Chapter Dynamic Modeling and Control of the Miniature UAV Helicopter In this chapter, we present the work on the dynamic modeling and control law design for our miniature size UAV helicopter. The helicopter has inherently unstable, complicated, and nonlinear dynamic, under the significant influence of exogenous disturbances and parameter perturbations. The system has to be stabilized using a feedback controller. To design a reliable automatic control law, an accurate model which could accurately capture the inputoutput dynamics is necessary. Due to the complexity of the helicopter dynamics, there have been efforts to apply nonmodel-based approaches such as PID (proportional-integral-derivative) tuning, fuzzy-logic control, neural network control, or a combination of these, etc. [53] While these approaches are attractive because no identification is required, they not guarantee closed-loop stability while they are being tuned or learnt. The stabilizing controller may also be designed by the model-based mathematical approach. The mathematical model-based approach assumes the availability of a linear or nonlinear system model for the controller design. In this case, the system identification process takes up a significant amount out of the whole research effort of building a UAV helicopter. 73 CHAPTER 4. 74 Fortunately during the construction of HeLion UAV helicopters, we have obtained a lot of experience, which enables us to conquer various problems in modeling nd control of the miniature UAV helicopter, BabyLion. In order to increase the fidelity and application scope of the developed model, we have sequentially implemented a variety of model derivation methods, including: time-domain system identification and frequency-domain system identification [11]. These two methods are for linear model identification. By conducting and comparing the different methods, we have successfully obtained a high-fidelity linear model based on the helicopter dynamics for the purpose of control law design and implementation. The remaining content of this chapter is organized as follows. Section 4.1 presents the data collection and preprocessing for the modeling and Section 4.2 introduces the model structure for the helicopter. Section 4.3 focus on first principle modeling technique based on the MIMO model. Section 4.4 presents the complete procedure for deriving an implementable model which is suitable for small UAV helicopters and with minimum complexity based on classical modeling method. The implementation results and necessary comparisons between the two methods are also addressed in Section 4.5.3. In Section 4.5, we draw some concluding remarks. 4.1 Data Collection and Preprocessing Data collection is a state-of-the-art process and with great importance. Without high-quality data, it could be never possible to identify the dynamic models accurately and reliably. To ensure the quality of collected flight test data, we sequentially carry out the following three sub-steps, including, (1) select the input signals, (2) collect flight test data via suitably conducted experiments, and (3) preprocess the raw data. 4.1.1 Select the Input Signals Frequency sweep and symmetric doublets are selected as the input signals for the purposes of model identification and model validation, respectively. CHAPTER 4. 75 Frequency sweep is a class of control inputs that has a quasi-sinusoidal shape of increasing frequency [56]. One typical frequency-sweep input signal is described by (4.1) and visually shown in Fig. 4.1. To persistently excite the linear dynamics which is dominant within the desired frequency range for the small-scale UAV helicopter, the issued frequencysweep signal needs to meet a series of requirements. 1. Initial frequency: The initial frequency is required to be as low as possible. Considering the pilot’s maneuverability, we choose an initial frequency as 0.2 Hz (e.g., 5s long period); 2. Highest frequency: The highest frequency for the frequency sweep input is also limited to avoid introducing unnecessary nonlinearities and structural vibrations. For the miniature scale UAV helicopter, the upper limitation is set as Hz; 3. Frequency-increasing progression: To ensure the low frequency range (0.2 ∼ 0.5 Hz) is fully excited, the pilot will issue two concatenated long-period ( ∼ sec) input at the beginning of each perturbation. After that, the input frequency is required to increase smoothly to the highest threshold value (2 Hz); 4. Amplitude: It is not necessary to keep the amplitude constant during the whole frequency sweep perturbation period. Typically the adjusting range is ±10 − 20%. However, the aerodynamics should be guaranteed linear. u(t) = asin(π(f0 + f (t))t) (4.1) where t is the time, u(t) is the generated frequency sweep signal with respect to time t, a is the adjustable amplitude, f0 and f (t) is the initial and time-increasing frequencies in Hz. Symmetric doublet, which is illustrated in Fig. 4.2, is an easy-to-issue input signal which could provide a clear visualization of key dynamic characteristics and model perfor- CHAPTER 4. 76 Figure 4.1: Typical frequency sweep input signal. mance [56]. Such signal is commonly used for model validation on both full- and small-scale helicopters [55, 43, 17]. 4.1.2 Collect Flight Test Data The overall data collection procedure involves two tasks, namely data collection for model identification, and data collection for model validation. The collection of flight data has been conducted under human control of the helicopter. Special flight maneuvers were carried out while the avionics system was running and logging the flight data which consists of all the helicopter outputs and the measured RC inputs. Regarding the input signal, we choose the frequency sweep signal, which has a quasi-sinusoidal shape of increasing frequency in the interested range, for the model identification and a doublet signal for model validation since they have been commonly used in numerous full-scale and small-scale UAV helicopter system identification [10]. CHAPTER 4. 77 Figure 4.2: Typical doublet input signal. The procedure of data collection is as follows. Firstly the helicopter is brought to a hover and input trim values are set for each of the control input to keep it at a desired operating point. When such a flight condition is achieved, the pilot is required to issue the frequency sweep for each of the four input channels which include aileron, elevator, collective pitch and rudder, while keeping the basic hovering flight condition. The resultant input and output data are then recorded for the purpose of identification. This experiment was also iterated several times until enough high quality flight data was obtained for the model validation. A visual illustration of one set of inputs and outputs collected in a rudder channel perturbation experiment is shown in Figures 4.3 to 4.7. Data collection for model validation is conducted at the end of each flight test. After achieving certain trimmed flight status, the symmetric doublet input signal is injected to each of the four input channels. Since the doublet signal is much easier to issue compared with frequency sweeps, only two to three set of experiment results are recorded per flight test. CHAPTER 4. 78 (−1~1) 0.1 0.05 δ lat −0.05 555 560 565 570 575 555 560 565 570 575 555 560 565 570 575 555 560 565 Time (s) 570 575 (−1~1) 0.05 δ lon −0.05 (−1~1) −0.14 −0.18 δ col −0.16 δped (−1~1) −0.2 0.2 −0.2 Figure 4.3: Input signals in the yaw channel perturbation experiment. x (m) 6.5 5.5 555 560 565 570 575 555 560 565 570 575 555 560 565 Time (s) 570 575 10.5 y (m) 10 9.5 z (m) 21 20 19 18 Figure 4.4: Position outputs in the yaw channel perturbation experiment. CHAPTER 4. 79 u (m/s) 0.5 −0.5 555 560 565 570 575 555 560 565 570 575 555 560 565 Time (s) 570 575 v (m/s) 0.5 −0.5 −1 w (m/s) 0.2 −0.2 −0.4 −0.6 Figure 4.5: Velocity outputs in the yaw channel perturbation experiment. p (rad/s) 0.2 −0.2 −0.4 555 560 565 570 575 555 560 565 570 575 555 560 565 Time (s) 570 575 0.4 q (rad/s) 0.2 −0.2 −0.4 r (rad/s) −2 Figure 4.6: Angular rates in the yaw channel perturbation experiment. CHAPTER 4. 80 Φ (rad) 0.1 0.05 −0.05 555 560 565 570 575 555 560 565 570 575 555 560 565 Time (s) 570 575 0.1 Θ (rad) 0.05 −0.05 Ψ (degree) 160 140 120 100 80 60 Figure 4.7: Euler angles in the yaw channel perturbation experiment. 4.1.3 Preprocessing of the Raw Dataset The raw dataset is required to be preprocessed such that the side effects caused by the nonzero trimming values, the piloted feedback input, and noises or disturbances are minimized. For the time-domain dataset, we can only conduct the basic preprocessing, which includes: 1. Data range selection: From the continuously recorded raw dataset, we need to pick out the meaningful slots related to frequency sweep and doublet perturbation. This is easy to realize since input shapes of frequency sweep and doublet are characteristic. 2. Data detrending: Conducting data detrending is necessary due to the following two reasons. Firstly, the aerodynamics is perturbed using frequency sweeps based on the trim flight conditions. as listed in Table 4.1. These trim values for both inputs and outputs are regarded as CHAPTER 4. 81 Table 4.1: Trim values for the tested flight conditions States u0 (m/s) v0 (m/s) w0 (m/s) p0 (rad/s) q0 (rad/s) r0 (rad/s) φ0 (rad) θ0 (rad) δlat0 (-1∼1) δlon0 (-1∼1) δcol0 (-1∼1) δped0 (-1∼1) Hovering 0.0330 -0.0350 0.0840 -0.0004 -0.0002 -0.0008 0.0630 -0.005 -0.0070 0.0190 -0.2879 0.0030 Forward m/s 1.0500 -0.2700 0.3720 -0.0031 0.0021 0.0005 0.0450 -0.1250 0.0025 0.0678 -0.2762 0.0065 Backward m/s -0.9367 -0.0875 0.1200 0.0021 0.0034 -0.0009 0.0710 -0.0844 -0.0121 0.0221 -0.2810 0.0005 Heave 1m/s 0.1619 -0.2006 -1.1220 0.0101 0.0020 0.0045 0.0880 -0.0204 0.0367 0.0227 -0.3275 0.0007 Side slip 1m/s 0.1976 1.3044 -0.3056 -0.0093 -0.0003 -0.0164 0.0962 0.0087 0.0127 0.0310 -0.2234 -0.0037 constant biases and need to be detrended prior to model identification. Secondly, the data collection is piloted-issued with unavoidable trimming or trending shift, which should be corrected such that the fidelity of the identification result is not affected. 3. Data filtering: The high frequency vibrations caused by the engine, main rotor and tail rotor may corrupt the data quality and thus should be further attenuated. For BabyLion, we apply a first order low-pass filter with the cut-off frequency with -3 dB at 10 Hz to both input and output channels. Note that using the identical filters for both inputs and outputs is compulsory to avoid extra time-delay caused by the filtering procedure. 4.2 Helicopter Aerodynamics Model Structure The model structure specifies the order and form of the differential equations which describe the dynamics. The model used in this project is adapted directly from the UAV group in CHAPTER 4. 82 the National University of Singapore Control and Simulation Laboratory [9]. This model structure was established through the analysis of physical effects using the first principles. Specifically the model structure consists of three parts, namely, the 6-DOF rigid-body dynamics, coupled rotor flapping dynamics, and yaw rate gyro dynamics 4.2.1 Degree of Freedom (DOF) Rigid-body Dynamics The 6-DOF rigid-body dynamics is described using Newton-Euler equation 4.2. Note that the tip-path-plane (TPP) flapping angles as and bs are involved due to the strong coupling between the main rotor and the fuselage of the small-scale UAV helicopters. Therefore,  u˙       v˙         w˙   p˙        q˙      r˙ = (−w0 q + v0 r) + Xu u − gθ + Xas as = (−u0 r + w0 p) + Yv v + gφ + Ybs bs = (−v0 p + u0 q) + Zw w + Zr r + Zcol δcol = Lu u + Lv v + Las as + Lbs bs = Mu u + Mv v + Mas as + Mbs bs = Nv v + Nr r + Nrf b rf b + Nped δped (4.2) where X(), Y() , Z() , L(), M() and N() are the lumped unknown dynamic derivatives which need to be identified, u0 , v0 , and w0 are the trimmed speed values of any designated flight condition listed in Table 4.1. In the near hovering flight, these values can be set all zeros and g is the gravity. 4.2.2 Coupled Rotor Flapping Dynamics Recalling the “hybrid model” structure developed in [7, 30] for the full-scale helicopters, the main rotor dynamics is described by two coupled first-order equations which represent the lateral and longitudinal flapping motions of the assumed tip-path-plane. Physically, the tippath-plane is the top of the corn-shaped rotor, defined by the first-harmonic representation of the main blade flapping motion [43]. An illustration of the TPP, which can also be found CHAPTER 5. 150 Figure 5.18: Flight result, (a) Horizontal plane. Red – virtual leader’s trajectory; Blue – follower trajectory; Cyan – follower reference. Figure 5.19: Flight result, (b) Longitude and Latitude. Dashed line – follower trajectory; Solid line – follower reference. CHAPTER 5. 151 Leader−Follower Two UAV Formation Flight X−Y Plot follower reference leader 10 lattitude: m −5 −10 −5 10 15 longitude: m 20 25 Figure 5.20: XY plot of flight data for leader-follower formation flight. members are set at: lc = m, fc = 0m, and hc = m. This implies that the follower UAV tracks the VL data from meters behind, meters to the right and meters above. As shown in Fig. 5.20 and 5.21, the tracking errors are between -1m to 2m exclusive a predefined lateral clearance of m. Videos of the two UAVs in formation flight can be seen at our website: http://vlab.ee.nus.edu.sg/ uav/videos.htm. The results clearly show that the UAV helicopter with the formation flight control law has successfully completed the preset formation flight missions. 5.5 conclusion In this chapter, we presented a formation dynamic model for the follower UAV in a leaderfollower formation flight scheme. In the formation control algorithm, the leader UAV periodically broadcasts its position and velocity to the other UAVs to maintain a fixed distance from the leader while following a prescribed mission trajectory. A special leader-follower experiment has been completed successfully with good performance. Moreover, we also developed a formation relative theory, such as formation reconfiguration, collision avoidance, CHAPTER 5. 152 lateral clearance: m 8.5 7.5 6.5 120 140 160 180 200 time: s 220 240 260 280 Figure 5.21: Tracking separation between the leader and follower UAVs etc Chapter Conclusions This dissertation presented the development of an autonomous low-cost miniature UAV helicopter platform and two UAVs were used to implement a leader-follower formation flight. Firstly, we built a reliable hardware system and configured the software system. Specifically, the UAV avionics is capable of communicating with the ground station, sending sensor information and collecting commands. Through the servo controller on IMU, it is also capable of controlling the servos on helicopter. Next, we calibrated sensor as well as developed attitude determination and GPS enhancement algorithm. This is because of the coarse information provided by the low-cost IMU and lack of attitude reference, and poor GPS resolution. We also obtained the high-fidelity robust dynamic model, which is the starting point for control law design based on the experimental data and theoretical analysis. We designed a suitable flight control law to realize automatic control and integrate the control algorithm in to the on-board embedded process. In real flight experiments, the UAV is able to control the helicopter for various autonomous flying trajectories. We also developed the leader follower formation flight theory and related research topics such as multiple UAV reconfiguration and collision avoidance, etc We constructed another identical miniature UAV helicopter using the same control law as the first UAV. Finally, both UAVs were used to realize a leader-follower formation flight. Subsequent real formation flight experiments 153 CHAPTER 6. 154 were completed successfully with good performance. 6.1 Contributions The specific contributions of this thesis to these topics are described as follows. Firstly, we have successfully constructed a reliable low-cost miniature UAV platform for research. This includes hardware integration and software modification. UAV platform is of high efficiency of the weight less than kg , and cost about 1/10 of the larger size UAV. The helicopter is a RC model easily available in the market. We choose a small size embedded computer and compact IMU. The UAV is capable of full functions as the other larger size UAVs. The onboard computer can collect sensor data from IMU, can communicate with the ground station via wireless communication, which including sending sensor information to the ground station and receive command or control parameter from the ground station, etc It can implement flight control law, drive the actuators on the helicopter. It also has a function of mode switching, which enables the human pilot to switch between manual and auto mode when needed. We have considered the CG problem and designed a suitable supporting structure to protect the on-board system. Secondly, we have studied the filtering algorithm. We also proposed a robust H∞ filtering algorithm for attitude determination and the GPS navigation enhancement. As the low-cost IMU doesn’t provide the attitude reference, we have to develop our own algorithms for attitude reference. Compared with classical Extended Kalman Filtering, the H∞ filtering algorithm has a wide application capability when the disturbance and measurement noise are not limited to be white. Test and real flight implementation results show that the algorithm is better than the EKF filtering algorithm. Thirdly, the modeling and control design work for our miniature UAV helicopters have been conducted comprehensively in Chapter 4. Classical modeling and control has a higher requirement for the sensor accuracy, which may not be applicable for our low-cost IMU CHAPTER 6. 155 sensor. As such, most research groups are keen on the Raptor 90 size (with 1.6m rotor span, 5kg weight), or larger scale UAV helicopter platforms with a higher resolution IMU or peripheral components. We propose a robust cascaded modeling and control method, in which the complicated modeling problem is reduced to 3-axes attitude loop identification. The system model to be identified is reduced to sixth orders from eleventh orders system. The velocity and position controls are in the outer loop, which is natural and good for the safety reason such as the GPS information loss, etc Our numerous flight experiments have proven that this method is robust and reliable with good performance. We also successfully controlled a second UAV based on this method. Finally, we developed a general leader-follower formation flight scheme, which is also suitable for fix-wing UAV or UGV. We constructed a second UAV based on the abovementioned platform. With these two UAVs, we successfully implemented a leader-follower formation flight experiment. We also proposed a general collision avoidance method for multiple-UAV teams. Obstacle avoidance method is also included. To sum up, we have successfully carried out the comprehensive study on the low-cost miniature UAV helicopters and studied the formation flight control problem based on our self-instrumented helicopter platform. The overall research procedure and achievements have been sequentially documented in Chapters to 5. 6.2 Future Work Although we have carried out comprehensive study on the development, control, and formation flight of the low-cost miniature UAV helicopters, these are only the beginning of the research in this area. Considering the requirements on various practical implementations, it should be meaningful to extend the miniature size UAV research in the following directions. CHAPTER 6. 156 Urban Area Surveillance, Search and Rescue Urban area is generally with limited space, complicated environment, and various uncertainties. As such, miniature size UAV helicopter is the most suitable platform for practical implementation like surveillance, search and rescue . The main challenge is rescue where the UAV can be used to help identify an area map or current situation in the area, where it is unsafe for human to intervene without any information or complete information about that area. The UAV can help locate survivors and detect structural damage in hard to reach areas. This is very useful for any rescue organization all around the world to minimize search and analysis time thus enabling the operation to run smoothly without any errors. To facilitate the implementation in this aspect, we need to combine the current miniature size UAV helicopters with research in other areas such as visual-based navigation and obstacle avoidance. Indoor Implementation In an indoor environment, the weak reception of GPS signals and limited space which restricts the avoidance options have posed additional challenges to the operations of the UAV. Moreover, due to size constraints, payload capabilities are also diminished and the UAV can only carry minimal amounts of payload in order to achieve the mission objectives. Indoor implementation for miniature size UAV helicopters is of great potential, but still at the very initial stage. Currently, the whole NUS UAV research team is undergoing challenging development of suitable UAV helicopter platforms and practical navigation methodology for future research and implementation. Smart UAV Swarming Although a single UAV is now capable of tracking complex trajectory or even large attitude manuvering, and we have proven that two UAVs can fly in a simple formation flight. It is still CHAPTER 6. 157 hard to conclude that the UAV control technique nowadays is smart. A smart UAV should be able to execute a task in unknown complicated environment, which increases the difficulty of the control problem. When multiple smart UAVs execute one task cooperatively, it means enabling groups of vehicles to sense and respond automatically, and more problems may emerge. The swarming technologies offer a new way to approach mission design, mission risk and mission requirements. The UAVs would communicate relevant information and reconfigure themselves, autonomously changing direction in response to sensor input to complete the mission at hand. For example, if 100 UAVs collect sensor inputs over a field of operation and five of them have engine failure or are shot out of the sky, the remaining ninety-five UAVs would reconfigure themselves to collect the required data and complete the mission. As such, swarming of multiple smart UAVs will definitely be a interesting topic in next stage. Currently, our NUS research group is exploring the potential in this area via combining the achievements on single small-scale UAV helicopters and hybrid control algorithms. Appendix A Publication/Submitted Paper List A. Journal Papers 1. B. Yun, G. Cai, A. Cai, B. M. Chen, “A robust cascaded controller law for micro-size low cost UAV helicopter,” IEEE Transactions on Industry Applications, 2010, submitted 2. B. Yun, B. M. Chen, K. Y. Lum, T. H. Lee, “Design and implement a leader-follower formation flight control scheme for UAV helicopters.” Journal of Control Theory Application 8, (1), pp. 61-68, 2010. 3. B. Yun, G. Cai, B. M. Chen, K. Peng and K. Y. Lum, “GPS signal enhancement and attitude determination for a mini and low-cost unmanned aerial vehicle,” Transactions of the Institute of Measurement and Control, in press. B. Conference Papers 4. B. Yun, B. M. Chen, K. Y. Lum, T. H. Lee “A leader-follower formation flight control scheme for UAV helicopters,” Proceedings of the IEEE International Conference on Automation and Logistics, Qingdao, China, September , pp 39-44, 2008. 158 APPENDIX A. PUBLICATION/SUBMITTED PAPER LIST 159 5. B. Yun, K. Peng, B. M. 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[...]... range of (−0.3 ∼ 0.3) Such results indicate that our designed controller is applicable for the BabyLion UAV s practical automatic flight control Next, we implement the designed control law on our BabyLion UAV helicopter to realize automatic hovering Specifically, the pilot took off the BabyLion UAV helicopter manually and made it hover at a designated point with suitable altitude and good sight After achieving... + Bas as + Blat δlat + Blon δlon (4.3) where Abs and Bas are coupled rotor flapping derivatives, ()lat and ()lon are control derivatives of aileron and elevator servos, respectively, and τ is the effective rotor time constant 4 .2. 3 Yaw Rate Gyro Dynamics The recently developed RC helicopters are commonly equipped with a yaw rate gyro to facilitate pilot to control the yaw rate and heading angle In the... low- cost small-size IMU, we need further pre-process the available data in su-step 1 The angular velocity data is taken as measurement information We introduce a low- pass filter to pre-process these measurement data Meanwhile the servo input data-taken as control input, also pass the same low- pass filter to reduce the delay ef- CHAPTER 4 88 Table 4.3: Estimated parameters through system identification,... (-1∼1) Physical Meaning Body frame X-axis velocity Body frame Y-axis velocity Roll angular rate (rad/s) Pitch angular rate Roll angle Pitch angle Lateral tip-path plane flapping angle Longitudinal tip-path plane flapping angle Body frame Z-axis velocity Yaw angular rate Yaw rate feedback Aileron servo input Ellevator servo input Collective pitch servo input Rudder servo input off-axis or correlated inputs... core loop of the autopilot control system In the attitude control loop, horizontal plane control is most important for hovering and near hovering mode flight In our system model, only the two horizontal control channels (longitudinal channel and lateral channel) are coupled As such, we can design control laws separately For the CHAPTER 4 1 02 horizontal plane control channels, the control law is designed... control law design which will presented in the next section CHAPTER 4 100 2 p simulation p: rad/s 1 0 −1 2 120 130 140 150 160 170 180 190 20 0 21 0 22 0 130 140 150 160 170 180 190 20 0 21 0 22 0 130 140 150 160 170 time: s 180 190 20 0 21 0 22 0 2 q: rad/s 1 0 −1 2 120 4 r: rad/s 2 0 2 −4 −6 120 a Angular rate and simulation 10 φ simulation φ: deg 5 0 −5 −10 −15 120 130 140 150 160 170 180 190 20 0 21 0 130... Hovering Flight Test The cascaded control method can easily adjust the control parameter, thus the dependence on rate gyro is greatly relieved The flight experiment can be processed in three stages In the first stage, the focus is on the attitude control of the UAV Attitude control parameters CHAPTER 4 108 are adjusted according to the flight performance, and a good flight performance means that the UAV can... BabyLion UAV helicopter to realize automatic hovering Similarly, the pilot took off the BabyLion UAV helicopter manually and made it hover at a designated point with suitable altitude and good line of sight After achieving manual hovering, the pilot switched the BabyLion to automatic hovering by sending a switching signal through the wireless RC link The actual responses are shown in Figures 4.17 (a) ... replace the GPS signal when a GPS loss or failure occur 4.4.1 Cascaded Modeling Lateral and longitudinal fuselage dynamics We noted that PID controller also works (see e.g [6]), which means physically that the attitude channels (roll, pitch and yaw) of the helicopter from the control surface deflection to the attitude angular rate is first-order by ignoring the actuator dynamics This is true in our manual... secondary rotor with much smaller aerodynamic surface and 90o phase difference to effectively dampen the inputs of aileron and elevator servos (δlat and δlon ) Based on [19, 43, 44], the stabilizer bar dynamics can be lumped into the bare main rotor dynamics The final first-order coupled rotor flapping dynamics is represented in 4.3 as ˙ = −q − as τ + Abs bs + Alatδlat + Alon δlon ˙ bs = −p − b1 τ + Bas as . (rad/s) q (rad/s) Pitch angular rate φ (rad) Roll angle θ (rad) Pitch angle a s (rad) Lateral tip-path plane flapping angle b s (rad) Longitudinal tip-path plane flapping angle w (m/s) Body frame Z-axis. by the model-based mathematical ap- proach. The mathematical model-based approach assumes the availability of a linear or nonlinear system model for the controller design. In this case, the system. B a s a s + B lat δ lat + B lon δ lon (4.3) where A b s and B a s are coupled rotor flapping derivatives, () lat and () lon are control deriva- tives of aileron and elevator servos, respectively, and τ