Table 2.2: Experimentally determined aircraft inertias [kg-m21 Aircraft Roll Axis I Pitch Axis Yaw Axis N 0.46 0.65 0.61 0.63 Mean 0.30 0.42 0.63 Std Dev.I 0.029 0.011 0.021 Using the small angle approximation for a since L R, and substituting known values from Eqs. 2.1 and 2.2, Eq 2.3 reduces to Irao L which is characterized by the undamped natural frequency, w, and period of oscilla Tp R2 (2.5) 2 By finding averaged values for the period of oscillation, Tp, in each of the pitch, roll and yaw axes, the inertia about each axis can be approximated. This experiment neglects aerodynamic and other forms of damping as well as the cross-axis inertias (e. g,, Ixz, Iy2). The experimental results are summarized for each of the axes and three different aircraft in the same configuration in Table 2.2, showing agreement between different vehicles used in the tests. The largest variation was found in the roll axis due to the difficulty of mounting the aircraft through the center of gravity, which is essential in this experimental setup 2.1.2 Actuator Models The servos used on the aircraft have saturation limits. limited bandwidth and limited slew rates which are captured in the actuation models of the Cloud Cap hardware-
� Table 2.2: Experimentally determined aircraft inertias [kgm2] Aircraft Roll Axis Pitch Axis Yaw Axis No. Ixx Iyy Izz 1 0.28 2 0.30 3 0.33 Mean 0.30 Std Dev. 0.029 0.46 0.44 0.47 0.42 0.011 0.65 0.61 0.63 0.63 0.021 Using the small angle approximation for α since L � R, and substituting known values from Eqs. 2.1 and 2.2, Eq. 2.3 reduces to ¨ mgR2 Ixxφ + φ = 0 (2.4) L which is characterized by the undamped natural frequency, ω, and period of oscillation, Tp ω = mgR2 IxxL (2.5) 2π Tp = ω (2.6) By finding averaged values for the period of oscillation, Tp, in each of the pitch, roll, and yaw axes, the inertia about each axis can be approximated. This experiment neglects aerodynamic and other forms of damping as well as the crossaxis inertias (e.g., Ixz, Iyz). The experimental results are summarized for each of the axes and three different aircraft in the same configuration in Table 2.2, showing agreement between different vehicles used in the tests. The largest variation was found in the roll axis due to the difficulty of mounting the aircraft through the center of gravity, which is essential in this experimental setup. 2.1.2 Actuator Models The servos used on the aircraft have saturation limits, limited bandwidth, and limited slew rates which are captured in the actuation models of the Cloud Cap hardware 35
wn.2 Actar h s2 +2*z'wnstwn'2 Actuator Actator Transfer Fcn Rate Limiter Figure 2-3: Actuator models used the Cloud Cap Hardware-in-the-loop sim- ulation n-the-loop simulator. As shown in Ref. 26, the actuator transfer function, Gact(s) is given by specifying the bandwidth limit, Bw 2丌B1w 8) =c=0.707 (29) where the damping ratio, S, is selected at the critical value to set the actuator band width equal to the natural frequency (wb Wn). The aileron, elevator, and rudder channels all respond with approximately the same characteristics(Bw= 10 Hz),but the throttle is modeled with less dynamic range(Bw= 2 Hz) as the engine RPM equires added time to ramp up to produce thrust. The input/output saturation and slew rate limits are determined as per manufacturer specifications(++60, 2 Hz respectively), and applied as shown in Figure 2-3 2.1. 3 Sensor noises The Cloud Cap hardware-in-the-loop simulator includes detailed sensor models based on information from the manufacturer to corrupt the simulation measurements. For the purposes of simulation, noises on the pressure, rate gyros and accelerometers onboard the aircraft are modeled using band-limited white noise and specified drift rates26. Although the same noise and drift models could be applied to GPS position and velocity measurements, this information is typically assumed to be perfect in the HWIL tests. The values used to parameterize the Piccolo pressure sensors, the CristaTM IMu angle-rate sensors. and the accelerometers are shown in Table 2.3
Figure 23: Actuator models used the Cloud Cap Hardwareinthelo op simulations intheloop simulator. As shown in Ref. [26], the actuator transfer function, Gact(s), is given by specifying the bandwidth limit, BW Gact(s)= ω2 n s2 + 2ζωns + ω2 n (2.7) ωn = 2πBW (2.8) ζ = ζc = 0.707 (2.9) where the damping ratio, ζ, is selected at the critical value to set the actuator bandwidth equal to the natural frequency (ωb = ωn) . The aileron, elevator, and rudder channels all respond with approximately the same characteristics (BW = 10 Hz), but the throttle is modeled with less dynamic range (BW = 2 Hz) as the engine RPM requires added time to ramp up to produce thrust. The input/output saturation and slew rate limits are determined as per manufacturer specifications (±60◦, 2 Hz respectively), and applied as shown in Figure 23. 2.1.3 Sensor Noises The Cloud Cap hardwareinthelo op simulator includes detailed sensor models based on information from the manufacturer to corrupt the simulation measurements. For the purposes of simulation, noises on the pressure, rate gyros and accelerometers onboard the aircraft are modeled using bandlimited white noise and specified drift rates [26]. Although the same noise and drift models could be applied to GPS position and velocity measurements, this information is typically assumed to be perfect in the HWIL tests. The values used to parameterize the PiccoloTM pressure sensors, the CristaTM IMU anglerate sensors, and the accelerometers are shown in Table 2.3. 36
Table 2.3: Crista IMU HWIL Sensor noise models PDynamic PStatic G unit P Pa] [deg/s] [m/s/s) Resolution unit 3.906 200|1.6E-46.0E-3 Min unit 300 0.0-5.20|-1000 4000110.0005.20 100.0 20.0 Butterworth Order 2 2 2 2 BW Cutoff Freq. Hzl 11.0 11020.0 20.0 Drift Rate unit 0.05 1.0 1.5E-42.0E-3 Max Drift value [unit] 15.0 100.00.01 0.20 Drift Update Rate [ s 5.0 5.0 1.0 1.0 2.1.4 Dryden Turbulence Stochastic turbulence disturbances are required for accurate HWIL simulations, as eal world experiments are characterized by unpredictable winds acting on the vehicle The Dryden turbulence model is one of the accepted methods for including turbulence in aircraft simulations 28. By applying shaped noise with known spectral properties as velocity and angle rate perturbations to the body axes of the vehicle, the effect of turbulence is captured during discrete time simulations. The noise spectrum for each of the perturbations is predominantly described by a turbulence scale length parameter L e airspeed reference velocity, Vo, and the turbulence intensity,o The selection of these parameters allows for the turbulence to be modeled accordin to the prevailing wind conditions The spectral frequency content for generalized aircraft turbulence have been well studied (29, 28 and are given for each of the aircraft body axes r1+(u)2 Suq(w) L。1+3(÷)2 (211) +( n()=9L.1+3({) (1+(÷=u)2
Table 2.3: Crista IMU HWIL Sensor Noise Models Sensor PDynamic PStatic Gyro Accel. [unit] [Pa] [Pa] [deg/s] [m/s/s] Resolution [unit] Min [unit] Max [unit] Noise Gain Butterworth Order BW Cutoff Freq. [Hz] Drift Rate [unit/s] Max Drift value [unit] Drift Update Rate [s] 3.906 300 4000 20.0 2 11.0 0.05 15.0 5.0 2.00 0.0 110,000 20.0 2 11.0 1.0 100.0 5.0 1.6E4 6.0E3 5.20 100.0 5.20 100.0 0.10 0.0 2 2 20.0 20.0 1.5E4 2.0E3 0.01 0.20 1.0 1.0 2.1.4 Dryden Turbulence Stochastic turbulence disturbances are required for accurate HWIL simulations, as real world experiments are characterized by unpredictable winds acting on the vehicle. The Dryden turbulence model is one of the accepted methods for including turbulence in aircraft simulations [28]. By applying shaped noise with known spectral properties as velocity and angle rate perturbations to the body axes of the vehicle, the effect of turbulence is captured during discrete time simulations. The noise spectrum for each of the perturbations is predominantly described by a turbulence scale length parameter, L, the airspeed reference velocity, Vo, and the turbulence intensity, σ. The selection of these parameters allows for the turbulence to be modeled according to the prevailing wind conditions. The spectral frequency content for generalized aircraft turbulence have been well studied [29, 28] and are given for each of the aircraft body axes: Sug(ω)= 2σ2 uLu πVo · 1 1 + ( Lu Vo ω)2 (2.10) Svg(ω)= σ2 vLv πVo · 1 + 3( Lv Vo ω)2 � 1 + ( Lv Vo ω)2 �2 (2.11) Swg(ω)= σ2 wLw πVo · 1 + 3( Lw Vo ω)2 � 1 + ( Lw Vo ω)2 �2 (2.12) 37
0.8(m) Lm1+( S9() (214) 15 + where w is the spectral frequency of the turbulence and the aircraft wingspan, b, is used as a parameter in the angle rate filters to scale the effect of rotation on the main lifting surface. The subscripts u, U, w and p, g, r refer to the familiar body frame aircraft wind velocities and angle rates, respectively, thereby allowing independent classification of the turbulence in each axis. These spectral shaping functions are used to form shaping filters to give the body axis noise transfer functions 30 1 H 丌V1+-ms (216) H9(s) 1+√3{m (218) + H (1+(#) (220) Hrg(s) H (221) (#) The block diagram for the full 6 DOF Dryden turbulence model is shown in Figure 2-4 Note the cross axis couplings of the angle rate filters gg and ro Example turbulence perturbations values are plotted in Figure 2-5 as a function of the scale lengths and intensities for each of the body axes. The same 4 x l white noise input was used for each trial set. Larger scale lengths, L, increase the time constant of
S σ2 0.8 � πLw �1/3 pg(ω)= w 4b �2 (2.13) LwVo · 1+ � 4b ω πVo � � ω 2 Vo Sqg(ω)= �2 · Swg(ω) (2.14) 1+ � 4b ω πVo � ω �2 Vo Srg(ω)= �2 · Svg(ω) (2.15) 1+ � 3b ω πVo where ω is the spectral frequency of the turbulence and the aircraft wingspan, b, is used as a parameter in the angle rate filters to scale the effect of rotation on the main lifting surface. The subscripts u, v, w and p, q, r refer to the familiar body frame aircraft wind velocities and angle rates, respectively, thereby allowing independent classification of the turbulence in each axis. These spectral shaping functions are used to form shaping filters to give the body axis noise transfer functions [30] � Lu 1 Hug(s)= σu 2 1+ Lu (2.16) πVo · s 1+ √3 Lv Vo s � Lv Vo � � 1+ Lv 2 Hvg(s)= σv (2.17) πVo · s Vo s � Lw 1+ √3 Lw Vo � � 1+ Lw 2 Hwg(s)= σw (2.18) πVo · s Vo � �1/6 0.8 � π 4b (2.19) 1/3 Hpg(s)= σw Vo Lw � 1+ � 4b s �� πVo s Hqg(s)= � Hwg(s) (2.20) Vo 4b � · 1+ s πVo s Hrg(s)= � Hvg(s) (2.21) Vo 3b � · 1+ s πVo The block diagram for the full 6 DOF Dryden turbulence model is shown in Figure 24. Note the cross axis couplings of the angle rate filters qg and rg. Example turbulence perturbations values are plotted in Figure 25 as a function of the scale lengths and intensities for each of the body axes. The same 4×1 white noise input was used for each trial set. Larger scale lengths, L, increase the time constant of 38
Filters an velocities ngular Rates Hg→ Hos Figure 2-4: Block diagram for the 6 DOF Dryden Turbulence model. The velocity perturbations ug, Ug, wg are independent outputs of the filtered values of the turbulence scale lengths, L, intensity values g and the white noise input sources. The principle axis coupling of the aircraft is taken into account through the inputs to the angle rate perturbation filters. the turbulence seen for a given airspeed, while larger o values increase the deviation about zero. L and o typically vary with altitude in the lower atmosphere as ground effects become more prominent, but for HWIL simulations they are usually fixed The frequency response of the Dryden filters are shown for the same three cases in Figure 2-6. The filter cutoff frequency is determined by the ratio of the scale length to airspeed, and this effectively produces lower bandwidth filters for larger scale lengths The scale length parameter is chosen according to one of several specifications, all of which take into account the variation of L with altitude. The military reference MIL-F-8785C provides one such model of the scale length at low altitudes, h, which
Figure 24: Block diagram for the 6 DOF Dryden Turbulence model. The velocity perturbations ug, vg, wg are independent outputs of the filtered values of the turbulence scale lengths, L, intensity values, σ and the white noise input sources. The principle axis coupling of the aircraft is taken into account through the inputs to the angle rate perturbation filters. the turbulence seen for a given airspeed, while larger σ values increase the deviation about zero. L and σ typically vary with altitude in the lower atmosphere as ground effects become more prominent, but for HWIL simulations they are usually fixed. The frequency response of the Dryden filters are shown for the same three cases in Figure 26. The filter cutoff frequency is determined by the ratio of the scale length to airspeed, and this effectively produces lower bandwidth filters for larger scale lengths. The scale length parameter is chosen according to one of several specifications, all of which take into account the variation of L with altitude. The military reference MILF8785C provides one such model of the scale length at low altitudes, h, which 39