Natural Frequency Damping Period Halving/Doubling* time Longitudinal
Short period 17.06 rad/s 0.48 0.42 s 0.09 s Phugoid 0.89 rad/s 0.08 7.02 s 10.28 s Latero-directional
Dutch roll 6.15 rad/s 0.13 1.03 s 0.89 s
Roll - 1 - 0.05 s
Spiral* - 1 - 69.24 s
Fortran elaboration package. The Fortran application performs the aerodynamic analysis of the lifting surfaces according to the extended lifting line theory, 2D aerodynamics of airfoils is provided as input in tabular format. Fuselage aerodynamics is achieved by superposition of potential flow, friction drag and cross-flow effects. The overall aerodynamic configuration is obtained by adding all the separate contributions. The aerodynamic coupling of the fuse- lage and the empennages with the wing is also considered. Aircraft control derivatives are computed with the lifting line theory, stability and damping derivatives are also evaluated. Among other properties, the Fortran application is also able to estimate neutral and ma- neuvering points, elevator and stick force gradients, hinge moments, mass distribution and moments of inertia.
Fig. 3.8: MH850 UAV model in ACI software tool
The accessory software package Endran, which includes a complete motor model, realizes a performance analysis for the obtained aircraft model. The analysis includes h-V diagram,
level flight range-endurance limitations, climbing and turning flight. Flight tests have vali- dated the estimated performance with a good degree of accuracy. The aircraft model obtained with ACI is therefore thought to be reliable enough to guarantee a satisfying understanding of the aircraft dynamical characteristics. This allows to perform with confidence an initial assessment of the controlled motion and to have a preliminary tuning the controller gains before the flight.
3.6.2 MH850 UAV actuators model
Small UAVs require compact, lightweight and responsive actuators. The actuators for these applications evolved consistently during last years, increasing their performances and becom- ing more suitable to UAV applications. Servos are small boxes that contain a DC electric motor, gears with an output shaft, a position-sensing mechanism, and a control circuitry. Most servos require a power supply between 4.8 V and 6.0 V. The higher the voltage, the faster the servo will move and the more torque it will have. A standard RC radio receiver sends Pulse Width Modulation (PWM) signals to the servo. The electronics inside the servo translate the width of the pulse into a position. When the servo is commanded to rotate, the motor is powered until the potentiometer reaches the value corresponding to the commanded position. The length of the pulse indicates the position to take. Nominally, when the pulse width is 0.6-2.4 ms the servo angular position is ∓45 deg. A pulse width of 1.5 ms sets the servo to central position. Increasing the pulse width by 10 µs results in about a degree of movement on the output shaft. The servo expects a pulse every 20 ms in order to gain correct information about the angle.
An analog sub-micro servo produced by GWS, see Fig. 3.9 is studied to identify typical properties of traditional servos employed in small UAVs applications. The analog servo model GWS IQ-100 at 6 V produces a torque of 0.084 Nm and has a time response to 60 degrees equal to 0.09 seconds. Its weight is only 5.5 grams. Static and dynamical performances are analyzed, and, in particular, an estimate of the system transfer function is performed. This is later introduced in the MH850 aircraft model that will be used as test for the controllers proposed in this project. The data for this analysis are based on a previous work where wind tunnel experiments tested the behavior of various servo configurations at different airspeed and frequency inputs. More details about the experimental setup are available in [83].
The servo is installed on a wing and tested at four different airspeeds: 0, 2.5, 5 and 7.5 m/s. Large frequency sweeps ranging from 0.1 to 4 Hz over a time interval up to 150 s are sent as input, they represent approximately 50% of the full stick range. The decomposition of a video sequence recorded at 25 Hz, sampling time interval 0.04 s, allows storing a time
Fig. 3.9: GWS IQ-100 analog servo.
domain input and output series of points. Fig. 3.10 illustrates a time series for 7.5 m/s where input and output data are normalized and the average value is subtracted. In order to reduce the noise influence on the model, a first order Butterworth lowpass filter is applied to the data with a cut off frequency equal to half the Nyquist frequency. The study is performed at V = 7.5 m/s as this speed is closer to the flight conditions encountered by the MH850 UAV.
Fig. 3.10: Time domain normalized input and output series for V = 7.5 m/s.
The experimental time series is elaborated with Matlab to estimate the system transfer function. The selected approach relies on the Prediction-Error Minimization Method (PEM). This algorithm estimates a discrete-time state space model using the subspace method, then it refines it by minimizing the prediction error generated from an optimally determined predictor [84]. In this case the error is numerically minimized through the scalar cost function
VN(G(z), H(z)) = N
X
t=1
err2(t)
where err(t) is the vector containing the error calculated for each of the N time steps. The higher the value of N , the more accurate the prediction is. For a linear SISO model the error is proportional to the difference between the measured output y(t) and the predicted output G(z)uc(t)
err(t) = H−1(z) (y(t) − G(z)uc(t))
Note that z is the discrete variable, G(z) and H(z) are the transfer functions of the estimator and uc(t) is the input. A state space model which fits the experimental data is estimated
and the related continuous time transfer function S(s) is easily computable as
S(s) = 9.311s + 8.241 s2+ 21.99s + 53.97
Fig. 3.11 shows the Bode plot of the servo transfer function.
3.6.3 C172P aircraft model
The Cessna 172P, see Fig. 3.12, is a single combustion engine aircraft with standard config- uration including high-wing and fixed tricycle landing gear. Technical data are summarized in Table 3.4. The aircraft is powered by a Lycoming O-320-D2J engine able to produce 160 hp and to guarantee a cruise speed of 55 m/s. The control surfaces include aileron, elevator and rudder. The choice of this vehicle is motivated by two reasons: i) it is a popular aircraft with much technical data available; and ii) the aircraft is available in FlightGear simulator which is employed for preliminary hardware-in-the-loop simulations.
Fig. 3.12: Cessna 172P aircraft