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

4.3 Tire Vertical Force 63damper system along the z-axis, with one side fixed to the ground and the other one attachedto a local frame (x,y) moving with respect to a fixed frame (X,Y), as depicted in Figure 4.2.Figure 4.2: A scheme of a spring-mass-damper system moving along the vertical direction.Let us consider the local frame start moving at a velocity v iz (t) with respect to the fixedframe, expressed in the local frame. As we consider the contact point not moving withrespect to the fixed frame, it is straightforward define the reaction force perceived from thelocal frame as follows:f iz = N i = −D z v iz (t) − K z∫v iz (t)dt (4.15)where the wheel center velocity v iz is related to the robot linear and angular velocity byequation (2.55).We notice that in (4.15) the gravitational force does not appear because the contribution ofthe gravity is already included in the robot dynamics determining the wheel vertical velocityv iz . Moreover, although we considered the contact point not moving with respect to the fixedframe, it must noticed that, in reality, it is not fixed with the ground. Thereby, even if we willbe considering in the following that the Roll and Pitch oscillations never make the wheel lift,the normal reaction force in (4.15) must be constraint to N i ≥ 0. Such a constraint will berepresented, in the simulation, as a saturation on N i , and therefore it introduces a nonlinearitywhich can cause the system instability.