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

5.2 Identification of the Tire Dynamic Parameters 76ω nx = 38.0031 radsω ny = 19.4807 radsω nz = 47.4008 rads≈ 38 rads , ξ x = 0.0526 ≈ 0.05≈ 19.5 rads , ξ y = 0.1797 ≈ 0.18≈ 47.4 rads , ξ z = 0.0949 ≈ 0.09Finally, by substituting the results obtained above in (5.5) and considering the tire elasticitycoefficients K x , K y , K z identified in Section 3.3, the moments of inertia around the principalaxes and the tire damping coefficients can be calculated as follows:I x = 4h2 K y + 4c 2 K z= 9.82, Dω 2 x = I zω nz ξ z − I y ω ny ξ yn x2(c 2 − h 2 )= 1596I y = 4h2 K x + 4a 2 K zω 2 n y= 19.83, D y = 2ξ z√Iz (4c 2 K x + 4a 2 K y ) − 4c 2 D x= 22844a 2(5.10)I z = 4c2 K x + 4a 2 K yω 2 n z= 0.97, D z = 2ξ y√Iy (4h 2 K x + 4a 2 K z ) − 4h 2 D x4a 2 = 4693By comparing the values of the identified moment of inertia in (5.10) with the inertiatensor in (5.1), we notice that only I z is close to value estimated using the Solidworks model,while the values of I x , I y are nearly one order greater than the estimated ones. This is probablydue to the approximation we have done by considering pure Roll and Pitch motion, i.e.v x , v y , v z = 0, as the robot was not constraint to move only around the two correspondingaxes. In particular, while the initial condition for the Yaw motion is provided by slightlyturning the robot around the z-axis, providing nearly the same initial tire stretching conditionalong the x, y-axis, the initial condition for the Roll and Pitch motion is provided by tiltingthe robot on only one side, providing different initial tire stretching conditions along they, z-axis and x, z-axis respectively. Moreover, the difference between K y , K X and K z makesthe tire stretch more easily along the x, y-axis rather than the z-axis, therefore the robot isinduced to move horizontally rather than vertically.The equations describing the Roll and Pitch motion can be found by either considering respectivelyv y , v z , ω x 0 and v x , v z , ω y 0 and including the equations describing ˙v x , ˙v y , ˙v z in(5.2), or by simply substituting respectively v x , ω y , ω z = 0 and v y , ω x , ω z = 0 in (2.65). Byconsidering the ¨small¨angles and therefore approximating sin θ ≈ θ, sin ψ ≈ ψ, cos θ, cos ψ =0, we obtain in both cases a fourth order linear system which can be represented by the followingequations in the Laplace domain (impulse responses):