For the sake of generality, the kinetic equations of motion are here developed assuming a sloped floor as indicated in the figure below:
The 11-element vector {u} devotes a redundant set of coordinates in terms of which the motion can be described. Here, we use:
The Lagrangian in this case is given by:
However, there are several independent algebraic (holonomic) constraints for the wheels:
and, as indicated in the figure below, two holonomic constraints for the mass center.
Likewise, there are three independent nonholonomic constraints:
which represent the "no-forward-slip" condition for each wheel, and
which represents the "no-lateral-slip" condition.
Beginning with the initial eleven coordinates, we find nine independent constraints. This (correctly) leaves a two-degree-of-freedom system. Note that the number of degrees of freedom of the system (2) is actually smaller than the number of independent scalars (5) necessary to specify fully the "configuration" of the system at any instant. This is true of systems with nonholonomic constraints.
It is straightforward, in this problem, to express the virtual work in terms of variations in the selected coordinates 
.
Hence, in view of the nine constraints, Langrange's equations become:
where, as noted above, only two of the eleven 
are different from zero, where 
are time-varying Lagrange multipliers, and where 
are coefficients in the differential form of the 
constraint:
This leaves 20 algebraic and differential equations:
which, together with initial conditions and given 
and 
, can be integrated forward numerically in time. For all of the usual reasons, such integrals will not be reliable indicators of position in practice. The above equations are useful, however, for assessing closed-loop system performance/stability.
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