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

3.1 The accelerometer 35where m a , D a , K a are respectively the mass, the damping and elasticity coefficient of thespring.The device usually measures the displacement of the proof mass (x a ), for instance bymeasuring the voltage between the two faces of a capacitor, and then it computes the secondderivative of the measured signal, producing the signal ẍ a as output. To better understand therelation between the measured acceleration and the real body acceleration, we can computethe Laplace transform of the spring position X a (s) by solving equation (3.1) in the frequencydomain, and its second time derivative by multiplying it for s 2 :s 2 X a (s) = s 2 1s 2 + D am as + K am aA(s) = T a (s)A(s) (3.2)where X a (s) and A(s) are the Laplace transforms of x a and a respectively, and T a (s) representsthe transfer function of the system. We can also define the natural angular frequency√Kof the system in (3.1) as ω na = am a, and the damping ratio as ξ a =D a2 √ K a m a.In order to avoid undesired oscillations in the measured signal x a , the parameters m a , D a , K aare always set such that the spring-mass-damper system defined by (3.1) is never underdamped(ξ a ≥ 1). Moreover, as the measured acceleration must be as close and as fast aspossible to the body acceleration, such a system is usually critically damped (ξ a = 1) andthe natural angular frequency is relatively small, depending on the physical structure of thedevice. As outlined by the Bode diagram depicted in Figure 3.1(b), the transfer functionT a (s) behaves like a second order derivative for ω > ω na , leavingunaltered the medium and high frequency components of a, except for a delay due toits phase. Moreover, as the exact differentiation is not physically possible, the accelerationẍ a is provided by employing electrical devices, like operational amplifiers, which introducein T a (s) a double pole at ω c , cutting off also the high frequency components of a. For thisreason, we can consider that, for a very large range of frequencies (usually ω naω c< 10 −3 and> 10 3 ) and with relatively small delay, the measured acceleration is equal to the realbody acceleration, i.e. ẍ a = a. As a matter of notation, it is usually considered as body ofinterest the body containing the spring-mass-damper, i.e. the physical accelerometer, and asits positive direction the direction of the y-axis as depicted in Figure 3.1(a), i.e. the oppositedirection of the spring axis which the position measurement is provided along. Thereby, a1-axis accelerometer can be considered a black-box behaving as a band-pass filter measuring