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

6.1 Simulink Model 85as explained in Section 4.1. Such a dynamics of the contact point when the tire is releasingis so complex that it would require an ¨ad hoc¨ study, which is not provided in this thesis.In order to overcome this issue, we consider N ic = m t g and we assume that the conditionson F icx given in (4.13) still hold. However, when the value of F icx switches from F icx 0 toF icx = 0, the value for v icx could not be zero as it should be. In such a case, as the contactpoint dynamics switches from m t˙v icx 0 to m t˙v icx = 0, the velocity v icx will stay stable tothe value it had before switching, so that also the tire reaction force will keep its stable valueinstead of performing the expected stretching/releasing behavior. The solution to this issuecan be found by simply rewriting (4.13) as follows:f ix = −D x (∆v ix − v icx ) − K x∫(∆v ix − v icx )dtm t˙v icx + D x v icx + µ kx N ci sign(v icx ) = F icx′⎧⎪⎨ 0, for |FF icx ′ ix ′=| < µ sxN i ∧ f i (t − ∆t) = 0⎪⎩ Fix ′ , for |F′ ix | > µ kxN i ∧ f i (t − ∆t) 0∫F ix ′ = m t˙v ix + D x v ix + K x (v ix − v icx )dt(6.4)where f i = 1, 0 is a flag indicating whether the contact point is moving or not.We notice that, in this case, when the value of F icx switches from F ′ icx 0 to F′ icx= 0, thesecond equation in (6.4) represents an asymptotically stable system instead of a simply stablesystem, therefore v icx will tend to zero independently from the value it had before switching.Briefly, as we do not know the dynamics of N ci allowing the contact point to stop, we areimposing v icx = 0 when F ′ icx= 0. However, it must be noticed that the overall dynamicsdoes not change from the theoretical one, except for the fact that the contact point does notstop when Ficx ′ = 0 as for the theoretical case, but it stops some time later. Moreover, as weit will be shown in the next Section, the system represented by the second equation in (6.4)is over-damped, i.e. D x > 2 √ m t K x , and the corresponding time constant is relatively small,i.e.√mtK x≪ 0.1. As a consequence, the contact point stops immediately after F ′ icx= 0. Itis worth noticing also that such approximation of the contact point dynamics does not takeinto account the case in which v ix is relatively high, or N ci relatively low, so that v icx couldnever reach zero, as explained in Section 4.1. However, we are considering only the caseω z < 64 degs≈ 1 rad,that is v s ix < 0.15 m , and the acquired data from the force sensor alwaysspresents the saw-tooth shape when pulling the robot up to 0.1 m , therefore we can assumes