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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2.1 State <strong>of</strong> the Art 18Finally, by combining equation (2.13) with (2.14), it is straight<strong>for</strong>ward to obtain the followingcondition, which will be useful in the treatment <strong>of</strong> the dynamic model <strong>of</strong> a SSMR:S T (q)A T (q) = 0. (2.16)2.1.2 Dynamic ModelIn this subsection, we present the dynamic model <strong>of</strong> SSMRs presented in [10], [4], [5],[11], [11], [15]. Because <strong>of</strong> interaction between the wheels and the ground, the dynamicproperties <strong>of</strong> SSMRs play a very important role. It should be noted that, if the robot ischanging its orientation, reactive friction <strong>for</strong>ces are usually much higher than <strong>for</strong>ces resultingfrom inertia. As a consequence, even <strong>for</strong> relatively low velocities, dynamic properties<strong>of</strong> SSMRs influence motion much more than <strong>for</strong> other vehicles <strong>for</strong> which non-slipping andpure rolling assumption may be satisfied. However, in this section only simplified dynamics<strong>of</strong> SSMRs, which will be useful <strong>for</strong> trajectory control purpose, are introduced. In order tosimplify this model, we assume that the mass distribution <strong>of</strong> the vehicle is almost homogeneous,the kinetic energy <strong>of</strong> the wheels and drives can be neglected, and detailed description<strong>of</strong> tire which can be found, <strong>for</strong> example, in [17] are omitted.The dynamic equation <strong>of</strong> 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 <strong>for</strong>ces vector without considering any external<strong>for</strong>ce and L = E − U is the Lagrangian <strong>of</strong> the system defined as the sum <strong>of</strong> kinetic andpotential energy. Due to Assumption 6, the gravitational potential energy is U = 0, there<strong>for</strong>ein our case the Lagrangian coincides with the kinetic energy <strong>of</strong> 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 <strong>of</strong> inertia <strong>of</strong> 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 <strong>of</strong> the wheels is neglected <strong>for</strong> a sake <strong>of</strong> simplicity.