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

2.1 State of the Art 19the model by setting U = mgZ.By applying the Euler-Lagrange equation (2.17) to the Lagrangian defined in (2.18), weobtain:⎡ ⎤m 0 0Γ = 0 m 0 ¨q = M ¨q (2.19)⎢⎣0 0 I⎥⎦Furthermore, the vector Γ can be calculated by taking into account that the active forceF ai exerted on the i th wheel/ground contact point, expressed on the local frame, is related tothe wheel torque τ i by the following relation:⎡τ ir⎤F ai = ⎢⎣0⎥⎦(2.20)To obtain the total active force and torque, firstly we compute the resultant force on thelocal frame, by summing all the active forces F aiby summing all the torques provided by each force F aii = 1, . . . , 4, and the resultant torque,with respect to the CoM. Then, weproject the resultant force and torque on the inertial frame, obtaining the following expressionof Γ:⎡⎡F x ∑ 4Γ = F y =⎢⎣τ⎤⎥⎦g i=1 F aiR l⎡⎢⎣ ∑ 4i=1 τ⎤⎥⎦ = 1 cos θ ∑ ⎤4i=1 τ isin θ ∑ 4i=1 τairi⎢⎣z c(−τ 1 − τ 2 + τ 3 + τ 4 )⎥⎦(2.21)Due to Assumption 4, we can simplify the notation of (2.21) by considering the followingcontrol input at dynamic level:⎡τ Lτ = ⎢⎣τ⎤⎥⎦ = τ 1 + τ 2⎡⎢⎣R τ 3 + τ⎤⎥⎦4leading to the following expression of Γ:(2.22)⎡cos θΓ = 1 sin θr⎢⎣−c⎤cos θsin θcτ = B(q)τ (2.23)⎥⎦In order to complete the dynamic model of a SSMR, we must add the reaction forces andtorques to the right term of the dynamic equation in (2.19) and include the non-holonomic