R centered at the point C on a horizontal plane. The plane of the disk is always vertical. Let l ⃗denote the unit vector along the axis of attachment of the rollers, τ⃗ the unit vector lying in the wheel plane tangent to the rim at the point of contact. The kinematic  constraint  relation  for  a  Mecanum  wheel  implies  that  the  vector       of

velocity ⇀v

of the point M of contact of the wheel with the plane points along   the

line perpendicular to the axis of the roller, i.e., the projection of the velocity of the point M onto the roller axis is equal to zero (see Fig. 1, right). The kinematic constraint relation has the form

v⃗M ⋅ l = 0. ð1Þ

The velocity  v⃗M  is defined by the equation

⇀ ——!

v⃗M = v⃗C + ω × CM, ð2Þ

where v⃗C is the velocity of the center C and ω⃗ is the angular velocity of the disk. Let φ be the angle of rotation of the disk about the axis passing through the point C perpendicular to the plane of the disk (ω = φ̇). The kinematic relation (1) becomes

ðv⃗C  − Rφ̇ τ⃗Þ ⋅ l = 0 ð3Þ

or

v⃗C  ⋅ l =

Rφ̇ sin δ, τ⃗ ⋅ l =

sin δ. ð4Þ

Here  δ  is  the  angle  between  the  vector  of  velocity  ⇀v between the normal to roller axis and the vector  τ⃗).

and  the  vector  τ⃗  (angle

On the basis of the analysis of the kinematic constraints of type (4) it is shown that if a mechanical system is based on n Mecanum wheels in such a way that

(a) n ≥ 3; (b) not all vectors li⃗ are parallel to each other; (c) the points of contact of the wheels with the plane do not lie on one line, then it is always possible to find

control functions φi (i = 1, ... , n) that implement any prescribed motion of the system’s center of mass (see e.g. Martynenko and Formal’skii   2007).

Fig. 2 A vehicle with four Mecanum wheels

Consider a model of a four-wheeled vehicle with Mecanum wheels (see Fig. 2). Let m 0 be the mass of the body, m 1 the mass of each of the wheels, J0 the mass moment of inertia of the body about the vertical axis passing though the center of mass, J1 the mass moment of inertia of each wheel about its axis of rotation, and J2 the moment of inertia of each wheel about the vertical axis passing through the center of the wheel.

The coordinates of the center mass of the system C in the reference frame (inertial system) fO, e⃗x, e⃗y, e⃗zg are xC , yC , and R is the radius of the wheels; the

.   !. .   !.

quantities .C——O 1. = .C——O 2. = d are the distances from the center of mass to the axes

. . . .

of the wheel pairs, and 2 l is the distance between the centers of the wheels of one axis. Let ψ  be the angle between the vector O——1—O 2  and the vector e⃗x   (i.e. angle   ψ

describes the orientation of the vehicle). The corresponding kinematic relations are obtained on the basis of Eq. (4) for each wheel in the   form

ẋC sinðψ + δ1Þ − ẏC cosðψ + δ1Þ − ψ ̇ðl sin δ1 + d cos δ1Þ = Rφ̇1 sin δ1,