Unsteady Aerodynamic Parameter Estimation for Multirotor Helicopters*

Trans. Japan Soc. Aero. Space Sci. Vol. 62, No. 1, pp. 32–40, 2019 DOI: 10.2322/tjsass.62.32

Hung Duc NGUYEN,† Yu LIU, and Koichi MORI

Department of Aerospace Engineering, Nagoya University, Nagoya, Aichi 464–8603, Japan

Today, multirotor helicopters (MRHs) play an important role in a broad range of applications such as transportation, observation and construction, and the safety of MRH flight is a matter of great concern. This study contributes to clarifying an aerodynamic aspect that enables the prediction of MRH behavior in unsteady conditions through flight tests. In working towards a comprehensive mathematical model that determines unsteady aerodynamics, a quadcopter is equipped with a data acquisition system to gather flight data including acceleration, angular rates, flow angles, airspeed and rotational speed. Based on the data collected, the combined blade element momentum theory is utilized to calculate steady and unsteady aerodynamic parameters. It is found that the experimental aerodynamic coefficients agree well with the theoretical results for steady forward flight. However, the conventional theory was insufficient to model the aerodynamic parameters under unsteady conditions. A new model to predict aerodynamic parameters under unsteady flight is proposed and validated on the basis of the flight data.

Key Words: Multirotor Helicopter, Unsteady Aerodynamics, Flight Experiment, Load Factor

Nomenclature p: bias of roll rate q: bias of pitch rate a: 2-D lift curve slope r: bias of yaw rate ax: acceleration along xb axis x: bias of acceleration ax ay: acceleration along yb axis y: bias of acceleration ay az: acceleration along zb axis z: bias of acceleration az A: rotor disk area : advance ratio B: tip loss factor : air density c: blade chord : solidity CD: drag coefficient : roll angle CH: horizontal force coefficient : pitch angle CT: thrust coefficient avg: rotor average pitch angle h: altitude : yaw angle H: horizontal force !: rotational speed n: load factor Subscripts N: number of blades i: inertial frame p: roll rate exp: experiment q: pitch rate r: yaw rate 1. Introduction rl: fraction of blade span from axis (¼ l=R) R: rotor radius Today, multirotor helicopters (MRHs) play an important T: thrust role in a broad range of applications such as transportation, 1,2) u: translational velocity along xb axis observation and construction. For the safety of MRH v: translational velocity along yb axis flight, unsteady the aerodynamic response of MRHs to vi: induced velocity abrupt steering and/or wind gusts is a matter of great con- V: total velocity cern. V1: free-stream velocity In literature, a few studies have been devoted to MRH 3–5) w: translational velocity along zb axis aerodynamics. The interference of the front rotor on oper- : angle of attack ation of the rear rotor due to the wake generated by the front : sideslip angle rotor was examined in Hung et al.3) The effect of rotor blade : lock number flapping on attitude control was explored by Hoffmann et :inflow ratio al.4) A comparison of fixed and variable pitch actuators was presented through a series of experiments conducted © 2019 The Japan Society for Aeronautical and Space Sciences 5) + by Cutler et al. However, the unsteady aerodynamic charac- Received 18 September 2017; final revision received 16 June 2018; accepted for publication 17 July 2018. teristics of MRHs are still not completely understood. †Corresponding author, [email protected] Achieving a precise model of external forces acting on

32 Trans. Japan Soc. Aero. Space Sci., Vol. 62, No. 1, 2019

MRHs under maneuverable flight conditions leads to the re- element theory.8) In the momentum theory, the flow is as- quirement that considerable flight experiments need to be sumed to be incompressible and inviscid, and blade-loading carried out. In our previous study,3) a wind-tunnel was used is assumed to be distributed uniformly over the blade. The to investigate the steady aerodynamics of a quadrotor. How- blade element theory is based on the lifting-line assumption, ever, that was not sufficient. Unsteady conditions with time- neglecting stall.8) The advance ratio ®,inflow ratio , and varying parameters occur frequently in actual flight due to solidity · are defined as pilot operation and/or wind gusts. The load factor, n, has V1 cos been used conventionally to characterize the unsteady aero- ¼ ð1Þ !R dynamic load in structural design and maneuverable capabil- 6) ity of conventional helicopters. For safety when designing V1 sin þ vi ¼ ð2Þ the airframe and automatic control, it is indispensable to !R study the unsteady aerodynamic response of MRHs. Nc For this study, outdoor flight experiments were conducted ¼ : ð3Þ R to measure the aerodynamic forces and incoming flow vector using on-board sensors and a data acquisition system so that In the conventional theory, the thrust coefficient of a rotor unsteady aerodynamics could be examined. For investigating in forward flight with a linearly twisted pitch angle is 8) the aerodynamic characteristics of a quadrotor helicopter in ¼ 0 þ twrl,

fl actual ight, this study addresses two primary interests: 1 1 1 1 fl fl C ¼ a B3 þ B4 þ B2 Steady forward ight and transition from forward ight to T 2 3 0 4 tw 2 0 ffi fl descent. Moreover, the thrust coe cient in transient ight ! ð4Þ is formulated as a function of the load factor, and is validated 1 1 þ B22 B2 based on the experimental data collected. The methodology 4 tw 2 proposed will have a strong impact on the future studies of flight dynamics, flight simulation and adaptive control sys- or, constant twist, ¼ avg ! tem design of MRHs. 1 1 1 1 C ¼ a B3 þ B2 B2 ð5Þ T 2 3 avg 2 avg 2 2. Mathematical Model and thrust, 2.1. Coordinate system T ¼ C !RðÞ24R2: ð6Þ It is necessary to define the different coordinate systems T for the following reasons7): The thrust coefficient based on the momentum theory is6): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : Aerodynamic forces act on a quadrotor, such as the drag 2 2 ð7Þ CT ¼ 2ðÞ tan þ : force described in the wind axes system and the thrust force described in the body axes system. By equating the right-hand sides of Eqs. (5) and (7), the : On-board sensors such as accelerometers and gyroscopes formula of the inflow ratio can be derived as: measure acceleration and angular rates with respect to ! 2 1 1 the sensory axes system. ¼ B3 þ B2 fi B2 3 avg 2 avg The coordinate systems are de ned as: Body axes system, ð8Þ Oxbybzb; wind axes system, Oxwywzw; and sensory axes sys- 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ox y z ðÞ tan 2 þ 2: tem, i i i; as shown in Fig. 1. B2a 2.2. Aerodynamic modeling for forward flight This model is based on the momentum theory and blade The Newton-Raphson procedure can be used to solve for iteratively. The drag force can be split into three components: induced drag, profile drag and parasite drag. The horizontal force coefficient is8): " ! C a 3 ÀÁ C ¼ d0 þ 0 þ þ tw þ H 4 2 3 1c 2 4 1c # ð9Þ 3 1 1 ÀÁ þ þ þ 2 þ 2 : 4 1c 6 0 1s 4 0 1c

The first term in Eq. (9) is the profile drag and the second term is the induced drag. Different from most conventional helicopters, stifffixed- Fig. 1. Coordinate system definitions. pitch rotor blades are used on MRHs. In this study, a thin air-

Dexp foil is utilized and the approximate calculation of rotor blade Cexp ¼ total : ð20Þ flapping is based on the formula derived for helicopter rotor D !RðÞ24R2 blades9): "# 1 ÀÁ M It should be noted that Eqs. (5) and (15) are respectively ¼ 0:5 1 þ 2 w ð10Þ ffi ffi 0 2 4 3 I!2 theoretical thrust coe cient and drag coe cient in steady ! forward flight, whereas Eqs. (19) and (20) are respectively 4 experimental thrust coefficient and drag coefficient applied ¼ 2 ð11Þ 1c 3 0:5 for all flight regimes including hovering, forward, descent and transient flight. 4 ¼ ð12Þ 2.4. Unsteady aerodynamic parameter estimation using 1s 3 0 the recursive least squares (RLS) method 2 where, £ is the lock number, Mw ¼ mbladegR =2 is the mo- On the basis of Eq. (5), the thrust coefficient in steady for- 3 ment caused by the rotor blade and I ¼ mbladeR =3 is the in- ward flight can be written in polynomial form as the sum of ertia moment of the rotor blade. More precise calculations of the static term and derivatives of inflow ratio and advance ra- the flapping coefficients for stiff, fixed-pitch rotor blades tio as: were provided by Johnson.8) Then, the horizontal force H be- Cexp ¼ C þ C þ C 2: ð21Þ comes: T T0 T T2 By equating the right-hand sides of Eq. (5) and Eq. (21), H ¼ C !RðÞ24R2: ð13Þ H the expressions of parameters in steady forward flight are: The measurement of drag force is necessary to know the 1 maximum forward speed that the quadcopter can achieve C ¼ aB3 ð22Þ T0 6 avg for a specific angle of attack and rotational speed (RPM). For this purpose, the total drag force of the quadcopter is ex- 1 C ¼ aB2 ð23Þ pressed as: T 4 D ¼ T sin H cos D : ð14Þ 1 total parasite C ¼ aB : ð24Þ T2 4 avg The total drag coefficient of the quadcopter is finally formulated as: There are some aspects that should be noticed. First, the fi ffi D nal form of the thrust coe cient depends on the theories ap- C ¼ total : ð15Þ plied theories, as blade element theory (Eq. (5)) or momen- D !RðÞ24R2 tum theory (Eq. (7)), in which several approximations are At a sufficiently high advanced ratio, reverse flow is gen- made to derive those equations; such as approximating the erated in a small circular region on the rotordisk. The diame- lift coefficient of the blade section, approximating blade flap- ter of the region of reverse flow corresponds to ®, and it is ping angle, and approximating the twist angle of the blade known that Eq. (9) is valid for ® less than 0.4 where the ef- section. If the reference twist angle is taken to be included fect of reverse flow is negligible.9) in the azimuth angle term,8) then the first-order term of ® will 2.3. Parameter estimation from flight data appear in thrust coefficient expressions Eqs. (5) and (21). Thrust is represented along the zb direction of the body Actual flight experiments often consist of steady and un- axes and points up, and is therefore directly measured by steady flights, and it is therefore necessary to extend conven- fl fl T exp ¼ mai : ð16Þ tional models by accounting for the in uence of vertical ow z (w) and horizontal flow (v)orinflow angles (¡, ¢) explicitly Total drag force is aligned and opposite the velocity vec- for some large amplitude maneuvers, transient flights or rap- tor, therefore the acceleration measured should be first trans- id excursions from the steady flight conditions.7) Therefore, i i i formed to body axes ax ¼ ax, ay ¼ay, and az ¼az, and we propose new models for thrust coefficient and drag coef- then transformed to wind axes using the directional cosine ficient that can be applied to all flight regimes: W exp 2 matrix HB : C ¼ C þ C þ C þ C v þ C w þ C 2 3 T T0 T T Tv Tw T2 cos cos sin sin cos ð25Þ 6 7 HW ¼ 4 cos sin cos sin sin 5: ð17Þ B Cexp ¼ C þ C þ C þ C þ C D0 D D D D sin 0 cos D ð26Þ þ C 2 þ C 2 þ C 2: As a result, D2 D2 D2 Dexp ¼ mai cos cos ai sin ai sin cos ; ð18Þ For the purpose of calculating aerodynamic forces in real- total x y z time, it is better to directly use Eqs. (19) and (20), whereas ffi Cexp ffi Cexp and thrust coe cient, T , and drag coe cient, D , are: Eqs. (25) and (26) are useful for modeling aerodynamic T exp forces acting on a quadcopter, which will be used in comput- Cexp ¼ ð19Þ ing the torques of roll, pitch and yaw motions for flight con- T !RðÞ24R2 troller design.

fl C C Table 1. Rotor parameters. When applying to all ight regimes, parameters T0 , T , C T , etc. in Eqs. (25) and (26) are time-varying and can be Parameters Value specified using the RLS method. Equation (25) can be writ- Rotor radius (m), R 0.12 ten in the general form using the vector and matrix notation Blade chord (75%R, m), c 0.018 Solidity, · 0.095 y ¼ xT ð27Þ Lock number, £ 7.541 Blade inertia moment, I 3.45 10¹6 ¹4 where, Blade moment, Mw 4.24 10 ÂÃ 2-D lift slope factor, a 5:9 0:2 y ¼ Cexp; x ¼ 1 vw2 T T Rotor average pitch (deg), avg 15 1:5 hifi ffi C 0:02 0:005 T Pro le drag coe cient, d0 ¼ C C C C C C : Tip loss factor, B 0.9 T0 T T Tv Tw T2 Cexp A similar way can be applied for D . The estimated value of is denoted by ~. Using RLS, the time variance of aerodynamic derivative parameters can be examined so that the influence of each parameter to the aerodynamic characteristics of the quadcopter in transient mode can be investigated. ~ In general, k is obtained each time by adding a correction ~ term to k1. The correction term depends on the current T ~ measurement and a predicted value, y~k ¼ xk k1. The deri- vation of RLS can be found in the study by Young.10) ÀÁ ~ ~ T ~ k ¼ k1 þ Kk yk xk k1 ð28Þ ÀÁ T 1 ð29Þ Kk ¼ Pkxk ¼ Pk1xk I þ xk Pk1xk ÀÁ T 1 T ð30Þ Pk ¼ Pk1 Pk1xk I þ xk Pk1xk xk Pk1: The RLS algorithm requires an initial value of parameters Fig. 2. Schematic diagram of DAQ. . 0 is chosen based on the least squares (LS) algorithm.

3. Experimental Method : Altimeter sensor for measuring quadcopter altitude (h). : Five-hole pitot tube for airspeed (V) and angles relative to 3.1. Facilities the flow (¡, ¢) information. The quadcopter platform utilized in this experiment has a : Four hall-effect sensors to record the rotational speeds 0.45-m diameter and a mass of 1.3 kg. The characteristic pa- (RPM) of the four rotors. rameters of the rotor, such as rotor radius, R, blade number, : Additional global positioning unit (GPS) necessary for N, blade chord, c, and solidity, ·, were directly measured as tracking the flight path. tabulated in Table 1. The 2-D lift slope factor, a, rotor aver- The data obtained was stored on a SD card at an update fi ffi C age pitch, avg, and pro le drag coe cient, d0 , were deter- rate of 30 Hz. mined on the basis of a wind tunnel experiment. The rotor 3.2. Flight test procedure possesses a typical static thrust coefficient of CT ¼ 0:013, The general procedure for flight tests is described as fol- measured when hovering. lows. Prior to each flight test, the onboard sensor system A five-hole pitot tube was used to measure the velocity was calibrated and the whole system was checked first to vector of the inflow to the rotor. The probe was calibrated avoid failure. The quadcopter then took-off from the ground to determine its sensitivity to the angle of attack and the side- and climbed to a height of 4–5 m. Before performing maneu- slip angle during a uniform flow in the wind tunnel. ver flight, hovering was carried out for 5–10 s. Data was col- The aim of the flight test program was to identify aerody- lected during maneuver flight. The pilot commanded the namic parameters such as thrust coefficient and drag coeffi- quadcopter to land after the flight test scenario was com- cient. To achieve this purpose, the experimental quadcopter pleted. The flight test trajectory is depicted in Fig. 3. is equipped with a data acquisition system (DAQ) for gather- The raw acceleration and angular rates obtained from the ing flight data, including acceleration, angular rates, atti- IMU were noisy, and therefore had to be filtered to get appro- tudes, inflow angles, airspeed, altitude, and rotational speed. priate data. The time history of the data collected after filter- The architecture of the DAQ is depicted in Fig. 2. ing is shown in Fig. 4. The centerpiece of the DAQ is a Mbed microcontroller hosting an ARM Cortex-M3 processor with multiple I/O 4. Results and Discussion protocol and an onboard sensor system including: : 6 DOF inertia measurement unit (IMU) 6050MPU to 4.1. Phase of flight measure linear acceleration (ax, ay, az) and angular rates The maneuver flight was comprised of three main phases: (p, q, r). Forward (F), transition (T) and descent (D). The forward

Fig. 3. Flight trajectory for maneuver phase. Fig. 6. Quadcopter in a climbing maneuver.

Fig. 4. Time history of ﬁltered acceleration.

Fig. 7. Demonstration of transition at the time of high blade loading and low airspeed.

w, manifests the climbing motion. The flight data are subject to noise and random measurement errors. Noise is sup- pressed by the Kalman filter. The smooth values of roll attitude, º, pitch attitude, ª, yaw attitude, ¼, airspeed, V, altitude, h, angle of attack, ¡, and sideslip angle, ¢, after removing the Fig. 5. Representation of transition from forward to descent phases. noise will be used to calculate the aerodynamic parameters. The inflow ratio and advance ratio flight data recorded are shown in Fig. 9 and Fig. 10, respectively. phase was from the time of beginning to t ¼ 20 s. The max- 4.3. Results of parameter estimation imum altitude reached at t ¼ 20 s corresponds to w ¼ 0. The comparison between experimental (red dotted line) Transition from the forward phase to the descent phase and theoretical (black solid line) thrust coefficient values is was during the time t ¼ 20–22 s. The transient behavior shown in Fig. 11. These values can be validated by compar- was as decelerating climb at which time the thrust vector ing the thrust coefficient when hovering, which is 0.013. The changed from a forward tendency to a rearward tendency, large discrepancy between theoretical result (Eq. (5)) and ex- as described in Fig. 5. From t ¼ 22 s, the quadcopter per- perimental result (Eq. (19)) happens in the transient phase formed a descent represented by a positive value of w. (“T” region) and descent phase (“D” region). For the descent A quadcopter may be required to perform maneuvers con- phase, this is because of the velocity induced, and therefore, sisting of high-load factor turns and climbing to avoid ob- the inflow ratio needs to be recomputed. stacles. The experimental quadcopter performed a climbing In the descent phase, a quadcopter operated in the vortex maneuver during the transient phase as demonstrated in ring state corresponds to the condition 0

Fig. 8. State and output estimation using the Kalman ﬁlter.

Fig. 9. Time history of inflow ratio. Fig. 10. Time history of advance ratio. order polynomial (i.e., in the current study, it is the first-order ing Eq. (5) with the following modification to inflow ratio: of w=vh), thus it can be implemented for MRHs of different w v ¼ i : ð32Þ configurations and sizes under different flight conditions. !R The thrust coefficient in the descent phase is calculated us-

Table 2. Validation of parameters of thrust coeﬃcient in Eq. (25). Parameter Eqs. (22)–(24) LS C T0 0.017 0.018 C ¹ ¹ T 0.11 0.04 C T2 0.03 0.09 C / ¹ Tv n a 0.001 C / Tw n a 0.001

Fig. 11. Experimental thrust coeﬃcient vs. theory.

Fig. 13. Experimental thrust coeﬃcient vs. LS reconstruction.

C t ¼ 18– C T , increases. From 20 s, T increases due to the decrease in . Advance ratio, ®, contributes a positive incre- Fig. 12. Experimental drag coefficient vs. theory. C C ment of T, represented by a positive value of T . The vertical flow, w, has an influence on CT, especially in the C Using Eqs. (5), (31) and (32), calculating the thrust coef- transient phase represented by an increase in the value of Tw ficient during descent (D) has been improved, as seen in after t ¼ 20 s. Fig. 11 (green solid line). Validation of the RLS method for calculating parameters ffi C C The experimental and theoretical drag coe cients are T and T in the transient phase, based on load factor, n, shown in Fig. 12. C is positive in forward flight, faster in is as follows: D ÀÁ ÀÁ movement the large CD is. CD is negative in transition from nC j ¼ C j ð33Þ T forward T trans forward flight to descent because of the change in direction ÀÁ ÀÁ nC j ¼ C j : ð34Þ of the thrust vector. T forward T trans ffi Equations (5) and (21) are insu cient to model the thrust As a result, ÀÁ coefficient in the transient phase (“T” region). The reason is C j C j ¼ n T forward ð35Þ Eqs. (5) and (21) do not account for the influence of vertical T trans j fl w ÀÁtrans ow, , in the transient phase. Since the quadcopter also per- C j formed sideways flight, both v and w are taken into account. C j ¼ n T forward : ð36Þ T trans j Therefore, to apply for all flight regimes including transient trans phase, Eq. (25) should be used. There exists the difference in By substituting the average values of n ¼ 1:5, C ¼0:01 ¼ 0:09 C ¼ 0:02 ¼ 0:125 comparing LS and analytical equations when calculating the T , , T , for for- parameters of Eq. (25), as tabulated in Table 2. Figure 13 ward; ¼ 0:05, and ¼ 0:075 for transition gives ffi C j ¼0:03 0:03 shows the experimental results for the thrust coe cient and T trans , compared to using RLS; C j ¼ 0:05 its reconstruction from Eq. (25) based on the LS method. T trans , in comparison with 0.07 using RLS (see The correlation coefficient, or ‘goodness of fit,’ of the LS Fig. 14). model is 0.7. Equation (25) and its parameters are deter- The influence of vertical flow on thrust coefficient during mined by minimizing the error, and therefore, these parame- the deceleration climbing transition is formulated using ters do not reflect the aerodynamic sense. Nevertheless, the Eq. (39). For a steady vertical rate, CT and mass flux, m_ a approximation equations (Eqs. (25) and (26)) are useful for are computed as: flight controller design. m_ w C ¼ a ð37Þ The time variance in parameters in Eq. (25) are also esti- T 4R2ðÞ!R 2 mated using the RLS method, and therefore, investigating m_ ¼ 4R2ðÞw þ v : ð38Þ what the influential variables might be in the transient mode a i is required. The static and derivative parameters of the thrust As a result, coefficient using the RLS method are shown in Fig. 14. @C w þ vi C ¼ 0:018 which is close to the hovering condition. C de- C j ¼ T ¼ : ð39Þ T0 T Tw trans @w ðÞ!R 2 creases inflow ratio, , represented by a negative value of

Table 3. Validation of parameter estimation during transient ﬂight. Proposed RLS Data set Parameter Eqs. (35), (36), (39) (Fig. 14) n ¼ 1:5 C j ¹ ¹ Set 1 ( ) T trans 0.03 0.03 C j T trans 0.05 0.07 C j Tw trans 0.001 0.001 n ¼ 1:3 C j ¹ ¹ Set 2 ( ) T trans 0.013 0.015 C j T trans 0.041 0.045 C j Tw trans 0.0015 0.001 n ¼ 1:2 C j ¹ ¹ Set 3 ( ) T trans 0.013 0.015 C j T trans 0.128 0.110 C j Tw trans 0.0015 0.0013

In addition, the flight test vehicle used in this study is a small-scale quadrotor. The Reynolds number of flow passing through the rotor is 103 to 104, while it is around 106 for conventional helicopters. It is well known that flows of such low-Reynolds numbers are different from flows of higher Reynolds numbers since the effects of viscous forces are dominant in low Reynolds number flow regimes, which may cause the laminar flow to separate.11) Under certain cir- cumstances, a separated flow causes negative effects on aerodynamic performance. These negative effects may increase drag and decrease lift. In our previous study,3) flow separa- tion caused a discontinuity in the mass flow through the rotor disk, which happens during high climb rate conditions and high angle of attack, in which the assumption of inviscid flow is invalid.

5. Conclusions

On the basis of the flight tests of a quadrotor helicopter, the following conclusions are made: 1. In steady forward flight, the experimental results agree well with the conventional theory throughout the investigated inflow ratio range: from 0 to 0.12, advance ratio less than 0.2 and angle of attack from 0 to ¹30 . 2. The external forces measured based on acceleration data were interpreted as a linear combination of aerodynamic parameters manifests the correlation of the data collected. 3. The effects of inflow ratio, advance ratio and vertical flow on thrust coefficient are represented by influential pa- ffi C C C fl Fig. 14. Estimation of aerodynamic derivative of thrust coe cient using rameters T , T and Tw . In transient ight, those parame- RLS. ters are larger in magnitude when compared to those of steady forward flight. Vertical flow should be taken into ac- Similarly, by substituting for the average values of count explicitly in transient flight. w ¼ 1 m/s, vi ¼ 2 m/s, ! ¼ 440 rad/s, and R ¼ 0:12 m 4. Conventional theory was insufficient to model the thrust C j ¼ 0:001 ff ffi gives Tw trans , in a comparison with a di erence coe cient during a decelerating climbing transient maneu- of 0.001 using RLS. ver. The influential parameters in transition were formulated Several flight tests were performed at different load factor using the method proposed in this study and validated using values, n. Table 3 presents the results of estimating the pa- the RLS method. The method proposed provides results that rameters in transient flight by applying two methods: RLS are similar to those of the RLS method. The time variance in and analytical Eqs. (35), (36), and (39). aerodynamic parameters, in the sense of flight mode chang- Although the aerodynamic parameters listed in Table 2 ing, is useful for the adaptive control system design of multi- can only be true for the multicopter used in our experiment, rotor helicopters. the method proposed in this study can be applied for MRHs of different configurations and sizes under different flight conditions.

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