 = sin(0 +0 t) 0 0 0 1 0 S1 = 1 0 0 , S2 = 0 010 1
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k= - sin(0 +0 t) 0 cos(0 +0 t) 0 , 0 1 1 0 1 0 0 1 , S3 = 0 1 00 00
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(t) = Sk Q0 (t),

1, 2 , 3 ,

0 0 , 1

(3.4)

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where Sk are the permutation matrices of the principal axes. Note that for these solutions the friction force at the point of contact vanishes and the momentum integral is directed vertically: M = Ik 0 ez . A natural question arises: can any singular (critical, dissipation-free etc.) particular solutions or, in a more general case, invariant submanifolds different from (3.4) be found for the system (3.1)? To solve the problem at hand, we note that since in the case of slipping of the point of contact the friction force is F = 0, the energy E decreases according to (3.3) for all motions except for those for which Vp = 0. Therefore, if the motion starts under the condition Vp = 0, then, due to boundedness of E below, the energy tends to some limit E : E (t) - E --
t

min E (V , , Q).

Thus, we see: ­ if the system (3.1) admits singular invariant submanifolds, they belong only to some limit levels of the energy function; ­ for each such invariant manifold on a fixed level of energy E (x) = E , which we denote by Mi , the condition is obviously satisfied that if x0 Mi , then E (x0 ) = 0. E E Using the above-mentioned properties, one can propose two approaches in this case in order to solve the problem at hand. 1. Find a set of critical points of the energy integral on the fixed level surfaces of the momentum integral (3.2), i.e. Mc = x E |M =const = 0 . Obviously, it fol lows from the relation E (x) = xi E xi that E = 0.
i xi Mc

Although in the general case the set of critical points consists of isolated points of the extremum of the function E , it can in many systems (and, in particular, in this case) degenerate and lead to nontrivial critical invariant submanifolds . The merit of this approach is that it can be applied for an arbitrary law of friction; on the other hand, it can turn out that not all invariant submanifolds lie on the critical levels of E , or not the entire set of critical points is invariant relative to the flow. Furthermore, if no additional first integrals are known, then not all critical submanifolds can be found. Thus, for example, in [5] Gallop presents only one additional integral -- a vertical component of the momentum vector M . Minimizing the energy (3.3) under the condition Mz = const, he found only the solutions (3.4), which, as will be shown below, do not exhaust all possible critical solutions.

2. Single out the maximal invariant submanifolds inside the submanifold in phase space for which the velocity of the point of contact is zero by successively differentiating the relation Vp = 0 by virtue of the system (3.1).
The advantage of this approach is that it allows all possible invariant submanifolds to be found in principle. On the other hand, we need to know the dependence of F on phase variables, i.e. the law of friction; depending on the law of friction the invariant submanifolds differ substantially -- for example, for dry friction the invariant submanifold includes tra jectories of rolling without slipping, when the force F lies inside the cone of friction. A mo dification of this metho d for the case of viscous friction (more precisely, the averaging pro cedure) was used by Moshchuk [15].

Remark 3. For the case of viscous friction the limit E cannot be reached within a finite time (due to analyticity of the right-hand sides of the equations of motion [12]). For the case of dry friction, the velocity of the point of contact can, as a rule, vanish after a finite interval of time. If the friction force satisfies the condition |F | < mg (i.e. the reaction lies inside the cone of friction), then the ball will roll without slipping until this condition is violated.
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3.1. Critical Periodic Solutions and their Stability Since the angular momentum relative to the point of contact (3.2) is preserved during the motion, we first restrict the system to the five-dimensional integral submanifold MM 0 = {x | M (x) = M 0 }. On this manifold we shall search for a set of critical points for the corresponding restriction of the energy function E M 0 :
M

Mc 0 = {x | dE M

MM

0

( x ) = 0} .

(3.5)

Prop osition 1. On each integral manifold MM 0 in the phase space the set of critical points of energy Mc 0 constitutes three pairs of circles given by the relations M (i) = e
i± 0 0 0 My Mz Mx , , , Ii + D Ii + D Ii

V
2-
1 2

(i)

= R â (i) , ± cos i± n
(i) (i)

n

(i)

=

(i) , |(i) |
(i)

i = 1, 2 , 3 , â (n
(i)

= ±n(i) , e
k±

e

j±

( ( = (nxi) )2 +(nyi) ) 2-
1 2

â ez - sin i± n
(i)

â ez ) , (3.6)

( ( = (nxi) )2 +(nyi) )

± sin i± n

â ez +cos i± n

â (n

(i)

â ez ) ,

where i± [0, 2 ) are the angles parametrizing each of the circles and i, j , k correspond to a cyclic permutation of numbers. Proof. We parametrize the orientation matrix of the ball Q by the Euler angles: cos cos - cos sin sin cos sin +cos cos sin sin sin Q = - sin cos - cos sin cos - sin sin +cos cos cos cos sin sin sin - sin cos cos , (3.7)

where is the nutation angle, is the angle of proper rotation and is the angle of precession. As mentioned previously, the vectors e1 , e2 , e3 are principal axes, for which Iei = Ii ei . We shall search for the critical manifold (3.5) using the undetermined multipliers. This leads to the system of equations E - V Ms = mV - m V Ms E - s = 0 Mx = Mx , s R â = 0, E - s Ms = I - I, ... ,

1 (, I ) - (, I ) = 0, 2 0 0 My = My , Mz = Mz ,

where = (x ,y ,z ) are the undetermined multipliers corresponding to the components of the vector integral (1.6), and the dots denote equations corresponding to differentiation with respect to the angles , . The first two (vector) equations yield V = R â , = . (3.8) Substituting this into the remaining equations, we obtain (, I ) = (, I ) = (, I ) = 0. In order to solve these equations, we make use of the following relations I e1 = (I1 - I3 )sin e3 , I e2 = (I2 - I3 )cos e3 , I e3 = (I1 - I3 )sin e1 +(I2 - I3 )cos e2 , I e3 = 0, I e1 = (I1 - I2 )e2 , I e2 = (I1 - I2 )e1 , (3.9)

I e1 = (I1 - I2 )cos e2 - (I1 - I3 )sin cos e3 ,

I e2 = (I1 - I2 )cos e1 +(I2 - I3 )sin sin e3 ,

I e3 = (I1 - I3 )sin cos e1 +(I2 - I3 )sin sin e2 .
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Hence, we conclude that there exist three solutions of Eqs. (3.9) (i) = ei , i = 1, 2 , 3 ,

where is some scalar. Consequently, (i) is the eigenvector of the tensor of inertia directed along one of the principal axes: I(i) = Ii (i) . Using this and the relations (3.8), on the given level set of the angular momentum integral M = M 0 we find
( xi) = 0 Mx , Ii + D ( yi) = 0 My , Ii + D ( zi) = 0 Mz , Ii

D = ma2 .

(3.10)

(i)

The relations (3.8) and (3.10) completely determine the angular and linear velocities of the ball , V (i) = R â (i) and two possible directions of the corresponding principal axis ei (ei (i) or ei (i) ), whereas the angle of rotation about it remains arbitrary, i.e. the critical level of the energy function E on the integral submanifold M = M 0 is degenerate. Combining (3.8) and (3.10) and expressing the matrix of rotation about the given axis gives (3.6). Thus, on each level set M = M 0 the critical manifold constitutes three pairs of different circles, which are parametrized by the angle of rotation about the corresponding principal axis ei .

Substituting the relations (3.6) into the equations of motion (3.1), we find for the set of critical points that mV
(i)

= 0,

I(

i) ·

= Ii (i) = 0,

(i) (i) Q± = Q± (i) = 0.

Consequently, these critical levels constitute the system's periodic tra jectories for which the friction force F = 0, and the body rolls along the straight line in such a way that one of the principal axes preserves a constant angle with the vertical line, see Fig. 2. By analogy with the case of integrable systems [1], we call these solutions critical periodic solutions .

Fig. 2. Rolling of a ball along a straight line in the case

ei .

The dependence of the critical values of energy on the value of the angular momentum is given by the relation E
(i)

=

2 2 M2 1 Mx + My +z. 2 Ii + D Ii

(3.11)

Since by the change of time dt dt, = const we can always get M 2 = 1, it is convenient to choose the following quantities (see [2, 3]) as variables 2ED = h [0, ), M2 b2 2 i g, 1 - bi M z = g [-1, 1], M2 D ,i = 1, 2, 3. Ii + D
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then on the plane of these quantities the relation (3.11) defines three bifurcation curves: i : h = bi + bi =

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Fig. 3. Bifurcation curves, the region of possible motions (shown in grey) and the pro jections of the tra jectories of the system on the plane of variables g , h (with I1 > I2 > I3 ). In the shaded region the motion without slipping is impossible.

Fig. 3 shows these curves under the condition I1 > I2 > I3 . They coincide with the corresponding bifurcation curves in the problem of the Chaplygin ball rolling without slipping [2]. Since the energy (3.3) is limited below, the smallest of the critical values (3.11), which corresponds to rotation about the largest axis, determines the absolute minimum of energy on the surface MM , M = const. It corresponds to the lower of the bifurcation curves 1 in Fig. 3, and the region of possible values of g , h is situated above this curve (in the figure it is shown in grey). The pro jections of the tra jectories onto the plane of these variables are vertical straight lines. As is well known [2], in the case of the Chaplygin ball rolling without slipping the region of possible values of the variables g , h is bounded above by the curve 3 (which corresponds to rotation about the smallest axis). Consequently, for al l values of (g, h) above the curve 3 the motion of the bal l without slipping is impossible, i.e. Vp = 0. Prop osition 2. For each level surface MM , where M ez , the periodic solution (3.5) corresponding to rotation about the largest axis (Ii = Imax ) is stable. Proof. Firstly, as was mentioned above, the energy of the ball (3.3) dissipates everywhere or remains constant: E 0. Secondly, for the above-mentioned solutions E takes the absolute minimum on MM , and since the energy is an analytic function it is not a constant in a neighborhood of these solutions. Hence, E is a Lyapunov function of these solutions on MM , and by the Lyapunov theorem these solutions are stable with respect to perturbations belonging to MM . Moreover, since these solutions also depend analytically on the value of the integral M , they remain stable with respect to its small perturbations. We note that in the case of dry friction (i.e. |F | = mg for Vp = 0 and |F | mg for Vp = 0) the stability of these solutions usually cannot be asymptotic, because in their neighborhood the velocity of the point of contact can vanish within a finite time, after which the ball will roll without slipping, as in the Chaplygin ball problem. At the same time, under the condition of viscous friction (i.e. F = -v ) the stability will be asymptotic. To show this, we consider the case in more detail.
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3.2. An Invariant Submanifold with Zero Velocity of the Point of Contact We show that for the case of viscous friction the second method of searching for invariant manifolds mentioned above (p. 150) yields the same result as the first one. Prop osition 3. If in the system (1.2) in the case of zero velocity of the point of contact Vp = 0 the friction force vanishes (F = 0), then on each fixed level set M = M 0 the maximal invariant submanifold for which Vp = 0 coincides with the critical Mc 0 , see (3.5). M Proof. Differentiating the condition Vp = V + â R = 0, using the system (1.2) and noting that F = 0, we obtain â R = I-1 (I â ) â R = 0, R = -aez . Hence, we conclude that for the variables , Q the invariant submanifold can be given by one of the following relations I â = 0, or I â = Iez .

The first relation is equivalent to the fact that is the eigenvector of the tensor I, i.e. = ei , i = 1, 2, 3. By the proof of the previous proposition we obtain the invariant submanifold Mc 0 for M each fixed level set M = M 0 . Let us assume that the second relation is satisfied with = 0. Performing scalar multiplication of this relation by and I, we obtain (I, ez ) = (M , ez ) = 0, (I2 , ez ) = (IM , ez ) = 0. The first of these equations obviously defines the invariant relation, i.e. (M , ez )· 0. We show that the second equation is not an invariant relation. Since M ez , we choose the fixed axes such that M = M 0 ex and (IM , ez ) = M 0 I13 = 0. On the other hand, using (1.5), we obtain (IM , ez )· = I(M â ) - (IM ) â , e
z -1

=
2 2 I12 (I11 I33 + I22 I33 - I33 - 2I23 ) = (det I) -1

= (det I)

I12 (I).

The condition I12 = 0, along with the equation I13 = 0, implies that M and are the eigenvectors of the tensor I, which, as was mentioned above, determines the submanifold Mc 0 . M Consequently, it is necessary to examine the case I12 = 0 and to show that under the condition I13 = 0 the function (I) is separated from zero. To do this, we make use of the definition I = QT IQ, where I = diag(I1 ,I2 ,I3 ) is the mass moment of inertia tensor, and parametrize Q by the Euler angles (3.7). From the equation I
13

= 0 we find tg = (I1 - I2 )sin cos . cos (I1 sin2 + I2 cos2 - I3 )
2 3

Using this, we find 0 (, ) = (I)
I13 =0 2 2 = - sin2 sin2 I1 - sin2 cos2 I2 - cos2 I

+sin2 I1 I2 +(1 - sin2 sin2 )I2 I3 +(1 - sin2 cos2 )I1 I3 . Since this is a bounded function, its maximum and minimum values are determined by its values at critical points. It is straightforward to show that at critical points 0 (, ) takes one of three possible values: I1 (I2 + I3 - I1 ), I2 (I3 + I1 - I2 ), I3 (I1 + I2 - I3 ).
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For real bodies (balls), due to the inequality of the triangle Ii + Ij > Ik , all three values are positive, therefore, the function 0 (, ) > 0. From this proposition we conclude that in a neighborhood of the periodic solutions (3.6) in the case of viscous friction the strict inequality E < 0. is satisfied. By the Lyapunov theorem we obtain Prop osition 4. In the case of viscous friction the periodic solutions (3.6) corresponding to rotation about the largest axis (i.e. Ii = Imax ) are asymptotical ly stable. Remark. We note that we have proved the stability of absolute motion of the ball. That is, when the ball rolls, it tends to preserve the position of the rotation axis in absolute space, and the departure from the original tra jectory on the plane can be no more than linear in time For rotations about the middle and the largest axes the following is valid Prop osition 5. In the case of viscous friction the periodic solutions (3.6) corresponding to rotation about the midd le and the smal lest axes of inertia are unstable. Proof. We write Eqs. (3.1) in view of (2.1) in terms of the variables V , L = I, , , L = -R â (V + â R), 1 (1 sin - 2 cos ), = 3 - ctg (1 sin - 2 cos ), = 1 cos +2 sin , = sin (3.12) -1 L and I is expressed in terms of the Euler angles using the relation (1.3). We restrict where = I the system (3.12) to the joint level set of the momentum integrals M = const by eliminating the variables V1 , V2 and L3 from this system via (3.2). As a result, we obtain a system of five differential equations L1 = (M1 - L1 - ma2 1 ), L2 = (M2 - L2 - ma2 2 ), m m 1 (1 sin - 2 cos ), = 3 - ctg (1 sin - 2 cos ), = 1 cos +2 sin , = sin (3.13) in which the values of the integrals M1 , M2 , and L3 = M3 are included as parameters. Without loss of generality, we assume the value of the integral M2 to be equal to zero: this can be done by choosing the axes of the fixed coordinate system such that M ey is satisfied. For the system (3.13) it is convenient to represent the periodic solutions (3.6) as rotations about the axis e3 . Different solutions will correspond to appropriate rearrangements of the moments of inertia: rotation about the largest axis of inertia -- I1 < I2 < I3 , rotation about the middle axis -- I2 < I3 < I1 , and rotation about the smallest axis -- I3 < I1 < I2 . In view of the equalities M2 = 0 and M3 = L0 these periodic solutions are 3 L1 = L0 = 1 = 0 = , 2 1 M1 , 1+ a3 L0 1 , L0 3 L 2 = L 0 = 0, 2 0 = (L0 )2 +(L0 )2 1 2 , I3 (3.14) mV = -(V + â R),

= 0 = arctan

= 0 t,

where the notation ai = D is introduced. We denote the value of energy for these dissipation-free Ii motions by E0 = 213 (1 + a3 )(L0 )2 +(L0 )2 . 1 3 I To prove the instability of the solution (3.14), we make use of the Lyapunov instability theorem [11]. As an appropriate Lyapunov function we choose the difference FL = E - E0 (3.3). As was proved above, its derivative is a function of fixed sign FL < 0 everywhere outside the
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solutions (3.14). We show that the function FL itself alternates in sign in a neighborhood of (3.14) for rotations about the middle and the smallest axes. To do this, we consider the series expansion of FL near the solution (3.14) up to the second order: FL = 1 z , B()z + ..., 2D (3.15)

where z = (L1 - L0 ,L2 , - 0 , - 0 ), and B() is a symmetric 4 â 4 matrix with entries 1 periodically dependent on . To determine the signs of the eigenvalues of B(), we make use of the following assertion [6]. Theorem 1 (Jacobi's theorem). If al l main minors 1 ... n of the symmetric matrix B are different from zero, the number of positive eigenvalues of the matrix B is equal to the number of sign preservations, and the number of negative eigenvalues is equal to the number of sign reversals in the sequence 1, 1 ,... , n . The main minors of the matrix B() can be represented as (L 1 2 = 0 2 (L1 ) +(L0 ) 3 3 = 1 =
0 )2 1

1 +(L0 ) 3
2

2

(1 + a3 )(L0 )2 +(a1 sin2 + a2 cos2 +1)(L0 ) 1 3

2

,
2

(a1 cos2 + a2 sin2 +1)(L0 )2 +(1+ a1 )(1 + a2 )(L0 ) 1 3
2

, (3.16)

(1 + a3 )(L0 )2 1 (1 + a3 ) (a1 - a3 )cos2 +(a2 - a3 )sin2 (L0 ) 1 (L0 )2 +(L0 )2 1 3
2

+ (a1 - a3 )(1 + a2 )cos2 +(a2 - a3 )(1 + a1 )sin2 (L0 ) 3

,

4 = (L0 )2 (L0 )2 +(L0 )2 (1 + a3 )2 (a1 - a3 )(a2 - a3 ). 1 1 3 Next we consider two cases separately.

1. Rotation about the smallest axis of inertia (a3 > a1 , a3 > a2 ). In this case, for all values of the constants L0 , L0 and the angles the following inequalities 1 3 for the minors (3.16) are satisfied 1 > 0, 2 > 0, 3 < 0, 4 > 0.

Hence, it follows that two eigenvalues of the matrix B() are positive and two negative. Thus, in a neighborhood of the solution (3.14) the energy is an alternating function, and by the Lyapunov instability theorem the rotations about the smal lest axis of inertia are unstable. 2. Rotation about the middle axis of inertia (a1 0, 2 > 0, 4 < 0,

and 3 can, depending on the value of , take both positive and negative values. Nevertheless, the number of sign reversals does not change. Therefore, for all values of three eigenvalues of B() are positive and only one is negative. Thus, by the Lyapunov theorem the rotation about the midd le axis of inertia is also unstable .
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Fig. 4

DISCUSSION In conclusion, we discuss some open issues in the problem of control of a spherical robot with rotors as described in this paper (in particular, the issues of practical realization of the proposed algorithms). To verify the results of this paper and the previous one, an experimental model of the ball with the control of rotors was created in the "Laboratory of Nonlinear Analysis and the Design of New Types of Vehicles" (see the diagram and the photo in Fig. 4). A relevant demonstration video can be found at http://lab.ics.org.ru/videomaterialy/ 19-12-2012/eksperimentalnyy-obrazec-sfericheskogo-robota-s-rotorami/. The experimental results show that the motion and control of the ball are substantially influenced by rol ling friction. If all friction forces are taken into account, the equations of motion of the ball with rotors can be represented as: mV = F , (I + K )· = R â F + M r + M s , where M r = -r is the rolling friction torque, which is parallel to the plane of support and opposes the pro jection of the angular velocity onto this plane [18], and M f is the spinning friction torque directed vertically. In the case of dry friction : z , M s = -s mg ez , M r = -r mg | | |z | for viscous friction : M r = -r , M s = -s z ez , where r , s are the corresponding coefficients of rolling friction and spinning friction (positive quantities). This system does not admit a momentum integral relative to the point of contact (and other first integrals), therefore, the solution of the control problem for it becomes much more complex. This problem has not been solved until now and requires an additional study. We also point out a number of technical problems, which also need to be solved for a practical realization of the control of a spherical robot: 1. measurement of the moments of inertia of a real spherical robot; 2. fine balancing of the ball (i.e. making the center of mass coincide with the geometrical center of the sphere); 3. calculation of the coefficients of friction (as the experiments show, the behavior of the ball varies greatly depending on the types of surfaces). Furthermore, to achieve a sufficiently fast and well-controllable motion of a spherical robot, it is necessary that the moment of inertia of each rotor form a considerable part of the general moment of inertia of the ball, but in the case of three rotors this is technically difficult to realize. For the above reasons it seems to us that the use of the rotor as the main propulsion unit is ineffective. At the same time the use of rotors for maneuvering looks quite promising.
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ACKNOWLEDGMENTS This research was supported by the Presidential grant of leading scientific schools NSh2519.2012.1 and Target Programmes for 2012­2014 (State contract 1.1248.2011, 1.7734.2013, 1.7734.2013). A. A. Kilin's research was supported by the grant of the President of the Russian Federation for the Support of Young Russian Scientists­Doctors of Science (MD-2324.2013.1). REFERENCES
1. Bolsinov, A. V., Borisov, A. V., and Mamaev, I. S., Topology and Stability of Integrable Systems, Uspekhi Mat. Nauk, 2010, vol. 65, no. 2, pp. 71­132 [Russian Math. Surveys, 2010, vol. 65, no. 2, pp. 259­318]. 2. 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. 17, no. 2, pp. 170­190. 3. 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. 4. Chaplygin, S. A., On a Ball's Rolling on a Horizontal Plane, Math. Sb., 1903, vol. 24, no. 1, pp. 139­168 [Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 131­148]. 5. Gallop, E. G., On the Rise of a Spinning Top, Proc. Cambridge Philos. Soc., 1904, vol. 19, no. 3, pp. 356­ 373. 6. Gantmacher, F. R., The Theory of Matrices, Moscow: Nauka, 1967 [Providence, RI: AMS, 1998] 7. Ishikawa, M., Kitayoshi, R., and Sugie, T., Volvot: A Spherical Mobile Robot with Eccentric Twin Rotors, in Proc. of the IEEE Internat. Conf. on Robotics and Biomimetics (Phuket, Thailand, Dec. 7­11, 2011), pp. 1462­1467. 8. Kilin, A. A., The Dynamics of Chaplygin Ball: The Qualitative and Computer Analysis, Regul. Chaotic Dyn., 2001, vol. 6, no. 3, pp. 291­306. 9. Karapetyan, A. V., The Stability of Steady Motions, Moscow: Editorial URSS, 1998 (Russian). 10. Karapetyan, A. V., Global Qualitative Analysis of Tippe Top Dynamics, Izv. Akad. Nauk SSSR Mekh. Tverd. Tela, 2008, no. 3, pp. 33­41 [Mech. Solids, 2008, vol. 43, no. 3, pp. 342­348]. 11. Malkin, I. G., Theory of Stability of Motion, Moscow: Nauka, 1952 [Ann Arbor, MI: Univ. of Michigan Library, 1958]. 12. Markeev, A. P., Dynamics of a Rigid Body that Col lides with a Rigid Surface, 2nd ed., Moscow­Izhevsk: R&C Dynamics, Institute of Computer Science, 2011 (Russian). 13. Markeev, A. P., Dynamics of a Rigid Body that Col lides with a Rigid Surface, Moscow: Nauka, 1992. 14. Martynenko, Yu. G., Motion Control of Mobile Wheeled Robots, Fundam. Prikl. Mat., 2005, vol. 11, no. 8, pp. 29­80 [J. Math. Sci. (N. Y.), 2007, vol. 147, no. 2, pp. 6569­6606]. 15. Moshchuk, N. K., On the Motion of Chaplygin's Sphere on a Horizontal Plane, Prikl. Mat. Mekh., 1983, vol. 47, no. 6, pp. 916­921 [J. Appl. Math. Mech., 1983, vol. 47, no. 6, pp. 733­737]. 16. Moshchuk, N. K., Motion of a Heavy Rigid Bo dy on a Horizontal Plane with Viscous Friction, Prikl. Mat. Mekh., 1985, vol. 49, no. 1, pp. 66­71 [J. Appl. Math. Mech., 1985, vol. 49, no. 1, pp. 53­57]. 17. Otani, T., Urakubo, T., Maekawa, S., Tamaki, H., and Tada, Y., Position and Attitude Control of a Spherical Rolling Robot Equipped with a Gyro, in Proc. of the 9th IEEE Internat. Workshop on Advanced Motion Control, 2006, pp. 416­421. 18. Painlev´ P., Le¸ e, cons sur le frottement, Paris: Hermann, 1895. 19. Svinin, M., Morinaga, A., and Yamamoto, M., On the Dynamic Mo del and Motion Planning for a Class of Spherical Rolling Robots, in Proc. of the IEEE Internat. Conf. on Robotics and Automation, 2012, pp. 3226­3231. 20. Svinin, M., Morinaga, A., and Yamamoto, M. An Analysis of the Motion Planning Problem for a Spherical Rolling Robot Driven by Internal Rotors, in Proc. IEEE/RSJ Internat. Conf. on Intel ligent Robots and Systems, 2012, pp. 414­419. 21. Svinin, M., Morinaga, A., and Yamamoto, M., On the Dynamic Mo del and Motion Planning for a Spherical Rolling Robot Actuated by Orthogonal Internal Rotors, Regul. Chaotic Dyn., 2013, vol. 18, nos. 1-­2, pp. 126-­143.

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Nos. 1­2

2013