Flight Trajectory Simulation and Aerodynamic Parameter ...

Research ArticleFlight Trajectory Simulation and Aerodynamic ParameterIdentification of Large-Scale Parachute

Yihua Cao and Ning Wei

School of Aeronautical Science and Engineering, Beihang University, Beijing, China

Correspondence should be addressed to Ning Wei; [email protected]

Received 18 December 2019; Revised 18 March 2020; Accepted 16 May 2020; Published 25 August 2020

Academic Editor: Giacomo V. Iungo

Copyright © 2020 Yihua Cao and Ning Wei. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

In this paper, the multibody parachute-payload system is simplified and analyzed. A six-degree-of-freedom rigid body flightdynamic model is established to calculate the flight trajectory, attitude, velocity, and drop point of the parachute-payloadsystem. Secondly, the random interference factors that may be encountered in the actual airdrop test of the parachute system areanalyzed. According to the distribution law of the interference factors, they are introduced into the flight dynamic model. TheMonte Carlo method is used to simulate the target and predict the flight trajectory and landing point distribution of theparachute system. The simulation results can provide technical support and theoretical basis for the parachute airdrop test.Finally, the genetic algorithm is used to identify the aerodynamic parameters of the large-scale Disk-Gap-Band parachute. Thesimulation results are in good agreement with the test results, which shows that the research method proposed in this paper canbe applied to study practical engineering problems.

1. Introduction

Because of its high deceleration efficiency and reliability, aparachute is an important decelerator in the recovery system[1], which is widely used in spacecraft return, Mars probelanding, and other missions [2]. Flexible parachutes arevulnerable to the complex and varied external environmentduring the recovery process. Therefore, it is necessary tothoroughly study the performance of the parachute systemunder various possible conditions in order to ensure the highreliability and high safety requirements of the aerospace engi-neering tasks [3].

Common methods for studying the performance ofparachute systems include wind tunnel test, airdrop test,and computer simulation [4]. There are many limitations inwind tunnel test methods, which are mainly manifested inthe small size of the wind tunnel, the high pressure, and thedensity of the test airflow. The parachute with scale modelcannot guarantee the similarity of geometry and stiffnessfor a real parachute system. Therefore, accurate parachuteparameters required for engineering development cannot beobtained by wind tunnel testing alone. Airdrop test methods

can obtain some parachute performance parameters moreaccurately. However, their shortcomings are high test cost,long cycle, and limited numbers of tests, and they are subjectto meteorological conditions. So, it is difficult to examine thecharacteristics of the system under various extreme condi-tions. Moreover, the installation difficulty of the measuringequipment makes it impossible to measure some importantparachute system parameters. The working process of theparachute system is greatly affected by random environmen-tal factors. It is difficult to comprehensively evaluate theperformance of the parachute system only through testmethods. However, the simulation method can make up forthe deficiencies of the experimental methods. It is of greatsignificance for improving the design level, reducing thenumbers of tests, shortening the development cycle, savingdesign cost, and ensuring the safety of airdrop [5].

In this paper, a general trajectory simulation model of theparachute system airdrop test with random interferencefactors is established, and various deviation factors thatmay occur in the actual test are analyzed. A sufficient numberof simulation target practices with the Monte Carlo methodwere carried out to check the strength of the parachute and

HindawiInternational Journal of Aerospace EngineeringVolume 2020, Article ID 5603169, 13 pageshttps://doi.org/10.1155/2020/5603169

https://orcid.org/0000-0002-3985-640X

https://creativecommons.org/licenses/by/4.0/






https://doi.org/10.1155/2020/5603169

to verify the accuracy and dispersion of the random fallpoints. Therefore, it can provide experimental gist forimproving the precision of airdrop, reducing the dispersionof the fall point, and correcting flight trajectory. On the otherhand, measurable information for large parachutes is scarce,and their aerodynamic coefficients are difficult to obtain.Therefore, based on the actual airdrop test, the genetic algo-rithm is used to identify the aerodynamic parameters oflarge-scale Disk-Gap-Band parachute. This identificationmethod can be used to study practical problems inengineering.

2. Modeling

2.1. Parachute-Payload System Dynamic Model. Beforeestablishing a six-degree-of-freedom trajectory model of theparachute-payload system, the following assumptions aremade:

The parachute-payload system is composed of parachuteand payload body which are rigidly connected. The aerody-namic force of the parachute is at the geometric center ofthe parachute canopy. The drag area of the main parachuteincreases with time and is a function of time [6]. Ignoringthe inflation process of the guide parachute, only the aerody-namic parameter identification of the test parachute isconsidered.

Figure 1 is a schematic diagram of a parachute-payloadsystem, in which the parachute body coordinate system andthe geodetic coordinate system are defined. The parachutecoordinate system is oxyz, where the origin o is the geometriccenter of the parachute canopy, namely, the aerodynamicpressure center of the parachute. PointC is the center of massof the parachute-payload system, and the distance from coor-dinate origin o to point C isXg. The origin Od of the geodeticcoordinate system coincides with the projection point of thecenter of mass of the projectile at the initial time on the localhorizontal plane. The OdYd axis is vertically upward. TheOdXd and OdZd axes are in the local horizontal plane, point-ing north and east, respectively.

2.2. Dynamic Equation. Using the Lagrangian method, thebasic equations of motion of the parachute-payload system

can be obtained. The general form is as follows:

ddt

∂T∂V

� �+Ω × ∂T

∂V

� �= F, ð1Þ

ddt

∂T∂Ω

� �+Ω × ∂T

∂Ω

� �+V × ∂T

∂V

� �=M: ð2Þ

The above formulas are also known as the Kirchhoffequation, where T is the kinetic energy of the system, F andM are the external forces and moments acting on the para-chute system, respectively, and V and Ω, respectively, arethe system velocity and the angular velocity vector. In thebody coordinate system of the parachute system, formulas(1) and (2) are composed of the following forms, where(vx, vy, vz) and (ωx, ωy , ωz) are the components of the veloc-ity and angular velocity of the parachute system on each axisof its body coordinate system, respectively, [Ixx , Iyy, Izz] is therotational inertia of the parachute system on each axis, andmis the total mass of the system.

z

yd

xdod

zd

o

x

c

y y

xg

Figure 1: Coordinate system diagram.

F =

m + α11ð Þ vx:− m + α33ð Þ vyωz − vzωy

� �− mxg + α26� �

ω2y + ω2

z

� �

m + α33ð Þ vy:− vzωx

� �+ m + α11ð Þvxωz + mxg + α26

� �ωz: + ωxωy

� �m + α33ð Þ vz

: + vyωx

� �− m + α11ð Þvxωy − mxg + α26

� �ωy:− ωzωx

� �

26664

37775,

M =

Ixx ωx:

Izz + α66ð Þωy: + Ixx − Izz − α66ð Þωzωx − mxg + α26

� �vz:− vxwy + vyωx

� �+ α11 − α33ð Þvzvx

Izz + α66ð Þωz: + Ixx − Izz − α66ð Þωyωx − mxg + α26

� �vy: + vxwz − vzωx

� �− α11 − α33ð Þvyvx

2664

3775:

ð3Þ

2 International Journal of Aerospace Engineering

3. Method

3.1. Fall Point Prediction Method

3.1.1. Trajectory Deviation Factor. The parachute system issusceptible to various random environmental factors duringits operation, which results in some state parameters of thesystem deviation greatly from the design state. The extremeconditions caused by these deviations can maximize the peakopening force and dynamic pressure. Peak opening force isone of the important indexes to ensure the safety and reli-ability of the parachute system. It directly affects whetherthe parachute system can be opened normally or not andis related to the success or failure of the whole recoverymission [7].

Because the airdrop process includes many stages andinvolves many influencing factors, it is necessary to establisha multistage simulation model for analysis [8]. According tothe airdrop test procedure, the airdrop process is simplifiedto several stages as shown in Figure 2.

Due to the influence of various random factors, variousdeviations may occur in the design parameters of the system.These deviations will cause the deviation of trajectory, thusaffecting the landing position of the projectile. The distur-bance factors include initial condition deviation, experimen-tal model deviation, control parameter deviation, parachutedrag area deviation, and wind influence. The deviation of ini-tial conditions is composed of throwing speed and heighterror. Experimental model deviations include mass, inertia,and centroid bias. Control time and dynamic pressure errorsare contained in the parachute control condition deviation.Parachute drag area deviation involves the drag area devia-tion of the guided parachute, test parachute, reefing mainparachute, and full inflation main parachute. For randomwind, it is assumed that the wind speed obeys normal distri-bution and the wind direction is uniformly distributed in thehorizontal plane. Each interference factor in the simulation isa random variable that follows a normal distribution. There-fore, the standard normal distribution function randn can beused to obtain random variable samples with normal distri-bution in simulation.

3.1.2. Monte Carlo Method. Because of the large number ofinterference models involved, it is difficult to directly obtainthe influence of the variation factors on the trajectory withintheir deviation range by the analytical method.

Using the Monte Carlo method, the target test can besimulated many times by computer simulation. Each simu-lated target is considered as an actual airdrop test. The resultswith good convergence can be obtained by the simulationmethod, which provides a basis for the formulation of a testscheme and system design.

The Monte Carlo method, also known as a random sim-ulation method or random sampling method, is a classicalstatistical test method. Its basic theory is the law of largenumbers and the central limit theorem [9]. According tothe law of large numbers, the accuracy of the evaluationimproves as the number of simulations increases [10, 11].In parachute airdrop test trajectory simulation, it is usuallynecessary to determine the standard values and deviationsof each random factor and then use the random number gen-erator to generate the corresponding pseudorandom numbersequence. Then, it is substituted into the established parachutesystem dynamic model for calculation. Finally, by statisticalprocessing of the calculated results, the relatively accuratesimulation results of disturbed trajectory and landing pointprediction can be obtained. Using the Monte Carlo methodto simulate target, the following steps can be taken [9]:

(1) Analyzing the dynamic characteristics of theparachute-payload system and establishing the tra-jectory simulation mathematical model of the para-chute system

(2) Determining all kinds of random deviations andrandom interference factors in the process of an air-drop. The pseudorandom number sequence of eachrandom variable is generated according to the distri-bution type and statistical characteristics of randomfactors

(3) The Monte Carlo method is realized by usingMATLAB programming. The sampling values ofrandom variables are substituted into the dynamic

Initialposition Guided

parachuteseparation Test

parachuteejection Main

parachuteopening

Landing

Figure 2: Airdrop process.

3International Journal of Aerospace Engineering

0 50 100Longitudinal distance X (m)

150 200 250

-0.50

Lateral distance Z (m)0.5

-500

0

500

1000

1500

2000

Hei

ght Y

(m)

2500

3000

3500

Figure 3: Three-dimensional trajectory curve.

Longitudinal distance X (m)

Hei

ght Y

(m)

0 50 100 150 200 250−500

0

500

1000

1500

2000

2500

3000

3500

Figure 4: Longitudinal plane trajectory curve.

0 50 100 150 200 250 3005

10

15

20

25

30

35

40

45

50

Time (s)

Spee

d (m

/s)

Figure 5: Speed-time diagram.


model of the parachute system. A trajectory curveand a landing coordinate can be obtained by runningthe program once

(4) According to the required number of tests, the pro-gram is cyclically run n times so as to obtain samplingsimulation of random trajectory parameters, that is,simulated target

(5) The simulation results are statistically processed, andthe mean and variance of the distance between eachfall point and the ideal undisturbed landing pointare calculated to evaluate the scattering of the fallingpoint

3.2. Parameter Identification Method. Dynamic parameteridentification is essentially a complex nonlinear constraintoptimization problem. If the traditional optimization algo-rithm is used to solve, it will involve a series of problems,such as the gradient calculation of the function and the selec-tion of the initial value of the iteration. In addition, the large-scale Disk-Gap-Band parachute used for Mars explorationhas almost no measurable data, and its explicit identificationequation is difficult to establish. As a nondeterministic quasi-natural algorithm, a genetic algorithm is an effective methodto solve such optimization problems. It is a global heuristicoptimization algorithm with adaptive probability searchand low sensitivity to initial values. Because of its advantagessuch as simplicity, easy operation, and strong robustness, ithas been widely used in aerodynamic parameter identifica-tion of various aircraft. In this paper, it is applied to the iden-tification of the aerodynamic parameters of the large-scaleDisk-Gap-Band parachute, which solves the problem thatthe aerodynamic force of the parachute is difficult tocalculate.

The genetic algorithm is an iterative algorithm. It has aset of solutions at each iteration. This set of solutions is ini-tially randomly generated. At each iteration, a new set ofsolutions is generated by genetic operations that simulateevolution and inheritance. Each solution is evaluated by an

objective function, and one iteration becomes a generation.Typical algorithm steps are as follows:

(1) Initialize the population

(2) Calculate the fitness of each individual in thepopulation

(3) Select individuals who will enter the next generationaccording to a certain rule determined by the individ-ual fitness value

(4) Cross operation according to probability pc

(5) Perform mutation operation according to probabilitypm

(6) If some stopping conditions are not met, go to (2),otherwise, go to the next step

Time (s)

Hei

ght (

m/s

)

0 50 100 150 200 250 300-500

0

500

1000

1500

2000

2500

3000

3500

Figure 6: Height-time diagram.

Time (s)

Pitc

h an

gle (

°)

0 50 100 150 200−90

−80

−70

−60

−50

−40

−30

−20

−10

0

10

Figure 7: Pitch angle-time diagram.


(7) Output the chromosome with the best fitness in thepopulation as a satisfactory or optimal solution tothe problem

4. Simulation Results

4.1. Fall Point Prediction Results

4.1.1. Ideal Trajectory. In this paper, the airdrop test of a pro-jectile with Disk-Gap-Band parachute is taken as an exampleto simulate. The projectile is a cone-shaped head with a cen-tral cylindrical section and a cross-shaped wing at the tail.The aerodynamic data are obtained by CFD numerical simu-lation [12]. Given each characteristic parameter and a set of

standard launching conditions, the disturbance factors areset to zero to simulate, and the states of the parachute are cal-culated (or obtained) by solving the differential equations ofmotion. The ideal trajectory curve of the parachute-payloadsystem and the relationship between the velocity, height,pitching angle, and trajectory inclination with time areobtained, as shown in the following figures.

As can be seen from Figures 3 and 4, the deviation of theideal windless trajectory in the lateral distance (Z direction)is very small and can be neglected, so it can be consideredas a two-dimensional trajectory in the longitudinal symmetryplane. Figures 5 and 6 are graphs of the speed and heightchanges with time during descent, respectively. From thepitch angle curve (Figure 7), it can be seen that the pitch

Time (s)

Traj

ecto

ry in

clin

atio

n an

gle (

°)

0 50 100 150 200−90

−80

−70

−60

−50

−40

−30

−20

−10

0

Figure 8: Trajectory inclination angle-time diagram.


Late

ral d

istan

ce Z

(m)

−3000 −2000 −1000 0 1000 2000 3000−400

−300

−200

−100

0

100

200

300

400

o Ideal landing point Simulation landing point

Figure 9: Scattering map of the fall points.


attitude of the projectile changes more dramatically when theparachute is full, because the effect of the parachute on theprojectile is equivalent to applying a drag on the projectile’stail, thereby increasing the pitch moment of the projectile.The parachute opening process has a great influence on theattitude change of the projectile; but with the parachuteopening, the influence decreases gradually. Throughout the

process, the pitching attitudes of the projectile change fromhorizontal to near to vertical direction as shown in Figure 8.

4.1.2. Disturbed Trajectory. The random disturbance value ofeach deviation factor is added to the ideal trajectory for sim-ulation. The simulation result of each trajectory is equivalentto the actual trajectory of the airdrop test. After 1000

−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500

0.0010.003

0.010.020.050.10

0.25

0.50

0.75

0.900.950.980.99

0.9970.999

Prob

abili

ty

Normal probability plot


Normal distribution line+ Simulation landing point

Figure 10: Fall point longitudinal coordinate normality test.

-300 −200 −100 0 100 200 300

0.0010.003

0.010.020.050.10

0.25

0.50

0.75

0.900.950.980.99

0.9970.999

Prob

abili

ty


Later al distance Z (m)

Simulation landing point +Normal distribution line

Figure 11: Fall point lateral coordinate normality test.


simulations were repeated, the scattering map of the fallpoints is obtained as shown in Figure 9. The horizontal andvertical coordinates of each fall point are statistically com-pared with the ideal fall point to get the scattering rule ofthe fall point. As can be seen from the figure, the maximumvertical scattering does not exceed 3000 meters, and the max-imum lateral scattering does not exceed 400 meters. The sim-ulation trajectory landing points are evenly distributedaround the ideal trajectory landing point.

The estimated value of the circular error probability(CEP) after 1000 simulated targets was calculated as CEP =779:0536m.

The longitudinal (x-axis) position circular error proba-bility of the simulated fall points is CEP = 794:5499m. Themean value of the longitudinal deviation Δx between the sim-ulated trajectory landing points and the ideal landing point is300m, and the mean square error is 1084m. The normal dis-tribution of the longitudinal coordinates X of the falling

Simulation serial number

0 100 200 300 400 500 600 700 800 900 10000

500

1000

1500

2000

2500

Land

ing

dista

nce d

evia

tion

(m)

Figure 12: Deviation of landing distance.

Landing distance deviation

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0.0010.003

0.010.020.050.10

0.25

0.50

0.75

0.900.950.980.99

0.9970.999

Prob

abili

ty


+Normal distribution lineLanding distance deviation

Figure 13: Falling-point deviation normality test.


points is tested, and the results are shown in Figure 10. It canbe seen that the longitudinal positions of the falling pointsconform to the normal distribution well.

The lateral (z-axis) position circular error probability ofthe simulated fall points isCEP = 115:2857m. The mean valueof the lateral deviation Δz between the simulated trajectorylanding points and the ideal landing point is 3m, and themean square error is 153m. The normal distribution of the lat-eral coordinates Z of the falling points is tested, and the resultsare shown in Figure 11. It can be seen that the lateral positionsof the falling points conform to the normal distribution well.

Calculating the deviation between the falling points ofeach simulation and the ideal falling point and plot the

falling-point deviations of 1000 simulations on the samepicture are plotted as shown in Figure 12.

It can be seen from Figure 12 that the distance deviationsbetween the simulated falling points and the ideal fallingpoint are mainly concentrated within 500 meters, the maxi-mum deviation is about 2000 meters, the average value ofthe distances is 892.5002m, and the standard deviation is641.1984m. The normality test is performed on the falling-point deviations, as shown in Figure 13, and it can be seenthat the deviations of the landing points are in good agree-ment with the normal distribution law.

As mentioned above, the ejection condition of the testparachute is that the dynamic pressure reaches a fixed value,

0 10 20 30 40 50 60 70 80 900

100

200

300

400

500

600

Angle of attack (º)

Tim

es

Figure 14: Angle of attack distribution histogram when the test parachute is ejected.

0−40

−35

−30

−25

−20

−15

−10

Velo

city

in Y

dire

ctio

n (m

/s) −5

0

5

2 4 6 8 10Time (s)

12 14 16 18 20

Test dataSimulation results

Figure 15: Velocity of test parachute in Y direction.


and the flight angle of attack of the parachute system affectsthe measurement of the dynamic pressure by the pitot tube.Therefore, it is necessary to check the distribution of theflight angle of attack of the parachute system when it reachesthe dynamic pressure of the ejection. Figure 14 is the angle ofattack distribution histogram when dynamic pressure is750Pa measured by the pitot tube.

It can be seen from the figure that in about 90% of 1000simulations, the angle of attack at the time of the test para-chute ejection is less than 20°. In rare cases, the angle of attackis even greater than 80°. If the angle of attack is too large, thepitot tube measurement may not be accurate, so theparachute-payload system model needs to be furtherimproved.

4.2. Parameter Identification Results. The aerodynamicparameter identification of parachute is to estimate the aero-dynamic coefficient of the parachute by establishing the sys-tem mathematical model based on the test data obtainedfrom the airdrop test. In this paper, based on the abovedynamic model, the parameters of test parachute are identi-fied by stages based on a genetic algorithm [13].

4.2.1. Steady Descent Stage. During the steady descent stage,the test parachute is fully opened. For simplicity, assumingthat its drag coefficient is a constant, written as

CT = const = p 1ð Þ, ð4Þ

where CT is the drag coefficient and pð1Þ is the parameter tobe identified. The initial parameters of the genetic algorithmare set as follows: the population size is 100, the generationnumber is 100, the crossover probability is 0.7, and the muta-

tion probability is 0.001. Taking the Y-direction velocity asthe observation measurement, the identification result is

CT = p 1ð Þ = 0:7616: ð5Þ

The identification result is substituted into the programfor simulation, and comparing it with the test data, as shownin Figure 15.

It can be seen from Figure 15 that the simulation results ina steady descent stage are basically consistent with the testdata, which can verify that the identification model is accurate.

2−2

0

2

4

6

8

10

12

14

16× 104

2.5 3 3.5 4Time (s)

Pull

of re

ar ca

bin

4.5 5 5.5 6


Figure 16: Pull of rear cabin-time diagram.


120

1

2Pull

of re

ar ca

bin

3

4

5

6× 104

13 14 15 16Time (s)

17 18 19 20



4.2.2. Test Parachute Inflation Stage. During inflation, theaerodynamic coefficient of the test parachute is expressed asstatic aerodynamic coefficient multiplied by an inflation fac-tor KI , which is written as [14]

KI = p 2ð Þ × t − tLStFI − tLS

� �p 3ð Þ: ð6Þ

In the formula, tLS is the moment of rope straightening,tFI is the moment of full inflation, and pð2Þ and pð3Þ arethe parameters to be identified. Taking the pull of rear cabinas the observation measurement, the identification result is

p 2ð Þ = 1:191,p 3ð Þ = 4:393:

ð7Þ


It can be seen from Figure 16 that the simulation resultsof aerodynamic force in the inflation stage are basically con-sistent with the test, which proves that the identified inflationfactor has a reference value.

4.2.3. Unloading Stage. During the unloading stage, the pro-jectile nose suddenly separated from the rear cabin, and theload weight of the test parachute decreased momentarily,which caused the drag coefficient of the test parachute to sud-denly decrease and then gradually recover. The aerodynamiccoefficient in this process is expressed as the static aerody-namic coefficient multiplied by an unloading factor Koff andwritten as

Koff = 1 − 1 − p 4ð Þð Þe− t−tsepð Þ/p 5ð Þ, ð8Þ

where tsep is the separation time and pð4Þ and pð5Þ are theparameters to be identified. Taking the pull of rear cabin as theobservation measurement, the identification result is

p 4ð Þ = 0:747,p 5ð Þ = 1:093:

ð9Þ


It can be seen from Figure 17 that the unloading stagelasted for about 2 seconds, and the simulation results of thepull are basically consistent with the test data.

The five parameters identified above were used to simu-late the entire process of the test parachute from ejection tounloading, and compared with the test data, the results areas shown in Figures 18 and 19.

As can be seen from Figures 18 and 19, the simulationresults agree well with the experimental data, which verifiesthe accuracy of the identification results.

5. Conclusion

(1) Based on the simplified analysis of the multibodyparachute-payload system, a six-degree-of-freedomrigid body flight dynamic model and a landing-point calculation model are established. Thesemodels can satisfy the requirements of solving thetrajectory, attitude, velocity, and landing-point dis-persion of the multibody parachute-payload system

3350

3300

3250

3200

3150

Hig

ht (m

)3100

3050

3000

2950

29000 2 4 6

Time (s)8 10 12


Figure 18: Height-time diagram.


(2) Aerodynamic data of the parachute-payload systemare obtained by CFD numerical simulation. Randominterference factors of the parachute-payload systemin an airdrop test are synthetically analyzed. Basedon the MATLAB simulation tool and Monte Carlomethod, a method of flight trajectory simulation andmodel construction under interference effect is pro-posed, which can effectively solve the dispersion oflanding point under effects of various deviation factors

(3) The numerical simulation method of parachute-payload system flight trajectory proposed in thispaper can provide simulation means for the researchand verification of parachute-payload system flighttrajectory and precise landing at expected points. Italso has reference significance for general parachuteairdrop technology and multibody parachute systemsimulation

(4) A method for identifying aerodynamic coefficients oflarge Disk-Gap-Band parachute based on a geneticalgorithm is proposed. This method can estimatethe aerodynamic parameters of the parachute basedon the airdrop test data and solves the problem thatthe aerodynamic force of the parachute is difficultto calculate. This identification model can solve thecomplex problem of fluid-structure coupling calcula-tion and can be used in engineering practice

Nomenclature

T : Kinetic energyF: External forcesM: External momentsx, y, z: Components of the position

vx, vy, vz : Components of the velocityωx, ωy, ωz : Components of the angular velocityIxx, Iyy, Izz : Rotational inertiaα: Additional massm: Total massCEP: Circular error probability.

Data Availability

The data used to support the findings of this study are avail-able from the corresponding author upon request.

Conflicts of Interest

The authors declare no conflict of interest.

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