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

4.1 Tire Lateral Force 600 ≤ µ iy (ω i , ∆v ix , v iy ) ≤ 1A qualitative representation of 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 therefore 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, thereforewe 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 of µ kmin for 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 force doesnot present the saw-tooth behavior as for 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 , thereforethe 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 for theconcrete floor, we have µ(0.36, −0.36, 0.001) = µ k minµ kmax≈ µ s minµ smax≈ 0.12, therefore we obtainF smin − F kminthe force sensor.= (µ smax − µ kmax )µ(0.36, −0.36, 0.001)mg ≈ 2 N, which is nearly the accuracy ofThe 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, for instance when the robot is moving along a straightline with an external lateral force, 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 force decreases when the robot is moving along a straight line, and in particularis proportional to the ratio v iyω i.The aforementioned characteristic of 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 of all,we observe that, by combining (2.28) with (2.8),(2.10), the following relation holds: