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

2.1 State of the Art 18Finally, by combining equation (2.13) with (2.14), it is straightforward to obtain the followingcondition, which will be useful in the treatment of the dynamic model of a SSMR:S T (q)A T (q) = 0. (2.16)2.1.2 Dynamic ModelIn this subsection, we present the dynamic model of SSMRs presented in [10], [4], [5],[11], [11], [15]. Because of interaction between the wheels and the ground, the dynamicproperties of SSMRs play a very important role. It should be noted that, if the robot ischanging its orientation, reactive friction forces are usually much higher than forces resultingfrom inertia. As a consequence, even for relatively low velocities, dynamic propertiesof SSMRs influence motion much more than for other vehicles for which non-slipping andpure rolling assumption may be satisfied. However, in this section only simplified dynamicsof SSMRs, which will be useful for trajectory control purpose, are introduced. In order tosimplify this model, we assume that the mass distribution of the vehicle is almost homogeneous,the kinetic energy of the wheels and drives can be neglected, and detailed descriptionof tire which can be found, for example, in [17] are omitted.The dynamic equation of a SSMR can be obtained using Euler-Lagrange principle with Lagrangemultipliers to include non-holonomic constraint. The Euler-Lagrange equation isdefined as:Γ = d dt∂L∂ ˙q − ∂L∂q(2.17)where Γ denotes the active torques and forces vector without considering any externalforce and L = E − U is the Lagrangian of the system defined as the sum of kinetic andpotential energy. Due to Assumption 6, the gravitational potential energy is U = 0, thereforein our case the Lagrangian coincides with the kinetic energy of the robot 4L = E = 1 2 (mVT V + Iω 2 z) = 1 2 [m(Ẋ2 + Ẏ 2 ) + I ˙θ 2 ]. (2.18)where m and I represent respectively the mass and the moment of inertia of the robotabout the CoM. It must be noticed that Assumption 6 limits the motion to flat grounds andit has been made only to simplify the computations, although it is not difficult to generalize4 Note that the kinetic energy of the wheels is neglected for a sake of simplicity.