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

2.2 Generalized Modeling 29⎤00⎡mI 3 0 3Γ = ⎢⎣0 3 I⎤⎥⎦ ¨q + mg 1= M g ¨q + G (2.49)g 00⎡⎢⎣ ⎥⎦0where I 3 indicates a three dimensional identity matrix and G = −m[g T0 T 3 ] T is a sixdimensionvector representing the contribution of gravity.As Assumption 4 is still considered valid, also because our robot has only one motor foreach side’s two wheels, we will be considering the vector τ defined in (2.22) as controlinput at dynamic level. Thereby, similarly to the planar case, we can calculate the vectorΓ by considering the vector of the active forces and torques exerted on the i th wheel/-ground contact point expressed on the local frame as F ia= [ τ ir0 0 0 0 0] T . Toobtain the total active force and torque applied to the CoM and expressed in the inertialframe, first we compute the resultant active force and torque vector on the local frame,F a = [F ax F ay F az M ax M ay M az ] T , and then we multiply it by the Plücker transformfor spatial force vectors [18], [19]. In particular, as we are considering the frame centered atthe i th contact point parallel to the local frame, we can calculate the active force and torquevector in the local frame as:F a =⎡4∑ I 3 0 3⎢⎣ˆp i I⎤⎥⎦ F ia =3i=1Thereby, we can express Γ also as follows:⎡⎢⎣τ L +τ Rr000−h(τ L +τ R )r−cτ L +dτ Rr⎤⎥⎦(2.50)Γ = (J −1 ) T (q)F a = (J −1 ) T (q)T f τ = B(q)τ (2.51)where