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

2.2 Generalized Modeling 32where we used the transformation law of the inertia tensor M g = (J −1 ) T MJ −1 [19], withM representing the inertia tensor expressed in the local frame.Moreover, it can be proved that the term J −1 ˙ J, that appears in the expression of ¯C, does notdepend on q, ˙q but on ˙V, therefore on ˙η and ṗ ICR , as it can be rewritten as:J −1 ˙ J = ˆV =⎡ ⎤0 ˆv⎢⎣ˆv ˆω⎥⎦(2.60)A simple proof can be obtained by considering η = V, therefore T = I, and rewritingequation (2.58) with the resultant of all forces and torques on the right side:M ˙V + MJ −1 ˙ JV = F totBy using Newton-Euler equation, we can also write the F tot as:⎡⎤⎡ ⎤m˙v + ω × mvF tot = ⎢⎣I ˙ω + ω × Iω⎥⎦ = M 0 ˆv˙V + M ⎢⎣ˆv ˆω⎥⎦ VFinally, by equating the two equations, we obtain the relation in (2.60) so that we canrewrite equation (2.58) as:T T MT ˙η + (T T MṪ + T T M ˆVT)η + T T F r + T T J −1 G = T T T f τ. (2.61)where the matrix ˆV is calculated by substituting the coordinates of V with the transformedcoordinates Tη.It is worth noticing that the dynamic model defined in (2.61) requires the knowledge of thecoordinates of the ICR. Let us imagine that the robot linear and angular velocities can bemeasured, for instance by employing an Inertial Measurement Unit (IMU). Then, from theknowledge of V, the coordinates of p ICR can be calculated by solving the equations in (2.44).Although the dynamic model in (2.61) can be useful for control issues, it can not be directlyemployed for simulation purposes, because it is not possible to determine the coordinates ofp ICR without knowing all the values of V, which can not be computed as the output of themodel. For this reason, we rewrite the dynamic model by substituting in (2.57) the kinematicmodel in (2.40) and its time derivative:T T MT ˙V + T T M ˆVV + T T F r + T T J −1 G = T T T f τ. (2.62)