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

4.2 Tire Longitudinal Force 61tan θ i =v iyv ix − rω i(4.11)Furthermore, we notice that the instable behavior, due to the difference between the staticand kinetic friction coefficient, provided by the parameter α s in (2.30),(2.31) [4], was alreadyincluded in the previously discussed dynamics of f iy , because of the tire stretching and releasing.Thereby, we can consider, for a sake of simplicity, α s = 0 so that we can rewrite(2.30),(2.31), after normalizing by µ smax , as follows:where k s = 1λ max.⎧⎪⎨ k s |λ i |, for |λ i | ≤ λ maxµ ix (ω i , ∆v ix ) =(4.12a)⎪⎩ 1, for |λ i | > λ maxµ iy (ω i , ∆v ix , v iy ) = tan θ i µ ix (ω i , ∆v ix ) (4.12b)As a consequence, only the identification of the parameter λ max isrequired, instead of the two parameters λ m , α s .We observe that the value of µ iy coincides with θ i when |λ i | > λ max , while we obtain µ iy =k sv ixrω iwhen |λ i | ≤ λ max , which corresponds to the expected behavior as described above. Itis worth noticing that, because of the saturation of µ ix , µ iy can assume values greater than 1,therefore the saturation to 1 must be guaranteed also for µ iy .4.2 Tire Longitudinal ForceWe can consider the tire modeled as the spring-mass-damper system in Figure 4.1 alsofor the reaction force along the x-axis. However, as the tire is not dragged on the groundbut it rolls, in this case, the contact point is not dragged by the wheel center velocity asfor the lateral motion. Conversely, we can consider the longitudinal slip velocity (∆v ix ) asdriving velocity for the contact point. Thereby, under the same assumptions done for thelateral reaction force, the longitudinal reaction force perceived from the local frame can bedescribed by the following equation: