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

= −p ax A z ω 2 n ze −ξ zω nz sin( √ 1 − ξ 2 z ω nz t + B z + Φ ) (5.8)5.2 Identification of the Tire Dynamic Parameters 71a x = g sin(ψ) ≈ gψa y = −g sin(θ) ≈ −gθa y = −p ax ¨φ= −p ax A z ω 2 n ze −ξ zω nz[(2ξ 2 z − 1) sin ( √ 1 − ξ 2 z ω nz t + B z)− 2ξz√1 − ξ 2 z cos ( √ 1 − ξ 2 z ω nz t + B z) ]where g is the gravitational acceleration and the first equality come from the linear propertyof trigonometric functions such that: a sin x + b cos x = √ a 2 + b 2 sin(x + Φ), wheretan Φ = b . The approximations come from the first order Taylor expansion of the sin() func-ation, which makes sense for ¨small¨ values of θ, ψ.By looking at the relations in (5.8), we can claim, with a good approximation, that the signalsmeasured from the accelerometers differ from the free responses in (5.7) only by their amplitudesand eventually phases. Thereby. the Roll, Pitch and Yaw angular natural frequencyω inand damping ratio ξ i can be estimated by looking at the frequency of the oscillationsmeasured from the accelerometers and the rate at which they decay. To this purpose, thedata from the accelerometers are acquired eight times per each motion (Roll, Pitch and Yaw)and analyzed by using the Matlab functions defined in Appendix A. In particular, the datacontain 2 12 samples and are acquired with a sample frequency of 1 kHz, but only the first 2 11samples are analyzed (tRange = ′ beginnig ′ ), in order to cut off possible disturbs present inthe last part of the data where the system has already reached the equilibrium. Moreover, thedata are lowpass-filtered at 20 Hz (fRange = [1, 20]) by using the function filterData()and the peak frequency is found by using the function findPeakFreq().Figure 5.4, 5.5, 5.6 depict three examples of the acquired data and their FFT respectively forthe Roll, Pitch and Yaw motion.Tab.5.2 shows all the peak frequencies found for each recorded data. As the standarddeviation is relatively small in all the three cases ( std( f peak)mean( f peak )