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

2.1 State of the Art 15where v L , v R denote respectively the longitudinal coordinates of left and right wheel velocities,while v F , v B are the lateral coordinates of front and rear wheels velocities. Thesevelocities can be expressed in matrix form by combing equations (2.5) and (2.3):⎡v Lv Rv F⎢⎣⎤⎥⎦v B⎡⎤1 −c⎡1 c v x=0 −x ICRG + b⎢⎣ω⎤⎥⎦z⎢⎣⎥⎦0 −x ICRG − a(2.6)Although the above relations are often used to obtain the kinematic model of SSMR, theassumption of non longitudinal slipping is highly restrictive for the accuracy of the model asthe slipping between the wheels and the ground usually is not negligible.We define the longitudinal wheel slip λ i at each wheel as the ratio between the difference ofthe wheel velocity and its center velocity, and the wheel velocity, namely:λ i = rω i − v ixrω i= − ∆v ixrω i, i = 1, . . . , 4 (2.7)where r is the so called effective radius of the wheels, ω i is the angular velocity of the i thwheel and ∆v ix = v ix − rω i is also called the longitudinal slip velocity of the i th wheel/groundcontact point P i . We note that λ 1 = λ 2 = λ L and λ 3 = λ 4 = λ R as ω 1 = ω 2 = ω L andω 3 = ω 4 = ω R , due to Assumption 4. It is also observed that under the above definition,λ i ∈ [0, 1] if the i th wheel is under traction (∆v ix < 0), and λ i ∈ (− inf, 0] if the wheel isunder braking (∆v ix > 0), which is undesirable for uniformly modeling the wheel/groundfriction under traction and braking cases. To avoid such a problem, using the same treatmentas in [4], we restrict the magnitude of λ i to a maximum magnitude of 1, i.e. we take λ i = −1if λ i < −1 under braking.Let ICR l = (x ICRl , y ICRl , 0) and ICR r = (x ICRr , y ICRr , 0) denote the instantaneous center ofrotation expressed in the local frame of the left-side and right-side wheel/ground contactpoints, respectively. It is known that ICR l , ICR r and ICR G lie on a line parallel to the y-axis [4], [5], [10], [16] (Figure 2.3). From the aforementioned discussion and equation (2.3),we define the longitudinal ICR location s as follows:s = x ICRl = x ICRr = x ICRG = − v yω z(2.8)