5441336907

516

J. Giergiel, W. Żylski

(2.1) are coefficients for yariations Se{. From the principle of virtual work, these coefficients can be defined by the relation (Giergiel et al., 2004)

'£e_iSe_i = j2^5ei'tc_ijQ_j    (2.3)

i = l    i = 1    j = l

The given Maggie’s eąuations can be used to describe motion of a two-wheeled mobile robot. Assuming that the mobile robot, whose model is shown in Fig. 1, moves in one piane, the uniąue formulatiion of the model reąuires giving the position of the point A, i.e. coordinates of this point xa and pa and the frame orientation, i.e. the instantaneous angle of rotation (3 as well as positions describing nodal lines of driving wheels denoted respectively by aj and ct2- In this case, yectors of generalized coordinates and generalized yelocities have the form

(2.4)

9 = \xA,HA,P,ai,aJi^T

q = [XA,yA,P,ótl,ót2]^T

while determining the external forces effecting the analysed system, one should take into account also the unknown dry friction forces existing in the piane of contact of the driving wheels with the road. Figurę 2 shows distribution of these forces, where Tio and T20 stand for circumferential dry friction forces, and T\p and T2p for transverse dry friction forces.

Fig. 2. Friction forces actong on driving wheels

The described system has two degrees of freedom. Assuming that positions of nodal lines of the driving wheels c*i and a: 2 are independent coordinates, the decomposition of velocity yectors of the points A, B and C results in the following relationship to yelocities

/3 = xr(Ai - d₂)    ŻA = ń(“i + d₂)cos/3

(2.5)

Zl 1    z

VA = ^ (dl + 0:2) sin/?