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ISSN 1560-3547, Regular and Chaotic Dynamics, 2015, Vol. 20, No. 6, pp. 716-728. c Pleiades Publishing, Ltd., 2015.

Spherical Rob ot of Combined Typ e: Dynamics and Control
Alexander A. Kilin1* , Elena N. Pivovarova1** , and Tatyana B. Ivanova2
2

***

Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia M. T. Kalashnikov Izhevsk State Technical University, ul. Studencheskaya 7, Izhevsk, 426069 Russia

1

Received Septemb er 17, 2015; accepted Novemb er 05, 2015

Abstract--This paper is concerned with free and controlled motions of a spherical robot of combined type moving by displacing the center of mass and by changing the internal gyrostatic momentum. Equations of motion for the nonholonomic mo del are obtained and their first integrals are found. Fixed points of the reduced system are found in the absence of control actions. It is shown that they correspond to the motion of the spherical robot in a straight line and in a circle. A control algorithm for the motion of the spherical robot along an arbitrary tra jectory is presented. A set of elementary maneuvers (gaits) is obtained which allow one to transfer the spherical robot from any initial point to any end point. MSC2010 numbers: 37J60, 70E18, 70F25 DOI: 10.1134/S1560354715060076 Keywords: spherical robot, control, nonholonomic constraint, combined mechanism

INTRODUCTION Over the last twenty years, starting with the work of Halme [1], the development and investigation of rob ots in the form of a sphere has attracted the attention of b oth many researchers in the area of mechanics and mechanical engineers. The advantage of such devices is that the spherical rob ots are more mobile and maneuverable than wheeled vehicles, and the spherical form protects well the fragile and moving parts of the rob ots against external damage. Also, the b ehavior of such rob ots is describ ed by mathematical models providing a detailed study of their dynamics and allowing control strategies for them to b e develop ed. Indeed, the investigation of the dynamics and controllability of spherical rob ots is a completely new and p opular model problem relating to a rapidly developing field, called the dynamics and control of nonholonomic systems. This problem is a tool for testing the methods of controlling dynamical systems in the presence of nonholonomic constraints. These nonholonomic constraints arise since slipping at the p oint of contact is neglected (model of an absolutely rough surface). Elementary control problems concerning the stabilization of motion were discussed as early as 1972 in the classical monography [2]. But with the advent of various mathematical software packages allowing computer simulations to b e carried out and with the development of new approaches in dynamical systems theory, new interesting results were obtained in the dynamics and control problem. In particular, we p oint out that the b ehavior of nonholonomic systems essentially differs from standard Hamiltonian mechanics. Nonholonomic systems can display prop erties that are typical of b oth Hamiltonian systems and dissipative and reversible systems and can exhibit b oth interesting regular [3-9] and complex chaotic b ehavior [10-12]. The free dynamics of the nonholonomic system we consider here are apparently nonintegrable and require a separate study. To date, a huge numb er of prototyp es of spherical rob ots have b een develop ed for a variety of applications ranging from different toys to devices for the exploration of other planets. The main difference b etween the existing designs of spherical rob ots lies in the internal drive mechanism.
* ** ***

E-mail: aka@rcd.ru E-mail: archive@rcd.ru E-mail: tbesp@rcd.ru

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There have b een many literature reviews concerning the description of the existing models and their technical realization (see, e.g., [13-16] and references therein), so we present here only some of the most recent results in this area (as compared to the review [17]). A detailed review of the principles of motion of spherical rob ots is made in [13]. Here we mention two main typ es of spherical rob ots: those moving by displacement of the center of mass [15, 18-22] and those set in motion by producing a variable gyrostatic momentum [17, 23]. We consider in more detail one of the most p opular (most studied) forms of locomotion, namely, displacement of the common center of mass of the whole rob ot, which causes it to move. This mechanism is implemented most commonly using a p endulum [18, 19] or an internal moving platform [15]. In [18], the dynamics of a spherical rob ot with a spherical p endulum is studied. The authors prop ose a path planning algorithm for a limited contact area of the spherical shell with the plane. They also consider the controlled motion "acceleration - deceleration" in a straight line. We note that the forcing actions describ ed by the authors for such a maneuver are analogous to the controls obtained in [19] for a ball with the Lagrange gyroscop e. The pap er [20] is devoted to the path planning for a spherical rob ot with a p endulum mechanism within the framework of the kinematic and dynamic models. We also mention the recent pap er [21], where the local controllability of a mobile spherical rob ot with a two-dimensional (spherical) p endulum is investigated. In [15], the dynamics and control of the motion of a spherical rob ot with an internal omniwheel platform is discussed. In the same pap er, particular solutions of the system are found, their stability is analyzed and control algorithms are presented for the motion along an arbitrary tra jectory and using elementary maneuvers (gaits). Also in [15], the theoretical results obtained are compared with the exp erimental results. The dynamics of spherical rob ots controlled by changing the internal gyrostatic momentum is considered, for example, in [17, 23]. In [17], the controlled motion of a dynamically asymmetric balanced ball by means of three noncoplanar rotors is investigated and the construction of an explicit algorithm for controlling the ball along a given tra jectory is presented. Related problems for the case of shortage of control actions (planning of paths realized by means of two rotors placed on orthogonal axes) are discussed in [23]. Unfortunately, in practice this model exhibits unsatisfactory dynamical b ehavior due to the influence of spinning friction and rolling friction. Despite a large numb er of models of spherical rob ots and their technical realizations, the question remains op en as to what typ e of propulsion device is b est-suited to ensure a simple control and efficiency of maneuvers. Exp erimental investigations of the dynamics of spherical rob ots with different internal propulsion devices (p endulum, rotors, omniwheel platform) have shown that a mechanism combining the ab ove-mentioned effects may b ecome the most promising for controlled motion. Motivated by this, we study the dynamics and controllability of a spherical rob ot of combined typ e that uses for its motion b oth the displacement of the center of mass and the change of gyrostatic momentum. 1. EQUATIONS OF MOTION AND FIRST INTEGRALS Consider the motion of a spherical rob ot of combined typ e rolling without slipping on a horizontal, absolutely rough plane (Fig. 1). The spherical rob ot is a spherical shell of radius Rs to the center of which an axisymmetric p endulum (gyroscopic p endulum) is attached. We shall model the gyroscopic p endulum by a weightless rod at the end of which a massive rotor is installed. The rotor is an axisymmetric b ody (disk) rotating ab out the symmetry axis coinciding with the rod (see Fig. 1). The technical design of the spherical rob ot has b een realized so that the p endulum is capable of executing oscillations only in a given plane related to the shell, which we shall call the plane of rotation of the pendulum in what follows. The spherical rob ot is set in motion by forced oscillations of the p endulum and by the rotation of the rotor by means of two motors. To describ e the dynamics of the spherical rob ot, we define two coordinate systems. The first one, O , is a fixed (inertial) coordinate system with the basis vectors , , . The second one, C e1 e2 e3 , is a moving coordinate system with the basis vectors e1 , e2 , e3 , the axes of which are attached to the p endulum so that the basis vector e1 is p erp endicular to the plane of rotation of the p endulum and the basis vector e3 is directed along its symmetry axis. The origin of the moving coordinate system coincides with the geometric center of the shell, C (see Fig. 1). In what follows, unless otherwise stated, we shall refer all vectors to the moving coordinate system C e1 e2 e3 .
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g aO b

e3 e1 C e2

Fig. 1. Schematic model of a spherical rob ot of combined typ e.

We shall sp ecify the p osition of the system by the coordinates of the center of the sphere r = (x, y , 0), by the angles of rotation, and , of the p endulum relative to the axes e1 and e3 , resp ectively, and by the matrix Q of transition from the fixed coordinate system to the moving coordinate system, the columns of which contain the coordinates of the fixed vectors , , referred to the moving coordinate system C e1 e2 e3 1 1 1 Q = 2 2 2 . 3 3 3 Thus, the configuration space of the system considered is the product N = {(r , ,, Q)} = R2 × T2 × SO(3). The absence of slipping at the p oint of contact of the shell with the plane is describ ed by the nonholonomic constraint F = v - Rs × = 0, (1.1) where v and are, resp ectively, the velocity of the center and the angular velocity of rotation of the shell. The kinetic energy of the sphere+p endulum system can b e represented as 1 1 1 2 ms v 2 + Is 2 + mb vb 2 2 2 where ms and Is are the mass and the moment of iner diag(Ibc1 ,Ibc1 ,Ibc3 ) are the mass and the central tensor of of the center of mass of the p endulum vb and its angular T= vb = v - Rb × e3 , where Rb Let us equations detailed d 1 + (, Ibc ), 2 tia of the spherical shell, mb and Ibc = inertia of the p endulum, and the velocity velocity are given by (1.2)

= + e1 + e3 ,

is the distance from the center of the sphere to the center of mass of the p endulum. write the dynamical equations of the system in the form of the D'Alemb ert - Lagrange of genus 2 in quasi-velocities with undetermined multipliers and forcing actions (for a erivation, see the App endix) L L
ž ž

-

L = K , L v + e1 , × L
T

-

L L + e1 , ×
ž

+ e1 , v ×

= K , (1.3) ,

L

+ + e1 L v
ž

L L L +v× + × = × v F v
T

F

L = + + e1 × v

,
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where is the vector of undetermined multipliers, K and K are the moments of external forces (control actions) applied to the p oint of attachment of the p endulum to the ball and to the p oint of attachment of the rotor to the rod of the p endulum, L = T - U is the Lagrangian function, and U = -mb Rb g( , e3 ) is the p otential energy of the system. For some asp ects of the theory of nonholonomic systems, see [2]. Supplementing Eqs. (1.3) with the kinematic relations describing the motion of the center of the spherical rob ot and the rotation of the moving axes relative to the fixed axes Q = Q + AQ, where the matrices and A have the form 3 - 2 0 = - 3 0 1 , 2 - 1 0 r = Q v, (1.4)

0 0 0 A = 0 0 1 , 0 -1 0

we obtain a closed system of equations completely describing the rolling motion of the spherical rob ot on the plane. The resulting system admits obvious geometric integrals 2 = 1, 2 = 1, 2 = 1, (, ) = ( , ) = ( , ) = 0.

After reduction to these integrals, Eqs. (1.3) and (1.4) define the flow in the twelve-dimensional phase space with two control actions. Thus, the system under consideration is an underactuated system. We recall that systems in which the numb er of controls is smaller than the numb er of degrees of freedom are called underactuated systems. A description of control with similar systems can b e found, for example, in [23, 24]. The explicit expression for the Lagrangian of the system considered has the form L= 1 (ms + mb )v 2 - mb Rb (v , × e3 )+ 2 1 + Ib1 2 1 (, (Is + Ib ))+ (, e1 )Ib1 +(, e3 )Ib3 2 1 2 + Ib3 2 - mb Rb (v , e1 × e3 ) + mb Rb g( , e3 ), (1.5) 2

2 2 where Ib = diag(Ib1 ,Ib1 ,Ib3 ) = diag(Ibc1 + mb Rb ,Ibc1 + mb Rb ,Ibc3 ) is the tensor of inertia of the p endulum relative to the center of the sphere. Since the Lagrangian L is indep endent of , and r , the equations of motion for , , , decouple from the complete system after substituting (1.5) into (1.3) and eliminating the undetermined multipliers, and take the following form:

Å e3 , Ib ( + e3 ) = K , Å e1 , Ib ( + e1 ) - mb Rb Rs e3 ×(× + × ) - e1 ,mb Rb Rs (× ) × ( + e1 ) × e3 + e1 , × mb Rb Rs (× ) × e3 +(Is + Ib ) + Ib1 e1 + Ib3 e3 + mb Rb g (e1 , × e3 ) = K ,

Å Å mb Rb Rs (× + × ) × e3 +(Is + Ib ) + Ib1 e1 + Ib3 e3 - mb Rb Rs (× ) × ( + e1 ) × e3 +( + e1 ) × mb Rb Rs (× ) × e3 +(Is + Ib ) + Ib1 e1 + Ib3 e3 + mb Rb g × e3 Å = Rs (ms + mb )Rs (× + × ) - mb Rb ( + e1 ) × e3 × +R
s

( + e1 ) × (ms + mb )Rs × - mb Rb ( + e1 ) × e3

× , (1.6)

= ×( + e1 ).

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Equations (1.6) govern the dynamics dimensional phase space M = {(, reconstructed from the solution of Eqs. condition (1.1).

of the reduced system and define the flow in the eight , , )}. The dynamics of the complete system can b e (1.6) with the help of the quadratures (1.4) and the no-slip

2. DYNAMICS OF THE FREE SYSTEM Consider the dynamics of the free motion of the spherical rob ot. By the free motion we mean the motion in the absence of control actions (K = K = 0). The free system admits, along with the geometric integral 2 = 1, another two integrals of motion - the integral linear in the angular velocities F1 = ( + e3 , e3 ) = 3 + ; - the energy integral 1 1 1 1 2 E = ms v 2 + Is 2 + mb vb + (, Ib ) - mb Rb g( , e3 ). 2 2 2 2 (2.2) (2.1)

The reduced system (1.6) needs three integrals and an invariant measure to b ecome completely integrable by the Euler - Jacobi theorem1) . It can b e shown that the system does not admit the existence of an invariant measure dep ending only on the p ositional variables and additional first integrals linear in the velocities. Hence, apparently, the system is integrable. Moreover, the absence of an invariant measure can lead to complex chaotic b ehavior. In particular, the system may exhibit strange attractors, as, for example, in rattleback dynamics [10]. In this pap er we restrict the study of the dynamics of the free system to analysis of its simplest particular solutions, namely, fixed p oints of the reduced system (1.6). Fixed p oints of the reduced system corresp ond to steady-state solutions of the complete system (1.3)-(1.4), which can b e of practical interest. The exp eriments conducted with different models of spherical rob ots have shown that such steady-state solutions can b e technically realized by means of constant forcing actions. For this to b e done, constant velocities of rotation of the control elements must b e given and the ball must b e brought to a certain initial p osition by means of the motors. ÅÅ Let us find fixed p oints of the reduced system. Setting in (1.6) the derivatives , , , and K , K to b e equal to zero, we obtain the following system of algebraic equations: e1 , × mb Rb Rs ( × ) × e3 +(Is + Ib ) + Ib1 e1 + Ib3 e3 - e1 ,mb Rb Rs ( × ) × ( + e1 ) × e3 + mb Rb g (e1 , × e3 ) = 0, (2.3)

mb Rb Rs ( × ) × ( + e1 ) × e3 - mb Rb g × e3 + mb Rb Rs ( × ) × e3 +(Is + Ib ) + Ib1 e1 + Ib3 e3
2 - (ms + mb )Rs ( × ) - mb Rb Rs ( + e1 ) × e3 × × + e1 = 0,

× ( + e1 ) = 0. We present all p ossible solutions of this system. 1) Two two-parameter families of fixed p oints v0 e1 , = +e3 , = Rs
1)

v0 =- , Rs

= 0 ,

(2.4)

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where v0 and 0 are the free parameters of the family. In absolute space, to fixed p oints of family there corresp ond motions of the ball in a straight line with constant velocity v0 where p endulum is in a vertical p osition and the rotor rotates with constant angular velocity 0 . e3 the sign is chosen dep ending on whether the p endulum is in the upp er or lower p osition. parameters of the family are related to the values of the first integrals by 0 = F1 ,
2 v0 = 2 2E - Ib3 F1 - 2mb Rb g 2 Rs . 2 Is +(ms + mb )Rs

this the For The

2) Three-parameter family of fixed p oints v0 v0 v0 , = (sin , 0, cos ), = - e1 , = Rs 3 Rs 3 (2.5) 2 2 -Rs ((ms +mb ) + mb Rb 1 ) - Rs 3 (mb Rb +1 (Ib1 - Ib3 )) + Is v0 +mb Rb gRs 2 1 =- , Rs v0 Ib3 1 where , v0 , are the parameters of the family. In absolute space, to this family there corresp onds a motion in a circle of radius with velocity v0 where the plane of rotation of the p endulum is inclined through the angle relative to the vertical. Thus, the reduced system of equations (1.6) describing the free motion of the spherical robot (K = K = 0) admits three families of fixed points: 1) two two-parameter families (2.4), which in absolute space correspond to the motion of the bal l with an arbitrary velocity in a straight line with an arbitrary angular velocity of rotation of the rotor; 2) the three-parameter family (2.5), to which in absolute space there corresponds the motion of the spherical robot with an arbitrary velocity v0 in a circle of arbitrary radius . In this case the inclination angle of the plane of rotation of the pendulum is constant and can be arbitrary too. Remark. It follows from the relations v0 = Rs 3 and |3 | 1 that the angular velocity of the p endulum, , is larger than or equal to the quantity v0 /Rs . As 3 0 (as the plane of rotation of the p endulum tends to the horizontal p osition), the angular velocity of the p endulum tends to infinity. This means that the larger the angle of inclination of the plane of rotation of the p endulum, the faster the p endulum must rotate relative to the shell to ensure that the spherical rob ot moves with the same velocity v0 in a circle of the same radius . Also, the absolute angular velocity of the moving coordinate system + e1 remains constant. 3. CONTROLLED MOTION 3.1. Control along an Arbitrary Tra jectory Consider the following version of the problem of controlled motion of the spherical rob ot. Suppose that at the initial instant of time we are given the position of the spherical robot r (0), Q(0), the velocity of rotation of the shel l about the vertical (0) and the angular velocities of rotation of the pendulum (0), (0). Can the control actions K and K be chosen such that during the interval t [0,T ] the spherical robot (its center) moves along a given trajectory r (t) = (x(t),y (t))? A similar problem in the case where the pro jection of the angular velocity (t) is given along with the tra jectory was solved in [17] for the Chaplygin ball with rotors and in [15] for a ball with an internal omniwheel platform. In the case we consider here the general solution of the control problem with an arbitrarily given function (t) apparently cannot b e obtained. This is due to a smaller numb er of control actions in the model considered. Therefore, in this section we restrict ourselves to controlling only a part of the variables, that is, we sp ecify only the dep endence x(t), y (t), and the pro jection of the angular velocity onto the vertical (t) remains an unknown function of time. Below we present an algorithm that allows us to solve this problem.
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1. Write the the angular velocity of the shell as = (t) + (t) + , where (t) and (t) are expressed from (1.4) using the constraint equation (1.1) as (t) = and are known functions 2. Substitute the exp The resulting equation, moving axes (1.4), forms , , , , , : y(t) , Rs (t) = - x(t) , Rs (3.1)

of time. ression for the angular velocity (3.1) into the third equation of (1.6). combined with the kinematic relations describing the rotation of the a closed, explicitly time-dep endent system of equations in the variables

Å Å mb Rb Rs (× + × )×e3 +(Is + Ib ) + Ib1 e1 + Ib3 e3 - mb Rb Rs (× )× ( + e1 )×e3 +( + e1 ) × mb Rb Rs (× ) × e3 +(Is + Ib ) + Ib1 e1 + Ib3 e3 + mb Rb g × e3 Å = Rs (ms + mb )Rs (× + × ) - mb Rb ( + e1 ) × e3 × +R
s

( + e1 ) × (ms + mb )Rs × - mb Rb ( + e1 ) × e3 = ×( + e1 ), = ×( + e1 ),

× , (3.2)

= ×( + e1 ),

where should b e viewed as the function (, , , ,t).

3. Solve this system numerically under the given initial conditions (0), (0), (0), (0), (0), (0). This yields the time dep endence of the vectors , , , of the vertical pro jection of the angular velocity of the ball, , and of the angular velocities of the p endulum, and . 4. Substitute the dep endencies obtained into the first two equations of (1.6) and find the explicit time dep endence of the control torques K , K . Example. As an example, we consider the motion of a ball in a circle of radius R0 in time T . We sp ecify the law of motion of the center of the ball along the tra jectory in the form x(t) = R0 cos s(t), y (t) = R0 sin s(t), (3.3)

s(t) = wt - sin wt,

where w = 2 . Figure 2 shows the resulting dep endencies of the comp onents of the vector , of the T angular velocities and and of the control torques K and K for the motion of the ball in a circle of radius R0 = 0.5 in time T = 78. The dep endencies of the control torques and angular velocities have the form of oscillations near zero with pronounced maxima and minima, and so Figs. 2b and 2c are represented on a modified (signed) logarithmic scale, i.e., instead of the functions f (t) the graphs arcsh(5 ž f (t)) are presented. Here and in the sequel, the numerical simulation was carried out for the following system parameters: mb = 1, ms = 0.1, Rs = 1, Rb = Rs /4, Ib1 = 0.3, Ib3 = 0.47, Is = 0.07. (3.4)

Remark. The problem of construction of control actions is not solvable for all tra jectories of the spherical rob ot. Indeed, equations (3.2) can b e reduced to the form A(z ) = f (z ,t), where z = (, , , , , ), and A(z ) and f (z ,t) are, resp ectively, a matrix and a vector that dep end on the variables z and time. The system under consideration contains a singularity on the manifold given by the degeneracy condition of the matrix A det A(z ) = 0.
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1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 (a) 2.00 0.73 0.23 0 -0.23 -0.73 -2.00 -5.46 10 20 30 10 20 30 40 50 60 70

14.84 5.46 2.00 0.73 0.23 0 -0.23 -0.73 -2.00 -5.46 (b) 10 20 30 40 50 60 70

40

50

60

70

(c )
Fig. 2. Time dep endence of the comp onents of the vector (a), of the angular velocities and (b) and of the control torques K and K (c) for the motion of the ball in a circle (3.3). The dep endence graphs (b) and (c) are represented on a modified logarithmic scale.

If the tra jectory of the system with given initial conditions passes near this manifold, then the velocities z take large values. Such a motion is technically not realizable since the velocities of rotation of the rotor and the p endulum are b ounded by the parameters of the motors. But if the tra jectory enters this manifold in finite time, then the construction of control for larger times is imp ossible even theoretically. Thus, the presence of the ab ove-mentioned singularities imp oses restrictions on p ossible tra jectories of the system and on the maneuverability of the model considered. 3.2. Control Using Gaits Ab ove we have shown that the system under consideration admits two kinds of steady-state solutions: the motion in a straight line (2.4) and that in a circle (2.5). Obviously, combining these solutions, one can realize the motion from any initial p oint to any end p oint. Moreover, since the steady-state solutions (particularly the stable ones) are the most applicable in practice, we consider here the problem of definition of control torques for the realization of transition from one steadystate solution to another. Using standard terminology, we shall refer to the elementary maneuvers realizing such transitions as gaits. Rotation ab out a fixed p oint. The sp ecial features of the design of the spherical rob ot we consider here allow it to move in a straight line only in a direction parallel to the plane of rotation of the p endulum. Consequently, b efore the start of the motion (acceleration), it is necessary to execute rotation ab out the vertical axis so that the direction of motion coincides with the plane of
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rotation of the p endulum. Therefore, as the first gait we consider the rotation of the spherical rob ot by a given angle 0 ab out the vertical axis passing through the fixed p oint in time T . Obviously, this gait connects two steady-state solutions corresp onding to the state of rest. The rotation ab out the vertical lies on the invariant manifold of Eqs. (1.4) and (1.6), which is given by (3.5) = (0, 0, 3 ), = 0, = (1 ,2 , 0), = (1 ,2 , 0), = (0, 0, 1), where 3 = with b eing the precession angle of the spherical shell. On this invariant manifold, Eqs. (1.6) take the simple form Is + Ib3 Å Å , K = -Is . (3.6) =- Å Ib3 Integrating the first equation of (3.6) and noting the initial conditions corresp onding to the state of rest (0) = (0) = 0, (0) = (0) = 0, (3.7) we obtain Is + Ib3 . (3.8) =- Ib3 Now, defining (t) in the form of an arbitrary function satisfying the initial conditions (3.7) and the required b oundary conditions (T ) = 0, (T ) = 0 , we obtain, from (3.8) and the second equation of (3.6), the dep endencies (t) and K (t) which realize the rotation by the angle 0 . Example. As an example, we consider the case where the angle is given as 0 (wt - sin wt), = 2 where w = 2 . In this case, it is easy to find an explicit form of the control T Is 0 w2 sin wt. 2 Acceleration in a straight line. We now consider the acceleration (deceleration) maneuver in the case of motion in a straight line. Since the acceleration is p ossible only in a direction parallel to the plane of rotation of the p endulum, we assume in what follows that this plane now coincides with the required direction of motion. Supp ose that at the initial instant of time the ball moves in a straight line according to the solution (2.4) with initial velocity v0 . After completing the maneuver (at time t = T ) it keeps moving according to the steady-state solution (2.4) in the same straight line, but now with velocity vT . Let us find the control torque K connecting these two solutions. The motion under consideration lies on the invariant manifold of (1.4) and (1.6), which is given by (3.9) = (1 , 0, 0), = 0, = (1, 0, 0), = (0,2 ,3 ), = (0,2 ,3 ). After parameterization of the vector by the angle = (0, sin , cos ), (3.10) K = - Eqs. (1.6) on the manifold (3.9) take the form 1 = - Å = mb Rb Rs sin Ib1 ( +1 )2 +mb Rb g cos -K (mb Rb Rs cos -Ib1 )
22 Ib1 I0 -m2 Rb Rs cos2 b

,

mb Rb sin Rs (1 +)2 (Ib1 -mb Rb Rs cos )+g(mb Rb Rs cos -I0) +K (I0 +Ib1 -2mb Rb Rs cos )
22 Ib1 I0 -m2 Rb Rs cos2 b = +1 ,

,

(3.11)
2 where the notation I0 = Is +(ms + mb )Rs is used for brevity.

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When t < 0 and t > T , the spherical rob ot moves in a straight line according to the steady-state solution (2.4), hence, at the initial and the final instant of time the p endulum must b e in the lower vertical p osition, and its velocity must b e equal to zero. The corresp onding b oundary conditions for the system (3.11) have the form (0) = (T ) = 0, v0 vT (0) = -(0) = , (T ) = -(T ) = . Rs Rs (3.12)

Moreover, it follows from (3.12) and the third equation of (3.11) that the function (t) must satisfy the additional condition (0) = (T ) = 0. (3.13) Eliminating from the first two equations of (3.11) using the third one, we obtain K =
22 2 Å Å m2 Rb Rs cos2 + I0 Ib1 + mb Rb sin mb Rb Rs cos + gI0 b

mb Rb Rs cos - I0 Å mb Rb sin (g + Rs 2 ) - (mb Rb Rs cos - Ib1 ) . 1 = mb Rb Rs cos - I0

, (3.14)

We now choose some dep endence (t, p) satisfying the conditions (3.12) and (3.13), where p is some parameter of the maneuver. Substituting this dep endence into (3.14), we obtain an explicit expression for K (t) and 1 (t), which realize the maneuver we consider here. Integrating the second equation of (3.14) over time, we obtain the value of the angular velocity of the spherical rob ot (T ) at the final instant of time T . Now, using the constraint equation (1.1), it is easy to obtain the value of linear velocity vT at the final instant of time and hence the change in the velocity v (p) dep ending on the maneuver parameter. Inverting this function, we obtain the dep endence p(v ), which will allow us to determine the maneuver parameters for acceleration to a given velocity. Example. As an example, we consider a controlled motion where the spherical rob ot with parameters (3.4) accelerates from rest with the initial conditions (0) = 0, (0) = (1, 0, 0), (0) = (0, 1, 0), (0) = (0, 0, 1), (0) = 0 to a given velocity vT . Let us choose the dep endence (t) in the form t , (3.15) T where p is the oscillation amplitude of the p endulum. Assume that it is necessary to accelerate the ball in time T = 3 to the velocity vT = 0.04. Integrating the second equation of (3.14) using (3.15), we obtain the (numerical) dep endence vT (p). Inverting this dep endence, we find that the acceleration to vT = 0.04 is p erformed with the amplitude p = 0.1. Substituting the resulting value of p and the dep endence (3.15) into the first equation of (3.14), we obtain an explicit form of the control torque K (t). Figure 3 shows this time dep endence of the control torque K and the pro jections of the vector onto the axis e3 and of velocity v . The figure shows that after completion of the maneuver returns to the vertical p osition, and the further motion is a rolling motion in a straight line with new velocity vT . (t) = p sin2 Combining the gaits that realize acceleration (deceleration), rotation ab out a fixed p oint and steady motion in a straight line (2.4), one can execute the motion of the spherical rob ot along an arbitrary tra jectory. But in this case a stop is necessary b efore each change in the direction of motion. To eliminate this limitation, it would b e interesting to construct other gaits, for example, those connecting the motions along straight lines having different directions, without an intermediate stop. Examples of such gaits for a ball with the Lagrange p endulum are given in [19]. In the case at hand the construction of such controls is much more complicated and is beyond the scope of this paper.
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Fig. 3. Example of the time dep endence K (a) and 3 , v (b) during the acceleration of the ball in a straight line to a given velocity.

CONCLUSION In conclusion, we highlight some op en problems which we b elieve to b e of particular interest. It would b e interesting to examine in more detail the p ossibility (or imp ossibility) of controlled motion along some curve r (t) from the p oint of view of solvability of the system (3.2). This would allow one to determine the maneuverability of the model considered and the class of "p ermissible" tra jectories along which a controlled motion can b e executed. Another op en problem is that of construction of gaits realizing the rotation without a stop, in particular, the construction of gaits connecting steady motions in straight lines located at an angle to each other, and of gaits connecting the motions in a straight line and in a circle. Also of interest is an exp erimental testing of the results obtained: realization of steady-state solutions by sp ecifying constant angular velocities of the control elements; realization of controlled motion in a straight line and of rotation ab out a fixed p oint using the gaits obtained. Based on such an exp erimental testing, it would b e p ossible, in particular, to determine the scop e of applicability of the model considered. APPENDIX e- We present here a derivation of equations that are a generalization of the Poincar? Hamel equations [25] in quasi-coordinates to systems with nonholonomic constraints. Consider the equations of motion of a Lagrangian dynamical system defined by the generalized coordinates qi = {Q, r , ,} and the quasi-velocities wi = {, v , , }, which are expressed in terms of the generalized velocities qi by the formulas
k

qi =
s=1

s vi (q )ws ,

i = 1 ... 8,

or in the explicit form Q = Q + AQ, r = Q v, = , = .

To derive the equations of motion, it is necessary to find the velocity comp onents of the system in the nonholonomic basis of the vector fields s vi (q ) . (A.1) vs = qi
i

To do so, we write explicitly the total time derivative of some function f : f = = + f = ( + A)ik Qkj + Qji vj + + f + ri + Qij ri Qij ri + Aik Qkj f = (i i + vi i + + )f, + Qji vj + + Qkj ik Qij ri Qij Q
ij

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where the vector fields have the form i = -ij k j + j + j k k + j - 1jk j = k

+ i + i , i = i k r1 r2 r3 + j = = + 1 , , k k

, (A.2)

here ij k is the Levi-Civita symb ol, i, j, k = 1 ... 3. Then the nonholonomic basis of the vector fields has the comp onents v = (1 , 2 , 3 , 1 , 2 , 3 , , ). The equations of motion in quasi-velocities with undetermined Lagrange multipliers have the form [26] d dt L wi =
r,s

cs wr ri

L + v i (L)+ ws

j
j

Fj , wi

i = 1 ... 8,

(A.3)

where the differentiation along the vector fields v i is defined using (A.1) and cs are the commutators ri of the vector fields (Poincar? parameters): e [v r , v i ] = cs (q )v s , ri where [ ž , ž ] is the Lie bracket of the vector fields. The commutation relations of the vector fields (A.2) have the form [i , j ] = ij k k , [i , j ] = ij k k , [i , ] = -1ik k , [i , ] = -1ik k ,

[i , j ] = [ , ] = [i , ] = [i , ] = [ , ] = 0. Adding the forcing actions K , K to the explicit form of (A.3), we obtain Eqs. (1.3). ACKNOWLEDGMENTS The work of A. A. Kilin was supp orted by the RFBR grant no. 15-08-09261-a. The work of E. N. Pivovarova was carried out within the framework of the grant of the President of the Russian Federation for the Supp ort of Young Candidates of Science (MK-2171.2014.1). The work of T. B. Ivanova (Section 3) was supp orted by the RSF grant no. 14-19-01303. REFERENCES
1. Halme, A., Schonberg, T., and Wang, Y., Motion Control of a Spherical Mobile Robot, in Proc. of the Å 4th IEEE Internat. Workshop on Advanced Motion Control (Mie, Japan, 1996): Vol. 1, pp. 259-264. 2. Neimark, Ju. I. and Fufaev, N. A., Dynamics of Nonholonomic Systems, Trans. Math. Monogr., vol. 33, Providence, R.I.: AMS, 1972. 3. Bizyaev, I. A., Nonintegrability and Obstructions to the Hamiltonianization of a Nonholonomic Chaplygin Top, Dokl. Math., 2014, vol. 90, no. 2, pp. 631-634; see also: Dokl. Akad. Nauk, 2014, vol. 458, no. 4, pp. 398-401. 4. 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. 170-190. 5. Bolsinov, A. V., Borisov, A. V., and Mamaev, I. S., Rolling of a Ball without Spinning on a Plane: The Absence of an Invariant Measure in a System with a Complete Set of Integrals, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 571-579. 6. Bizyaev, I. A., Bolsinov, A. V., Borisov, A. V., and Mamaev, I. S., Topology and Bifurcations in Nonholonomic Mechanics, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2015, vol. 25, no. 10, 21 pp. 7. Borisov, A. V. and Mamaev, I. S., Isomorphism and Hamilton Representation of Some Nonholonomic Systems, Siberian Math. J., 2007, vol. 48, no. 1, pp. 26-36; see also: Sibirsk. Mat. Zh., 2007, vol. 48, no. 1, pp. 33-45. 8. Borisov, A. V. and Mamaev, I. S., Rolling of a Non-homogeneous Ball Over a Sphere Without Slipping and Twisting, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 153-159. 9. 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. 557-571.
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10. Borisov, A. V., Kazakov, A. O., and Kuznetsov, S. P., Nonlinear Dynamics of the Rattleback: A Nonholonomic Mo del, Physics-Uspekhi, 2014, vol. 57, no. 5, pp. 453-460; see also: Uspekhi Fiz. Nauk, 2014, vol. 184, no. 5, pp. 493-500. 11. Borisov, A. V., Jalnine, A. Yu., Kuznetsov, S. P., Sataev, I. R., and Sedova J. V., Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Mo del of the Rattleback, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 512-532. 12. Borisov, A. V., Kilin, A. A., and Mamaev, I. S., New Effects in Dynamics of Rattlebacks, Dokl. Phys., 2006, vol. 51, no. 5, pp. 272-275; see also: Dokl. Akad. Nauk, 2006, vol. 408, no. 2, pp. 192-195. 13. Chase, R. and Pandya, A., A Review of Active Mechanical Driving Principles of Spherical Robots, Robotics, 2012, vol. 1, no. 1, pp. 3-23. 14. Crossley, V. A., A Literature Review on the Design of Spherical Rol ling Robots, Pittsburgh, Pa., 2006. 6 pp. 15. Karavaev, Yu. L. and Kilin, A. A., The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform, Regul. Chaotic Dyn., 2015, vol. 20, no. 2, pp. 134-152. 16. Ylikorpi, T. and Suomela, J., Ball-Shaped Robots, in Climbing and Walking Robots: Towards New Applications, H. Zhang (Ed.), Vienna: InTech, 2007, pp. 235-256. 17. 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. 3-4, pp. 258-272. 18. Svinin, M., Bai, Y., and Yamamoto, M., Dynamic Mo del and Motion Planning for a Pendulum-Actuated Spherical Rolling Robot, in Proc. of the 2015 IEEE Internat. Conf. on Robotics and Automation (ICRA), pp. 656-661. 19. Ivanova, T. B. and Pivovarova, E. N., Dynamics and Control of a Spherical Robot with an Axisymmetric Pendulum Actuator, arXiv:1511.02655 (2015). 20. Zhan, Q., Motion Planning of a Spherical Mobile Robot, in Motion and Operation Planning of Robotic Systems, G. Carbone, F. Gomez-Bravo (Eds.), Cham: Springer, 2015, pp. 361-381. 21. Ga jbhiye, S. and Banavar, R. N., Geometric Mo deling and Lo cal Controllability of a Spherical Mobile Robot Actuated by an Internal Pendulum, Int. J. Robust Nonlinear Control, 2015. 22. Borisov, A. V. and Mamaev, I. S., Two Non-holonomic Integrable Problems Tracing Back to Chaplygin, Regul. Chaotic Dyn., 2012, vol. 17, no. 2, pp. 191-198. 23. Morinaga, A., Svinin, M., and Yamamoto, M., A Motion Planning Strategy for a Spherical Rolling Robot Driven by Two Internal Rotors, IEEE Trans. on Robotics, 2014, vol. 30, no. 4, pp. 993-1002. 24. Fantoni, I. and Lozano, R., Non-Linear Control for Underactuated Mechanical Systems, London: Springer, 2002. 25. Hamel, G., Die Lagrange-Eulerschen Gleichungen der Mechanik, Z. Math. u. Phys., 1904, vol. 50, pp. 1- 57. 26. Borisov, A. V. and Mamaev, I. S., Dynamics of a Rigid Body: Hamiltonian Methods, Integrability, Chaos, 2nd ed., Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).

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