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 ...
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:
2.1 State of the Art 21P i = F i tan θ i i = 1, . . . , 4 (2.27)The terms tan θ i can be calculated with respect to the ICRs positions according to thegeometrical description depicted in Figure 2.3, obtaining:tan θ 1 = a − x ICR ly ICRl − ctan θ 2 = −b − x ICR ly ICRl − ctan θ 3 = −b − x ICR r−y ICRr − c(2.28)tan θ 4 = a − x ICR r−y ICRr − cIt must be noticed that the above definitions of tan θ i are always meaningful since it canbe proved that the conditions y ICRl c, y ICRr −c are always verified [16].The issue due to the non-smoothness of the term µ cix sign(∆v ix ) in equation (2.25) is overcomeby replacing it with a suitable function µ(λ i ), which represents a reasonable approximationof the sign() function. Here we present two approximating functions for µ(λ i ) [9], [4] withtheir advantages and disadvantages:2µ(λ i ) = µ cixπ arctan(k 2s∆v ix ) = −µ cixπ arctan(k srω i λ i )⎧−k s λ m + k sλ m −µ s1+λ m(λ i + λ m ), for λ i ∈ (−1, −λ m ]⎪⎨µ(λ i ) = k s λ i , for λ i ∈ (−λ m , λ m )(2.29a)(2.29b)⎪⎩ k s λ m − k sλ m −µ s1−λ m(λ i − λ m ), for λ i ∈ [λ m , 1)where k s is the friction stiffness coefficient, λ m is the longitudinal slip corresponding to themaximum wheel/ground friction coefficient, and µ s is the longitudinal wheel/ground slidingfriction coefficient.The function defined in (2.29a) derives directly from the conventional smooth approximationof the sign() function [9]. Its advantage is due to the dependency of only two identifiedparameter (µ cix , k s ), while the disadvantages are the non-linearity of the arctan() functionand the absence of an instability region due to the difference between the static and kineticfriction coefficient.Conversely, the function defined in (2.29b) represents a linear piece-wise function, which
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- Page 66 and 67: 4.1 Tire Lateral Force 56Figure 4.1
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2.1 State <strong>of</strong> the Art 21P i = F i tan θ i i = 1, . . . , 4 (2.27)The terms tan θ i can be calculated with respect to the ICRs positions according to thegeometrical description depicted in Figure 2.3, obtaining:tan θ 1 = a − x ICR ly ICRl − ctan θ 2 = −b − x ICR ly ICRl − ctan θ 3 = −b − x ICR r−y ICRr − c(2.28)tan θ 4 = a − x ICR r−y ICRr − cIt must be noticed that the above definitions <strong>of</strong> tan θ i are always meaningful since it canbe proved that the conditions y ICRl c, y ICRr −c are always verified [16].The issue due to the non-smoothness <strong>of</strong> the term µ cix sign(∆v ix ) in equation (2.25) is overcomeby replacing it with a suitable function µ(λ i ), which represents a reasonable approximation<strong>of</strong> the sign() function. Here we present two approximating functions <strong>for</strong> µ(λ i ) [9], [4] withtheir advantages and disadvantages:2µ(λ i ) = µ cixπ arctan(k 2s∆v ix ) = −µ cixπ arctan(k srω i λ i )⎧−k s λ m + k sλ m −µ s1+λ m(λ i + λ m ), <strong>for</strong> λ i ∈ (−1, −λ m ]⎪⎨µ(λ i ) = k s λ i , <strong>for</strong> λ i ∈ (−λ m , λ m )(2.29a)(2.29b)⎪⎩ k s λ m − k sλ m −µ s1−λ m(λ i − λ m ), <strong>for</strong> λ i ∈ [λ m , 1)where k s is the friction stiffness coefficient, λ m is the longitudinal slip corresponding to themaximum wheel/ground friction coefficient, and µ s is the longitudinal wheel/ground slidingfriction coefficient.The function defined in (2.29a) derives directly from the conventional smooth approximation<strong>of</strong> the sign() function [9]. Its advantage is due to the dependency <strong>of</strong> only two identifiedparameter (µ cix , k s ), while the disadvantages are the non-linearity <strong>of</strong> the arctan() functionand the absence <strong>of</strong> an instability region due to the difference between the static and kineticfriction coefficient.Conversely, the function defined in (2.29b) represents a linear piece-wise function, which
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