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

2.2 Generalized Modeling 28where T(d C ) is the matrix derived from the substitution of v y , v z , ω x , ω y and it is definedas follows:⎡1 0⎤The vector η = [v xV = T(d C )η =−⎢⎣y ICRx ICR− x2 ICR +y2 ICRx ICR z ICRy ICRx ICR z ICRx 2 ICR +y2 ICR −z2 ICRx 2 ICR z ICR+y 2 ICR z ICR+z 3 ICRx 2 ICR +y2 ICR +z2 ICRx ICRx 2 ICR y ICR+y 3 ICR +y ICRz 2 ICRx ICR z ICR− y2 ICR +z2 ICRx ICR z ICRx 2 ICR y ICR+y 3 ICR +y ICRz 2 ICRx 2 ∗z+y 2 ∗z+z 30 1⎥⎦η (2.47)ω z ] T , using the same notation for the planar case, represent the controlinput at kinematic level, therefore equation (2.46) represents the generalized kinematicmodel for SSMRs.Finally, by combining equation (2.45) with (2.46), it is straightforward to obtain the same relationas (2.16), but with the new connotation of the matrices A(q), S (q). Similarly as it wasdone for the dynamic model in the planar case, such relation will be useful in the treatmentof the generalized dynamic model of a SSMR expressed in the inertial and local frame.It can be also proved that, although the relations in (2.3) does not hold for the 3D case andthe non-holonomic constraint changes, the relations on wheel velocities defined in (2.5) stillhold and a relation similar to (2.11) can be found. For this reason, we can consider the samecontrol input η at kinematic level.The generalized dynamic equation of SSMRs can still be obtained using Euler-Lagrangeprinciple with Lagrange multipliers to include non-holonomic constraint. As Assumption6 does not hold in the general case and all linear and angular velocities are allowed, theLagrangian of the system is defined as:L = E − U = 1 2 (mẊT Ẋ + Ω T I g Ω) − mX T g = 1 2 ˙qT M g ˙q + mgZ (2.48)where g = [0 0 − g] T is the gravitational acceleration vector, with g = 9.8 m , and Is 2 g ∈R 3 , M g ∈ R 6 are respectively the inertia matrix and the generalized inertia tensor calculatedwith respect to the inertial frame.By applying the Euler-Lagrange equation in (2.17) to the Lagrangian in (2.48), we obtain: