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

4.1 Tire Lateral Force 57µ ky NV ic (s) = V iy (s) −(4.5)m t s 2 + D y s + K ywhere V ic (s) and V iy (s) are the Laplace transform of v ic and v iy respectively.By calculating the Laplace inverse transform of equation (4.5), we obtain the following relationin the time domain:where h(t) = L −1 1m t s 2 +D y s+K y.v ic (t) = v iy (t) − µ ky N i h(t) (4.6)We notice that, if the parameters m t , D y , K y are defined such that the system is stable, thecontribution of the friction in (4.6) initially increases, reaching its maximum, and then goesexponentially to zero, either oscillating or directly, depending on whether the system is underor over-damped. Thereby, if we suppose that v ic (t) > 0 ∀t, i.e. v iy (t) > µ ky N i h(t) ∀t,the contact point will keep moving with a velocity v ic (t) → v iy (t). Conversely, if we havev ic (t) < 0 for some values of t, equation (4.6) does not hold anymore, as the friction forcechanges its sign, and the nonlinearity of the sign() function can affect the system stability.However, it can be proved, by substituting the function sign(v ic ) with arctan(k s v ic ), k s ¨big¨,that the system is at least simply stable [21]. In particular, if |v ic |, |˙v ic | become relativelysmall, i.e. |v ic |, |˙v ic | < µ kyN ik s, we can consider as if the contact point stops. Thereby, the systemsuddenly switches to the static configuration, in which the friction coefficient switches fromthe kinetic to the static one, so that the contact point does not move until the elastic forcedoes not overcome the static friction force, i.e. until the condition |F ic | ≤ F smax is satisfied.The system describing the dynamics of the contact point presents, therefore, a sort of hysteresis.More precisely, when the condition |F ic | ≤ F smaxis not anymore satisfied the dynamicsswitches from v ic = 0 to (4.4), and when the condition |v ic |, |˙v ic | < µ kyN ik sfrom (4.4) to v ic = 0, and so on.is satisfied it switchesBy considering the local frame in Figure 4.1 as the frame fixed to the i th wheel and takinginto account the aforementioned considerations, the lateral reaction force perceived from thei th wheel can be written as:f iy = −D y (v iy (t) − v ic (t)) − K y∫(v iy (t) − v ic (t))dtwhere the dynamics of v ic is either v ic = 0 or (4.4), depending whether the i th contact pointis moving or not, and the wheel center velocity v iy is related to the robot linear and angular