− ẋC sinðψ − δ2Þ + ẏC cosðψ − δ2Þ + ψ ̇ðl sin δ2 + d cos δ2Þ = Rφ̇2 sin δ2, ð6Þ

− ẋC sinðψ − δ3Þ + ẏC cosðψ − δ3Þ − ψ ̇ðl sin δ3 + d cos δ3Þ = Rφ̇3 sin δ3, ð7Þ

ẋC sinðψ + δ4Þ − ẏC cosðψ + δ4Þ + ψ ̇ðl sin δ4 + d cos δ4Þ = Rφ̇4 sin δ4, ð8Þ

Equations (5)–(8) define four non-holonomic constraints. Note, that if translational motions (ψ ̇ = 0) or rotations about the center of mass (ẋC = ẏC = 0) are allowed, then the constraints become holonomic.

3 Dynamic Equations

The equations of motion of the robot can be obtained by using any method of non-holonomic mechanics, for example, by deriving Lagrange’s equations with multipliers or Appel’s equations. The mechanical system under consideration has three degrees of freedom, its configuration is characterized by 7 Lagrangian vari- ables, xC , yC , ψ , φ1, φ2, φ3, and φ4, subject to 4 constraints (5)–(8). For the  case

where all δi = π 4 (i = 1, ... , 4), the governing equations have the simplest form

.. p

xC ðmR2 + 4J1Þ + 4ẏCψ ̇J1 = R   ffi2ffiffiðM1 sinðψ + π 4̸ Þ

+ M2 cosðψ + π 4̸   Þ + M3 cosðψ + π 4̸   Þ + M4 sinðψ + π 4̸   ÞÞ,

ð9Þ

.. p

yC ðmR2 + 4J1Þ − 4ẋCψ ̇J1 = − R   ffi2ffiffiðM1 cosðψ + π 4̸ Þ

− M2 sinðψ + π 4̸   Þ − M3 sinðψ + π 4̸   Þ + M4 cosðψ + π 4̸   ÞÞ,

ð10Þ

.JCR2  + 4J1ðl + dÞ2. ψ ̈ = − Rðl + dÞðM1 − M2 + M3 − M4Þ. ð11Þ

Here  Mi  are   the   torques   applied   to   the   respective   wheels   (i = 1, ... , 4), m = m0 +4 m1 is the total mass of the system, JC = J0 + 4J2 + 4m1ðl2 + d2Þ is the moment of inertia of the entire system relative to the center of   mass.

If the torques Mi (i = 1, ... , 4) are defined as functions of time, then the angle of rotation of the robot about its center of mass ψ ðtÞ can be expressed in terms of quadratures. For the solutions that correspond to the rotation of the robot about its center of mass at a constant angular velocity (ψ ̇ = Ω = const), the torques applied to the wheels are related by

M1 + M3 = M2 + M4. ð12Þ

The motions with Ω = const involve, apart from the rotation about the center of mass, the translatory motion of the robot (Ω = 0, ψ = const). Such motions are of interest for applications.

In this case, the system of Eqs. (9) and (10) is a nonhomogeneous system of linear differential equations with constant coefficients and can be solved in an explicit form.

We assume that the torques developed by each of the four identical DC motors are defined by (see e.g. Gorinevsky et al.  1997)

MiðtÞ = cuUiðtÞ − cvφ̇iðtÞ,