Non-Propellant Eddy Current Brake and Traction in Space Using Magnetic Pulses

aerospace

Article Non-Propellant Eddy Current Brake and Traction in Space Using Magnetic Pulses

Yi Zhang †, Qiang Shen †, Liqiang Hou † and Shufan Wu *

School of Aeronautics and Astronautics, Shanghai Jiao Tong University, No. 800, Dongchuan Road, Minhang District, Shanghai 200240, China; [email protected] (Y.Z.); [email protected] (Q.S.); [email protected] (L.H.) * Correspondence: [email protected]; Tel.: +86-34208597 † These authors contributed equally to this work.

Abstract: The safety of on-orbit satellites is threatened by space debris with large residual angular velocity and the space debris removal is becoming more challenging than before. This paper explores the non-contact despinning and traction problem of high-speed rotating targets and proposes an eddy current brake and traction technology for space targets without any propellant consumption. The principle of the conventional eddy current brake is analyzed in this article and the concept of eddy current brake and traction without propellant is put forward for the first time. Secondly, according to the key technical requirements, a brand-new structure of a satellite generating artificial magnetic field is designed accordingly. Then the control mechanism of eddy current brake and traction without propellant is analyzed qualitatively by simplifying the model and conditions. Then, the magnetic pulse control method is proposed and analyzed quantitatively. Finally, the feasibility of the technology is verified by the numerical simulation method. According to the simulation results, the eddy current brake and traction technology based on magnetic pulses can make the angular speed of target decrease linearly without propellant during the process. This technology has huge advantages compared with conventional eddy current brake technology in terms of efficiency and Citation: Zhang, Y.; Shen, Q.; Hou, reduced propellant consumption. L.; Wu, S. Non-Propellant Eddy Current Brake and Traction in Space Keywords: eddy current; debris removal; electromagnetic traction; despinning Using Magnetic Pulses. Aerospace 2021, 8, 24. https://doi.org/10.3390/ aerospace8020024 1. Introduction Academic Editor: George Z.H. Zhu With the development of aerospace technology, human activities in space are becoming Received: 8 December 2020 more frequent. The space debris left by human beings in space, especially in the Earth’s Accepted: 18 January 2021 Published: 20 January 2021 orbit, is increasing at a tremendous rate because of the rapid deployment of large scale low earth orbit constellations, such as the plan of more than 20,000 satellites proposed by two

Publisher’s Note: MDPI stays neutral dozen companies [1]. Space debris mainly consists of abandoned satellites, rocket final with regard to jurisdictional claims in stage devices and collision debris [2]. According to the results of computer simulation, published maps and institutional affil- even if we were to stop all space activities and comply with the 25-year removal guidelines, iations. the amount of space debris will continue to grow, and the density of space debris will increase as well due to random collisions. This transitive phenomenon is called the “Kessler phenomenon” [3]. Actively removing orbital debris is the only feasible solution to control the spread of debris in the future. Unlike satellites that operate stably in orbit, the motion state of space debris is irregular Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. and it is likely that there is a large residual angular momentum due to passivation, collision, This article is an open access article environmental torque, etc. Existing contact debris despinning technologies such as flexible distributed under the terms and brush [4,5], mechanical pulse [6], space manipulator technology or deceleration brush conditions of the Creative Commons despinning technology relying on manipulators [7] require a relatively static situation Attribution (CC BY) license (https:// between the service satellite and the target. This requires extremely high accuracy in creativecommons.org/licenses/by/ modeling and control and is prone to collision risks. For irregular targets that rotate at high 4.0/). speed, there is no way to make safe contact. However, the more researched space rope

Aerospace 2021, 8, 24. https://doi.org/10.3390/aerospace8020024 https://www.mdpi.com/journal/aerospace Aerospace 2021, 8, 24 2 of 16

net method [8] also has unsafe factors such as excessive force at the moment of contact. Non-contact despinning technology is a hot topic that has emerged in recent years. Its application methods include the compressed gas method [9], ion beam method [10,11], electrostatic force method [12] and laser ablation method [13,14] and other new concepts and methods. Electricity, magnetism or jet-flow are usually adopted to generate despinning torque. Since there is no direct contact, the magnitude of the despinning torque is usually small, at the level of mN·m. Therefore, the target has little impact on the service satellite and robotic arm during the process of despinning and the time required is usually at the level of hundreds of hours. During this period, the position and attitude accuracy requirements are low, and the effective working distance has a large range of variation. In order to overcome the shortcomings of the existing contact and non-contact despinning technologies, it is necessary to seek a more effective non-contact despinning method. Electromagnetic eddy current despinning is a method of despinning based on the electromagnetic eddy current damping effect. It is a non-contact depinning method that uses the damping moment generated by the rotating conductor to cut the magnetic line of induction in an artificial magnetic field. This torque is also called the eddy current torque. The earliest systematic study of eddy current torque came from NASA’s study on the rotation characteristics of a near-Earth satellite in the Earth’s magnetic field in 1965 [15]. It was the first work to put forward the inherent characteristics of conductors-the concept of electromagnetic tensor, and give the eddy current torque expression. Due to the weak geomagnetic effect, the ultimate research goal was to explore the long-term effect of the average eddy current induced by the geomagnetism on the satellite. Later, researchers mostly carried out research and experiments on the basis of this theoretical model. The concept of eddy current despinning was formally proposed in the 1980s. Scientists conducted more in-depth research on the eddy current torque generated by rotating magnets induced by artificial electromagnetic fields. In 1995, the American scientist Kadaba [16] conducted an engineering feasibility study on eddy current despinning for spacecrft, and proposed an electromagnetic eddy current despinning scheme based on high-temperature superconducting coils. In the past decade, the need for space debris removal has become increasingly urgent, and new concepts including electromagnetic eddy current despinning methods have once again become research hotspots. Sugai studied an electromagnetic brake system based on a robotic arm [17], using a smart robotic arm to carry electromagnetic brake pads close to the target surface for damping braking, which is similar to the method of the robot arm decelerating brush proposed by the Japan Aerospace Agency in 2006 [7]. In 2019, Xie et al. designed new permanent magnet array brake pads and dual manipulator structure and established dynamic model further verifying the effectiveness of this approach [18]. However, the distance between the eddy current brake and the target is limited to a very small range in above method, which cannot really solve the collision safety problem of contact despinning. In 2015, Ortiz Gómez et al. at the University of Southampton used Ariane H10 final stage despinning as an application scenario, focusing on explaining the mechanism of eddy current depinning [19,20]. and they constructed a standardized eddy current force and eddy current torque model. In 2019, Liu et al. conducted dynamic modeling for eddy current brake in three-dimensional environment on the basis of existing researches, and designed nonlinear sliding model controller, which further verified the feasibility of eddy current despinning technology [21]. However, the existing eddy current despinning technology still relies on the thrust device of the service satellite to maintain the relative position, which often lasts as long as several days. Moreover, the efficiency depends on the target speed. When the target speed drops, the despinning efficiency will be significantly reduced. The above problems greatly limit the application process of eddy current despinning technology. In order to break through the bottleneck of the application of eddy current despinning technology, this paper develop a non-propellant eddy current despinning and traction technology based on magnetic pulses. Compared with [19–21], Aerospace 2021, 8, 24 3 of 16

the proposed method makes the despinning process get rid of the propellant limitation, and enables the target speed to decrease steadily. This article is outlined as follows. In the second section, the article introduces the technical details of the non-propellant eddy current brake and traction technology based on magnetic pulse, including basic theory (Section 2.1), device design (Section 2.2), analysis method (Section 2.3) And control strategy (Section 2.4). In the third section, the simulation results of this technology are shown. The effectiveness and superiority of the technology proposed in this paper are proved from three aspects of control strategy (Section 3.1), efﬁciency (Section 3.2) and traction control effect (Section 3.3).

2. Non-Propellant Eddy Current Brake and Traction Using Magnetic Pulse In this section, non-propellant eddy current brake and traction technology based on magnetic pulse is introduced in detail. Firstly, the origin of the concept of non-propellant eddy current brake is explained. Secondly, according to the key technical details, an overall plan to realize the technology is designed and the model is simpliﬁed in order to analyze the technical principles. Finally, a unique magnetic pulse control method is proposed.

2.1. Basic Theory Natalia Ortiz Gómez et al. analyzed the mechanism of eddy current despinning in detail in [19,20], and studied the traction problem under the conventional mode. According to the derivation results in the literature, the object rotating in a ﬁxed magnetic ﬁeld can be damped by the eddy current torque T. The eddy current torque calculation equation is shown as follows: T = µe f f M(w × BGt) × BGt (1) where, w is the relative rotation angular velocity vector between the target and serving satellite, M is the electromagnetic tensor of the target, which is the inherent electromagnetic property of the conductor target, determined by the material and structure of the conductor. The target selected in this paper is a metal spherical shell, and its electromagnetic tensor is:

4 M = 2πσtRt etE/3 (2)

where, M is the third-order diagonal matrix, E third-order unit matrix, σt is the conductivity of the target conductive material, Rt is the radius of the spherical shell, et is the thickness of the spherical shell. µeff is the effective factor of non-uniform magnetic field, used to correct the eddy current torque error caused by non-uniform magnetic field. The object rotating in a fixed magnetic field is also affected by the eddy current force F, so that the angular momentum conservation of the system can be satisfied. The eddy current force calculation equation is shown as follows:

F = µe f f MΛGt(w × BGt) (3)

where, ΛGt represents the Jacobian matrix of the magnetic ﬁeld, the matrix equation is shown as follows:

  −2 0 0   0 1 0  3µ0 Λ = m  0 1 0  + m⊥ 1 0 0  (4) Gt 4πr4 // 0 0 1 0 0 0

where, m is the magnetic moment of the coil, µ0 is permeability of vacuum, r is the distance between target and service satellite.

2.2. Device Design This section ﬁrst analyzes the principle of the conventional eddy current despinning method. Then, in order to further improve the efﬁciency and reduce propellant consump- Aerospace 2021, 8, x FOR PEER REVIEW 4 of 17

Aerospace 2021, 8, 24 where, m is the magnetic moment of the coil, μ0 is permeability of vacuum, r is the distance 4 of 16 between target and service satellite.

2.2. Device Design tion,This we proposedsection first the analyzes concept the principle of eddy currentof the conventional despinning eddy and current traction despinning based on a variable method. Then, in order to further improve the efficiency and reduce propellant consump- magnetic field. Finally, a new device was designed according to the requirements. tion, we proposed the concept of eddy current despinning and traction based on a variable magneticThe field. conventional Finally, a new eddy device current was designed despinning according mode to the is shown requirements. in Figure 1. The service satelliteThe appliesconventional a constant eddy current magnetic despinning field tomode the target,is shown so in that Figure the 1. target The service is subject to eddy currentsatellite applies damping a constant torque. magnetic Eddy field current to the forces target, areso that generated the target atis subject the same to eddy time, satisfying thecurrent angular damping momentum torque. Eddy conservation. current forces are However, generated the at the eddy same current time, satisfying force generated by thethe angular coupling momentum of the target conservation. angular However, velocity the and eddy the current applied force magneticgenerated by field the is regarded ascoupling a non-linear of the target interference angular velocity and needs and the to applied be offset magnetic by the fie attitudeld is regarded orbit controlleras a of the non-linear interference and needs to be offset by the attitude orbit controller of the service servicesatellite. satellite.

FigureFigure 1. 1. SchematicSchematic diagram diagram of conventional of conventional mode. mode.

AccordingAccording to toEquation Equation (3), the (3), eddy the current eddy current force is mainly forceis controlled mainly controlledby the relative by the relative angularangular velocity velocity and and magnetic magnetic field fieldstrength. strength. The relative The angular relative ve angularlocity is essentially velocityis essentially the angular velocity of the target relative to the static magnetic field. When the magnetic the angular velocity of the target relative to the static magnetic field. When the magnetic field is variable, the relative angular velocity is determined by the angular velocity of the fieldtarget is and variable, the angular the velocity relative of angular the magnetic velocity field.is As determined long as the hardware by the angular conditions velocity of the targetpermit, and the variable the angular magnetic velocity field is ofable the to magneticdominate the field. direction As long and magnitude as the hardware of the conditions permit,target eddy the current variable force, magnetic achieving field arbitrary is able control to dominate of the target. the direction and magnitude of the targetFigure eddy 2 currentshows our force, newly achieving designed despinning arbitrary controldevice. According of the target. to the concept of eddyFigure current2 brakeshows using our variable newly magnetic designed field, despinning the despinning device. device According needs to tobe the concept ofequipped eddy currentwith at least brake three using non-coplanar variable magnetic magnetic field field, generators the despinning to meet the needs device of needs to be generating arbitrary magnetic fields. In order to make the distance between the satellite equippedand the target with adjustable, at least the three despinning non-coplanar device needs magnetic to have field a telescopic generators structure. to meet Due the needs of generatingto the application arbitrary of superconducting magnetic fields. technology, In order light to makeand heat the shielding distance needs between to be the satellite anddesigned the targetfor the adjustable, magnetic generators, the despinning thereby devicereducing needs the power to have consumption a telescopic of the structure. Due tosatellite. the application In addition, the of superconductingstorage, transportation technology, and launch lightof satellites and heatneed to shielding meet ac- needs to be designedtual conditions, for theand magneticthe despinning generators, device should thereby also have reducing a foldable the design. power It can consumption main- of the satellite.tain the folded In addition, state during the the storage, storage, tr transportationansportation and and launch launch stages, of enhancing satellites the need to meet structural strength. actual conditions, and the despinning device should also have a foldable design. It can Aerospace 2021, 8, x FOR PEER REVIEW 5 of 17 maintain the folded state during the storage, transportation and launch stages, enhancing

the structural strength.

FigureFigure 2.2. Diagram of of service service star star structure. structure. The electromagnetic coil structure in the Figure 2 is able to generate any magnetic field at the target. The coil realizes the adjustment of the working range and the storage function through the adjustable telescopic rod. The triangular column structure in the picture can be retracted into the satellite star. The rotation nodes on the cylinder work with the slide rails together so that the coil structure can also be incorporated into the satellite star during launch. Figure 3 shows the fully folded service star structure.

Figure 3. Diagram of service satellite folding state.

In addition, the coil structure should also be specially treated as shown in Figure 4 so as to better shield the light and heat interference.

Figure 4. Diagram of the shielding structure. Located in the center of the connecting rod in the Figure 4 is the superconducting cable and the coolant channel. The wire is gray, and the coolant path and loop are green and blue respectively. The black part of the primary connecting rod is the supporting

AerospaceAerospace 2021 2021, 8,, x8 ,FOR x FOR PEER PEER REVIEW REVIEW 5 of5 of17 17

Aerospace 2021, 8, 24 5 of 16 FigureFigure 2. 2.Diagram Diagram of ofservice service star star structure. structure.

TheThe electromagnetic electromagneticelectromagnetic coil coilcoil structure structure structure in in the the FigureFigure Figure2 2is is2 able isable able to to generate togenerate generate any any magneticany magnetic magnetic ﬁeld fieldfieldat theat atthe target. the target. target. The The coil The coil realizes coil realizes realizes the adjustment the the ad adjustmentjustment of the of working ofthe the working working range range and range the and storageand the the storage function storage functionfunctionthrough through thethrough adjustable the the adjustable adjustable telescopic telescopic telescopic rod. The rod. rod. triangular The The triangular triangular column column column structure structure structure in the in picture inthe the pic- pic- can tureturebe can retracted can be be retracted retracted into the into satelliteinto the the satellite star.satellite The star. star. rotation The The rotation nodesrotation onnodes nodes the cylinderon on the the cylinder workcylinder with work work the with slidewith thetherails slide slide together rails rails together sotogether that theso so that coil that structurethe the coil coil struct canstruct alsoureure can be can incorporated also also be be incorporated incorporated into the satellite into into the the star satellite satellite during starstarlaunch. during during Figure launch. launch.3 shows Figure Figure the 3 shows3 fully shows folded the the fully servicefully folded folded star service structure.service star star structure. structure.

FigureFigure 3. 3.Diagram Diagram of of service service satellite satellite folding folding state. state.

InIn addition, addition, the the coil coil structure structure should should also also be be specially specially treated treated as as shown shown in ininFigure Figure 4 4 so4 so so asas to tobetter better shield shield the the light light and and heat heat interference. interference.

FigureFigure 4. 4.Diagram Diagram of ofthe the shielding shielding structure. structure.

LocatedLocated in in inthe thethe center center center of of ofthe the connecting connecting rodro rod in din the inthe the Figure Figure Figure4 is 4 the is4 isthe superconducting the superconducting superconducting cable cablecableand and the and the coolant the coolant coolant channel. channel. channel. The The wireThe wire wire is gray,is isgr gray, anday, and theand the coolant the coolant coolant path path path and and loopand loop loop are are green are green green and andandblue blue blue respectively. respectively. respectively. The The black The black partblack part of part the of primary ofthe the primary primary connecting connecting connecting rod is therod rod supportingis isthe the supporting supporting structure of the connecting rod, including the column core and the column shell. The thicker black part of the secondary connecting rod is the column shell, and the primary connecting rod is wrapped in the middle. The brown part of the connecting rod section in the picture and the blue outer cover of the coil are light and heat shielding layers.

2.3. Model Simplified Analysis 2.3.1. Simplification of Satellite Model Simplifying the problem in the three-dimensional space to the two-dimensional plane for research is a common problem analysis strategy. This section simplifies the despinning problem and mechanism, as shown in Figure5. AerospaceAerospace 2021 2021, 8, ,8 x, xFOR FOR PEER PEER REVIEW REVIEW 6 6of of 17 17

structurestructure of of the the connecting connecting rod, rod, including including th thee column column core core and and the the column column shell. shell. The The thickerthicker black black part part of of the the secondary secondary connecting connecting rod rod is is the the column column shell, shell, and and the the primary primary connectingconnecting rod rod is is wrapped wrapped in in the the middle. middle. The The br brownown part part of of the the connecting connecting rod rod section section in in thethe picture picture and and the the blue blue outer outer cover cover of of the the coil coil are are light light and and heat heat shielding shielding layers. layers.

2.3.2.3. Model Model Simplified Simplified Analysis Analysis 2.3.1.2.3.1. Simplification Simplification of of Satellite Satellite Model Model

Aerospace 2021, 8, 24 SimplifyingSimplifying thethe problemproblem inin thethe three-dimensionalthree-dimensional spacespace toto thethe two-dimensionaltwo-dimensional6 of 16 planeplane for for research research is is a a common common problem problem analysis analysis strategy. strategy. This This section section simplifies simplifies the the despinningdespinning problem problem and and mechanism, mechanism, as as shown shown in in Figure Figure 5. 5.

FigureFigure 5. 5. SimpliﬁedSimplified Simplified model. model.

TheThe service service satellitesatellite satellite has has twotwo two connectingconnecting connecting rodsro rodsds andand and magneticmagnetic magnetic fieldfield field generatorsgenerators generators placedplaced placed perpendicularperpendicular toto to eacheach each other,other, other, whichwhich which cancan can generategenerate generate aa a magneticmagnetic magnetic fieldfield field ofof of anyany any sizesize size andand and direction direction ininin the the xOy xOy plane plane at atat the thethe target. target.target. The The The target target target rotation rotation rotation axis axis axis is is isalways always always parallel parallel parallel to to theto the the z z-axis-axis z-axis di- di- direction.rection.rection. According According According to to to Equation Equation Equation (3), (3), it it can can be be seen seen that thatthat the thethe eddy eddyeddy current currentcurrent force force on on the the target target isisis inin in thethe the xOy xOy plane.plane. plane. WhenWhen When thethe the eddyeddy eddy currentcurrent current forceforce force FF F isis is given,given, given, itit it isis is notnot not difficultdifficult difficult toto to findfind find fromfrom from FigureFigure6 6 that6 that that thethe the magneticmagnetic magnetic fieldfield field needsneeds needs toto to rotaterotate rotate in in in a a specificaspecific specific direction direction direction at at at a a specifica specific specific angular angular angular velocityvelocity toto to meetmeet meet the the control control requirementsrequirements requirements accordingacco accordingrding toto to EquationEquation Equation (3).(3). (3). TheThe The magneticmagnetic magnetic fieldfield field atat at this time is a variable periodic function, and the period is ∆T . thisthis time time is is a a variable variable periodic periodic function, function, and and the the period period is is Δ ΔTiTi.i .

FigureFigure 6. 6. Diagram Diagram of of instantaneousinstantaneou instantaneous s eddy eddy current current force.force. force.

The magnetic field components in each direction have the following mathematical expressions:

= = ··· Bix Bi(θ) cos wBi t i 1, 2, 3 = = ··· (5) Biy Bi(θ) sin wBi t i 1, 2, 3

where: Bi θ = θ0 Bi(θ) = (6) 0 θ 6= θ0

2.3.2. Simplification of Superconducting Coil Loop The last section gives the expression form of the ideal magnetic field, and the period ∆Ti under ideal conditions can be infinitely small. In reality, charging and discharging of Aerospace 2021, 8, x FOR PEER REVIEW 7 of 17

The magnetic field components in each direction have the following mathematical expressions: B = B ()θ cos()w t i =1,2,3 ix i Bi (5) B = B ()θ sin()w t i =1,2,3 iy i Bi

where: θ =θ B 0 B ()θ = i i  θ ≠ θ (6) 0 0

2.3.2. Simplification of Superconducting Coil Loop The last section gives the expression form of the ideal magnetic field, and the period ΔTi under ideal conditions can be infinitely small. In reality, charging and discharging of Aerospace 2021, 8, 24 7 of 16 the inductance is limited because of the inductive reactance of the magnetic field generators. The expression of the ideal magnetic field in Equation (5) needs to be corrected to Equation (7): the inductance is limited becauseB = ofB the()θ inductivecos()w t reactancei =1,2,3 of the magnetic ﬁeld gener- ix i Bi ators. The expression of the ideal magnetic ﬁeld in Equation (5) needs to be corrected to(7) Equation (7): B = B ()θ sin()w t i =1,2,3 iy i Bi = = ··· Bix Bi(θ) cos wBi t i 1, 2, 3 where: = = ··· (7) Biy Bi(θ) sin wBi t i 1, 2, 3 where: θ −θ ≤ τ Bi 0 k wB ()θ = | − | ≤ i Bi Bi θ θ0 kτwBi (8) Bi(θ) = θ −θ > τ (8) 0 − >0 k wB 0  |θ θ0| kτwBi i

where,where, kkττwwBiBi representsrepresents the the angular angular displacement displacement in ink timesk times the thecharge charge and anddischarge discharge time, time,τ is theτ is time the constant time constant of the of charge the charge and disc andharge discharge when whenthe total the magnetic total magnetic pulse pulseis gener- is generated,ated, and w andBi isw theBi isangular the angular velocity velocity of the of magnetic the magnetic field. ﬁeld.As shown As shown in Figure in Figure 7, the7 width, the widthof pulse of pulseneeds needsto meet to the meet charging the charging and disc andharging discharging time of time the ofcoil, the and coil, the and angle the anglewidth widthof pulse of pulseθ shouldθ should not exceed not exceed a certain a certain radian. radian. Theref Therefore,ore, it is necessary it is necessary to further to further study study the exact relationship between w and the charge-discharge time constant τ. the exact relationship between wBi andBi the charge-discharge time constant τ.

FigureFigure 7. 7.Magnetic Magnetic pulse.pulse.

Aerospace 2021, 8, x FOR PEER REVIEW InIn thethe simplifiedsimplified model,model, thethe magneticmagnetic fieldsfields inin eacheach directiondirection cancan bebe linearlylinearly8 of super-17super- imposed,imposed, so so it it is is better better to to study study the the charge charge and and discharge discharge characteristics characteristics of of a a single single circuit circuit first.first. FigureFigure8 8shows shows a a simple simple loop loop with with a a single single coil. coil.

FigureFigure 8. Simplified 8. Simpliﬁed superconducting superconducting loop. loop.

In theIn figure, the ﬁgure, R is Rtheis high-current the high-current resistance resistance connected connected in series, in series, L is theL is supercon- the supercon- ductingducting coil, coil,E is Etheis power the power supply supply voltage voltage drop, drop, i representsi represents the thecurrent. current. According According to to − existingexisting material material data, data, the theworking working resistance resistance of the of thesuperconducting superconducting coil coil is 10 is− 1024 Ω24/cm,Ω/cm, whichwhich can be can negligible be negligible compared compared to the to 1 the Ω resistance 1 Ω resistance connected connected in series. in series. In other In other words, words, the superconductingthe superconducting coil coilin the in loop the loop can canbe simplified be simpliﬁed as an as ideal an ideal inductance. inductance. AccordingAccording to the to current the current char characteristicsacteristics of the of inductor: the inductor: di di E =E iR= +iRL+ L (9) (9) dt dt Solving Equation (9), we can get:

− t − t E τ τ = n − n + n = in (t) (1 e ) e in (0) n x, y, z (10) Rn

τ = Ln = n n x, y, z (11) Rn

where, the coil inductance Ln can be approximated by the following equation: kμ SN2 L = 0 n = x, y, z (12) n l where, k is the Nagaoka coefficient, S is the cross-sectional area of the coil, N is the number of turns, and l is the length of the coil. Here we take the x-axis as an example to study the charging and discharging process of the magnetic pulse so as to draw conclusions about ΔTi and τ. First, the derivative of Equation (1) in Equation (8) is obtained as follows:

B = −B w sin()w t ix i Bi Bi (13) so:  2B π 2B π  B ∈[]− B w , B w = − i , i (14) ix i Bi i Bi  Δ Δ   Ti Ti  Then we take the derivative of the current equation in the x direction in Equation (10):

R 1 − t i ()t = ()E − i(0)R ⋅e L (15) x L

where, E is the power supply voltage, and can be changed freely in range [−Em, Em]. Be- cause of B ∝ i , there is the following equation relationship: ixx

B = C i ix B x (16)

Aerospace 2021, 8, 24 8 of 16

Solving Equation (9), we can get:

En − t − t in(t) = (1 − e τn ) + e τn in(0) n = x, y, z (10) Rn

Ln τn = n = x, y, z (11) Rn

where, the coil inductance Ln can be approximated by the following equation:

kµ SN2 L = 0 n = x, y, z (12) n l where, k is the Nagaoka coefﬁcient, S is the cross-sectional area of the coil, N is the number of turns, and l is the length of the coil. Here we take the x-axis as an example to study the charging and discharging process of the magnetic pulse so as to draw conclusions about ∆Ti and τ. First, the derivative of Equation (1) in Equation (8) is obtained as follows:

. = − Bix BiwBi sin wBi t (13)

so: . 2B π 2B π ∈ − = − i i Bix BiwBi , BiwBi , (14) ∆Ti ∆Ti Then we take the derivative of the current equation in the x direction in Equation (10):

. 1 − R t i (t) = (E − i(0)R) · e L (15) x L

where, E is the power supply voltage, and can be changed freely in range [−Em, Em].

Because of Bix ∝ ix, there is the following equation relationship:

Bix = CBix (16)

where, CB is the constant coefficient. It’s necessary to make the actual magnetic field to keep up with ideal magnetic field, the following inequality can be obtained: . ≤ − min CBix BiwBi . (17) ≥ max CBix BiwBi

Namely: − ( + ) CB Em imaxR ≤ −B w L i Bi (18) CB(Em − imaxR) ≥ L BiwBi With further simpliﬁcation:

2πBi L ∆Ti ≥ (19) CB(Em − iminR)

Equation (19) gives the limiting condition of the control period ∆Ti. The charge and discharge time constant of the total magnetic pulse is set as follows: τ = max τx, τy, τz (20)

In the process of selecting magnetic pulse parameters, the angular velocity wBi of the magnetic ﬁeld needs to meet the constraints of Equation (19). The charging and discharging time constant τ of the total magnetic pulse satisﬁes Equation (20). The k in Equation (8) can Aerospace 2021, 8, x FOR PEER REVIEW 9 of 17

where, CB is the constant coefficient. It’s necessary to make the actual magnetic field to keep up with ideal magnetic field, the following inequality can be obtained: min()C i ≤ −B w B x i Bi (17) max()C i ≥ B w B x i Bi Namely: −C ()E + i R B m max ≤ −B w L i Bi (18) C ()E - i R B m max ≥ B w L i Bi With further simplification: 2πB L ΔT ≥ i i ()− (19) CB Em imin R

Equation (19) gives the limiting condition of the control period ΔTi. The charge and discharge time constant of the total magnetic pulse is set as follows: τ = []τ τ τ max x , y , z (20) Aerospace 2021, 8, 24 9 of 16 In the process of selecting magnetic pulse parameters, the angular velocity wBi of the magnetic field needs to meet the constraints of Equation (19). The charging and discharging time constant τ of the total magnetic pulse satisfies Equation (20). The k in beEquation adjusted (8) according can be adjusted to the pulse according width, to so the that pulse the angular width, displacementso that the angular of the magneticdisplace- ﬁeldment vectorof the magneticBi in a period field meets vector the Bi in control a period requirements meets the control of traction. requirements of traction. 2.4. Control Method Design 2.4. Control Method Design 2.4.1. Control Strategy 2.4.1. Control Strategy The core of the magnetic pulse control technology is the variable pulse magnetic ﬁeldThe described core of inthe Section magnetic 2.3.1 pulse. In ordercontrol to tec obtainhnology the is control the variable signal pulse used magnetic to resolve field the electromagneticdescribed in Section pulse, 2.3.1. the In system order introducesto obtain the Equation control (21),signal called used the to resolve reference the dynamic electromagnetic pulse, the system introduces Equation (21), called the reference dynamic Equa- Equation [20], achieving reference state rref, as shown in Figure9. tion [20], achieving reference state rref, as shown in Figure 9.

Figure 9.9. Magnetic pulse control loop.loop.

The reference state is used to generate the reference control signal Fref, which forms an open-loop control system. In order to suppress the divergence of the system, the state output r of the dynamic Equation is differentiated from the reference state rref and the result is introduced into control signals, forming a feedback control loop. The reference control signal Fref and the feedback control signal Ffb together form a continuous signal for magnetic pulse calculation. The original control signal is a continuous and needs to be discretized. Different from the conventional discretization method, we propose a method based on momentum, which is introduced in detail in Section 2.4.2:

.. h T i r = − µ I − 3 R · R r + mt·mc F re f R3 R R re f mt+mc ct .. (21) R = − µ R R3 In addition, after the system converges, the eddy current control force is greatly reduced, and the eddy current torque will also decrease simultaneously, which reduces the despinning efficiency. In order to avoid the reduction of the efficiency, the eddy current torque enhancement module is specially introduced and designed. By imposing periodic disturbance on the state of the observer, the eddy current torque is always maintained at an effective value, which can be verified later:

rRe f = rre f + A · sin(w · t) (22)

where, A represents the interference amplitude, and w represents the angular velocity of the sinusoidal interference signal. Aerospace 2021, 8, 24 10 of 16

2.4.2. Discretization Method of Continuous Control Signal As shown in Figure6, due to the uniqueness of the eddy current force form, conventional control ideas cannot be used directly here. It is not difficult to find that the instantaneous eddy current force has a certain periodic nature. The eddy current force at a certain moment can be regarded as a pulse in a variable period ∆Ti, where ∆Ti = 2π/wi. Since the target sways slightly near the predetermined position in the actual control process, there is no harm using the impulse within a variable period ∆Ti to characterize the control effect of the object. In order to facilitate the derivation and expression of the control force, it is stipulated that the calculation range of the impulse received by the target is a circle. that is, the time for a magnetic field to scan the target is ∆Ti, and the total impulse received by the target during this period is calculated. The curve of impulse, reduced impulse and equivalent impulse with respect to position after unfolding the target circle in the unit of circumference is shown in Figure 10. The left side of the picture is the actual demonstration diagram, and the right side is the schematic diagram of the circle expansion. The curve in the first coordinate system shows the distribution of impulse within ∆Ti of the applied control force at different positions in the unit of circle. The curve in the second coordinate system represents the impulse distribution curve at the position where the remaining impulse is positive after the impulse in the opposite direction is offset. The curve in the third coordinate system represents the Aerospace 2021, 8, x FOR PEER REVIEW 11 of 17 equivalent impulse of the vector sum of the impulses in ∆Ti, and the impulse pulse width is determined according to the coil inductance model parameters in Section 2.3.2.

FigureFigure 10. 10. DiagramDiagram of of impulse impulse model. model.

ItIt should should be be noted noted that that the the concept concept of equivalent of equivalent impulse impulse in Δ inTi ∆isT basedi is based on the on as- the sumptionassumption that thatΔTi is∆ Tsmalli is small enough enough and the and target the disp targetlacement displacement within the within variable the variableperiod isperiod small. is small.

2.4.3.2.4.3. Solution Solution of of Magnetic Magnetic Pulse Pulse Signal Signal TheThe first first step step of of magnetic magnetic pulse pulse control control is is to to find find the the control control force force at at any any time, time, and and thenthen through through the the discretization, discretization, the the relative relative angular angular velocity velocity ww andand the the central central magnetic magnetic fieldfield BBGtGt cancan be be inversely inversely solved solved by by Equation Equation (3). (3). As As shown shown in in Figure Figure 11, 11 the, the result result of of the the inverseinverse solution solution is is not not unique. unique. Therefore, Therefore, the the solution solution rules rules need need to to be be appointed. appointed.

Figure 11. Diagram of inverse solution result.

In the two-dimensional plane model, the magnetic angular velocity required in Re- sult 1 is much smaller than that in Result 2. A larger magnetic angular velocity may increase the difficulty of controlling the magnetic field generator. In addition, the eddy current torque in result 1 is opposite to the target speed, which plays a role of braking. The eddy current torque in result 2 is in the same direction as the target speed, which accelerates the target’s rotation speed. Therefore, when solving Equation (3), it is necessary to make the applied magnetic field speed as small as possible and the eddy current torque

Aerospace 2021, 8, x FOR PEER REVIEW 11 of 17

Figure 10. Diagram of impulse model.

It should be noted that the concept of equivalent impulse in ΔTi is based on the as- sumption that ΔTi is small enough and the target displacement within the variable period is small.

2.4.3. Solution of Magnetic Pulse Signal The first step of magnetic pulse control is to find the control force at any time, and Aerospace 2021, 8, 24 then through the discretization, the relative angular velocity w and the central magnetic11 of 16 field BGt can be inversely solved by Equation (3). As shown in Figure 11, the result of the inverse solution is not unique. Therefore, the solution rules need to be appointed.

FigureFigure 11.11.Diagram Diagram ofof inverseinverse solutionsolution result.result.

InIn thethe two-dimensional two-dimensional plane plane model, model, the the magnetic magnetic angular angular velocity velocity required required in Result in Re- 1sult is much 1 is much smaller smaller than thatthan in that Result in Result 2. A larger 2. A larger magnetic magnetic angular angular velocity velocity may increase may in- thecrease difficulty the difficulty of controlling of controlling the magnetic the magnetic field generator.field generator. In addition, In addition, the eddythe eddy current cur- Aerospace 2021, 8, x FOR PEER REVIEW 12 of 17 torquerent torque in result in result 1 is opposite 1 is opposite to the to target the ta speed,rget speed, which which plays aplays role ofa role braking. of braking. The eddy The currenteddy current torque torque in result in 2result is in the2 is samein the direction same direction as the target as the speed, target whichspeed,accelerates which acceler- the target’sates the rotation target’s speed.rotation Therefore, speed. Therefore, when solving when Equation solving (3),Equation it is necessary (3), it is tonecessary make the to appliedmake the magnetic applied fieldmagnetic speed field as small speed as as possible small as and possible the eddy and current the eddy torque current along torque the alongopposite the opposite direction direction of the targetof the angulartarget angular velocity velocity as much as asmuch possible, as possible, which which must bemust taken be astaken the as criterion. the criterion.

3. Numerical3. Numerical Study Study 3.1.3.1. Accuracy Accuracy Verification Veriﬁcation of Discretization of Discretization Method Method TheThe role role of the of the momentum momentum model model is to is ensure to ensure the the stability stability of the of the system system during during the the discretizationdiscretization of the of thecontinuous continuous control control signal. signal. The discrete The discrete process process without without the eddy the cur- eddy rentcurrent torque torque enhancement enhancement module module is shown is shown in Figures in Figures 12 and 12 and 13. 13When. When the the continuous continuous controlcontrol signal signal is input, is input, the the total total impulse impulse of the of the input input signal signal is calculated is calculated in units in units of cycles of cycles..

FigureFigure 12. 12.PeriodicPeriodic total total impulse impulse curve. curve.

Figure 13. Pulse control curve.

Applying the continuous control signal to the model, as shown in Figure 14, the control effect does not change significantly compared with the above discrete control. There- fore, when ΔTi is small, the momentum-based discrete method proposed in the article does not affect the control performance of the system.

Aerospace 2021, 8, x FOR PEER REVIEW 12 of 17

along the opposite direction of the target angular velocity as much as possible, which must be taken as the criterion.

3. Numerical Study 3.1. Accuracy Verification of Discretization Method The role of the momentum model is to ensure the stability of the system during the discretization of the continuous control signal. The discrete process without the eddy current torque enhancement module is shown in Figures 12 and 13. When the continuous control signal is input, the total impulse of the input signal is calculated in units of cycles.

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Figure 12. Periodic total impulse curve.

FigureFigure 13. 13. PulsePulse control control curve. curve.

ApplyingApplying the the continuous continuous control control signal signal to the to model, the model, as shown as shown in Figure in Figure14, the con-14, the control Aerospace 2021, 8, x FOR PEER REVIEWtroleffect effect does does not not change change significantly signiﬁcantly compared compared with with the theabove above discrete discrete control. control. There-13 of 17 Therefore, Aerospace 2021, 8, x FOR PEER REVIEW 13 of 17 fore,when when∆T ΔisTi small,is small, the the momentum-based momentum-based discrete discrete methodmethod proposed in in the the article article does not i doesaffect not the affect control the control performance performance of the of the system. system.

FigureFigure 14. 14.RelativeRelative distan distancece curve. curve. Figure 14. Relative distance curve. 3.2.3.2. Performance Performance Verification Verification of Eddy of EddyCurrent Current Torque TorqueEnhancement Enhancement Module Module 3.2. Performance Verification of Eddy Current Torque Enhancement Module TheThe eddy eddy current current force force control control curve curveis shown is shownin Figure in 13. Figure When 13 the. When system the is stable, system is stable, The eddy current force control curve is shown in Figure 13. When the system is stable, thethe control control force force is significantly is significantly reduced. reduced. According According to Equation to Equation (1) and Equation (1) and Equation(2), it (2), it canthe becontrol seen thatforce there is significantly is a synchronous reduced. relationship According between to Equation force (1) and and torque Equation changes. (2), it cancan be be seen seen that that there there is a issynchronous a synchronous relationship relationship between between force and force torque and changes. torque changes. Theoretically,Theoretically, the theeddy eddy current current torque torque will also will be also significantly be significantly reduced, reduced,and the control and the control effectTheoretically, will be greatly the eddy reduced, current as torqueshown willin Figures also be 15 significantly and 16. reduced, and the control effecteffect will will be be greatly greatly reduced, reduced, as shown as shown in Figures in Figures 15 and 15 16. and 16.

Figure 15. Eddy current torque curve in conventional mode. FigureFigure 15 15.. EddyEddy current current torque torque curve curve in conventional in conventional mode. mode.

Figure 16. Target speed curve in conventional mode. Figure 16. Target speed curve in conventional mode.

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Figure 14. Relative distance curve.

3.2. Performance Verification of Eddy Current Torque Enhancement Module The eddy current force control curve is shown in Figure 13. When the system is stable, the control force is significantly reduced. According to Equation (1) and Equation (2), it can be seen that there is a synchronous relationship between force and torque changes. Theoretically, the eddy current torque will also be significantly reduced, and the control effect will be greatly reduced, as shown in Figures 15 and 16.

Aerospace 2021, 8, 24 13 of 16 Figure 15. Eddy current torque curve in conventional mode.

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FigureFigure 16. 16.Target Target speed speed curve curve in in conventional conventional mode. mode. AccordingAccording to tothe the simulation simulation results, results, when when the thesystem system is stable, is stable, the control the control torque torque is almostis almost zero, zero,and the and control the control effect effectcan be can ignored. be ignored. In order In to order despin to despinthe target, the an target, eddy an currenteddy torque current enhancement torque enhancement mechanism mechanism must be must introduced. be introduced. Considering Considering that the that non the- propellantnon-propellant eddy current eddy current torque torque is essentially is essentially a concomitant a concomitant product product of the of theeddy eddy current current controlcontrol force. force. As Aslong long as sufficient as sufficient eddy eddy current current control control force force is always is always generated, generated, the ex- the istenceexistence of a stable of a stable elimination elimination torque torque can canbe guaranteed. be guaranteed. As Asshown shown in Figu in Figurere 9, 9this, this paper paper addsadds periodic periodic disturbances disturbances to tothe the state state output output of ofthe the full full-dimensional-dimensional state state observer. observer. The The availableavailable disturbance disturbance signals signals include include sine sine waves waves and and square square waves. waves. In Inthis this section, section, sine sine waveswaves are are selected selected as astorque torque-enhanced-enhanced disturbance disturbance signals signals for for simulation simulation experiments. experiments. TheThe eddy eddy current current torque torque curve curve and and target target speed speed curve curve after after adding adding the the enhancement enhancement modulemodule are are shown shown in Figures in Figures 17 and 17 and18, respectively. 18, respectively. According According to the tonumerical the numerical exper- ex- imentperimentalal results, results, it is not it difficult is not difficult to find tothat find the that pulsed the eddy pulsed current eddy torque current is torque stably ismain- stably tainedmaintained in the working in the working range. The range. target The angular target angularvelocity velocityshows a showslinear downward a linear downward trend, andtrend, the despinning and the despinning effect is significantly effect is significantly improved improved than before than the before torque the is increased. torque is in- Thecreased. conventional The conventional electromagnetic electromagnetic despinning despinning curve in the curve literature in the literature[20] is shown [20] isin shownFig- urein 19. Figure Compared 19. Compared with the with method the method proposed proposed in this paper, in this paper,it can be it canconsidered be considered that the that magneticthe magnetic pulse pulse type type non-propellant non-propellant despinning despinning technology technology with with torque torque enhancement enhancement modulemodule is more is more stable. stable. The The efficiency efficiency of ofthe the new new technology technology has has a weak a weak connection connection with with thethe change change of ofthe the target target rate, rate, which which has has obvious obvious performance performance advantages advantages compared compared with with conventionalconventional electromagnetic electromagnetic despinning despinning methods. methods.

FigureFigure 17. 17. EnhancedEnhanced eddy eddy current current torque torque curve. curve.

Figure 18. Target speed curve after torque enhancement.

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According to the simulation results, when the system is stable, the control torque is almost zero, and the control effect can be ignored. In order to despin the target, an eddy current torque enhancement mechanism must be introduced. Considering that the non- propellant eddy current torque is essentially a concomitant product of the eddy current control force. As long as sufficient eddy current control force is always generated, the ex- istence of a stable elimination torque can be guaranteed. As shown in Figure 9, this paper adds periodic disturbances to the state output of the full-dimensional state observer. The available disturbance signals include sine waves and square waves. In this section, sine waves are selected as torque-enhanced disturbance signals for simulation experiments. The eddy current torque curve and target speed curve after adding the enhancement module are shown in Figures 17 and 18, respectively. According to the numerical experimental results, it is not difficult to find that the pulsed eddy current torque is stably maintained in the working range. The target angular velocity shows a linear downward trend, and the despinning effect is significantly improved than before the torque is increased. The conventional electromagnetic despinning curve in the literature [20] is shown in Fig- ure 19. Compared with the method proposed in this paper, it can be considered that the magnetic pulse type non-propellant despinning technology with torque enhancement module is more stable. The efficiency of the new technology has a weak connection with the change of the target rate, which has obvious performance advantages compared with conventional electromagnetic despinning methods.

Aerospace 2021, 8, 24 14 of 16 Figure 17. Enhanced eddy current torque curve.

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FigureFigure 18. 18. TargetTarget speed speed curv curvee after after torque torque enhancement. enhancement.

Figure 19. Torque curve of conventional eddy current brake with different initial conditions. FigureFigure 19. 19. TorqueTorque curve curve of of conventional conventional eddy eddy current current brake brake with with different different initial initial conditions conditions.. 3.3.3.3. Verification Veriﬁcation of ofEddy Eddy Current Current Force Force Traction Traction Control Control Effect Effect 3.3. Verification of Eddy Current Force Traction Control Effect TheThe relative relative distance distance between between the the service service satellite satellite and the target isis shownshown inin Figure Figure 20 . The relative distance between the service satellite and the target is shown in Figure 20When. When the the torque torque enhancement enhancement module module is not is not introduced, introduced the, the system system quickly quickly converges con- 20. When the torque enhancement module is not introduced, the system quickly con- vergeands stabilizesand stabilize at thes at predetermined the predetermined position. position Then. less Then eddy less current eddy force current is needed force isto verges and stabilizes at the predetermined position. Then less eddy current force is neededmaintain to maintain satellite formation, satellite formation, which results which in results the reduction in the reduction in eddy current in eddy torque current and needed to maintain satellite formation, which results in the reduction in eddy current torquedespinning and despinning efﬁciency. efficiency. While the While target the swings target for swings a long for period a long near period the predeterminednear the pre- torque and despinning efficiency. While the target swings for a long period near the pre- determinedposition with position the torque with enhancementthe torque enhancement module, the module, eddy current the eddy force current and torque force always and determined position with the torque enhancement module, the eddy current force and torqueexist, always which isexist, shown which in Figure is shown 17. Accordingin Figure 17 to. According the simulation to the results, simulation it can results, be thought it torque always exist, which is shown in Figure 17. According to the simulation results, it canthat be thethought technology that the of technology non-propellant of non despinning-propellant and de tractionspinning based and traction on magnetic based pulse on can be thought that the technology of non-propellant despinning and traction based on magneticcan well pulse realize can non-propellant well realize non traction-propella of spacent traction conductor of space targets. conductor targets. magnetic pulse can well realize non-propellant traction of space conductor targets.

FigureFigure 20. 20. RelativeRelative distance distance curve. curve. Figure 20. Relative distance curve. 4. Conclusions 4. Conclusions This paper studies the despinning method of the space rotating target, proposes a method ofThis despinning paper studies and tractionthe despinning based on method magnetic of the pulse space without rotating propellant target, proposes, and designs a method a newof despinning despinning and device. traction In order based to realizeon magnetic the proposed pulse without despinning propellant and traction, and designsmethod, a a newdiscretization despinning method device. of In controlorder to signal realize based the proposed on momentum despinning model and is tractionstudied method,in this paper.a discretization This method method grasps of thecontrol relatively signal static based characteristics on momentum of modelthe formation is studied between in this thepaper. active This and method passive grasps sides andthe relativelydiscretizes static the continuous characteristics control of the signal formation by calculating between thethe total active impulse and passive of the eddysides currentand discretizes force in athe small continuous period. Then,control the signal simplified by calculating induct- ancethe totalmodel impulse is used of to the derive eddy the current magnetic force pulse in a insmall the twoperiod.-dimensional Then, the simplified simplified model, induct- obtainingance model constraints is used to on derive key parameters the magnetic such pulse as thein the electromagnetic two-dimensional pulse simplified period Δ model,Ti. In addition,obtaining an constraints eddy current on torquekey parameters enhancement such module as the electromagnetic that uses periodic pulse interference period Δ Tsig-i. In nalsaddition, is designed an eddy so currentas to solve torque the enhancementproblem of the module despinning that uses torque periodic disappearing interference in the sig- steadynals is state designed of the so system. as to solve the problem of the despinning torque disappearing in the steady state of the system.

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4. Conclusions This paper studies the despinning method of the space rotating target, proposes a method of despinning and traction based on magnetic pulse without propellant, and designs a new despinning device. In order to realize the proposed despinning and traction method, a discretization method of control signal based on momentum model is studied in this paper. This method grasps the relatively static characteristics of the formation between the active and passive sides and discretizes the continuous control signal by calculating the total impulse of the eddy current force in a small period. Then, the simplified inductance model is used to derive the magnetic pulse in the two-dimensional simplified model, obtaining constraints on key parameters such as the electromagnetic pulse period ∆Ti. In addition, an eddy current torque enhancement module that uses periodic interference signals is designed so as to solve the problem of the despinning torque disappearing in the steady state of the system. According to the simulation results, the discretization method based on momentum model proposed in this paper doesn’t affect the control stability of the system under the limitation of electromagnetic pulse parameters. By comparing the changes of the target angular velocity before and after the introduction of the eddy current torque enhancement module, it is not difficult to find that the eddy current torque enhancement module introduced in this paper really solve the problem of the torque disappearing in the steady state. What’s more, the efficiency of the conventional eddy current despinning method decrease with the decrease of the target speed, and it is necessary to consume propellant for a long time to maintain the formation. The eddy current despinning and traction method based on magnetic pulse does not need propellant, and the efficiency is not affected by the target state. Compared with conventional methods, the method proposed in this paper has certain advantages. Apart from the field of despinning, this technology can also be applied to the field of passive traction of space targets, etc. The theoretical research and numerical experimental data in this paper can provide an effective reference for the development of related fields.

Author Contributions: Conceptualization, Y.Z.; Investigation, Y.Z.; Methodology, Y.Z.; Project administration, Q.S., S.W. and L.H.; Writing—review and editing, Q.S., S.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported partially by National Natural Science Foundation of China under Grant U20B2056, Natural Science Foundation of Shanghai under Grant 20ZR1427000, Shanghai Sailing Program under Grant 20YF1421600. Informed Consent Statement: Informed consent was obtained from all subjects involved in the study. Acknowledgments: The author would like to thank the reviewers and the editors for their valuable comments and constructive suggestions that helped to improve the paper significantly. Conflicts of Interest: The authors declare no conflict of interest.

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