:4 2 5 6 7 8 9 Set 2 (a) Velocity perturbations 0.2 67 2 5 0 Time [s] (b) Angle rate perturbations Figure 2-5: The output of the Dryden velocity and angle rate filters for different selections of the intensity and scale lengths. Set 1 L=150.σ=0.5.Set2:L=1500 0.5.Set3:L=150
0.5 0 −0.5 Vu Vv 0 1 2 3 4 5 6 7 8 9 1 0 −1 0 1 2 3 4 5 6 7 8 9 −1 −0.5 0 0.5 Vu Set 1 Set 2 Set 3 Qr Qq Qp 0 1 2 3 4 5 6 7 8 9 Time [s] (a) Velocity perturbations 0.4 0.2 0 −0.2 −0.4 0 1 2 3 4 5 6 7 8 9 0.2 0 −0.2 0 1 2 3 4 5 6 7 8 9 0.1 0 −0.1 −0.2 0 1 2 3 4 5 6 7 8 9 Time [s] (b) Angle rate perturbations Figure 25: The output of the Dryden velocity and angle rate filters for different selections of the intensity and scale lengths. Set 1: L = 150,σ = 0.5, Set 2: L = 1500, σ = 0.5, Set 3: L = 150, σ = 1.5 40
is valid up to 1000 feet 29 h (0.177+0.000823h)1 (223) The turbulence intensity is a gain factor that scales the magnitude plots in Fig- ure 2-6 to values appropriate for different wind levels(.e, light, moderate, severe) The intensity level has been defined for low altitude fight according to MIL-F-8785C 0.1 (224) (0.177+0.000823b)04 (225) where w20 is the wind speed as measured at 20 it in altitude. According to MIL F-8785C, W20 15 knots is classified as "light"turbulence, W20 N 30 knots moderate, and W20>45 knots is "heavy". Other military specifications such as MIL-HDBK-1797 exist for the low altitude cases29), and different types of models are used for other regions of the atmosphere. For the purposes of the UAV application the low altitude models are sufficient The utility of the Dryden turbulence model is that it allows the expected turbu- lence levels to be described for an aircraft flying at a given reference speed for more realistic HWIL simulations. Turbulence is applied to the vehicle body axes consistent with the known parameterized values for scale length and intensity, which effectively defines the appropriate filters with cutoff frequencies and magnitudes needed for sim- ulation. Note that in addition to turbulence, wind is also usually modeled with a static component, w, that represents a prevailing magnitude and direction in an inertial axis. Together these define an arbitrary three-axis wind vector W=W+sl where dw is the effective Dryden wind turbulence in each axis after being rotated through the appropriate body to inertial transformation direction cosine matric For small scale aircraft, the static wind component is usually a gross disturbance elative to the aircraft airspeed, and it can have a large effect on high level planning 41
is valid up to 1000 feet [29]. Lw = h (2.22) h Lu = Lv = (2.23) (0.177 + 0.000823h)1.2 The turbulence intensity is a gain factor that scales the magnitude plots in Figure 26 to values appropriate for different wind levels (i.e., light, moderate, severe). The intensity level has been defined for low altitude flight according to MILF8785C as σ σw = 0.1W20 (2.24) w σu = σv = (2.25) (0.177 + 0.000823h)0.4 where W20 is the wind speed as measured at 20 ft in altitude. According to MILF8785C, W20 < 15 knots is classified as “light” turbulence, W20 ≈ 30 knots is “moderate”, and W20 > 45 knots is “heavy”. Other military specifications such as MILHDBK1797 exist for the low altitude cases [29], and different types of models are used for other regions of the atmosphere. For the purposes of the UAV application, the low altitude models are sufficient. The utility of the Dryden turbulence model is that it allows the expected turbulence levels to be described for an aircraft flying at a given reference speed for more realistic HWIL simulations. Turbulence is applied to the vehicle body axes consistent with the known parameterized values for scale length and intensity, which effectively defines the appropriate filters with cutoff frequencies and magnitudes needed for simulation. Note that in addition to turbulence, wind is also usually modeled with a static component, W, that represents a prevailing magnitude and direction in an inertial axis. Together these define an arbitrary threeaxis wind vector W = W + δWI (2.26) where δWI is the effective Dryden wind turbulence in each axis after being rotated through the appropriate body to inertial transformation direction cosine matrix. For small scale aircraft, the static wind component is usually a gross disturbance relative to the aircraft airspeed, and it can have a large effect on high level planning 41
Bode Diagram 0---:---如 Set 1 Set 2 Set 3 (a) Velocity filter for wg Bode Diagram (b)Angle rate filter for qg Figure 2-6: The Bode plots of the Dryden velocity and angle rate filters inputs to Hug(s) in(a)and Hag(s). Hugs in(b). Various selections of the intensity and scale lengths are shown in different sets. Set 1: L= 150, 0=0.5, Set 2: L= 1500 a=0.5,Set3:L=150,a=15
Bode Diagram 20 10 0 −10 −20 Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) −30 −40 −50 −600 −45 −90 Set 1 Set 2 Set 3 −4 −3 −2 −1 0 1 10 10 10 10 10 10 Frequency (rad/sec) (a) Velocity filter for wg Bode Diagram 0 −20 −40 −60 −80 −100 −120 −140 −160 −180 −200 0 −45 −90 −135 −180 −3 −2 −1 0 1 2 3 10 10 10 10 10 10 10 Frequency (rad/sec) (b) Angle rate filter for qg Figure 26: The Bode plots of the Dryden velocity and angle rate filters, given white noise inputs to Hvg(s) in (a) and Hqg(s) · Hwg(s) in (b). Various selections of the intensity and scale lengths are shown in different sets. Set 1: L = 150,σ = 0.5, Set 2: L = 1500, σ = 0.5, Set 3: L = 150, σ = 1.5 42
lgorithms. The effect of this type of disturbance on the planning system and aircraft dynamics is discussed in more detail in Chapters 4 and 5 2.2 Open Loop aircraft modeling 2.2.1 Longitudinal Dynamics A common model for the longitudinal motion of the aircraft is (27, 31] u mu nw ng ne 0010 A+ bu (228) where the state variables a Ju w q 0 refer to the longitudinal velocities, u and w, the pitch rate, g and the angle of inclination, 6. The elements of the A matrix in Eq 2.27 represent the concise form aerodynamic stability derivatives referring te the airplane body axis. Tables of values relating the concise form derivatives to the dimensionless or dimensional derivatives are available in numerous sources [27, 32 The control input u= de is the elevator defection angle with the engine thrust fixed and is input to the dynamics through the aerodynamic control derivative matrix B The Longitudinal Dynamics in Eq. 2.27 are typically resolved into two distinct phugoid and short period modes, which represent dynamics of the aircraft on different timescales. The short period is characterized by high frequency pitch rate oscillations and can have high or low damping, depending on the dynamic stability of the aircraft In contrast, the phugoid mode is characterized by lightly damped, low frequency oscillations in altitude and airspeed with pitch angle rates, q, remaining small Short period mode A simple approximation for the short period mode of the aircraft can be obtained by assuming the speed of the aircraft is constant over the timescale of the short perio dynamics(i =0), that the aircraft is initially in steady level fight and that the
algorithms. The effect of this type of disturbance on the planning system and aircraft dynamics is discussed in more detail in Chapters 4 and 5. 2.2 Open Loop Aircraft Modeling 2.2.1 Longitudinal Dynamics A common model for the longitudinal motion of the aircraft is [27, 31] ⎡ ⎢ ⎢ ⎡ ⎢ ⎢ ⎤ ⎥ ⎥ ⎡ ⎢ ⎢ ⎤ ⎥ ⎥ ⎡ ⎢ ⎢ ⎤ ⎥ ⎥ ⎤ ⎥ ⎥ u˙ x x x x u w q θ u xδe ⎢ w˙ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ ⎥ z z z z u w q θ w zδe = + δe (2.27) q˙ θ ˙ m m m m q θ u w q θ mδe ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ 0 0 1 0 0 x˙ = Ax + Bu (2.28) where the state variables x = [uwqθ] T refer to the longitudinal velocities, u and w, the pitch rate, q and the angle of inclination, θ. The elements of the A matrix in Eq. 2.27 represent the concise form aerodynamic stability derivatives referring to the airplane body axis. Tables of values relating the concise form derivatives to the dimensionless or dimensional derivatives are available in numerous sources [27, 32]. The control input u = δe is the elevator defection angle with the engine thrust fixed and is input to the dynamics through the aerodynamic control derivative matrix B. The Longitudinal Dynamics in Eq. 2.27 are typically resolved into two distinct phugoid and short period modes, which represent dynamics of the aircraft on different timescales. The short period is characterized by high frequency pitch rate oscillations and can have high or low damping, depending on the dynamic stability of the aircraft. In contrast, the phugoid mode is characterized by lightly damped, low frequency oscillations in altitude and airspeed with pitch angle rates, q, remaining small. Short Period Mode A simple approximation for the short period mode of the aircraft can be obtained by assuming the speed of the aircraft is constant over the timescale of the short period dynamics ( ˙u = 0), that the aircraft is initially in steady level flight and that the 43
derivatives refer to a wind-axis system(0=a=0). The equations of motion then reduce te (229) Following further approximations shown in 27 which make assumptions about the relative size of the mg, zg and zu derivatives, the transfer functions for the two short term equations describing the response to elevator are in(s+voz △kn(s+1/Ta) 6(s)(s2-(m+xm)s+(mq2-mn)s2+25s4s+u2 (230) qs △kq(s Se(s)(s2-(mg iu)s+(mgtw-muuVo)) 32+25. (2.31) where kq, ku, Te, Ta, Ss, and ws represent approximate values for the short period mode and Vo is the vehicle reference speed One of the most accurate ways to obtain models for the aircraft data is to use actual fight data. Identification algorithms such as those in the Matlab System Iden tification Toolbox 33 can be used to used to obtain open loop models of the system from fight data collected d dels can then be used to validate the hWIl simulation environment as well as to help determine the gain settings for the autopilot control loops as shown in Section 2.3. Input-output data was collected by disengaging all of the autopilot loops and performing a series of maneuvers to measure the aircraft response to deflections from the elevator and aileron control surfaces. Example data from two experiments are shown in Figures 2-7(a)-(b)depicting the longitudinal and lateral modes, respectively To capture the longitudinal dynamics, the bank angle was held fixed at zero degrees, while a series of pitch oscillations were commanded using the elevator. Figure 2-7(a)shows a sample of data that was collected on one run of the pitch test. From the plot it is clear that the longitudinal modes are being excited due to input from the elevator, while the lateral motions in the roll and yaw axes are essentially fixed Sample data from a roll excitation run is plotted in Figure 2-7(b ). This plot also early shows coupling in the yaw axis due to the dihedral angle of the wing
derivatives refer to a windaxis system (θ = α = 0). The equations of motion then reduce to ⎡ w˙ ⎤ ⎡ zw zq ⎤ ⎡ w ⎤ ⎡ zδe ⎤ ⎣ ⎦ = ⎣ ⎦ ⎣ ⎦ + ⎣ ⎦ δe (2.29) q˙ mw mq q mδe Following further approximations shown in [27] which make assumptions about the relative size of the mq, zq and zw derivatives, the transfer functions for the two short term equations describing the response to elevator are: w(s) zn � s + Vo mδe � � kw(s + 1/Tα) (2.30) zδe = δe(s) (s2 − (mq + zw)s + (mqzw − mwVo)) s2 + 2ζsωss + ωs 2 q(s) mn(s − zw) � kq(s + 1/Tθ) = (2.31) δe(s) (s2 − (mq + zw)s + (mqzw − mwVo)) s2 + 2ζsωss + ωs 2 where kq, kw, Tθ, Tα, ζs, and ωs represent approximate values for the short period mode and Vo is the vehicle reference speed. One of the most accurate ways to obtain models for the aircraft data is to use actual flight data. Identification algorithms such as those in the Matlab System Identification Toolbox [33] can be used to used to obtain open loop models of the system dynamics from flight data collected during experiments. These models can then be used to validate the HWIL simulation environment as well as to help determine the gain settings for the autopilot control loops as shown in Section 2.3. Inputoutput data was collected by disengaging all of the autopilot loops and performing a series of maneuvers to measure the aircraft response to deflections from the elevator and aileron control surfaces. Example data from two experiments are shown in Figures 27(a)(b) depicting the longitudinal and lateral modes, respectively. To capture the longitudinal dynamics, the bank angle was held fixed at zero degrees, while a series of pitch oscillations were commanded using the elevator. Figure 27(a) shows a sample of data that was collected on one run of the pitch test. From the plot it is clear that the longitudinal modes are being excited due to input from the elevator, while the lateral motions in the roll and yaw axes are essentially fixed. Sample data from a roll excitation run is plotted in Figure 27(b). This plot also clearly shows coupling in the yaw axis due to the dihedral angle of the wing. 44