Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://ics.org.ru/upload/iblock/b6d/195-dynamics-and-control-of-an-omniwheel-vehicle_ru.pdf Äàòà èçìåíåíèÿ: Wed Oct 28 19:31:35 2015 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 00:08:23 2016 Êîäèðîâêà:

ISSN 1560-3547, Regular and Chaotic Dynamics, 2015, Vol. 20, No. 2, pp. 153­172. c Pleiades Publishing, Ltd., 2015.

Dynamics and Control of an Omniwheel Vehicle
Alexey V. Borisov1,
2*

, Alexander A. Kilin1** , and Ivan S. Mamaev1,
1

3 ***

Udmurt State University ul. Universitetskaya 1, Izhevsk, 426034 Russia 2 Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, 141700 Russia 3 Kalashnikov Izhevsk State Technical University, ul. Studencheskaya 7, Izhevsk, 426069 Russia
Received January 14, 2015; accepted March 2, 2015

Abstract--A nonholonomic mo del of the dynamics of an omniwheel vehicle on a plane and a sphere is considered. A derivation of equations is presented and the dynamics of a free system are investigated. An explicit motion control algorithm for the omniwheel vehicle moving along an arbitrary tra jectory is obtained. MSC2010 numbers: 70F25, 70E18, 70E55, 70E60 DOI: 10.1134/S1560354715020045 Keywords: omniwheel, roller-bearing wheel, nonholonomic constraint, dynamical system, invariant measure, integrability, controllability

INTRODUCTION The dynamics of various vehicles (carriages) with conventional wheels are described in many publications (for more details and an extensive list of references on this topic, see, e.g., [30]). Nevertheless, even in this sub ject, which is far from being new, there remain a great number of open problems and problem statements for which the motion has not been completely explored. The omniwheel (also called rol ler-bearing wheel in the Russian-language literature) was invented not very long ago, and there has been little published research on the dynamics of various omniwheel vehicles so far. This concerns both free dynamics analysis and control problems [42]. We refer the reader to the review article [19] for a more detailed treatment of the kinematics of omniwheel systems. In this paper we consider the dynamics of a free and controlled motion of the omniwheel vehicle on a plane and a sphere in the nonholonomic setting. An analogous problem is formulated in [31, 32], where a fairly particular case of the vehicle on a plane is studied; in [45, 46] equations for a more general case are obtained and the dynamics of a reduced system for a special configuration of wheels of the vehicle are analyzed. We note that in [45, 46] the equations of motion are derived using the procedure [39], which, in our opinion, involves unjustifiably cumbersome calculations. We use here equations in quasi-velocities with undetermined multipliers which correspond to nonholonomic constraints. This is a much more convenient way to derive equations of motion. In particular, after fairly simple calculations we obtain equations of motion for the omniwheel vehicle on a sphere. The dynamics of an omniwheel vehicle on a sphere are of considerable interest from the viewpoint of dynamics and applications. This gives rise to many intriguing problems closely related to problems of controlling mobile robots of a new generation. In particular, this system is interesting as a more advanced modification of mobile devices which are propelled using various mechanisms installed inside (wheel carriages, pendulums, gyroscopes etc.; see, e.g., [1, 5, 6, 9, 10, 14, 28, 29, 38, 43, 44]) or outside a ball (the so-called Ballbots [34, 35]). The omniwheels used in such devices provide a higher maneuverability.
* ** ***

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

153


154

BORISOV et al.

In this paper we present, therefore, a preliminary analysis of the dynamics of the omniwheel vehicle on a sphere. We also note an analogy of the equations of motion obtained with those of another nonholonomic problem, which was investigated previously in connection with the presence of algebraic first integrals [17] (see also [13, Chapter 3]). A related study concerned with the dynamics and control of one of the modifications of the spherical robot is presented in [26]. We also explore the controlled motion of the omniwheel vehicle on a plane. We note that modern treatments of the controllability of systems with nonholonomic constraints are, as a rule, restricted to constructing kinematic models based on analysis of nonholonomic distributions and ignoring the dynamics of systems considered [2, 3, 28, 29, 33]. Among studies concerned with analysis of dynamical controllability we mention [31, 32], where a particular case of a threewheel omnidirectional vehicle is discussed. In this paper we consider a more general problem of controllability of a vehicle with an arbitrary number and arrangement of omniwheels. In doing so, we use general methods of control theory (Rashevsky ­ Chow theorem) and present constructive control planning algorithms for motion along a given tra jectory. 1. THE NONHOLONOMIC MODEL OF AN OMNIWHEEL We recall that such a wheel has rollers fastened on the periphery (outer rim), so that only one of the rollers of the wheel is in contact with the supporting surface. Each roller rotates freely about the axis which is fixed relative to the plane of the disk, and the wheel can roll in a straight line which makes a fixed angle with the plane of the wheel. For Mecanum wheels the roller axis is fastened at an angle of 45 to the plane of the wheel, whereas for most omniwheels the roller axis lies in the plane of the wheel. The simplest nonholonomic model of an omniwheel is a flat disk for which the velocity of the contact point with the supporting surface is directed along a straight line which forms a constant angle with the plane of the wheel (see Fig. 1).

Fig. 1. The schematic and the nonholonomic model of an omniwheel.

Let and n be the tangent and normal vectors to the plane of the wheel at the point of contact, such that the vector â n is directed vertically upwards, and let be the unit vector along the axis of attachment of the rollers. Then the constraint equation is (V , ) = 0, Q (1.1) where V is the velocity of the point of contact. If VO is the velocity of the center of the wheel Q and is the angle of the wheel's rotation (measured counterclockwise when viewed from the tip of the vector n), then the constraint equation can be rewritten as (V + h , ) = 0, O where h is the radius of the wheel. It is convenient to solve this equation for , whence 1 (1.2) = - (V , ), s = (, ) = sin . O sh Remark. Since the vector is defined up to the sign, one may assume without loss of generality that s 0.
REGULAR AND CHAOTIC DYNAMICS Vol. 20 No. 2 2015


DYNAMICS AND CONTROL OF AN OMNIWHEEL VEHICLE

155

2. AN OMNIWHEEL VEHICLE We obtain the equations of motion for an omniwheel vehicle rolling on a horizontal plane and consisting of a platform (framework), to which an arbitrary number of wheels is attached in such a way that their axes are fixed relative to the platform (Fig. 2).

Fig. 2. Schematic of an omniwheel vehicle.

Choose a moving coordinate system Oe1 e2 rigidly attached to the platform. Let x, y , and denote the coordinates of the origin O and the angle of rotation of the axes e1 and e2 about the fixed coordinate system, respectively. The vectors ri , i , i ,i = 1, ... , n, specifying the position, plane and direction of the roller axis of each wheel are constant in the moving axes. Let v = (v1 ,v2 )denote the velocity of the origin O of the moving coordinate system which is referred to the moving axes, and let be the angular velocity of the platform. Then the constraint equations (1.2) defining the direction of the velocities of the contact points can be written as fi = i + Gi , 1 Gi = (v + Jri , i ), si hi 0 -1 . J= 10

(2.1)

Remark. The vector Jri in the three-dimensional space corresponds to the vector product of the form â ri , where = (0, 0, ). With no constraints imposed, the kinetic energy of the system consists of the kinetic energy of the platform 1 1 T0 = m0 v 2 + I0 2 + m0 (JRoc , v ), 2 2 where m0 is the mass of the framework, I0 is its moment of inertia relative to the point O, and Roc is the vector OC (C is the center of mass of the platform), and of the kinetic energy of each of the wheels 1 1 2 1 Ti = mi (v + Jri , v + Jri )+ Ii i + Ii 2 , 2 2 2 where mi is the mass of the i-th wheel, Ii and Ii are the central moments of relative to the axis and the diameter, respectively. Here and in the sequel centers of mass of the wheels lie on their axes, and the mass distribution of axial symmetry, i.e., the central tensor of inertia of the wheel has the form Ii
REGULAR AND CHAOTIC DYNAMICS Vol. 20 No. 2 2015

inertia of the wheel we assume that the the wheel possesses = diag(Ii , Ii , Ii ). For


156

BORISOV et al.

the total kinetic energy we finally obtain
n

T = T0 +
i=1

1 1 1 Ti = mv 2 + I 2 + m (Jrc , v )+ 2 2 2

2 Ii i ,
i

m = m0 +
i

mi is the total mass of the system,

I = I0 + rc = m
i -1

2 mi ri + Ii is the total moment of inertia relative to the point O,

(m0 Roc +
i

mi ri ) is the center of mass of the entire system.

Remark. In this approach we neglect the inertia of the rollers. We write for this system the equations of motion in quasi-velocities with undetermined multipliers in the form [13] T v T
· ·

+ J

T = v

k
k

fk = v

k
k

Gk , v Gk ,

T + Jv , v T i
·

=
k

fk = k

k
k

= i + Mi ,

i = 1, ... , n,

where Mi is the moment of forces applied to the axes of the wheels. From the last equation and the constraint equations (2.1) we find the undetermined multipliers in the form Ii (v + Jri , i ) - Mi . (2.2) i = -Ii Gi - Mi = - si hi Substituting them into the first two equations and adding the kinematic relations governing the motion of the moving axes Oe1 e2 , we obtain a complete system of equations of motion in the form ( + mE)v + m (Jrc + R)+ m J(v + Jrc ) = -
i

i i ,

^ I + m(Jrc + R, v )+ m (v , rc ) = -
i

i (Jri , i ), (2.3)

x = v1 cos - v2 sin , y = v1 sin + v2 cos , = , Ii Ii R = m -1 = 2 h2 i i , 2 h2 (Jri , i )i , si i si i i i ^ I=I+
i

Ii 2h si i

(Jri , i )2 ,

i =

Mi , si hi

where E is the identity matrix. Remark. Recall that the tensor product of the vectors a and b is defined as follows: a b = ai bj . Note that the matrix is degenerate if and only if all vectors i are parallel to each other, otherwise is nondegenerate. If the moments of forces i applied to the wheel axes have the form i = i , where i are independent of v and , then Eqs. (2.3) admit the singular invariant measure [45] = 1 dv1 dv2 d dx dy d.
Vol. 20 No. 2 2015

REGULAR AND CHAOTIC DYNAMICS


DYNAMICS AND CONTROL OF AN OMNIWHEEL VEHICLE

157

Arbitrariness in the choice of the origin and the rotation of the moving axes can be used in order to simplify either the equations of motion or the integrals of the system (if any). 3. FREE MOTION Consider a free motion of the omniwheel vehicle. Set i = 0,i = 1, ... , n in Eqs. (2.3). Note that the simplest stable solutions of the free system are used in controlling the system by the method of gaits [36, 37]: the required complex motion of the system is achieved by the simplest motions of the free system, which are "conjugated" by means of controllable motions. Choose a moving coordinate system such that the following relations hold: Jrc + R = 0, The equations of motion are represented as ^ ( + mE)v = m 2 rc - m Jv , I = -m (v , rc ), x = v1 cos - v2 sin , y = v1 sin + v2 cos , = . This system admits an energy integral and a linear integral: E= 1 2 2 ^ (1 + m)v1 +(2 + m)v2 + I 2 , 2 ^ F = I +(1 + m)2 v1 - (2 + m)1 v2 ,
1

= diag(1 , 2 ).

(3.1)

where rc = (1 ,2 ). Since is a sum of positive definite symmetric matrices, the eigenvalues and 2 are nonnegative.

Remark. For this choice of moving axes the energy of the system is given by the diagonal quadratic form, whereas in the system of the center of mass it is not diagonal. Reduced system. Consider separately a reduced system governing the evolution of the variables v1 ,v2 , and . There are three different cases where the system can be reduced (by nondimensionalization) to the simplest form. 1) 1 = 0, 2 = 0. After the substitution v1 = 2 u1 , v2 = -1 u2 the equations are represented as a1 u1 = ( - u2 ), a1 = and the integrals become F = a1 u1 + a2 u2 + b , 2) 1 = 0, 2 = 0. After the substitution v1 = 2 u1 , v2 = -2 u2 we obtain a1 u1 = -u2 , a1 = and the integrals are F = a1 u1 + b ,
REGULAR AND CHAOTIC DYNAMICS Vol. 20

a2 u2 = - ( - u1 ),

(1 + m)2 , m1

b = - (u1 - u2 ), ^ (2 + m)1 I a2 = , b= , m2 m1 2 1 E = (a1 u2 + a2 u2 + b 2 ). 1 2 2

(3.2)

a2 u2 = - ( - u1 ), a2 = 2 + m , m

1 + m , m

b = u2 , ^ I b= 2, m2

(3.3)

1 E = (a1 u2 + a2 u2 + b 2 ). 1 2 2
No. 2 2015


158

BORISOV et al.

3) 1 = 2 = 0. The system has the form a1 v1 = v2 , a2 v2 = -v1 , = 0, 1 + m 2 + m a1 = , a2 = , m m and the integrals are F = , 1 2 2 E = (a1 v1 + a2 v2 ). 2

Fig. 3. Intersection of the surfaces of constant energy E = const and the linear integral F = const with a1 = 1.5, a2 = 1.4, b = 1.

Fig. 4. A typical view of the tra jectories of a reduced system in case 1 for a1 = 1.5, a2 = 1.4, b = 1.

Cases 1 and 2 are similar to each other, the tra jectories of both the reduced and the complete system form a two-parameter family defined by the values of the integrals F = f , E = h. Remark. The initial conditions for the variables x, y , and do not add any new parameters, because they can be excluded by choosing fixed axes. In the three-dimensional intersection of the "ellipsoid Figs. 3­5), and according to the reduced system. Thus, 4 space of variables u1 , u2 , and the tra jectories are given by the of energy" E = h with the plane of the linear integral F = f (see (3.2) and (3.3), the plane = 0 has been filled with fixed points of types of tra jectories of the reduced system can be distinguished:

type 1 -- fixed points in which the plane of the linear integral is tangent to the ellipsoid of energy,
REGULAR AND CHAOTIC DYNAMICS Vol. 20 No. 2 2015


DYNAMICS AND CONTROL OF AN OMNIWHEEL VEHICLE

159

Fig. 5. A typical view of the tra jectories of a reduced system in case 2 for a1 = 1.5, a2 = 1.1, b = 1.

type 2 -- fixed points lying in the plane = 0, type 3 -- asymptotic tra jectories given by curves which intersect the plane = 0 and from which the points of intersection with this plane are excluded, type 4 -- periodic tra jectories defined by curves which do not intersect the plane = 0. Bifurcation diagram. Consider the plane of values of the first integrals R2 = {(f, h)}; tra jectories of the reduced system of one of the above-mentioned types correspond to those po of this plane which correspond to possible values of the integrals E and F . The fixed points of system of type 1 in this case are the critical tra jectories of the system under consideration; on plane (f, h) they correspond to the bifurcation curve 1 which is defined by the equation h= f2 in case 1 2(a1 + a2 + b) or h= f
2

the ints the the

2(a1 + b)

in case 2.

This curve restricts below the region of fixed points on the plane = 0 (i.e., of typ consideration if the curve of intersection of them (see Figs. 4 and 5); they correspond to h=

possible values of the first integrals E and F . The e 2) are the critical tra jectories of the system under the integrals E = const and F = const is tangent to the bifurcation curve 2 given by the equation or h= f2 in case 2. 2a1

f2 in case 1 2(a1 + a2 )

This curve on the plane (f, h) separates the region of values of the integrals which correspond to periodic tra jectories of type 3 from the region corresponding to the tra jectories of type 4 (see Fig. 6).

Fig. 6. Bifurcation diagram of the system in cases 1 and 2. Grey indicates the region of possible values of the first integrals. The shaded area corresponds to the periodic tra jectories of the reduced system.

In the first two cases the equations of the reduced system can be easily integrated after rescaling time dt = d.
REGULAR AND CHAOTIC DYNAMICS Vol. 20 No. 2 2015


160

BORISOV et al.

Case 1 . The tra jectory of the system (3.2) is represented as Ab a1 + a2 Ab u2 () = C - a1 + a2 u1 ( ) = C - () = C + A sin(0 )+ B cos(0 ), Bb (sin(0 )+ a1 0 cos(0 )) - (cos(0 ) - a2 0 sin(0 )), a1 + a2 Bb (sin(0 ) - a1 0 cos(0 )) - (cos(0 )+ a1 0 sin(0 )), a1 + a2 a1 + a2 + b 2 0 = . a1 a2 b B , and C are related to the integrals F and E by 2E = (a1 + a2 + b) C 2 + b (A2 + B 2 ) . a1 + a2

The constants of integration A,

F = (a1 + a2 + b)C,

The tra jectory of the vehicle on the plane in this case is defined by the quadratures = (), x = 2 u1 ()cos + 1 u2 ()sin , y = 2 u1 ()sin - 1 u2 ()cos . Case 2 . The tra jectories of the system (3.3) are given by the relations () = C + A sin(0 )+ B cos(0 ), b u1 () = C - (A sin(0 )+ B cos(0 )), a1 u2 () = 0 b(A cos(0 ) - B sin(0 )), a1 + b 2 0 = , a1 a2 b where the constants of integration satisfy the relations F = (a1 + b)C, 2E = (a1 + b) C 2 + b (A2 + B 2 ) . a1

The corresponding quadratures for x, y , and have the form = (), x = 2 (u1 ()cos + u2 ()sin ), y = 2 (u1 ()sin - u2 ()cos ).

Case 3 (the most simple case): the tra jectories in the space of variables , v1 , and v2 are ellipses formed by the intersection of the elliptic cylinder E = h = const with the planes F = f = const. The motion along the tra jectory is governed by = 0 t + 0 , A sin(1 2 )+ B cos(1 2 ) A cos(1 2 ) - B sin(1 2 ) , v2 = , v1 = 2 1 0 = f, A2 + B 2 = 22 2 h, 0 = const, 12 where 1 = 1 1 , 2 = 1 2 , and A and B are the constants of integration, which can be regarded a a as multivalued integrals of the system. Integrating the relations (2.3) for x and y , we find
x - x0 = 1 20 1 1 y - y0 = 20 1
2

2

1 1 - 2 (A cos 1 - B sin 1 ) - 1 1 1 - 2 (A sin 1 + B cos 1 ) - - 1 1 = 1 + 1 2 , 2 = 1 - 1 -

+ 2 (A cos 2 + B sin 2 ) , 2 + 2 (A sin 2 - B cos 2 ) , 2 2.

As above, the quantity A2 + B 2 expressed from these equations is a single-valued periodic function in mod 2 ; it defines the invariant surface in the configuration space (x, y , ), which is diffeomorphic to a torus, and is given by the relation 2 2 2 2 1 2 2 )2 1 + (1 + 2 )2 2 = const, (1 + 1 2 1 = (x - x0 )cos +(y - y0 )sin , 2 = -(x - x0 )sin +(y - y0 )cos . G=
REGULAR AND CHAOTIC DYNAMICS Vol. 20 No. 2 2015


DYNAMICS AND CONTROL OF AN OMNIWHEEL VEHICLE

161

The pro jection of this surface onto the plane of variables (x, y ) defines the region in which the motion of point O occurs (this region has the form of a ring). Absolute motion. Consider a typical motion pattern of the vehicle, which corresponds to different types of tra jectories of the reduced system. The motion is described by the tra jectory of the origin of the moving axes (x(t),y (t)) and by the angle of rotation of the platform (t). 1. The fixed points of type 1 correspond to the solutions of the reduced system = (0) = const, v1 = V
(0) 1

= const,

v2 = V

(0) 2

= const.

Integrating the equations for x, y , and , we find x(t) - x0 cos = sin y (t) - y0
V1
(0) 2

(t) = (0) t + 0 ,
2 2

+ ( t ) + ( t )

- sin cos


2 + ( t 2 + ( t )

)



V1 (0) (0) V2 (0)

(0)

.

Thus, the vehicle uniformly rotates about the vertical axis, and the point O moves in a circle of radius
+ V2 ((0) )2
(0) 2

(the angular velocities of both motions coincide).
(0) (0)

2. The fixed points of type 2 (i.e., on the plane = 0) correspond to = 0, hence, = 0 = const, Vx = V1
(0)

v1 = V1

= const,

v2 = V2

= const,

x(t) = x0 + Vx t, cos 0 + V2
(0) (0)

y (t) = y0 + Vy t,

sin 0 = const, cos 0 = const,

Vy = -V1

(0)

sin 0 + V2

that is, the vehicle moves in a straight line; in the case where the center of mass is located in front of the origin of coordinates defined by (3.1), these motions are stable; otherwise they are unstable. Remark. The stability is considered with respect to variations of the initial velocities. 3. The asymptotic tra jectories of type 3 correspond to motions of the vehicle where it tends to rectilinear uniform motion as t ±. During the motion the vehicle rotates through some finite angle about the vertical axis. 4. The periodic tra jectories of type 4 correspond to the most complex motions of the vehicle. In this case the vehicle rotates about the vertical axis with variable angular velocity without changing the direction of rotation, and the tra jectory of the origin of the moving coordinate system is a complex curve in the circular domain on the plane and consists of repetitive segments rotating about some point on the plane.

Fig. 7. A typical view of tra jectories of the origin of coordinates, O, on the plane in the case of asymptotic (a) and periodic (b) motion of the reduced system (b = 1, a1 = 1.5, a2 = 1.1, 1 = 0, 2 = 1).

REGULAR AND CHAOTIC DYNAMICS

Vol. 20

No. 2

2015


162

BORISOV et al.

Examples. Consider three schemes of attachment of omniwheels to the platform which are found in the literature (the arrangement of wheels and the directions of the roller axes are shown in Fig. 8). In the first two vehicles, conventional omniwheels are used (the axis of the rollers lies in the plane of the wheel), in the last case Mecanum wheels are used (the axis of the rollers is attached to the rim at an angle of 45 ). In each of the cases considered we choose a coordinate system attached to the vehicle, as shown in Fig. 8. We shall assume that the coordinates of the center of mass in this system are
(0) rc = (c1 ,c2 ).

For each of the schemes the wheels are assumed to be identical (the first two vehicles have conventional omniwheels, while the last vehicle has Mecanum wheels). We define the quantity mw = Iw , s2 h2 ww

as the reduced mass of the wheel mw , where Iw is the moment of inertia of the wheel relative to the axis, hw is its radius, sw = 1 for the omniwheel and sw = 22 for the Mecanum wheel. Using (3.1), we find the position of the origin O of the coordinate system in which the energy of the system is diagonal, and denote the corresponding coordinates by (1 ,2 ). Calculating the position of the center of mass in the new coordinate system, we find the quantities (1 ,2 ). The corresponding expressions for these quantities are presented in Fig. 8.

4. CONTROLLED MOTION 4.1. Complete Controllability In this section we present an analysis of the controllability of this system based on the Rashevsky ­ Chow theorem. Theorem 1. If among the vector fields X1 ,... , Xm and among the fields composed of them by successive applications of the Lie bracket one can point out n vector fields Y1 ,... , Yn that are linearly independent at any point of the region G , where dim G = n, then one can come from any point of the region G to any other point by moving a finite number of times along the trajectories of the fields X1 ,... , Xm .

As shown in [9], if one passes to new vector fields using the linear transformation
m

Xk =
k =1

Skk (z )Xk ,

k = 1,... ,m,

where S is an (m â m)-matrix nondegenerate at each point z G , then the theorem of complete controllability remains valid also for the fields X1 ,... , Xm . Let us write Eqs. (2.3) in a standard form used in control theory z = X0 (z )+
i

Xi i ,
Vol. 20 No. 2

(4.1)
2015

REGULAR AND CHAOTIC DYNAMICS


REGULAR AND CHAOTIC DYNAMICS

DYNAMICS AND CONTROL OF AN OMNIWHEEL VEHICLE

Vol. 20

No. 2

(a)

(b)

(c)

2015 = diag

3 3 mw , mw 2 2 2mc1 2mc2 1 = , 2 = 2m +3mw 2m +3mw 3mw c1 3mw c2 1 = , 2 = 2m +3mw 2m +3mw

= diag(mw , 2mw ) mc1 mc2 + mw b 1 = , 2 = m +2mw m + mw 2mw c1 mw (b - c2 ) 1 = , 2 = - m +2mw m + mw

= diag(2mw , 2mw ) mc1 mc2 , 2 = 1 = m +2mw m +2mw 2mw c1 2mw c2 1 = , 2 = m +2mw m +2mw

Fig. 8. Possible schemes of omniwheel vehicles and the corresponding parameter values appearing in equations of motion.

163


164

BORISOV et al.

where z = (v1 ,v2 , ,x,y ,),

-m (v2 + 1 m (v1 - 2 ) m (v , rc ) -1 X0 = -S v1 cos - v2 sin v1 sin + v2 cos + mE m(Jrc + R)T S= 0 0 0



) ,





Xi = -S-

1

m(Jrc + R) 0 0 I 0 0 0 00 10 01 00

i1 i2 (Jri , i ) , 0 0 0 0 0 0. 0 1

(4.2)

Here the vector field X0 corresponds to the free motion of the vehicle and Xi are the "control" vector fields. In control theory, the terms X0 in (4.1) are called drift. To prove complete controllability for such systems, it is necessary not only to show the completeness of the vector fields (4.2) and their commutators (according to Theorem 1), but also to prove the Poisson stability of the drift, that is, to prove that for a free system in the phase space there exists everywhere a dense set of Poisson stable points. We first consider the question of completeness of the vector fields (4.2) and their commutators. As stated above, it suffices to consider the vector fields X0 = -SX0 , Xi = -SXi and their commutators Yi = X0 , Xi , Yij = Xi , Xj = 0.

It is straightforward to show that if the rank of the linear span of the vector fields Xi is equal to three, then the vector fields Xi and Yi form a complete basis in the entire phase space. The above condition for independence of the vector fields Xi leads to some restrictions on the structure of the system. Their physical meaning will be discussed below. We now consider the Poisson stability of the free system. As shown in [11], all motions of the reduced system (for the variables v1 , v2 , ) are broken down into periodic and asymptotic motions. Obviously, due to the existence of asymptotic tra jectories the free system is not Poisson stable. One can only point out regions in the phase space in which the condition of Poisson stability is satisfied. These regions coincide with the regions filled with periodic tra jectories of the reduced system, and in the space (v1 ,v2 , ) they form the interiors of two elliptic cones. 4.2. Explicit Control Along a Tra jectory Although the complete controllability of the system (2.3) has not been proved so far, one can present an explicit algorithm for developing a control for the motion along a predetermined tra jectory. Consider the control of the vehicle by choosing control torques i , i = 1,... ,n, in such a way that the vehicle moves in a prescribed manner (t) = (x(t),y (t),(t)). According to Eqs. (2.3), the following proposition turns out to be valid: Prop osition 1. If the vehicle rests on no less than three driving omniwheels for which the inequality
REGULAR AND CHAOTIC DYNAMICS Vol. 20 No. 2 2015


DYNAMICS AND CONTROL OF AN OMNIWHEEL VEHICLE





165

21 31 11 det 12 22 32 (Jr1 , 1 ) (Jr2 , 2 ) (Jr3 , 3

= 0, )

(4.3)

is satisfied, then one can always choose control torques i (t), i = 1,... ,n, in such a way that the prescribed motion (t) is realized. To prove the proposition, we rewrite the equations of the reduced system (for v1 , v2 , and ) as Sz
(3) (3)

= X0 (z

(3)

(3)

)+ G,

(4.4)

where z (3) = (v1 ,v2 , ), X0 = m (v2 + 1 ), -m (v1 - 2 ),m (v , rc ) , = (1 ,...,N ) is a vector composed of control torques, and G is a constant matrix depending on the design of the vehicle and having the form ... N 1 11 (4.5) GN = 12 ... N 2 . (Jr1 , 1 ) ... (JrN , N ) The system (4.4) can be solved for i , and thus we can obtain the control law (t) from the known dependencies z (3) (t). For the system (4.4) to be solvable, we need to impose restrictions on the structure of the vehicle such that the rank of the matrix (4.5) is equal to three. This requires that the condition (4.3) be satisfied. The condition (4.3) implies, in particular, that the controllability of the system does not admit parallelism of all vectors i . Moreover, if the condition i j rij = ri - rj holds for two wheels, this pair of wheels can be regarded as one wheel, since the corresponding columns in the matrix (4.5) will be equal, and the corresponding control torques will appear in (4.4) only as the sums i + j . We present a general algorithm for calculating the control torques for the motion along the predetermined tra jectory (t) = (x(t), y (t), (t)). 1. Calculate the time dependence of v1 , v2 , , and their derivatives from the formulae v1 = x cos + y sin , v2 = -x sin + y cos , = . 2. Solve the system (4.4) for any of the three control torques for which the condition (4.3) is satisfied. As a result, we obtain the dependence of these three torques on time and the remaining N - 3 torques. 3. The remaining torques can have an arbitrary time dependence. This arbitrariness can be used, in particular, for control optimization (for example, for minimization of power inputs for the motion). Further, we present specific examples for the control of motion along the simplest tra jectories for the vehicle design shown in Fig. 8c. The vectors ri , i and the quantities si are expressed as follows: r1 = -r3 = (a, b), r2 = -r4 = (-a, b), 11 11 1 1 = 3 = - , , 2 = 4 = , , s1,2,3,4 = . 22 22 2 Figure 9 shows an experimental model of such a vehicle developed at the Laboratory of Nonlinear Analysis and Design of New Types of Vehicles of the Udmurt State University. The numerical values of its geometric and mass characteristics are: m = 3 581 g, a = 24.44 cm, b = 15.22 cm, h1,2,3,4 = 4.35 cm, I = 78 346.51 gcm2 , Ii = 1 196.07 gcm2 , rc = (0, 0). Below we present the results of numerical calculations for the above values of the characteristics of the vehicle. We use arbitrariness in the choice of one of the control torques to minimize the general control torque 2 . In the case at hand this condition reduces to the equation 1 + 2 - 3 - 4 = 0. i
REGULAR AND CHAOTIC DYNAMICS Vol. 20 No. 2 2015


166

BORISOV et al.

Fig. 9. 3D-model of a Mecanum wheel vehicle.

Example 1. Motion in a straight line over distance l0 with rotation through an angle 0 : x(t) = 0, y (t) = l0 sin2 t, (t) = 0 sin2 t, t = [0, 1]. 2 2 The time dependences of the control torques for 0 = 2 , l0 = 1 m are shown in Fig. 10.

(4.6)

Fig. 10. Time dependence of the control torques for motion along the tra jectory (4.6).

Example 2. Motion along one period of the sine curve (the orientation of the vehicle coincides with the direction of instantaneous velocity): x(t) = l0 s, y (t) = R sin(2s), R (t) = arctg 2 cos(2s) , s = sin2 t, l0 2 (4.7)

t = [0, 1],

where R is the amplitude of the sine curve and l0 is the length of one period. An example of dependence of the control torques for the motion along a sine curve for R = 0.1, l0 = 1 is given in Fig. 11.

Fig. 11. Time dependence of the control torques for motion along the tra jectory (4.7).

REGULAR AND CHAOTIC DYNAMICS

Vol. 20

No. 2

2015


DYNAMICS AND CONTROL OF AN OMNIWHEEL VEHICLE

167

Fig. 12. Time dependence of the reactions i for motion along the tra jectory (4.7).

In conclusion, in Fig. 12 we present the time dependence of the reactions i (2.2) for motion of the vehicle along the sine curve (example 2). As seen in the figure, the value of the reactions is an order smaller than the value of the control torques and is very irregular. It would be interesting to compare the dependencies obtained with results of calculations, for example, within the framework of the dry friction model. 5. AN OMNIWHEEL VEHICLE ON THE SURFACE OF A SPHERE Suppose an omniwheel vehicle moves on the surface of a fixed sphere (in a gravitational field, see Fig. 13).

Fig. 13

In this case it is convenient to regard the vehicle as a rigid body with a fixed point, which is chosen to be the center of the sphere. We choose a moving coordinate system Oe1 e2 e3 with origin at the center of the sphere and with axes coinciding with the principal axes of inertia of the system. Let denote the angular velocity of the vehicle and i the total angular velocity of each wheel; if the angle of rotation of the wheel about the vehicle is i , the following relations hold: i = + i ni , i = 1,... ,n, where ni is the unit vector directed along the axis of the wheel. For the velocity of the contact point Qi of the wheel with the sphere we have VQi = â ri + i â hi ri , R - hi i = 1,...,n,

where R and hi are the radii of the sphere and the wheel, respectively. Using this relation, we represent the constraint equations (1.1) as 1 (ni â ri , i ) , ( , ri â i ) = 0, si = fi = i + si hi R where i is the unit vector along the roller axis at the point of contact. In the chosen coordinate system the vectors i , ni , and ri are constant.
REGULAR AND CHAOTIC DYNAMICS Vol. 20 No. 2 2015


168

BORISOV et al.

We shall assume that for each wheel the axis of rotation passes through the center of mass of the wheel and is its principal axis of inertia. Then the kinetic energy of the system can be represented as 1 1 mi ( â ri )2 +(i , Ii i ) T = ( , I0 )+ 2 2
i

1 = ( , I )+ 2 I = I0 +
i

1 Ji i ( , ni )+ 2

2 Ji i ,
i

(5.1)

2 mi (ri E - ri ri )+ Ii ,

Ji = (ni , Ii ni ),

where I0 is the tensor of inertia of the framework relative to the center of the sphere, Ii is the tensor of inertia of the wheel relative to its center of mass, and I is the tensor of inertia of the entire vehicle relative to the center of the sphere. Since the axis of rotation of the wheel coincides with its principal axis of inertia, the relation Ii ni = Ji ni holds. In this case the equations for quasi-velocities referred to the moving coordinate axes have the form T fi T · +â i = - mg r0 â ,
i

T k

·

=
i

fi i + Mk = k + Mk , k

k = 1,... ,n,

where r0 is the vector from the center of the sphere to the center of mass of the vehicle and is the unit vector of the vertical. Eliminating the undetermined multipliers i using the second equation and the constraint (5.1), we find I +
i

Ji 2 h2 si i

( , i )i -

Ji si h
i

( , i )ni +( , ni )i
i

+ â I -
i

Ji ( , i )n si hi

=-

i i - mg r0 â ,

i = ri â i , where i = sMii . Adding the kinematic relation governing the evolution of the fixed unit vector ih in the moving axes, we finally rewrite the equation of motion as I + â J = -
i

i i - mg r0 â ,

= â , , (5.2)

I=I+
i

Ji 2 h2 si i

i i -

Ji i ni + ni i si hi

J=I-
i

Ji ni i . si hi

If the matrices I and J the angular velocity , Without the control equations in rigid body

are diagonalized simultaneously (i.e., I, J = 0) and i are independent of then Eqs. (5.2) admit the standard invariant measure d d . torques (i = 0,i = 1,... ,n) Eqs. (5.2) are analogous to the Euler­Poisson dynamics: I = (J ) â - mg r0 â , = â . (5.3)

The system (5.2) can be regarded as a nonholonomic analog of the Euler ­ Poisson equations. It admits two additional integrals 1 F0 = 2 = 1, E = (I , ) 2
REGULAR AND CHAOTIC DYNAMICS Vol. 20 No. 2 2015


DYNAMICS AND CONTROL OF AN OMNIWHEEL VEHICLE

169

(the integral E coincides with the kinetic energy of the system calculated with the constraints taken into account). When the gravitational field is zero (g = 0), one has the third additional integral F1 = (I , J ). Remark 1. In Eqs. (5.3) the matrix I is symmetric and hence can always be diagonalized by an orthogonal transformation. After such a transformation the vector r0 in (5.3) has in general three nonzero components. We note that in nonholonomic mechanics there may exist specific restrictions on the system's parameters when an invariant measure exists (see, e.g., [16, 24, 41]). Therefore, the problem of existence (in the general case) of an invariant measure and two additional integrals in the system (5.2), which are necessary for integrability by the Euler ­ Jacobi theorem, remains open. At the same time it should be noted that in nonholonomic systems a hierarchy of possible types of dynamical behavior may arise, depending on combinations of tensor invariants (see, e.g., [4, 7, 16]). Moreover, the emergence of strange attractors [8, 25, 27] is possible. We consider in more detail the case in which there is no gravitational field, J and I are diagonal matrices and, moreover, their product JI is a positive definite matrix (in this case there exists a standard invariant measure and the integrals F0 , F1 , and E ). Then, if we define the momentum 1 M = (JI) 2 , then the equations of motion take the form M = M â AM , A= det J
1 2 1 2

= â BM , B = (JI)
-
1 2

J-1 ,

(5.4) .

det I To abbreviate some of our forthcoming formulae, we introduce the notation A = diag(a1 ,a2 ,a3 ) e­ B = diag(b1 ,b2 ,b3 ). The resulting system (5.4) also describes the limiting case of the Poincar´ Zhukovskii equations (see § 3, Chapter 3 in [13]). In [17, 18], A. V. Borisov and A. V. Tsygvintsev obtained a necessary condition for existence of an additional algebraic integral by using the Kovalevskaya ­ Painlev´ method e k 2 a23 a21 a13 - a32 b2 - a13 b2 - a21 b2 = 0, 1 2 3 aij = ai - aj , (5.5) where k = p , p, q Z. If k is odd and positive, the additional integral F2 is linear in , has degree q k in M and can be written in the form [17, 18] F2 = ( , S (M )), where the vector S (M ) can be represented as S (M ) = diag(M1 ,M2 ,M3 )k (M )T , here K and Cn are the matrices
1 k = (C-1 K)(C-1 K) ... (C--2 K), 1 3 k

K = diag(

2 M1

2 ,M2

2 ,M3

),

b3 -b2 -na32 Cn = -b3 -na13 b1 b2 -b1 -na21

,

and the vector T is the kernel of the matrix Ck . Under the condition B = k A, for which the condition (5.5) holds identically, the equations of motion (5.4) were integrated in terms of elliptic sigma functions in [22]. The problem of Hamiltonization of (5.4) remains unresolved. If we set b1 = 0 and b2 = 0 in (5.4), then 3 remains constant and, moreover, we obtain the integral F3 = 1 sin + 2 cos ,
REGULAR AND CHAOTIC DYNAMICS

=

b3 ln( a13 M1 + a32 M2 ). a13 a32
2015

Vol. 20

No. 2


170

BORISOV et al.

We note that nonalgebraic integrals also arise in the Suslov problem with the inhomogeneous constraint [23] (whose practical realization is unknown). Remark. If the matrices I and J are diagonalized simultaneously and, moreover, a pair of eigenvalues of each of the matrices I and J coincides, then the system (5.3) can be integrated by quadratures. Example. Consider in more detail a vehicle on is shown in Fig. 8b. If all wheels have the same according to Fig. 8b we obtain I11 I= 0 0 r1 = (a, 0,
2 Rw - a2 ),

a sphere for which a wheel attachment scheme radius h, then by choosing the axes e1 and e2 0 I22 I23 0 I23 , I33 3 = (1, 0, 0), r3 = (0,b,
2 Rw - b2 ),

1 = 2 = (0, 1, 0),

2 r2 = (-a, 0, Rw - a2 ), Rw = R - h.

In this case, i = ri â i = Rw ni , i = 1, 2, 3, and for the matrices I and J we obtain a2 2 1 - R2 0 0 w 2h 2 b2 b b2 I = I + m w Rw 1 - - Rw 1 - R2 , 0 1 - R2 w w Rw b b2 2a2 +b2 0 - Rw 1 - R2 2 Rw w a2 0 0 2 2 1 - Rw b2 b b2 J = I - mw hRw - Rw 1 - R2 . 0 1 - R2 w w b b2 2a2 +b2 0 - Rw 1 - R2 - R2
w w

The matrices J and I commute under the condition 2h = R, i.e., Eqs. (5.2) admit in this case an invariant measure with constant density. Remark. This relation coincides in form with the condition for existence of an integral in the problems of a ball rolling on a sphere [17, 18]. A topological analysis of the reduced system in the problem of a ball rolling without slipping on a sphere is presented in [15], while absolute dynamics remain unexplored (for example, analysis of the complete system for a ball on a plane is carried out in [12, 40]). ACKNOWLEDGMENTS The research concerned with controllability analysis was supported by the grant of the Russian Science Foundation (pro ject no. 14-19-01303). The investigation of free dynamics was carried out within the framework of the state assignment for institutions of higher education. REFERENCES
1. Alves, J. and Dias, J., Design and Control of a Spherical Mobile Robot, J. Syst. Control Eng., 2003, vol. 217, pp. 457­467. 2. Asama, H., Sato, M., Bogoni, L., Kaetsu, H., Mitsumoto, A., and Endo, I., Development of an OmniDirectional Mobile Robot with 3 DOF Decoupling Drive Mechanism, in Proc. of the 1995 IEEE Internat. Conf. on Robotics and Automation (Nagoya, Japan, 21­27 May 1995): Vol. 2, pp. 1925­1930.
REGULAR AND CHAOTIC DYNAMICS Vol. 20 No. 2 2015


DYNAMICS AND CONTROL OF AN OMNIWHEEL VEHICLE

171

3. Ashmore, M. and Barnes, N., Omni-Drive Robot Motion on Curved Paths: The Fastest Path between Two Points Is Not a Straight-Line, in AI 2002: Advances in Artificial Intel ligence, Lecture Notes in Comput. Sci., vol. 2557, Berlin: Springer, 2002, pp. 225­236. 4. Bizyaev, I. A., Nonintegrability and Obstructions to the Hamiltonianization of a Nonholonomic Chaplygin Top, Dokl. Math., 2014, vol. 90, no. 2, pp. 631­634; see also: Dokl. Akad. Nauk, 2014, vol. 458, no. 4, pp. 398­401. 5. Bizyaev, I. A., Borisov, A. V., and Mamaev, I. S., The Dynamics of Nonholonomic Systems Consisting of a Spherical Shell with a Moving Rigid Bo dy Inside, Regul. Chaotic Dyn., 2014, vol. 19, no. 2, pp. 198­ 213. 6. Bolotin, S. V. and Popova, T. V., On the Motion of a Mechanical System inside a Rolling Ball, Regul. Chaotic Dyn., 2013, vol. 18, nos. 1­2, pp. 159­165. 7. Bolsinov, A. V., Borisov, A. V., and Mamaev, I. S., Rolling of a Ball without Spinning on a Plane: The Absence of an Invariant Measure in a System with a Complete Set of Integrals, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 571­579; see also: Nelin. Dinam., 2012, vol. 8, no. 3, pp. 605­616. 8. Borisov, A. V., Kazakov, A. O., and Sataev, I. R., The Reversal and Chaotic Attractor in the Nonholonomic Mo del of Chaplygin's Top, Regul. Chaotic Dyn., 2014, vol. 19, no. 6, pp. 718­733. 9. 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; see also: Nelin. Dinam., 2012, vol. 8, no. 2, pp. 289­307. 10. Borisov, A. V., Kilin, A. A., and Mamaev, I. S., How To Control the Chaplygin Ball Using Rotors: 2, Regul. Chaotic Dyn., 2013, vol. 18, nos. 1­2, pp. 144­158; see also: Nelin. Dinam., 2013, vol. 9, no. 1, pp. 59­76. 11. Borisov, A. V., Kilin, A. A., and Mamaev, I. S., An Omni-Wheel Vehicle on a Plane and a Sphere, Nelin. Dinam., 2011, vol. 7, no. 4, pp. 785­801 (Russian). 12. Borisov, A. V., Kilin, A. A., and Mamaev, I. S., The Problem of Drift and Recurrence for the Rolling Chaplygin Ball, Regul. Chaotic Dyn., 2013, vol. 18, no. 6, pp. 832­859. 13. 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). 14. Borisov, A. V. and Mamaev, I. S., The Dynamics of the Chaplygin Ball with a Fluid-Filled Cavity, Regul. Chaotic Dyn., 2013, vol. 18, no. 5, pp. 490­496; see also: Nelin. Dinam., 2012, vol. 8, no. 1, pp. 103­111. 15. Borisov, A. V. and Mamaev, I. S., Topological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere, Regul. Chaotic Dyn., 2013, vol. 18, no. 4, pp. 356­371; see also: Nelin. Dinam., 2012, vol. 8, no. 5, pp. 957­975. 16. Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A., The Hierarchy of Dynamics of a Rigid Bo dy Rolling without Slipping and Spinning on a Plane and a Sphere, Regul. Chaotic Dyn., 2013, vol. 8, no. 3, pp. 277­ 328; see also: Nelin. Dinam., 2013, vol. 9, no. 2, pp. 141­202. 17. Borisov, A. V. and Tsygvintsev, A. V., Kowalewski Exponents and Integrable Systems of Classic Dynamics: 1, 2, Regul. Chaotic Dyn., 1996, vol. 1, no. 1, pp. 15­37 (Russian). 18. Borisov, A. V. and Tsygvintsev, A. V., Kovalevskaya's Metho d in Rigid Bo dy Dynamics, J. Appl. Math. Mech., 1997, vol. 61, no. 1, pp. 27­32; see also: Prikl. Mat. Mekh., 1997, vol. 61, no. 1, pp. 30­36. 19. Campion, G., Bastin, G., and d'Andr´ ea-Novel, B., Structural Properties and Classification of Kinematic and Dynamic Mo dels of Wheeled Mobile Robots, IEEE Trans. Robot. Autom., 1996, vol. 12, no. 1, pp. 47­62. ere, 20. Ferri` L., Campion, G., and Raucent, B., ROLLMOBS, a New Drive System for Omnimobile Robots, Robotica, 2001, vol. 19, pp. 1­9. 21. Fedorov, Yu. N. and Kozlov, V. V., Various Aspects of n-Dimensional Rigid Bo dy Dynamics, Amer. Math. Soc. Transl. Ser. 2, 1995, vol. 168, pp. 141­171. 22. Fedorov, Yu. N., Maciejewski, A. J., and Przybylska, M., The Generalized Euler ­ Poinsot Rigid Body Equations: Explicit Elliptic Solutions, J. Phys. A, 2013, vol. 46, no. 41, 415201, 26 pp. 23. Garc´ ia-Naranjo, L. C., Maciejewski, A. J., Marrero, J. C., and Przybylska, M., The Inhomogeneous Suslov Problem, Phys. Lett. A, 2014, vol. 378, nos. 32­33, pp. 2389­2394. 24. Garc´ ia-Naranjo, L. C., Marrero, J. C., Non-Existence of an Invariant Measure for a Homogeneous Ellipsoid Rolling on the Plane, Regul. Chaotic Dyn., 2013, vol. 18, no. 4, pp. 372­379. 25. Gonchenko, A. S., Gonchenko, S. V., and Kazakov, A. O., Richness of Chaotic Dynamics in the Nonholonomic Mo del of Celtic Stone, Regul. Chaotic Dyn., 2013, vol. 18, no. 5, pp. 521­538. 26. 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. 134­152. 27. Kazakov, A. O., Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane, Regul. Chaotic Dyn., 2013, vol. 18, no. 5, pp. 508­520. 28. Kilin, A. A. and Karavaev, Yu. L., The Kinematic Control Mo del for a Spherical Robot with an Unbalanced Internal Omniwheel Platform, Nelin. Dinam., 2014, vol. 10, no. 4, pp. 497­511 (Russian). 29. Kilin, A. A., Karavaev, Yu. L., and Klekovkin, A. V., Kinematic Control of a High Mano euvrable Mobile Spherical Robot with Internal Omni-Wheeled Platform, Nelin. Dinam., 2014, vol. 10, no. 1, pp. 113­126 (Russian).
REGULAR AND CHAOTIC DYNAMICS Vol. 20 No. 2 2015


172

BORISOV et al.

30. Lobas, L. G., Nonholonomic Models of Vehicles, Kiev: Naukova Dumka, 1986 (Russian). 31. Martynenko, Yu. G., Stability of Steady Motions of a Mobile Robot with Roller-Carrying Wheels and a Displaced Centre of Mass, J. Appl. Math. Mech., 2010, vol. 74, no. 4, pp. 436­442; see also: Prikl. Mat. Mekh., 2010, vol. 74, no. 4, pp. 610­619. 32. Martynenko, Yu. G. and Formal'skii, A. M., On the Motion of a Mobile Robot with Roller-Carrying Wheels, J. Comput. Sys. Sc. Int., 2007, vol. 46, no. 6, pp. 976­983; see also: Izv. Ross. Akad. Nauk. Teor. Sist. Upr., 2007, no. 6, pp. 142­149. 33. Muir, P. F. and Neuman, C. P., Kinematic Modeling for Feedback Control of an Omnidirectional Wheeled Mobile Robot, in Autonomous Robot Vehicles, I. J. Cox, G. T. Wilfong (Eds.), New York: Springer, 1990, pp. 25­31. 34. Nagara jan, U., Mampetta, A., Kantor, G. A., and Hollis, R. L., State Transition, Balancing, Station Keeping, and Yaw Control for a Dynamically Stable Single Spherical Wheel Mobile Robot, IEEE Internat. Conf. on Robotics and Automation (ICRA) (Kobe, Japan, 2009), pp. 998­1003. 35. Nagara jan, U., Kantor, G., and Hollis, R. L., Tra jectory Planning and Control of an Underactuated Dynamically Stable Single Spherical Wheeled Mobile Robot, IEEE Internat. Conf. on Robotics and Automation (ICRA) (Kobe, Japan, 2009), pp. 3743­3748. 36. Ostrowski, J. P., The Mechanics and Control of Undulatory Robotic Locomotion: Ph. D. Thesis, Pasadena, CA, California Institute of Technology, 1995. 149 p. 37. Ostrowski, J. P., Desai, J. P., and Kumar, V., Optimal Gait Selection for Nonholonomic Lo comotion Systems, Internat. J. Robotics Res., 2000, vol. 19, no. 3, pp. 225­237. 38. 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. 1­2, pp. 126­143. 39. Tatarinov, Ya. V., Equations of Classical Mechanics in New Form, Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 2003, no. 3, pp. 67­76 (Russian). 40. Tsiganov, A. V., On the Lie Integrability Theorem for the Chaplygin Ball, Regul. Chaotic Dyn., 2014, vol. 19, no. 2, pp. 185­197. 41. Tsiganov, A. V., One Invariant Measure and Different Poisson Brackets for Two Non-Holonomic Systems, Regul. Chaotic Dyn., 2012, vol. 17, no. 1, pp. 72­96. 42. Watanabe, K., Shiraishi, Y., Tzafestas, S. G., Tang, J., Fukuda, T., Feedback Control of an Omnidirectional Autonomous Platform for Mobile Service Robots, J. Intel l. Robot. Syst., 1998, vol. 22, nos. 3­4, pp. 315­330. 43. Yoon, J.-C., Ahn, S.-S., and Lee, Y.-J., Spherical Robot with New Type of Two-Pendulum Driving Mechanism, Proc. 15th IEEE Internat. Conf. on Intel ligent Engineering Systems (INES) (Poprad, High Tatras, Slovakia, 2011), pp. 275­279. 44. Zhan, Q., Cai, Y., and Yan, C., Design, Analysis and Experiments of an Omni-Directional Spherical Robot, IEEE Internat. Conf. on Robotics and Automation (ICRA) (Shanghai, China, 2011), pp. 4921­ 4926. 45. Zobova, A. A., Application of Laconic Forms of the Equations of Motion in the Dynamics of Nonholonomic Mobile Robots, Nelin. Dinam., 2011, vol. 7, no. 4, pp. 771­783 (Russian). 46. Zobova, A. A. and Tatarinov, Ya. V., The Dynamics of an Omni-Mobile Vehicle, J. Appl. Math. Mech., 2009, vol. 73, no. 1, pp. 8­15; see also: Prikl. Mat. Mekh., 2009, vol. 73, no. 1, pp. 13­22.

REGULAR AND CHAOTIC DYNAMICS

Vol. 20

No. 2

2015