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

4.3 Tire Vertical Force 62f ix (t) = −D x (∆v ix (t) − v icx (t)) − K x∫(∆v ix (t) − v icx (t))dtm t˙v icx (t) = F icx (t)⎧⎪⎨ 0, for |F ix (t)| < µ sx N i ∧ v icx (t − ∆t) = 0F icx (t) =⎪⎩ F ix (t), for |F ix | > µ kx N i ∧ v icx (t − ∆t) 0∫F ix (t) = m t˙v ix (t) + D x (v ix (t) − v icx (t)) + K x (v ix (t) − v icx (t))dt − µ kx N i sign(v icx (t))(4.13)with the same meaning of the variables and parameters as for the lateral reaction force,where µ kx is defined in (4.12a).It is worth noticing that the parameters D x , K x depend not only on the elasticity and dampingproperties of the tire along the longitudinal direction, but they also depend on the elasticityand damping properties of the motor belt. In particular, we can consider the spring-damperfor the tire and the one for the belt as they were in series, therefore the relations betweenK x , D x and the elasticity and damping coefficient of the tire (K tx , D tx ) and the motor belt(K b , D b ) can be written as follows:K x =K txK bK tx + K b, D x = D txD bD tx + D b(4.14)As K tx , D tx do not depend on the tire inflating properties, like for the lateral and verticaldeflection, but only on the tire’s rubber stretching properties, we usually have K tx >> K b andD tx >> D b . Thereby, by looking at relation (4.14) we obtain:K x ≈ K b ,D x ≈ D b4.3 Tire Vertical ForceLet us consider the robot rotating around the x-axis and y-axis. In such a case, it is easyto deduce that the normal reaction forces at each contact point do not exactly correspond tothe robot’s weight distribution as when the robot does not rotate. The normal reaction forceN i perceived at each wheel, in fact, depends on the tire vertical compression, which dependson the wheel vertical position and velocity.Similarly to what we have done for the tire lateral force, let us model the tire as a spring-