Advance Modeling of a Skid-Steering Mobile Robot for Remote ...

2.2 Generalized Modeling 33where we used the matrices transformation described in (2.59).Furthermore, we notice that the transformation matrix T defined in (2.47) is always fullcolumn-rank as the first and the last rows are linear independent ∀x ICR , y ICR , z ICR . Thereby,an optimal inverse solution of equation (2.47) can be always obtained by pre-multiplying itby the pseudo-inverse of T as follows:η = T # V = (T T T) −1 T T V (2.63)Similarly, we notice that the matrix T Tpseudo-inverse as:is full row-rank, therefore we can defined its(T T ) # = (TT T ) −1 T (2.64)Thereby, by pre-multiply equation (2.62) by (T T ) # , the dynamic model of SSMRs can berewritten considering V as control input:M ˙V + M ˆVV + J −1 G = T f τ + F r = F. (2.65)which represents, as expected, the generalized dynamic model of a rigid-body freely movingin a three-dimensional space, with the term M ˆVV representing the contribution of theCoriolis forces.It is worth noticing that the dynamic model in (2.61),(2.65) holds if and only if Assumption6 is satisfied. Conversely, if the wheels can lift from the ground surface, the impact model,relating the contact point velocity before and after the impact with the ground, must be addedto the model.