Advance Modeling of a Skid-Steering Mobile Robot for Remote ...
Advance Modeling of a Skid-Steering Mobile Robot for Remote ...
Advance Modeling of a Skid-Steering Mobile Robot for Remote ...
- No tags were found...
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
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 trans<strong>for</strong>m <strong>of</strong> v ic and v iy respectively.By calculating the Laplace inverse trans<strong>for</strong>m <strong>of</strong> 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 <strong>of</strong> 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 <strong>for</strong> some values <strong>of</strong> t, equation (4.6) does not hold anymore, as the friction <strong>for</strong>cechanges its sign, and the nonlinearity <strong>of</strong> 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 <strong>for</strong>cedoes not overcome the static friction <strong>for</strong>ce, i.e. until the condition |F ic | ≤ F smax is satisfied.The system describing the dynamics <strong>of</strong> the contact point presents, there<strong>for</strong>e, a sort <strong>of</strong> 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 a<strong>for</strong>ementioned considerations, the lateral reaction <strong>for</strong>ce 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 <strong>of</strong> 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