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

4.1 Tire Lateral Force 56Figure 4.1: A scheme of a spring-mass-damper system with friction between the mass and the ground.∫ t∆Y = Y ic − Y i = y ic = − v iy (τ)dτ (4.1)0where y ic denotes the position of the contact point with respect to the local frame.The mass does not move until the resultant of all the forces applied to the contact point (F ic )does not overcome the maximum static friction force F smax= µ sy N i , i.e. until the condition|F ic | ≤ F smax is satisfied, with N i representing the normal reaction to the ground. Conversely,when such a condition is not anymore satisfied, the mass starts moving and the contact pointposition is described by the following equation:and relation (4.1) becomes:m t ÿ ic + D y ẏ ic + K y y ic = F ky = −µ ky N i sign(v ic ) (4.2)∫ t∆Y = y ic = (v ic (τ) − v iy (τ))dτ (4.3)t kwhere v ic = Ẏ ic represents the velocity of the contact point with respect to the fixed frame,expressed in the local frame.It is worth noticing that relation (4.3) still holds if from t k the mass is not moving, i.e. v ic =0. By considering v iy as input of the system and substituting (4.3) in (4.2), we obtain thefollowing equation:m t (˙v ic (t) − ˙v iy (t)) + D y (v ic (t) − v iy (t)) + K y∫(v ic (t) − v iy (t))dt = −µ ky N i sign(v ic ) (4.4)Let us suppose that the contact point starts moving with v ic > 0, so that sign(v ic ) = 1. Bysolving equation (4.4) in the frequency domain, we obtain: