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

2.1 State of the Art 20constraint (2.13) by using the Lagrange multipliers.Let N, N i be the normal reaction force exerted from the ground surface to the whole vehicleand to each wheel/ground contact point, respectively. According to the geometric descriptiondepicted in Figure 2.3 and thank to the symmetry along the longitudinal axis, the followingrelations can be easily deduced:bN 1 = N 4 =2(a + b) N = b2(a + b) mg(2.24)aN 2 = N 3 =2(a + b) N = a2(a + b) mgLet F i , P i denote respectively the longitudinal and lateral reactive force vector at the i thwheel/ground contact point expressed on the local frame (Figure 2.2,2.3). These forces canbe thought only as friction forces according to [10]. Friction is difficult to model becauseof the high non-linearity related to the many variables that contribute to it. For sake ofsimplicity, the friction can be modeled as a superposition of viscous and Coulomb frictionsas:F f i = −µ ci sign(∆v i )N i − µ vi ∆v i i = 1, . . . , 4 (2.25)where F f i = [F i P i ] T is the total friction force on the i th wheel/ground contact point,µ ci , µ vi represent respectively the Coulomb and viscous friction coefficients and ∆v i is thewheel slip velocity (Figure 2.3).As SSMRs are vehicles which usually operate at low velocity, in particular during turns, wecan assume that µ ci N i >> |µ vi ∆v i |. Thereby, the term −µ vi ∆v i in (2.25) can be neglected,leading to simpler model:⎡−µ cix sign(∆v ix )N iF f i = −µ ci sign(∆v i )N i = ⎢⎣−µ ciy sign(∆v iy )N⎤⎥⎦ii = 1, . . . , 4 (2.26)We note that equation (2.26) can’t be directly employed for two reasons: first it requiresthe knowledge of ∆v iy which is not directly known, second it does not represent a smoothfunction since the function sign(x) is discontinuous, and therefore not differentiable, in x = 0.In order to overcome the first issue, we note that F i , P i are dependent on each other and, inparticular, they form a friction force circle, namely F i = |F f i | cos θ i , P i = |F f i | sin θ i , where θ iis the slip angle at the i th wheel/ground contact point (Figure 2.3). Thereby, we can rewritethe lateral friction force as: