Real-Time Wind Field Estimation and Pitot Tube Calibration Using an Extended Kalman Filter

mathematics

Article Real-Time Wind Field Estimation and Pitot Tube Calibration Using an Extended Kalman Filter

Qian Zhang 1, Yifan Xu 2 , Xueyun Wang 2,*, Zelong Yu 2 and Tianyi Deng 2

1 School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China; [email protected] 2 School of Instrumentation Science and Opto-Electronics Engineering, Beihang University, Beijing 100191, China; [email protected] (Y.X.); [email protected] (Z.Y.); [email protected] (T.D.) * Correspondence: [email protected]

Abstract: The airspeed is an important feedback signal for flight control, and its measurement accuracy is related to the safety of aircraft, especially for hybrid vertical takeoff and landing (VTOL) unmanned aerial vehicles (UAV) in the transition phase. However, offline calibration of the pitot tube cannot fully simulate the situation in real cases, and this is why online calibration after installation is necessary. In addition, the environmental wind field creates a high risk for the conversion flight of a hybrid UAV, thus real-time wind field measurement of the flight field has great significance for the flight path planning of the conversion process. In this article, an innovative method for online calibration of pitot tube and wind field estimation is proposed. After establishing the extended Kalman filter (EKF) for estimation, an analysis method is proposed to analyze the observability of EKF under different flight strategies. Then, the hovering flight is selected as the experimental flight trajectory. The laboratory computer simulation and flight experiment results validate the theory and prove that the proposed method could re-calibrate the scale factor of the pitot tube and estimate the wind field. Citation: Zhang, Q.; Xu, Y.; Wang, X.; Yu, Z.; Deng, T. Real-Time Wind Field Keywords: unmanned aerial vehicle; airspeed; calibration; extended kalman filter; wind field Estimation and Pitot Tube Calibration Using an Extended Kalman Filter. Mathematics 2021, 9, 646. https:// doi.org/10.3390/math9060646 1. Introduction During recent decades, unmanned aerial vehicles (UAVs) have been increasingly used Academic Editor: Yolanda Vidal in civil and military applications which include search and rescue operations [1], area mapping [2], weather monitoring [3], and agricultural operations [4]. UAV platforms can Received: 2 March 2021 be divided into two categories according to flight modes: fixed-wing UAV and rotorcraft Accepted: 16 March 2021 Published: 18 March 2021 UAV [5]. The fixed-wing UAV has advantages of high cruising speed and altitude. Fur- thermore, the push-to-weight ratio of the fixed-wing UAV is generally less than 0.1, which

Publisher’s Note: MDPI stays neutral gives the fixed-wing UAV high flight efficiency. However, fixed-wing UAV needs a runway with regard to jurisdictional claims in for reliable takeoff and landing, which imposes restrictions on its applications [6]. On the published maps and institutional affil- other hand, rotorcraft UAV can take off and land vertically. For stationary or low speed iations. applications, the rotorcraft UAV has a hovering capability. As the thrust of rotorcraft UAV has to overcome gravity, the push-to-weight ratio needs to be greater than 1, which is the main factor that limits the flight speed and time of rotorcraft UAV [7]. According to the characteristics of the above-mentioned two aircrafts, a new technical approach for hybrid vertical takeoff and landing (VTOL) UAV is proposed which integrates the advantages of Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. fixed-wing and rotorcraft [8,9]. This article is an open access article The hybrid VTOL has two independent propulsion systems for hover and level flight. distributed under the terms and During the vertical take-off and hover phases, the rotor lift system works like a quadrotor conditions of the Creative Commons and the push system turns off or remains in idle state. The attitude and position are Attribution (CC BY) license (https:// controlled by direct force of the rotor system. During the level flight phase, the rotor lift creativecommons.org/licenses/by/ system is turned off and the aircraft works like a fixed-wing. As the hybrid VTOL is a 4.0/). fusion of fixed-wing UAV and rotorcraft UAV, the conversion process of two flight modes

Mathematics 2021, 9, 646. https://doi.org/10.3390/math9060646 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 646 2 of 16

is accompanied by changes in the actuators and control law, and the control accuracy at this stage directly affects the safety of the hybrid VTOL [10,11]. At this phase, the switching criterion of two different control strategies depends on the change of airspeed. Accurate measurement of airspeed is critical to the safety of the conversion process. Generally, airspeed is measured by a pitot tube using the Bernoulli principle and the measurement accuracy is influenced by air density, installation error and operational environment [12]. In addition, the environmental wind field affects the conversion flight of the hybrid UAV by changing the airspeed and may cause the UAV to enter an irretrievable stall [13]. Therefore, the accurate calibration of the pitot tube and real-time wind field measurement of the flight field have great significance for the flight safety of the conversion process of the hybrid VTOL. Several pitot tube calibration and wind field estimation approaches have been studied. Cooper introduced a calibration method for the pitot tube using a doppler laser air-motion sensor; in this way, the pitot tube could be used independently of any other temperature sensor [14]. Am Cho proposed a Kalman filter method to calibrate the error of airspeed, baro-altitude and side-slip angle due to the position error of the pitot static tube, and the method was verified by being applied to data from a commercial air-data computer [15]. Conley made a low-cost system for measuring horizontal winds, based on a global positioning system (GPS) that provided aircraft heading and a ground-referenced velocity. Then, the horizontal wind velocity was estimated by comparing the ground velocity and standard true airspeed [16]. A real-time wind estimation method was proposed in [17]. In this article, the wind was estimated using on-board sensors only and did not need any additional airspeed sensor or dedicated anemometer. In order to improve the positioning accuracy of quadrotor, Waslander proposed a wind estimation method which relies on the use of the accelerometer data [18]. A single-antenna GPS receiver was used to estimate wind speed and to calibrate the airspeed in [19]. However, this article makes a very limited assumption in that the attack and sideslip angles are ignored and the wind field was assumed to be 2D. Therefore, it cannot be fully applied in actual flight and cannot estimate wind shear information. In this article, the measurements from an integrated navigation system and pitot tube are used, and a method based on an extended Kalman filter (EKF) for online calibration of airspeed tube and measurement of wind field is proposed. In this way, installation error and environmental error of the pitot tube, which are difficult to calibrate by wind tunnel test, can be calibrated precisely, and real-time measured wind field data is more informative. The algorithm was implemented and evaluated with a computer simulation experiment and actual flight verification. The article is organized as follows: Section2 presents the measuring principle of the pitot tube and offline calibration with wind tunnel, together with the experimental results and error analysis. Section3 introduces the method of wind field measurement and pitot tube calibration. In order to improve measurement accuracy, the flight strategy and observability analysis are discussed. Computer simulations and experimental results are provided and discussed in Section4, followed by conclusion in Section5.

2. Measurement Principle and Description of Problem 2.1. Measurement Principle of Pitot Tube The airspeed is generally obtained by measuring total and static pressure with a pitot tube. As shown in Figure1, the pitot tube has a total pressure hole and several static pressure holes. The oncoming airﬂow passing through the pitot tube is divided into two channels. Based on the Bernoulli equation, the relationship between pressure and velocity is: 1 1 P + ρ V2 = P + ρ V2 = const (1) 1 2 1 1 2 2 2 2 Mathematics 2021, 9, x FOR PEER REVIEW 3 of 16

Mathematics 2021, 9, 646 3 of 16 Mathematics 2021, 9, x FOR PEER REVIEW 3 of 16

P 1 P2 P3 V V V1 2 3 P 1 P2 P3 FigureV 1. SchematicV of pitotV tube.3 1 2 FigureFigure 1.As 1. Schematic Schematicthe diameter of of pitot pitot of tube. the tube. total pressure hole is small, the airflow across the total pres‐ sure hole is blocked, and the kinetic energy at this point is completely converted into AsAs the the diameter diameter of of the the total total pressure hole isis small,small, the the airflow airflow across across the the total total pressure pres‐ V2  0 sureholepressure, hole is blocked, is therefore,blocked, and the and kinetic the. Thekinetic energy pressure energy at this in at point the this static ispoint completely pressure is completely converted hole is converted the into same pressure, intoas the therefore,atmosphericV2 = environment.0. TheV  pressure0 Therefore, in the static Equation pressure (1) can hole be isrewritten the same as as the atmospheric pressure, therefore, 2 . The pressure in the static pressure hole is the same as the environment. Therefore, Equation (1) can be rewritten1 as atmospheric environment. Therefore, EquationP PV (1) can be2 rewritten as (2) 232 11 1 2 − =1 2 PP2 PPV3 ρ1V1 (2) Equation (2) represents that the23 airspeed2 11of flight is measured indirectly by(2) the 2 pressureEquation of the (2) relative represents airflow that [20], the airspeedand the calibration of flight is formula measured can indirectly be expressed by the as: pres- Equation (2) represents that the airspeed of flight is measured indirectly by the sure of the relative airflow [20], and1 the calibration formula can be expressed as: pressure of the relative airflow [20], andVK22 the calibration PCVV formula  can be expressed as: (3) 2 am a f am out 1 1 2 222 ρVamVK· Ka = ∆P PCVV= Cf Vam = Vout (3) V 2 am a f am out K (3) where am is the airspeed measured2 by pitot tube, is the air density, a is the scale V whereVVamPis the airspeed measured by pitotout tube, ρ is the air density,KKa is the scale wherefactor, am is theis the airspeed pressure measured difference by and pitot tube, is the outputis the air of density,the pitot tubea is sensor. the scale If the factor, ∆P is the pressure difference and Vout is the output of the pitot tube sensor. If the environmental errors are constant, the sensorV output is proportional to the square of the factor,environmental P is the errors pressure are difference constant, theand sensor out is output the output is proportional of the pitotC totube the sensor. square If of the the measured airspeed. Therefore, accuracy calibration of coefficient f will facilitate ac‐ environmentalmeasured airspeed. errors Therefore,are constant, accuracy the sensor calibration output of is coefficient proportionalCf will to the facilitate square accurate of the measurementcurate measurement of airspeed. of airspeed. C measured airspeed. Therefore, accuracy calibration of coefficient f will facilitate ac‐ curate2.2.2.2. Description Descriptionmeasurement of of Problem Problem of airspeed. BeforeBefore using using the the pitot pitot tube, tube, the the offline offline calibration calibration experiment experiment should should be be made made in in the the 2.2. Description of Problem windwind tunnel. tunnel. As As shown shown in in Figure Figure2 ,2, the the pitot pitot tube tube is is installed installed on on the the bracket bracket in in the the wind wind tunnel.tunnel.Before The The using upwind upwind the pitot angle angle tube, of theof thethe pitot offlinepitot tube tubecalibration can becan achieved be experiment achieved by adjusting by should adjusting the be bracket. made the inbracket. Then, the windtheThen, two tunnel. the pressure two As pressureshown pipes in are pipes Figure connected are 2, connectedthe to pitot the deferencetube to the is deferenceinstalled pressure on pressure sensor the bracket of sensor the in flight ofthe the controlwind flight tunnel.system.control The Bysystem. upwind collecting By collectingangle the pitot of the the tube pitotpitot sensor’s tubetube sensor’scan data be and achieved data the windand bythe speed adjusting wind in speed the the wind in bracket. the tunnel, wind Then, the two pressure pipes are connected to the deference pressure sensor of the flight coefficient Cf can beC calibrated.f Figure3 shows the fitting result. controltunnel, system. coefficient By collecting can thebe calibrated. pitot tube sensor’sFigure 3 datashows and the the fitting wind result. speed in the wind C tunnel, coefficient f can be calibrated. Figure 3 shows the fitting result.

Figure 2. The offline calibration experimental equipment. Figure 2. The ofﬂine calibration experimental equipment. Figure 2. The offline calibration experimental equipment.

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Mathematics 2021, 9, x FOR PEER REVIEW 50 4 of 16 Mathematics 2021, 9, 646 Collected data 4 of 16 Fitting result 40

50 30 Collected data Fitting result 40 20 Airspeed(m/s)

30 10

20 0

Airspeed(m/s) 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 The output voltage(V) 10 Figure 3. Fitting result of offline calibration.

0 2.5The output2.6 2.7 of the2.8 deference2.9 pressure3 3.1 sensor3.2 is 3.3given as voltage value. As shown in Figure 3, after fittingThe the output test data,voltage(V) the measured airspeed could be obtained by collecting FiguretheFigure 3.voltage Fitting 3. Fitting signalresult result ofof offline ofthe offline pitot calibration. calibration.tube sensor. The cost of calibration in the wind tunnel is very high, and the actual operation processTheThe output is output complicated. of the of the deference deference Meanwhile, pressure pressure after sensor being sensor iscalibrated, given is given as asvoltagethe voltage pitot value. tube value. willAs Asshown be shown installed in in FigureonFigure the 3, after3aircraft., after fitting fitting Due the theto test the test data, processing data, the the measured measured accuracy airspeed airspeedof the couldmounting could be be obtained base obtained and by mounting by collecting collecting er‐ therors,the voltage voltage the signalpitot signal tube of the of cannot the pitot pitot tubebe tubecompletely sensor. sensor. parallel to the aircraft’s longitudinal axis, and The cost of calibration in the wind tunnel is very high, and the actual operation theThe non cost‐parallel of calibration angle  inmay the change wind tunnelwith different is very aircrafthigh, and bodies. the Asactual shown operation in Figure process is complicated. Meanwhile, after being calibrated, the pitot tube will be installed process is complicated. Meanwhile, afterOXYZ being calibrated,O the pitot tube will be installedX on 4,onthe the the aircraft. body aircraft. frameDue Due to is tothe defined the processing processing as bbbb accuracy accuracy, where of of the the mounting mountingb is the center base andof gravity, mountingmounting with errors,er ‐ b rors,the the pitot pitot tube tube cannot cannot be be completely completelyY parallel parallel to to the the aircraft’s aircraft’sZ longitudinal longitudinal axis, axis, and and the pointing to the front of the aircraft, b pointing right and b pointing down. The re‐ non-parallel angle δ may change with different aircraft bodies. As shown in Figure4, the thelationship non‐parallel between angle body may frame change and withair frame different can beaircraft obtained bodies. with As attack shown angle in Figure  and body frame is defined as ObXbYbZb, where Ob is the center of gravity, with Xb pointing to  OXYZ O X 4, thesideslipthe body front offrame the aircraft, is definedYb pointing as bbbb right, andwhereZb pointingb is the down. center The of relationshipgravity, with between b body frame and air frame can be obtained with attack angle α and sideslip β Yb Zb pointing to the front of theuV aircraft, cos  cos pointing cos right  sin and sin pointing a down. The re‐ lationship between bodyu  frame cosand α aircos frameβ − coscan αbesin obtainedβ − sin withα  attackV  angle  and v  sin cos 0  0 a (4)  v = sin β cos β 0 0 (4) sideslip  w  sin cos sin  sin cos   0  w sin α cos β − sin α sin β cos α 0

uV cos cos cos  sin sin  a    v  sin cos 0  0 (4) w sin cos sin  sin cos   0   

 Yb   X p X b X Zb s Va  Yb   Figure 4. The relationship between air frameX and body frame. p X b X Then, the airspeed Zb component measured bys pitot tube is V a VamVuam= u coscosδ= V V a cos α cos β cos δ (5) Figure 4. The relationship between air frame and body frame. It is is clear clear that that the the measured measured airspeed airspeed is isaffected affected by by the the attack attack angle, angle, the thesideslip sideslip an‐ gle,angle,Then, and and thethe the airspeed installation installation component angle, angle, and andmeasured these these effects effectsby pitot cannot cannot tube beis be considered considered during during the offlineoffline calibration. Furthermore, theVu coefficientcos C Vf is cos the function cos cos of air density ρ and temperature am a (5) T, and it will vary with different aircraft operating conditions. Therefore, an online calibrationIt is clear method that the that measured avoids airspeed the high costis affected of offline by operationthe attack and angle, takes the into sideslip account an‐ the gle,actual and the operating installation conditions angle, ofand the these pitot effects tube should cannot be be proposed. considered during the offline

Mathematics 2021, 9, x FOR PEER REVIEW 5 of 16

C  calibration. Furthermore, the coefficient f is the function of air density and tem‐ Mathematics 2021, 9, 646 perature T , and it will vary with different aircraft operating conditions. Therefore,5 ofan 16 online calibration method that avoids the high cost of offline operation and takes into account the actual operating conditions of the pitot tube should be proposed. The accurate calibration of the pitot tube cannot be separated from the accurate The accurate calibration of the pitot tube cannot be separated from the accurate measurement of real‐time airspeed. However, the airspeed is closely related to ground measurement of real-time airspeed. However, the airspeed is closely related to ground speed and wind speed. As shown in Figure 5, two‐dimensional horizontal velocity is speed and wind speed. As shown in Figure5, two-dimensional horizontal velocity is considered. The yaw angle of the target airspeed vector changes at a constant angular considered. The yaw angle of the target airspeed vector changes at a constant angular speed over time and the absolute value is constant. Due to the presence of wind field, the speed over time and the absolute value is constant. Due to the presence of wind ﬁeld, the absoluteabsolute value value of of ground ground speed variesvaries fromfrom the the yaw yaw angle. angle. Therefore, Therefore, the the wind wind ﬁeld field affects af‐ fectsthe shapethe shape of the of speedthe speed triangle triangle and and needs needs to be to consideredbe considered during during the the calibration calibration of theof thepitot pitot tube. tube.

X n

  V Va  w Vg

Figure 5. The relationship between airspeed, ground speed and wind speed. Figure 5. The relationship between airspeed, ground speed and wind speed. 3. Pitot Tube Calibration and Wind Field Measurement 3. Pitot Tube Calibration and Wind Field Measurement The local-level north-east-down (NED) frame is selected as the navigation frame (n frame).The local The‐ three-dimensionallevel north‐east‐down speed (NED) triangle frame is is shown selected in Figureas the 6navigation and it satisﬁes frame the (n frame).following The formula three‐dimensional speed triangle is shown in Figure 6 and it satisfies the

following formula *2 *2 * * *2 Vg + Vw − 2 Vg Vw cos λ = Va (6) 22   2 VV2cos VV  V (6) * gw* gw a where V g is the ground speed and Vw is the wind speed. Considering Equation (3) and EquationV (5), Equation (6) can be rewrittenV as: where g is the ground speed and w is the wind speed. Considering Equation (3) and Equation (5), Equation (6) can be rewritten as: * * V V2 + V2 + V2 + V2 + V2 + V2 − 2V · V = out (7) gx gy gz wx wy wz g w 0 2 2222 2 2 C Vf (outcos α cos β) VVVVVVgx gy gz wx wy wz2 VV g  w 2 (7) 0 Cf cos cos  where C f is the coefﬁcient which includes the inﬂuence of installation error and operational * * * * environment.C V · V is the inner product of V and V . Because the attack angle and where f is theg coefficientw which includes the influenceg w of installation error and oper‐ sideslip angle in Equation  (7) are small values, they can be calculated by the speed in the body frame VVg  w Vg Vw ational environment. is the inner product w of and . Because the attack α = tan−1 (8) angle and sideslip angle in Equation (7) are smallu values, they can be calculated by the speed in the body frame v β = tan−1 √ (9) u21+wv2 + w2   tan  (8) u

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1 v Mathematics 2021, 9, 646   tan 6 of(9) 16 22 2 uvw

X n

  V Vw  a  Vg

On Yn

Figure 6. Three-dimensional velocity triangle. Figure 6. Three‐dimensional velocity triangle. The speed in body frame can be obtained by attitude transformation matrix and groundThe speed speed in body frame can be obtained by attitude transformation matrix and ground speed      u cos θ cos ψ cos θ sin ψ − sin θ Vgx uV cos cos cos  sin sin  gx  v  =  sin θ cos ψ sin φ − sin ψ cos φ sin θ sin ψ sin φ + cos ψ cos φ cos θ sin φ  Vgy  (10) vVsin cos sin sin  cos sin  sin sin  cos  cos cos  sin gy (10) w sin θ cos ψcosφ+ sin ψ sin φ sin θ sin ψ cos φ − cos ψ sin φ cos θ cos φ Vgz  wV sin cos cos sin  sin sin  sin cos cos  sin cos  cos    gz where the roll angle φ, pitch angle θ, yaw angle ψ and ground speed Vg∗ can all be obtained   V whereby the the integrated roll angle navigation , pitch system angle of  UAV., yaw angle and ground speed g* can all be obtained by the integrated navigation system of UAV. 3.1. Design of Extended Kalman Filter 3.1. DesignAccording of Extended to Equations Kalman (7)–(10), Filter an EKF method is proposed to estimate the parameter of the pitot tube and wind field information. The Kalman filter is a linear quadratic According to Equations (7)–(10), an EKF method is proposed to estimate the pa‐ estimation (LQE) algorithm. When a system has a series of measurements with errors, rameter of the pitot tube and wind field information. The Kalman filter is a linear quad‐ the Kalman filter can estimate unknown variables with these measurements and obtain ratic estimation (LQE) algorithm. When a system has a series of measurements with er‐ more accurate variables than those based on a single measurement alone [21,22]. For rors, the Kalman filter can estimate unknown variables with these measurements and computer applications, the Kalman filter is always expressed in discrete linear form as in obtain more accurate variables than those based on a single measurement alone [21,22]. the following equations For computer applications, the Kalman filter is always expressed in discrete linear form as in the following equations( Xk = Φk,k−1Xk−1 + Γk−1Wk−1 (11) = + + ZkX kkkkkkHkXk ,1VXWk 1 fk,ϕγ  1 1  (11) m  ZHXVfkkkkkn , where Zk ∈ R is the measure of the output, Xk ∈ R represents the system state, Φk,k−1 ∈ n×n n×r R is the onem step transfer matrix of system state, and nΓk−1 ∈ R is the noise matrix. Zr k  R m X k  R whereWk ∈ R and Vk ∈is Rtheare measure independent of the Gaussian output, white noise represents sequence the system state, Rnn Rnr kk,1 is the one step transfer matrixn of systemo state, and k 1 is the noise E{W } = 0, E W WT = Q δ (12) WR r VR m k k j k kj matrix. k and k are independent Gaussian white noise sequence n To E{Vk} = 0, E VkVj T = Rkδkj (13) EW kkjkkj 0, EWW Q (12)  

where δkj is Krone Nick function, and γ is a randomT vector which presents the size of the EV kkjkkj 0, EVV  R (13) fault. fk,ϕ is a piecewise function can be described as

1, k ≥ ϕ f = (14) k,ϕ 0, k < ϕ

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T h 0 i According to the Equation (7), the state vector is defined as x = Vwx, Vwy, Vwy, C f . As the UAV’s flight time is short, the wind field can be considered as constant. Then the one step transfer matrix can be described as

 1 0 0 0   0 1 0 0  Φ − =   (15) k,k 1  0 0 1 0  0 0 0 1

The measurement vector is the output of the pitot tube and it can be deduced from Equation (7)

2 2 2 2 2 2 ! 0 2 Vgx + Vgy + Vgz + Vwx + Vwy + Vwz Vout = C f (cos α cos β) (16) −2 VgxVwx + VgyVwy + VgzVwz

The linearized measurement matrix can be expressed as

h iT H = ∂h ∂h ∂h ∂h ∂Vwx ∂Vwy ∂Vwz ∂Cf  0 2  C f (cos α cos β) 2Vwx − 2Vgx  0 2  (17)  C f (cos α cos β) 2Vwy − 2Vgy  =  0 2   C f (cos α cos β) 2Vwz − 2Vgz    2 2 2 2 2 2 2 (cos α cos β) Vgx + Vgy + Vgz + Vwx + Vwy + Vwz − 2 VgxVwx + VgyVwy + VgzVwz

3.2. Design of Flight Scheme and Observability Analysis In this section, the observability analysis method is introduced to analyze the observable degrees of state estimation. In this way, the flight strategy can be designed in advance to make the state estimation more accurate and efficient. As the measurement matrix of the EKF is time-varying, the traditional observability analysis method cannot be used directly. The piece-wise constant system (PWCS) observability analysis method was first proposed by Goshen-Meskin in 1992 [23]. This method divides the time-varying system into several sections which can be approximately considered as constant systems in a short enough time interval. Then, a stripped observability matrix (SOM) is introduced to realize observability analysis. For the discrete system model shown in Equation (11), the PWCS can be expressed as:

( j Xk = Φ Xk−1 + Γk−1Wk−1 (18) j j Zk = H Xk + Vk + fk,ϕγ

where j = 1, 2, 3 ··· r are the different time periods. In each time period, Φj and Hj are all constant. Then, all observations can be expressed as functions of X1 Mathematics 2021, 9, 646 8 of 16

 Z1 = H1X  1 1  1 = 1 1  Z2 H F X1   .  .   1 1 1n−1  Zn = H F X1   2 2 1n−1  Zn = H F X1   2 2 2 1n−1 Zn+1 = H F F X1 (19)  .  .  .  2 2 2n−1 1n−1  Z = H F F X1  2n  n−1 n−1  Z3 = H3 F2 F1 X  2n 1  n−1 n−1  Z3 = H3F3 F2 F1 X  2n+1 1  .  . The SOM can be written as,

 T  h n−1 i H1 H1F1 ··· H1F1    h iT   2 2 2 2 2n−1   H H F ··· H F    QSOM =   (20)  .   .   T   n−1  Hk Hk Fk ··· Hk Fk

The SOM can analyze the observability of the system; however, the observability of each variable cannot be analyzed. Therefore, a singular value decomposition (SVD) analysis method is carried out to calculate the observable degree of each filter state [24]. Based on SVD theory, the SV reflects the observability of each state. The greater the SV value, the higher the observable degree of the corresponding state [25]. Therefore, by comparing the SVD of different flight strategies, the optimal flight trajectory can be obtained. Here, straight flight and circle flight are analyzed as two typical flight strategies. Figures7 and8 give the histograms of SVD analysis of straight flight and circle flight. 0 No.1, 2, 3 and 4 represent the singular value of Vwx, Vwy, Vwz, C f , respectively. It can be seen that each flight strategy is observable for the state vector, and the similar singular values represent similar observable degrees of state. These two figures also show that the observability of each state of circle flight is greater than that of straight flight. The singular values of Vwx for the two typical flight strategies are similar, because the installation of the pitot tube is parallel to the longitudinal axis of the aircraft, mainly to measure the velocity in this direction. The singular values of the other three states of straight flight are smaller than those of circle flight, especially for the singular values of vertical wind speed and scale factor, which means the observable degrees of straight flight are worse. Because the measurement direction of the pitot tube can be at any angle to the horizontal wind speed in the circle flight strategy, the observability of the state is improved. The observability analysis results prove that circle flight is better for state estimation. Therefore, the circle flight strategy is used in the next experimental section. Mathematics 2021, 9, x FOR PEER REVIEW 9 of 16 MathematicsMathematics 20212021, 9,, 9x, 646FOR PEER REVIEW 9 of 16 9 of 16

SV=36340.7427 SV=552.1542 SV=103.1413 SV=41.8593 1.4 SV=36340.7427 7 SV=552.1542 16 SV=103.1413 0.6 SV=41.8593 1.4 7 16 0.6 1.2 6 14 14 0.4 1.2 6 0.4 12 1 5 12 1 5 0.2 10 0.2 0.8 4 10 0.8 4 8 0 0.6 3 8 0 0.6 3 6 -0.2 0.4 2 6 -0.2 0.4 2 4 4 -0.4 0.2 1 -0.4 0.2 1 2 2 0 0 -0.6 0 -0.6 0 0 0 -0.2 -1 -2 -0.8 -0.2 1 2 3 4 -1 1 2 3 4 -2 1 2 3 4 -0.8 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

FigureFigure 7. Histograms of of singular singular valued valued decomposition decomposition (SVD) (SVD) analysis analysis of straight of straight ﬂight. flight. Figure 7. Histograms of singular valued decomposition (SVD) analysis of straight flight.

SV=50003.5762 SV=3828.8478 SV=3424.4537 SV=259.005 1.4 SV=50003.5762 1.6 SV=3828.8478 0.05 SV=3424.4537 0.5 SV=259.005 1.4 1.6 0.05 0.5 1.2 1.4 1.2 1.4 0 1.2 0 0 1 1.2 0 1 1 -0.5 0.8 1 -0.5 0.8 -0.05 0.8 -0.05 0.6 0.8 -1 0.6 0.6 -1 0.6 -0.1 0.4 -0.1 0.4 0.4 -1.5 0.4 -1.5 0.2 0.2 0.2 -0.15 0.2 -0.15 -2 0 0 -2 0 0 -0.2 -0.2 -0.2 -2.5 -0.2 1 2 3 4 -0.2 1 2 3 4 -0.2 1 2 3 4 -2.5 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

Figure 8. Histograms of SVD analysis of circle flight. Figure 8. Histograms of SVD analysis of circle flight. Figure4. Simulation 8. Histograms and Experiments of SVD analysis of circle flight. 4.1. Computer Simulation Results 4. Simulation and Experiments 4. SimulationAccording and to the Experiments above analysis, the computer simulation test is performed before the 4.1. Computer Simulation Results 4.1.actual Computer flight verification. Simulation The Results simulation is done with a Matlab2014.8.3.0 program on an IntelAccording Core i7 Duo to processor. the above In theanalysis, simulation the flightcomputer strategy, simulation the aircraft test makes is performed a circular before According to the above analysis, the computer simulation test is performed before thehovering actual motion flight verification. around a fixed The point simulation on the ground. is done If with there a is Matlab2014.8.3.0 no wind, the airspeed program on the actual flight verification. The simulation is done with a Matlab2014.8.3.0 program on anand Intel ground Core speed i7 Duo are consistent. processor. As In shown the simulation in Figure9, ifflight there strategy, is constant the wind aircraft at the makes a an Intel Core i7 Duo processor. In the simulation flight strategy, the aircraft makes a circularflight site, hovering the ground motion speed needsaround to bea fixed changed point to ensure on the airspeed ground. is constant.If there Figureis no wind,10 the circulardepicts the hovering profiles ofmotion airspeed around and ground a fixed speed. point on the ground. If there is no wind, the airspeed and ground speed are consistent. As shown in Figure 9, if there is constant wind airspeed and ground speed are consistent. As shown in Figure 9, if there is constant wind at the flight site, the ground speed needs to be changed to ensure airspeed is constant. at the flight site, the ground speed needs to be changed to ensure airspeed is constant. Figure 10 depicts the profiles of airspeed and ground speed. Figure 10 depicts the profiles of airspeed and ground speed.

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55 55 Airspeed AirspeedGround speed 50 Ground speed 50

45 45 40 40

Speed(m/s) 35 Speed(m/s) 35 30 30

25 25 0 50 100 150 200 0 50 100Time(s) 150 200 Time(s) Figure 9. The airspeed and ground speed during the hovering flight. FigureFigure 9. The 9. The airspeed airspeed and and ground ground speed during during the the hovering hovering ﬂight. flight.

50 50 30 30 10 10 -10 (m/s) -10 -30 (m/s) -30 -50 Air speed Ground speed Northward speed Northward -50 -70 Air speed Ground speed Northward speed Northward -70 0 50 100 150 200 0 50 100 150 200 50 50 30 30 10 10

(m/s) -10

(m/s) -10 -30 -30 Eastward speed -50 0 50 100 150 200 Eastward speed -50 0 50 100 150 200 2 2 1 1 0

0 (m/s)

(m/s) -1 -1 Vertical speed Vertical -2 Vertical speed Vertical -2 0 50 100 150 200 0 50 100Time(s) 150 200 Time(s) Figure 10. The speed profiles of airspeed and ground speed. Figure 10. The speed profiles of airspeed and ground speed. Figure 10.In The the computer speed profiles simulation, of airspeed the weight and ground of the speed. aircraft is 60 kg and horizontal cruising speedIn is the about computer 38 m/s. Itsimulation, is assumed the that weight the northward of the aircraft and eastward is 60 kg wind and speeds horizontal are cruis‐ 10ingIn m/s speedthe and computer is 5 m/s,about the simulation, 38 vertical m/s. It speed is the assumed weight is zero and thatof the the the aircraft scale northward factor is 60 is kg 0.8.and and The eastward horizontal wind speed wind cruis speeds‐ ingprofile arespeed 10 is m/sis described about and 5 38 m/s, in m/s. Figure the It vertical is 11 assumed. The speed hovering that is zero the radius northwardand isthe 350 scale m and and factor eastward it takes is 0.8. about windThe onewind speeds speed areminute profile10 m/s to andis circle. described 5 m/s, The the aircraft in vertical Figure starts speed with11. The zero is zerohovering yaw and and the circles radius scale clockwise isfactor 350 mis from 0.8.and zero The it time.takes wind As about speed one shown in Figure9, with the influence of the wind field, ground speed changes around the profileminute is described to circle. Thein Figure aircraft 11. starts The withhovering zero radiusyaw and is 350circles m andclockwise it takes from about zero one time. minuteairspeed to circle. with time. The Theaircraft estimation starts resultwith iszero shown yaw in and Figure circles 12 and clockwise it is obvious from that zero the time. windAs shown speeds in converge Figure 9, quickly with andthe influence the estimation of the result wind of thefield, scale ground factor isspeed also accurate.changes around As shownthe airspeed in Figure with 9, time. with Thethe influenceestimation of result the wind is shown field, inground Figure speed 12 and changes it is obvious around that the theairspeed wind withspeeds time. converge The estimation quickly and result the is estimation shown in Figureresult of 12 the and scale it is factorobvious is alsothat ac‐ the curate.wind speeds converge quickly and the estimation result of the scale factor is also ac‐ curate.

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12 11 1210 (m/s) 11 9

Northward wind 10 8

(m/s) 0 50 100 150 200 9 8

Northward wind 8 0 50 100 150 200 6 8

(m/s) 4 6

Eastward wind Eastward 2

(m/s) 4 0 50 100 150 200 2 Eastward wind Eastward 2 01 50 100 150 200 2 0 (m/s) 1-1

Vertical wind 0-2

(m/s) 0 50 100 150 200 -1 Time(s) Vertical wind -2 0 50 100 150 200 Figure 11. The wind speed set in computerTime(s) simulation.

Figure 11. The wind speed set in computer simulation. Figure 11. The wind speed set in computer simulation.

12 9 6 12 (m/s) 3 9 0 Northward wind Northward 6 0 50 100 150 200 (m/s) 3 0 6 Northward wind Northward 04 50 100 150 200 2 6 (m/s) 0 4

Eastward wind -2 2 0 50 100 150 200 (m/s) 0 -3 x 10 1 Eastward wind -2 0 50 100 150 200 0.5 -3 x0 10 1 (m/s) -0.5 0.5 Vertical wind Vertical -1 0 0 50 100 150 200 (m/s) -0.5

Vertical wind Vertical -1 1 0.80 50 100 150 200 0.6 10.4 0.80.2 0.6 0

Scale factor error Scale 0.4 0 50 100 150 200 0.2 Time(s) 0

Figure factor error Scale 0 12. The estimation50 results of100 wind speed and150 scale factor.200 Time(s) Figure4.2. Actual 12. The Flight estimation Veriﬁcation results of wind speed and scale factor.

To verify the dynamic performance of the system, a real flight experiment is carried out. 4.2. Actual Flight Verification FigureThe method 12. The estimation is verified onresults a hybrid of wind VTOL, speed HQ10. and scale The factor. experimental system is composed of an on-boardTo verify system the dynamic and a ground performance station system, of the assystem, shown a in real Figure flight 13 .experiment The flight control is carried 4.2.out.system Actual The with Flightmethod inertial Verification is measurementverified on a unithybrid (IMU) VTOL, and real-timeHQ10. The kinematic experimental (RTK) can system measure is com‐ posedtheTo accuracy verifyof an navigation theon‐ boarddynamic states,system performance and and give a ground the of control the stationsystem, command system, a real to HQ10.flight as shown experiment The ground in Figure displayis carried 13. The out.flightand The control control method system system is verified communicate with on inertial a hybrid with measurement the VTOL, flight controlHQ10. unit systemThe (IMU) experimental through and thereal system digital‐time radio.kinematic is com‐ posed of an on‐board system and a ground station system, as shown in Figure 13. The flight control system with inertial measurement unit (IMU) and real‐time kinematic

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(RTK) can measure the accuracy navigation states, and give the control command to Mathematics 2021, 9, 646 HQ10.(RTK) canThe measureground display the accuracy and control navigation system states, communicate and give with the thecontrol flight12 command of control 16 sys to‐ temHQ10. through The ground the digital display radio. and control system communicate with the flight control sys‐ tem through the digital radio.

On-board System IMU On-board System IMU Control RTK Control RTK Airspeed sensor Airspeed sensor Measure Measure Magnetometer Magnetometer Barometer Barometer Receiver Digital radio Receiver Digital radio

e C y t c d a o t n n e m y C e a S t

c d g t a o m r n m t n h m e e S a a g m n g i t m m r o m l d h E e F a C g m n i m o l d

E F C

Ground Station System Ground Station System

Figure 13. Block diagram of the experimental system. FigureFigure 13. 13.Block Block diagram diagram of the of experimentalthe experimental system. system. As shown in Figure 14, the HQ10 is equipped with four rotor motors with maximum AsAs shown shown in Figurein Figure 14, the14, HQ10the HQ10 is equipped is equipped with four with rotor four motors rotor withmotors maximum with maximum forceforce of of 30 30 kgf kgf and and a 150 a 150 cc gasoline cc gasoline engine. engine. The RTK The equipment RTK equipment and inertial and navigation inertial navigation force of 30 kgf and a 150 cc gasoline engine. The RTK equipment and inertial navigation systemsystem (INS) (INS) are are installed installed on the on airplanethe airplane and the and horizontal the horizontal positioning positioning accuracy accuracy is less is less thansystemthan one one meter.(INS) meter. Duringare Duringinstalled algorithm algorithm on the veriﬁcation, airplane verification, the and ground the the horizontal speed ground and attitudespeedpositioning and are obtained attitude accuracy are is lessob‐ bythantained the RTK-INSone by meter.the RTK integrated During‐INS integrated navigation algorithm systemnavigation verification, and thesystem the pitot ground and tube the data speed pitot is obtained tubeand dataattitude by the is obtained are ob‐ differentialbytained the differentialby pressure the RTK transducer pressure‐INS integrated transducer MS4525DO. navigation MS4525DO. system and the pitot tube data is obtained by the differential pressure transducer MS4525DO.

FigureFigure 14. 14.The The hybrid hybrid vertical vertical takeoff takeoff and landing and landing (VTOL) (VTOL) HQ10. HQ10. Figure 14. The hybrid vertical takeoff and landing (VTOL) HQ10. TheThe HQ10 HQ10 is commandedis commanded to fly to around fly around a fixed a pointfixed withpoint the with 350 the m radius 350 m at radius a at a constantconstantThe height. height.HQ10 The isThe EKFcommanded EKF is used is used when to when thefly aircraftaround the aircraft enters a fixed enters a fixed point a altitude fixed with altitude circle. the 350 Figure circle. m 15radius Figure at 15 a shows the roll, pitch and yaw angle throughout the flight. Due to the existence of the showsconstant the height. roll, pitchThe EKF and is yaw used angle when throughout the aircraft theenters flight. a fixed Due altitude to the existencecircle. Figure of the 15 windshows field, the the roll, aircraft pitch needs and to yaw periodically angle throughout adjust the roll the angle flight. when Due it hovers to the around existence a of the fixedwind point. field, the aircraft needs to periodically adjust the roll angle when it hovers around a fixedwind point.field, the aircraft needs to periodically adjust the roll angle when it hovers around a fixed point.

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50 30 1050

Roll anlge(°) -1030 0 50 100 150 200 10 Time(s)

Roll anlge(°) 12-10 0 50 100 150 200 10 Time(s) 128 10 Pitch anlge(°) 6 0 50 100 150 200 8 Time(s)

Pitch anlge(°) 4006 0 50 100 150 200 300 Time(s) 200 400 100300

Yaw anlge(°) 0 2000 50 100 150 200 100 Time(s)

Yaw anlge(°) 0 0 50 100 150 200 Time(s) Figure 15. Attitudes of the HQ10 during the hovering flight. Figure 15. Attitudes of the HQ10 during the hovering flight. FigureAs 15. shown Attitudes in Figure of the HQ1016, during during the the hovering flight. flight, the airspeed is maintained at a fixed valueAs shown and ground in Figure speed 16, during fluctuates the hovering with the flight, angle the of airspeed hovering is maintained flight based at on a the fixed value and ground speed fluctuates with the angle of hovering flight based on the airspeed.As shownThe estimation in Figure results 16, during are shown the hovering in Figures flight, 17 andthe airspeed18. The EKF is maintained estimator hasat a airspeed. The estimation results are shown in Figures 17 and 18. The EKF estimator has convergedfixedconverged value after and after 100 ground 100 s, s, and and speed the northward fluctuates and andwith eastward eastward the angle wind wind of speeds hovering speeds finally finallyflight converge basedconverge to on theto −airspeed.3.8− m/s3.8 m/s and The and 7.1 estimation 7.1 m/s, m/s, and and theresults the vertical vertical are shownwind wind isis in relativelyrelatively Figures small. 17small. and As As the18. the pitotThe pitot tubeEKF tube hasestimator been has been has calibratedconvergedcalibrated offline, after offline, 100 the the s,average average and the value northward of of scale scale factorand factor eastward is is approximately approximately wind speeds 1.2. 1.2. finally converge to −3.8 m/s and 7.1 m/s, and the vertical wind is relatively small. As the pitot tube has been calibrated35 offline, the average value of scale factor is approximately 1.2. Airspeed Ground speed 35 30 Airspeed Ground speed

30 25

Speed(m/s) 25 20 Speed(m/s)

20 15 0 50 100 150 200 Time(s) 15 Figure 16.0 The airspeed50 and ground speed100 during the actual150 ﬂight. 200 Figure 16. The airspeed and ground speed during the actual flight. Time(s)

Figure 16. The airspeed and ground speed during the actual flight.

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Mathematics 2021, 9, 646 14 of 16 2 0 -2 2 (m/s) -4 0 -6 Northward wind Northward -20 50 100 150 200 (m/s) -48 -6 Northward wind Northward 6 0 50 100 150 200 4

(m/s) 8 2 6

Eastward wind 0 40 50 100 150 200 (m/s) 0.042

Eastward wind 0 0.020 50 100 150 200

0.040 (m/s) 0.02 Vertical wind Vertical -0.02 0 50 100 150 200 0 (m/s) 1.8

Vertical wind Vertical -0.021.6 0 50 100 150 200 1.4 1.8 1.2 1.6 1 Scale factor error factor Scale 1.40 50 100 150 200 1.2 Time(s) 1 Scale factor error factor Scale 0 50 100 150 200 Figure 17. The actual estimation resultsTime(s) of wind speed and scale factor. Figure 17. The actual estimation results of wind speed and scale factor. Figure 17. The actual estimation results of wind speed and scale factor. 1 0.8 0.6 0.4 (m/s) 1 0.80.2 0 Northward wind 0.6 0 50 100 150 200 0.4 (m/s) 0.21 0.80 Northward wind 0.60 50 100 150 200 0.4

(m/s) 1 0.80.2

Eastward wind Eastward 0.60 0 50 100 150 200 0.4 (m/s) 0.21.2

Eastward wind Eastward 01 0 50 100 150 200 0.8

(m/s) 1.2 0.6 1 Vertical wind 0.4 0.80 50 100 150 200 (m/s) 0.60.4

Vertical wind 0.40.3 0 50 100 150 200 0.2 0.4 0.1 0.3 0 Scale factor error Scale factor 0.20 50 100 150 200 0.1 Time(s) 0 Figure error Scale factor 18.0 The state mean50 square error100 of extended150 Kalman ﬁlter (EKF).200 Time(s) Figure5. Conclusions 18. The state mean square error of extended Kalman filter (EKF).

In this article, an innovative method is proposed to estimate the calibration error of Figurethe pitot 18. The tube state and mean environmental square error of wind extended field. Kalman As the filter offline (EKF). calibration of the pitot tube using a wind tunnel does not consider the actual usage condition, the scale factor error needs to be re-calibrated in actual flight. Therefore, an extended Kalman filter is obtained by using the integrated navigation system and differential pressure transducer. Before experimental verification, observability analysis is researched to prove that hovering flight

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has a greater ability to estimate state variables. Computer simulation and experimental results show that the proposed method can estimate the scale factor of pitot tube and wind speed in circle flight. It can be seen from the actual flight results that the estimator can converge in about 100 s which is approximately equal to the time it takes for the aircraft to fly one lap. This method can optimize the measurement accuracy of the pitot tube on the basis of the offline calibration process and is more practical in combination with the actual flight conditions of the aircraft. However, the estimation algorithm discussed in this article is based on an accurate integrated navigation system. Therefore, the influence mechanism of navigation errors on wind field and scale factor estimation needs further research in future works.

Author Contributions: Conceptualization, Q.Z. and Y.X.; methodology, Q.Z.; software, Q.Z. and T.D.; validation, Q.Z., Y.X. and X.W.; formal analysis, Q.Z.; investigation, Q.Z.; resources, Y.X.; data curation, Y.X.; writing—original draft preparation, Q.Z.; writing—review and editing, Q.Z.; visualization, X.W.; supervision, Z.Y. and T.D.; project administration, X.W.; funding acquisition, X.W. All authors have read and agreed to the published version of the manuscript. Funding: The work described in this article was partially supported by Beijing Natural Science Foundation Project (4204103) and Aeronautical Science Foundation of China (2019ZC051009). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The authors are grateful to Xu Liang for providing the dataset used in the experiments. Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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