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

5.2 Identification of the Tire Dynamic Parameters 68tively defined as Roll, Pitch and Yaw motion. Consequently, the robot dynamics can bedescribed by considering planar motion respectively on the y − z, x − z and x − y plane, asdepicted in Figure 5.3.Under the assumption that the wheels are not spinning, the contact points are not movingand some external torques M z , M y , M x are applied to the robot CoM respectively for thethree rotations, we can consider the Roll, Pitch and Yaw motion described by the followingequations:I x ˙ω x =I y ˙ω y =I z ˙ω z =4∑(y i f iz − z i f iy ) + M xi=14∑(−x i f iz + z i f ix ) + M y(5.2)i=14∑(x i f iy − y i f ix ) + M zi=1where x i , y i , z i and f ix , f ix , f ix are respectively the coordinates and the reaction forces ofthe i th contact point. By substituting equations (4.13),(4.9),(4.15) in (5.2) and consideringonly ω x , ω y , ω z 0 respectively for three rotations, we can rewrite the equations describingthe Roll, Pitch and Yaw motion as follows:I x ¨ω x + (4h 2 D y + 4c 2 D z ) ˙ω x + (4h 2 K y + 4c 2 K z )ω x = M xI y ¨ω y + (4h 2 D x + 4a 2 D z ) ˙ω y + (4h 2 K x + 4a 2 K z )ω y = M y(5.3)I z ¨ω z + (4c 2 D x + 4a 2 D y ) ˙ω z + (4c 2 K x + 4a 2 K y )ω z = M zwhere, for a sake of simplicity, it was considered b = a.In order to identify the parameters in (5.3), we consider the free response of a second ordersystem. We can write the free response of equation (5.3) by setting the inputs M x = M y =M z = 0 and dividing by the moment of inertia I x , I y , I z , obtaining the equation:¨ω i + 2ξ i ω ni ˙ω i + ω 2 n iω i = 0 (5.4)where ω ni and ξ i , with i = x, y, z, correspond respectively to the natural angular frequenciesand the damping ratios of the systems in (5.4), and they are related to the parametersK i , D i by the following formulas: