4.1 Tire Lateral Force 600 ≤ µ iy (ω i , ∆v ix , v iy ) ≤ 1A qualitative representation <strong>of</strong> the function µ iy can be obtained by first considering that,when the wheel slip is relatively high (|λ i | > λ max ), i.e. the slip velocity is relatively highcompared to the wheel spinning, the friction coefficient is proportional to the ratio betweenthe wheel lateral velocity and the slip velocity. In fact, when v iy is relatively high withrespect to ∆v ix , we can consider as if the contact point is not moving along the longitudinaldirection and there<strong>for</strong>e we can consider µ iy≈ 1. Conversely, when v iy is relativelylow with respect to ∆v ix , we can consider as if the contact point is moving along the longitudinaldirection much faster than the wheel is moving along the lateral direction, there<strong>for</strong>ewe can consider µ iy ≈ µ min . The value µ min depends on the wheel lateral velocityand can be zero only if v iy = 0. An example <strong>of</strong> µ kmin <strong>for</strong> v iy = 1 mm is provided in Sec-stion 3.3 (Figure 3.5).We notice that in such a case the tire lateral reaction <strong>for</strong>ce doesnot present the saw-tooth behavior as <strong>for</strong> the case ∆v ix= 0. This is due to the fact that,when v iy is relatively low with respect to ∆v ix , the difference between µ sy and µ ky , there<strong>for</strong>ethe difference between F sy and F ky , become so small that the tire stretching/releasingbehavior is negligible, also because the saw-tooth would have such a high frequency thatthe robot structure would filter it out.In fact, if we consider the data acquired <strong>for</strong> theconcrete floor, we have µ(0.36, −0.36, 0.001) = µ k minµ kmax≈ µ s minµ smax≈ 0.12, there<strong>for</strong>e we obtainF smin − F kminthe <strong>for</strong>ce sensor.= (µ smax − µ kmax )µ(0.36, −0.36, 0.001)mg ≈ 2 N, which is nearly the accuracy <strong>of</strong>The proportionality to the ratio between the wheel lateral velocity and the slip velocity isnot sufficient to describe µ iy when the wheel slip is relatively low (|λ i | < λ max ). In fact, ifwe consider ∆v ix ≈ 0 but ω i , v iy 0, <strong>for</strong> instance when the robot is moving along a straightline with an external lateral <strong>for</strong>ce, by only considering µ iy ∝ v iy∆v ixand constraining it to 1 asmaximum value, we would obtain µ sy = µ smax , µ ky = µ kmax , which is the same result obtainedwhen the robot is not moving. We empirically verified that such a result is not true, as thetire lateral <strong>for</strong>ce decreases when the robot is moving along a straight line, and in particularis proportional to the ratio v iyω i.The a<strong>for</strong>ementioned characteristic <strong>of</strong> the function µ iy can be analytically expressed by consideringthe function proposed by Song et al. in [4], provided in (2.27),(2.30). First <strong>of</strong> all,we observe that, by combining (2.28) with (2.8),(2.10), the following relation holds:
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 <strong>of</strong> f iy , because <strong>of</strong> the tire stretching and releasing.Thereby, we can consider, <strong>for</strong> a sake <strong>of</strong> 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 |, <strong>for</strong> |λ i | ≤ λ maxµ ix (ω i , ∆v ix ) =(4.12a)⎪⎩ 1, <strong>for</strong> |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 <strong>of</strong> the parameter λ max isrequired, instead <strong>of</strong> the two parameters λ m , α s .We observe that the value <strong>of</strong> 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 <strong>of</strong> the saturation <strong>of</strong> µ ix , µ iy can assume values greater than 1,there<strong>for</strong>e the saturation to 1 must be guaranteed also <strong>for</strong> µ iy .4.2 Tire Longitudinal ForceWe can consider the tire modeled as the spring-mass-damper system in Figure 4.1 als<strong>of</strong>or the reaction <strong>for</strong>ce 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 as<strong>for</strong> the lateral motion. Conversely, we can consider the longitudinal slip velocity (∆v ix ) asdriving velocity <strong>for</strong> the contact point. Thereby, under the same assumptions done <strong>for</strong> thelateral reaction <strong>for</strong>ce, the longitudinal reaction <strong>for</strong>ce perceived from the local frame can bedescribed by the following equation: