Advance Modeling of a Skid-Steering Mobile Robot for Remote ...
Advance Modeling of a Skid-Steering Mobile Robot for Remote ...
Advance Modeling of a Skid-Steering Mobile Robot for Remote ...
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5.2 Identification <strong>of</strong> 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 <strong>for</strong> 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 <strong>for</strong>ces <strong>of</strong>the 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 <strong>for</strong> 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, <strong>for</strong> a sake <strong>of</strong> simplicity, it was considered b = a.In order to identify the parameters in (5.3), we consider the free response <strong>of</strong> a second ordersystem. We can write the free response <strong>of</strong> equation (5.3) by setting the inputs M x = M y =M z = 0 and dividing by the moment <strong>of</strong> 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 <strong>of</strong> the systems in (5.4), and they are related to the parametersK i , D i by the following <strong>for</strong>mulas: