Comparison of Solar Radiation Torque and Power Generation of Deployable Solar Panel Conﬁgurations on Nanosatellites

aerospace

Article Comparison of Solar Radiation Torque and Power Generation of Deployable Solar Panel Conﬁgurations on Nanosatellites

Syahrim Azhan Ibrahim * and Eiki Yamaguchi

Department of Civil Engineering, Kyushu Institute of Technology, Kitakyushu, Fukuoka 804-8550, Japan; [email protected] * Correspondence: [email protected] or [email protected]; Tel.: +81-80-8354-1457

Received: 26 February 2019; Accepted: 29 April 2019; Published: 2 May 2019

Abstract: Nanosatellites, like CubeSat, have begun completing advanced missions that require high power that can be obtained using deployable solar panels. However, a larger solar array area facing the Sun increases the solar radiation torque on the satellite. In this study, we investigated solar radiation torque characteristics resulting from the increased area of solar panels on board the CubeSats. Three common deployable solar panel configurations that are commercially available were introduced and their reference missions were established for the purpose of comparison. The software algorithms used to simulate a variety of orbit scenarios are described in detail and some concerns are highlighted based on the results obtained. The solar power generation of the respective configurations is provided. The findings are useful for nanosatellite developers in predicting the characteristics of solar radiation torques and solar power generation that will be encountered when using various deployable solar panels, thus helping with the selection of a suitable configuration for their design.

Keywords: CubeSat; solar radiation torque; deployable solar panel; solar power generation

1. Introduction For satellites in low Earth orbit (LEO), major sources of external disturbances that perturb their total angular momentum include the gravitational field, magnetic field, atmosphere, and solar radiation. At altitudes of 400 to 1000 km, the disturbances that affect those satellites the most are the gravitational and magnetic torques. Solar radiation can also exert an appreciable disturbance torque in which the level could match the magnitude of atmospheric drag at altitudes over 500 km [1]. The order of disturbance magnitude can significantly increase, especially for satellites with large surface areas. Therefore, the effects of solar radiation should also be a concern for nanosatellites, like CubeSats, as their missions have become more advanced and subsequently require more power, which has necessitated the use of deployable solar panels. With the addition of these deployable solar panels, the surfaces of the satellite facing the Sun increase and consequently change the characteristics of the total external disturbances on the satellite. Deployable solar panels on CubeSat can be used to optimize solar power generation and to accomplish specific missions. Many papers about the former application have been published, wherein novel solar panel configurations were designed and solar energy harnessing was studied [2–5]. For the latter application, in one mission example, the strong atmospheric drag at orbits below 500 km was manipulated to achieve attitude stability [6,7]. The effect of in-orbit temperature variations on the solar cells used by CubeSats has been examined [8], and a structure design was proposed to increase the thermal conductance between the solar array and the solid structure of CubeSats to reduce temperature fluctuations [9].

Aerospace 2019, 6, 50; doi:10.3390/aerospace6050050 www.mdpi.com/journal/aerospace Aerospace 2019, 6, 50 2 of 22

Most of these CubeSats have their own primary missions, and they rely on commercial suppliers for their deployable solar panels as far as reliability is concerned. Whenever deployable parts are involved, products with a flight heritage record are important for increasing the success rate of their primary missions. Although not explicitly specified, some prematurely failed CubeSats missions have been reported, caused by non-functional power subsystems and mechanical deployment systems [10]. We focused on solar radiation torque resulting from the increased area of solar cells on board the CubeSats. In Section2, solar radiation pressure is briefly explained. In Section3, three deployable solar panel configurations that are commercially available are introduced and their reference missions are established for comparison purposes. Formulations applied to obtain the solar radiation torque-related parameters are described. In Section4, the computation used in simulation software is described in detail. Subsequently, in Section5, significant results are presented and discussed. The solar power generation of the different configurations is shown. The work here is useful to help CubeSat developers select suitable deployable solar panel configurations for their CubeSats and to foresee possible disturbance patterns that could affect the satellite motion.

2. In-Orbit External Disturbances

2.1. Solar Radiation Pressure The solar radiation incident on a spacecraft’s surface produces a force that results in a torque about the spacecraft’s center of mass. The major sources of solar radiation pressure are (1) solar ﬂux from the Sun, (2) radiation emitted from the Earth and its atmosphere, and (3) solar radiation reﬂected by the surface and clouds of the Earth, i.e., Earth’s albedo. This radiation (photons) contains momentum that creates pressure on the lit surface of the spacecraft. The solar pressure is:

S p = (1) c where S is the mean solar flux at 1367 W/m2 and c is the speed of light. In the vicinity of Earth, the value of p is constant at 4.56 10 6 N/m2. Earth-emitted radiation can be assumed as uniform over the × − surface of the earth. The flux is: 400 q = (2) r 2 RE where r is the distance between the Earth’s center and the satellite’s altitude, and RE is the earth radius at 6378.165 km. The 400 W/m2 is based on the assumption that the Earth is a black body and has a temperature of 289.8 K. The third source, due to albedo, is complex because the Earth cannot be treated as a point source as the reflectivity varies over the surface. In this work, the reflection from the Earth’s surface and from clouds is assumed to be diffuse. The flux due to albedo can then be computed by integrating over the surface of the earth. The equation of the flux is:

S q = aF (3) r 2 RE where aF is the albedo factor, which is 0.33. Of these three sources, solar flux is the dominant source of solar radiation pressure. To model the solar radiation forces, incident radiation is assumed to be either absorbed, reflected specularly, reflected diffusely, or some combination thereof. In terms of the fractions of the incoming radiation, the following is true for a surface:

ρa + ρd + ρs = 1 (4) Aerospace 2019, 6, 50 3 of 22 where ρ (optical properties) stands for the fraction of photons that are absorbed, diffusely reflected, and specularly reflected. The force caused by solar radiation can be expressed as: → T T ρd T F = pAsˆ n 2 ρssˆ n + n + (ρa + ρ )sˆ for sˆ n > 0 (5) − 3 d where sˆ is the unit sun vector, T denotes its transpose, n is the unit normal to the surface, and A is the surface’s area. From Equation (5), the specular component produces the biggest force, followed by the diffuse component and the absorbed component [11]. sˆTn is the dot product, which is the cosine of the angle between sˆ and n. Its positive value means that the surface normal faces the sun direction. Later, we explain that the sˆ value is also interchangeable with the other two sources of radiation, which are the Earth’s radiation and albedo.

2.2. Other Disturbances To observe how the solar radiation torque level changes due to usage of the deployable solar panels, other major disturbance torques in LEO will also be accounted for in the study. The disturbances consist of the aerodynamics drag, gravity gradient, and the residual dipole. Aerodynamic force disturbance is due to the interaction between a planetary atmosphere and a spacecraft surface. The disturbance force model used in this study is:

1 →F = ρ C A →v →v (6) 2 atm D p where ρatm is the atmospheric density, CD is the drag coefficient, Ap is the projected area, and →v is the velocity vector. The projected area for a flat plate is Ap = Acosα, where α is the angle between the 12 3 surface normal and the velocity vector. The value of ρatm is 3.725 10 kg/m , based on the scale × − heights atmospheric model [12] (p. 820) whereas the CD is estimated as 2 [12] (p. 64). For LEO satellites, which have off-diagonal terms in their inertia matrix, they can experience the gravity gradient disturbance torque from the variation of the Earth’s gravitational force. In the case of nadir pointing satellites, the vector torque is modeled as:    Iyz  → 2 −  T = 3ωo Ixz  (7)    0  where ωo is the orbital natural frequency. The solar panels would cause off-diagonal terms, but are very small as they are just flat plates. The residual dipole disturbance torque results from the interaction of the magnetic field generated by current loops on the spacecraft with the Earth’s magnetic field. The torque direction, T, is normal to both the satellite’s residual dipole, M, and the Earth magnetic field vector, B, as shown in the equation below: →T = M→ →B (8) × Both M and B must be resolved into the body frame [11] (p. 137). In this study, the magnetic field model is based on a tilted dipole model [12] (pp. 783–784) whereas a 0.01 Am2 residual dipole strength along the z-axis would be a good estimation from the literature on CubeSats [13]. The total torque disturbances will be the summation of torques due to the solar radiation pressure, aerodynamics drag, gravity gradient, and the residual dipole.

3. Model Parameters In this section, the satellite surface parameters that consist of the unit normal to the surface, area of the surface, and the surface optical properties mentioned in Equation (5) are described. Then, Aerospace 2019, 6, 50 4 of 22 the formulations used to determine the unit sun vector are shown. Since an improved solar power generation is always the main purpose for using a deployable solar array, the method to compute solar power generation is also presented so that the results can be compared.

3.1. Satellite Conﬁguration

AerospaceA CubeSat 2019, 6, xhas FOR a PEER standard REVIEW built dimension of 10 cm3 for one unit (1U) size with no protuberant4 of 23 parts at launch. A 3U CubeSat is basically composed of three 1U CubeSats stacked lengthwise. For the presentdepicted analysis, in Figure three 1. The 3U CubeSat models u modelsse a simplified were configured model of with areas deployable and normal, solar and panels consist as depictedof a set of invertices Figure1 and. The faces models defining use a the simplified exterior model of the ofsatellites. areas and The normal, optical and property consist of of the a set body of vertices surfaces and that facesare mounted defining thewith exterior solar cells of the is satellites.assigned Theas a opticalsolar cell, property whereas of thethe bodysurfaces surfaces that have that areno mountedsolar cells withare assigned solar cells as is radiator assigned surface. as a solar All cell, three whereas models the can surfaces be considered that have as no common solar cells designs are assigned as they asare radiatorcommercially surface. available All three [14 models–16] and can similar be considered designs were as commonused for some designs CubeSat as they missions are commercially and studies available[2,5–7,17,18] [14–.16 ] and similar designs were used for some CubeSat missions and studies [2,5–7,17,18].

Inner Top

0.4 Outer

0.2

0 Bottom z 0 Nadir -0.2

-0.2

Velocity 0.2 0.2 vector Nadir 0.2 0 0.2 0 0 0 -0.2 -0.2 -0.2 -0.2 y x y x

(a) (b)

Top

0.1

0 z

-0.1

Bottom 0.2

Nadir 0.2 0 0

-0.2 y -0.2 x

(c)

FigureFigure 1. 1.CubeSat CubeSat geometrygeometry ofof ( a(a)) model model 1, 1, ( b(b) ) model model 2, 2, and and ( c()c) model model 3 3 with with the the surface surface normal normal indicatedindicated by by the the solid solid arrows. arrows.

TheThe cube cube faces faces have have six six surfaces, surfaces, which which create create the the satellite satellite body. body.Each Eachmodel modelhad hadfour four fixed, fixed, body-mountedbody-mounted solar solar panels panels on on the the long long section section surfaces. surfaces. Each Each deployable deployable solar solar panel panel had had a a length length of of 300300 mm, mm, a widtha width of of 100 100 mm, mm and, and a thickness a thickness of 3 of mm. 3 mm. Each Each panel panel is divided is divided intothree into partsthree withparts each with parteach having part having two surfaces: two surfaces Top and: T bottom.op and bottom. Therefore, Therefore, each deployable each deployable solar array solar has array six surfaces has six thatsurfaces can be that defined can be di definedfferently. differently. Altogether, Altogether, each model each has model 30 surfaces has 30 to surfaces be defined. to be Surfacedefined. normal Surface normal vectors are shown by the blue arrows in Figure 1. In the simulation code, both areas and surface normal vectors are gathered into their respective matrix and placed into the x-, y-, or z-axis. For ease of comparison, all the CubeSat models used have the same mission: A nadir-pointing mission. We assumed that their respective attitude control systems are able to maintain the pointing in the directions assigned. The state-of-the-art CubeSat technology has been found to be potentially compatible with some Earth observation missions [19]. For model 1, the satellite has four deployable solar panels attached 90 degrees at the +z-axis short edges of the satellite body. The top surfaces of the solar panels and four sides of the x-axis and y-axis satellite body are assigned as solar cells, whereas the bottom sides of the extendable panels and z-axis sides of the body are assigned as a radiator. The nadir pointing is on the −z-axis of the satellite body. Model 2 has its extendable solar

Aerospace 2019, 6, 50 5 of 22 vectors are shown by the blue arrows in Figure1. In the simulation code, both areas and surface normal vectors are gathered into their respective matrix and placed into the x-, y-, or z-axis. For ease of comparison, all the CubeSat models used have the same mission: A nadir-pointing mission. We assumed that their respective attitude control systems are able to maintain the pointing in the directions assigned. The state-of-the-art CubeSat technology has been found to be potentially compatible with some Earth observation missions [19]. For model 1, the satellite has four deployable solar panels attached 90 degrees at the +z-axis short edges of the satellite body. The top surfaces of the solar panels and four sides of the x-axis and y-axis satellite body are assigned as solar cells, whereas the bottom sides of the extendable panels and z-axis sides of the body are assigned as a radiator. The nadir pointing is on the z-axis of the satellite body. Model 2 has its extendable solar panels angle deployed at − fixed 30 degrees with respect to the +z axis of the satellite. This configuration is normally known as the space-dart configuration. The outer sides of the deployable solar panels are assigned solar cell surface properties, whereas the inner sides are assigned radiator surface properties. The remaining surfaces of the main satellite body part are assigned as model 1. This configuration was found to be capable in providing attitude aerodynamic stability for CubeSats that orbit at altitudes below 500 km [6,7]. To manipulate the aerodynamic drag for attitude stabilization, the short section of the satellite body that does not have deployable solar panels (in this case, the z-axis side) point in the direction of the − velocity of the satellite. We set one long section of the body to be fixed to the nadir pointing direction (in this case, the +x-axis side) to match the nadir pointing missions of the other two models to facilitate the solar radiation torque comparison study. Lastly, model 3 has two-double solar panels deployed along the long edge of the 3U CubeSat body. Like model 1, the top surfaces of the solar panels are solar cells and the bottom sides are radiators. The nadir pointing is also on the z-axis of the satellite body. − 3.2. Position of the Sun and Eclipse Condition To compute the Sun’s position with respect to the satellite for any given location at a given time, there are three steps to follow [20]: (1) Calculate the Sun’s position in the ecliptic coordinate system, (2) convert the result from step 1 to the Earth-centered inertial (ECI) frame, and (3) convert the result from step 2 to the satellite body coordinate system. For the first step, a simple algorithm is used to generate the position of the Sun in the ecliptic coordinate system to a precision of about 1 arcminute for dates between 1950 and 2050. First, the number of days, j, is computed from the epoch referred to as the Julian date, 2,451,545.0:

j = JD 2451545.0 (9) − where JD is the Julian date of interest. Then, the mean longitude of the Sun (L), mean anomaly of the Sun (g), and ecliptic longitude of the sun (λ) are computed:

L = 280.460◦ + 0.9856474◦ j (10)

g = 357.528◦ + 0.9856474◦ j (11)

λ = L + 1.915◦sing + 0.020◦sin2g (12) where all the values are in the range of 0 to 360◦. The distance of the Sun from the Earth (usun) in the unit meter can be approximated as follows:

usun = (1.00014 0.01671cosg 0.00014cos2g) 149600e3 (13) − − ∗ Aerospace 2019, 6, 50 6 of 22

Other than λ and usun, another parameter to form a complete position of the sun in the ECI frame is the ecliptic latitude, β. The Sun’s ecliptic latitude can be approximated by β = 0. Next, in step 2, the unit sun vector (uˆsun) in the equatorial coordinate system is computed as follows:    cosλ    uˆsun =  cos sinλ  (14)    sin sinλ  where is the obliquity of the ecliptic, which can be approximated by:

7 = 23.439◦ 4.00 10− j (15) − × Aerospace 2019, 6, x FOR PEER REVIEW 6 of 23 In step 3, the uˆsun must be converted to a unit vector from the satellite toward the Sun (sˆ), to be usedNext, in Equation the Sun’s (5). distance In a later and section, its position this conversion from the Earth will be can explained. be used to determine the conditions underNext, which the eclipses Sun’s distance occur. The and problem its position of calculatin from theg Earth the eclipse can be usedtimes toof determinea spacecraft the orbiting conditions the underEarth has which been eclipses studied occur.in depth The using problem various of method calculatings [12,21–23]. the eclipse In this times work, of athe spacecraft existing spherical orbiting theEarth Earth conical has shadow been studied model in given depth by using Wertz various [12] was methods used—the [12, 21atmospheric–23]. In this effects work, are the neglected. existing sphericalWe were only Earth concerned conical shadow about the model eclipse given seen by by Wertz objects [12 of] was negligible used—the sizes, atmospheric such as satellites. eﬀects The are neglected.variables for We eclipse were geometry only concerned are shown about in Figure the eclipse 2. seen by objects of negligible sizes, such as satellites. The variables for eclipse geometry are shown in Figure2.

Figure 2. Variables for eclipse geometry, wherewhere DD isis thethe distancedistance fromfrom thethe EarthEarth toto thethe Sun,Sun, RREE is the radius of of the the Earth, Earth, and and RRS Sisis the the radius radius ofof the the visible visible surface surface of the of the Sun. Sun. By simple By simple trigonometry, trigonometry, the thedistance distance from from the center the center of the of Eart theh Earth to the to apex the of apex the ofshadow the shadow cone, 𝐶 cone, = 1.385C = 1.385 × 10 km10 6andkm 𝜌and= × ρ0.264°c = 0.264. ◦. * To ﬁnd the condition when a satellite is in eclipse condition, let DS be𝐷 the⃑ vector from the satellite To find the* condition when a satellite is in eclipse condition, let be the vector from the tosatellite the Sun to andthe SunDE beand the 𝐷⃑ vector be the from vector the from satellite theto satellite the center to the of thecenter Earth. of the Referring Earth. Referring to Figure 3to, fromFigure the 3, satellite’sfrom the satellite’s position, position, the parameters the paramete to be determinedrs to be determined are the angular are the radiusangular of radius the sun, of theψS, thesun, angular 𝜓, the radiusangular of radius the Earth, of theψ Earth,E, and 𝜓 the, and angular the angular separation, separation,θ, between 𝜃, between the Sun the and Sun the and Earth. the TheseEarth. threeThese parameters three parameters are given are by:given by: R ψ = sin 1 S (16) S 𝑅− 𝜓 =𝑠𝑖𝑛 DS (16) 𝐷 1 RE ψ = sin− (17) E 𝑅 D E (17) 𝜓 =𝑠𝑖𝑛 1𝐷 θ = cos− Dˆ Dˆ E (18) S· (18) The necessary conditions for the𝜃=𝑐𝑜𝑠 total eclipse𝐷 occur∙𝐷 when: The necessary conditions for the total eclipse occur when: D < D < (D + C) and (ψE ψs) > θ (19) S − 𝐷𝐷 (𝐷+𝐶) and (𝜓 −𝜓) >𝜃 (19)

Satellite

ψE

ψS Earth Sun

Figure 3. Solar eclipse geometry. The parameters to be determined are the angular radius of the Sun,

𝜓, the angular radius of the Earth, 𝜓, and the angular separation, 𝜃, between the Sun and the Earth.

Aerospace 2019, 6, x FOR PEER REVIEW 6 of 23

Next, the Sun’s distance and its position from the Earth can be used to determine the conditions under which eclipses occur. The problem of calculating the eclipse times of a spacecraft orbiting the Earth has been studied in depth using various methods [12,21–23]. In this work, the existing spherical Earth conical shadow model given by Wertz [12] was used—the atmospheric effects are neglected. We were only concerned about the eclipse seen by objects of negligible sizes, such as satellites. The variables for eclipse geometry are shown in Figure 2.

Figure 2. Variables for eclipse geometry, where D is the distance from the Earth to the Sun, RE is the radius of the Earth, and RS is the radius of the visible surface of the Sun. By simple trigonometry, the distance from the center of the Earth to the apex of the shadow cone, 𝐶 = 1.385 × 10 km and 𝜌 = 0.264°.

To find the condition when a satellite is in eclipse condition, let 𝐷⃑ be the vector from the satellite to the Sun and 𝐷⃑ be the vector from the satellite to the center of the Earth. Referring to Figure 3, from the satellite’s position, the parameters to be determined are the angular radius of the sun, 𝜓, the angular radius of the Earth, 𝜓, and the angular separation, 𝜃, between the Sun and the Earth. These three parameters are given by:

𝑅 𝜓 =𝑠𝑖𝑛 (16) 𝐷

𝑅 (17) 𝜓 =𝑠𝑖𝑛 𝐷

(18) 𝜃=𝑐𝑜𝑠 𝐷 ∙𝐷 The necessary conditions for the total eclipse occur when: Aerospace 2019, 6, 50 7 of 22 𝐷𝐷 (𝐷+𝐶) and (𝜓 −𝜓) >𝜃 (19)

Satellite

ψE

ψS Earth Sun Figure 3. Solar eclipse geometry. The parameters to be determined are the angular radius of the Sun, Figure 3. Solar eclipse geometry. The parameters to be determined are the angular radius of the Sun, ψS, the angular radius of the Earth, ψE, and the angular separation, θ, between the Sun and the Earth. 𝜓, the angular radius of the Earth, 𝜓, and the angular separation, 𝜃, between the Sun and the Earth. 3.3. Solar Power Calculation

The power produced by the solar arrays mounted on the body and the deployable solar panels for any given location at a given time is [11]:

X16 T T Power = ηρaAiSnisˆ n f or sˆ n > 0 (20) i=1 where i is the solar cell number, η is the solar cell eﬃciency, and sˆTn is the dot product, equivalent to the cosine of the angle between sˆ and n. Whenever the resultant dot product is negative, it means that the particular surface does not face the Sun, and hence does not produce any power. In addition, since the solar cell mounted is normally smaller than the panel area, we set the A value in Equation (20) to 70% of the total panel surface area, for a more realistic result.

4. Simulation Program Flow The simulation software program was implemented using Princeton Satellite Systems CubeSat Toolbox [24], which is a MATLAB® Toolbox (The MathWorks Inc., Natick, MA, USA) for designing CubeSats and analyzing CubeSats missions. Figure4 shows the flow chart of the simulation software program. Only solar radiation torque computation will be explained since the same procedure is applied to other disturbances albeit with differences in the environment parameters. The first step is to define data for the CubeSat model. Data that are populated consist of the satellite’s center of mass (Cm), deployable solar panels dimension and configuration, surface vectors with respect to the origin (rFace), surface area (A), surface the outward unit normal (n), and the optical properties (ρ) of each surface. In the second step, simulation time is first set as a reference to pre-allocate the maximum amount of space required by arrays of data in the subsequent process. This can improve the code execution time. Next, the initial position and velocity vectors of the satellite are propagated based on defined Keplerian elements. An orbit can be described using six classical Keplerian orbital elements: Semi-major axis (a), eccentricity (e), right ascension of the ascending node (RAAN), inclination (i), argument of perigee, and mean anomaly. These elements can provide information on the orbit size, orbital plane, and the position of the satellite. However, it is easier to use Cartesian elements that consist of a position, →r , and velocity, →v , for propagating the orbit. By using small step sizes, a numerical integrator can determine the new position and the velocity of the satellite given the current position, velocity, and acceleration. Subsequently, acceleration from other forces can be added, such as the solar radiation pressure effect. The Cartesian coordinate frame used in step 2 is the Earth-centered inertial (ECI) coordinate frame, which has its origin at the center of the Earth and is inertially fixed (Figure5). The fundamental plane contains the equator, and the positive X-axis (XI) points in the vernal equinox direction. The Z-axis (ZI) points in the direction of the geographical north pole and the Y-axis (YI) Aerospace 2019, 6, x FOR PEER REVIEW 7 of 23

3.3. Solar Power Calculation The power produced by the solar arrays mounted on the body and the deployable solar panels for any given location at a given time is [11]: 𝑃𝑜𝑤𝑒𝑟 =𝜂𝜌 𝐴 𝑆𝑛𝑠̂ 𝑛𝑓 𝑜𝑟 ̂(𝑠 𝑛) > 0 (20) where i is the solar cell number, 𝜂 is the solar cell efficiency, and 𝑠̂ 𝑛 is the dot product, equivalent to the cosine of the angle between 𝑠̂ and n. Whenever the resultant dot product is negative, it means that the particular surface does not face the Sun, and hence does not produce any power. In addition, since the solar cell mounted is normally smaller than the panel area, we set the A value in Equation (20) to 70% of the total panel surface area, for a more realistic result.

(𝐶), deployable solar panels dimension and configuration, surface vectors with respect to the origin Aerospace(𝑟𝐹𝑎𝑐𝑒2019), 6surface, 50 area (A), surface the outward unit normal (n), and the optical properties (𝜌)8 ofof 22 each surface. In the second step, simulation time is first set as a reference to pre-allocate the maximum amount of space required by arrays of data in the subsequent process. This can improve the code completes the right-hand set of coordinate axes. The algorithm to convert between Kepler elements to execution time. Cartesian elements are well documented, for example, by Wertz [12] and Sidi [25].

1. Define a CubeSat structure

2. Set simulation time and define an orbit

3. Generate the quaternions that transforms

from ECI to LVLH and rotate the payload Aerospace 2019 , 6, x FOR PEER REVIEW 8 of 23 down to nadir

Next, the initial position and velocity vectors of the satellite are propagated based on defined

Keplerian elements. An orbit can be described using six classical Keplerian orbital elements: Semi- major axis (a), eccentricity (e), right ascension of the ascending node (RAAN), inclination (i), argument 4. Computation of Sun vector and eclipse of perigee, and mean anomaly. These elements can provide information on the orbit size, orbital plane, and the position of the satellite. However, it is easier to use Cartesian elements that consist of a position, 𝑟⃗ , and velocity, 𝑣⃗, for propagating the orbit. By using small step sizes, a numerical integrator can determine the new position and the velocity of the satellite given the current position, 5. Disturbance and power computation velocity, and acceleration. Subsequently, acceleration from other forces can be added, such as the solar radiation pressure effect. The Cartesian coordinate frame used in step 2 is the Earth-centered inertial (ECI) coordinate frame, which has its origin at the center of the Earth and is inertially fixed

(Figure 5). The fundamental plane contains the equator, and the positive X-axis (XI) points in the 6. End vernal equinox direction. The Z-axis (ZI) points in the direction of the geographical north pole and the Y-axis (YI) completes the right-hand set of coordinate axes. The algorithm to convert between Figure 4. Flowchart of disturbance computation. Kepler elements to Cartesian elemenFigurets are 4. Flowchart well documented, of disturbance for example,computation. by Wertz [12] and Sidi [25].

Figure 5. Coordinate frames for the nadir-pointing satellite. Figure 5. Coordinate frames for the nadir-pointing satellite. Although the simulator is able to consider various orbital parameters, some simplifications can be appliedAlthough when the studying simulator the solaris able radiation to consider torque vari inous LEO. orbital Firstly, parameters, since the amountsome simplifications of momentum can flux bethat applied is incident when onstudying an earth-orbiting the solar radiation satellite torque is roughly in LEO. the sameFirstly, within since thethe earth’samount vicinity of momentum (i.e., at 1 fluxastronomical that is incident unit) [on1], an the earth-orbiting altitude or the satellite semi-major is roughly axis can the be same fixed within at one the value. earth’s Then, vicinity the focus (i.e., is at 1 astronomical unit) [1], the altitude or the semi-major axis can be fixed at one value. Then, the focus is on the shape of the disturbance during the transition from eclipse to daylight and vice versa. The rapid changes in thermal loading initiated during the transitions have been known to cause thermally induced dynamics in satellite appendages, such as the deployable solar arrays [26–28]. Therefore, a class of orbits can be selected, such as the orbit occupied by the International Space Station (ISS). The orbit is essentially important for CubeSats since, from 2012 to 2018, more than 200 CubeSats have been launched from ISS, mainly through services provided by the Japan Aerospace Exploration Agency (JAXA) [29] and the National Aeronautics and Space Administration (NASA) [30]. Next, a circular orbit is assumed; therefore, eccentricity is always zero and the argument of perigee can be removed. This is sensible since most satellites in LEO have an eccentricity less than 0.01 [5]. The inclination (i) and RAAN define the orbit plane, which means that the Sun’s angle to the satellite depends on them. Therefore, the values of i and RAAN vary throughout the simulations.

Aerospace 2019, 6, 50 9 of 22 on the shape of the disturbance during the transition from eclipse to daylight and vice versa. The rapid changes in thermal loading initiated during the transitions have been known to cause thermally induced dynamics in satellite appendages, such as the deployable solar arrays [26–28]. Therefore, a class of orbits can be selected, such as the orbit occupied by the International Space Station (ISS). The orbit is essentially important for CubeSats since, from 2012 to 2018, more than 200 CubeSats have been launched from ISS, mainly through services provided by the Japan Aerospace Exploration Agency (JAXA) [29] and the National Aeronautics and Space Administration (NASA) [30]. Next, a circular orbit is assumed; therefore, eccentricity is always zero and the argument of perigee can be removed. This is sensible since most satellites in LEO have an eccentricity less than 0.01 [5]. The inclination (i) and RAAN define the orbit plane, which means that the Sun’s angle to the satellite depends on them. Therefore, the values of i and RAAN vary throughout the simulations. In the third step, the satellite is aligned to the local vertical local horizontal (LVLH) coordinate frame from the ECI frame. The LVLH is another coordinate frame that has its origin at the center of the satellite and fixes to the orbit (Figure5). It is commonly assigned to Earth /nadir-pointing satellites, where ZLVLH points towards the Earth, YLVLH is normal to the local plane with a negative direction, and XLVLH completes the right-handed orthogonal axis set. By completing the transformation from ECI to LVLH, the motion of the satellite can be described by the translational motion of the center of mass of the satellite around the center of the mass of the Earth. The transformation from ECI to LVLH can be performed following the procedure outlined by Paluszek et al. [11]. Let the ECI unit vector be uÎ, and let []u represent the unit operator on a vector, returning a vector of the same direction with length 1: T uÎ = M uˆLVLH (21) where M is a transformation matrix used to transform vectors from the ECI frame to the LVLH frame and uˆ is a unit vector defined in the LVLH frame. LVLH The transformation matrix, M, may be computed from the spacecraft ECI position →r and velocity →v vectors as follows:  T   x   T  M =  y  (22)    zT  h i h i where y = →v →r , z = →r , and x = [y z] . The transformation matrix above has nine × u − u × u elements, which complicates its application for the propagation of an object. A more efficient method involves using quaternion terminology, which has a mere four elements. The quaternion (in this case, the quaternion from the ECI to LVLH frame) is represented by a four-row vector:    q0     q  q =  1  = q + iq + jq + kq I/LVLH   0 1 2 3 (23)  q2    q3 where q0 is a scalar component and q1 3 are vector components. The conversion of the transformation matrix to quaternion can be obtained− using the procedure outlined by Sidi [25]. Next, we wanted to rotate the payload down to the nadir direction. As an example, for the model 1 CubeSat, the nadir pointing is at the z-axis. The pointing target vector can be achieved by rotating − the CubeSat 180◦ around the x-axis. The quaternion of this rotation (from the LVLH coordinate frame to the body coordinate frame), herewith named qLVLH/B, is:

qLVLH/B = cosθ + i(xsinθ) + j(ysinθ) + k(zsinθ) (24) Aerospace 2019, 6, 50 10 of 22 where θ is the angle of rotation and x, y, and z are vectors representing the rotation axes. Finally, the transformation from the ECI frame to the body frame can be obtained as follows:

qI/B = qI/LVLHqLVLH/B (25)

From here on, each set quaternion and its correspondence time and other properties defined in the first step are processed one at a time in the fourth and fifth steps. In step 4, the Sun distance, its unit vector in the ECI frame, and the position when the satellite is in daylight or eclipse are computed using the methods explained in Section 3.2. In the last step, solar pressure force due to the solar flux from the Sun, albedo, and Earth-emitted radiation in the satellite body frame are collected, and then the disturbance torques are calculated. For the surfaces that face the solar flux, to find the unit Sun vector, sˆ, in Equation (5), the unit Sun vector, uˆsun, obtained in Equation (10) must be converted to the unit vector from the satellite toward the Sun. This can be completed using quaternion transformation as follows:

= sˆ qI/BuˆsunqI∗/B (26) where qI∗/B is the conjugate of qI/B. For the surface not facing the Sun, the sˆ is replaced as follows: h i sˆ = q →r q∗ (27) I/B − u I/B The value of sˆ in Equation (23) will be used in solar ﬂux force and solar power calculations, whereas the value of sˆ in Equation (24) will be used for both albedo force and Earth-emitted radiation force calculation. The negative sign in Equation (27) indicates that the source of radiation is from the → nadir direction. The total force on the satellite body, F B, will be the summation of these three forces. The eclipse condition determined in the fourth step is also considered during the calculation. The force on a surface acts as a torque on the satellite body if the face is oﬀset from the center of mass, as: X → → → T = rFace Cm FB (28) − ×

5. Disturbance and Power Evaluation A series of simulations were conducted to evaluate the solar radiation disturbance on the CubeSat models in LEO. The data of this orbit are shown in Table1. The orbit used resembles the orbit occupied by the ISS. The simulation time was set to one year to demonstrate the effects of season change. One year is a reasonable period since most CubeSats are currently designated with a mission lifetime less than one year long. The orbital parameters were simplified to focus solely on the characteristics of solar radiation torque. Therefore, the variation in inclination from 0 to 90 degrees is shown to demonstrate the variation in solar radiation torque due to satellite orientation with respect to the Sun. The fluctuation range of the disturbance torques between positive and negative values are used to compare the performance among the models tested. Additionally, the solar power generation of the different configurations was compared. Details of the satellite specifications and disturbance parameters are provided in Tables2 and3. The simulation time of the six orbits of the eclipse fraction at 408 km altitude is depicted in Figure6. In each orbit period, the eclipse fraction lasts about 2180 s, whereas the remaining daylight period lasts around 3352 s. Accordingly, the fraction of solar flux follows suit (i.e., occurs only during the daylight period). As shown in Figure6, the Earth-emitted radiation is constant throughout the orbit, since we previously assumed that it is uniform over the surface of the earth, whereas the reflected radiation due to albedo varies with the Sun vector and satellite vector from the Earth. Aerospace 2019, 6, 50 11 of 22

Table 1. Orbital parameters.

Parameter Value Semi major axis (a) 408 km Orbit inclination (i) 51.64◦ Initial right ascension of the ascending node (RAAN) 0◦ Argument of perigee (ω) 0◦ Eccentricity (e) 0 Initial mean anomaly (M) 0◦ Initial Julian date 2,458,563

Table 2. CubeSats speciﬁcation.

Item Specification Value Aerospace 2019, 6, x FOR PEER REVIEW Center of mass (x, y, z) (0, 0, 0) 11 of 23 All three models Residual dipole (x, y, z) (0, 0, 0.002 Am2) Table 3. Disturbance parameters. Absorbed (ρa) 0.75 Solar cell surface properties [24] Diffuse (ρ ) 0.08 Parameter d Value Specular (ρs) 0.17 Solar flux (S) 1367 Wm-2 Absorbed (ρ ) 0.15 Earth radiation a 400 Wm-2 Radiator surface properties [24] Diffuse (ρd) 0.16 Albedo factor (aF) Specular (ρs)0.33 0.69 Atmospheric density ( ) 3.725 × 10−12 kg m Drag coefficient ( ) 2 3 Table 3. Disturbance𝜌𝜌𝑎𝑎𝑎𝑎𝑎𝑎 parameters. ⁄ 𝐷𝐷 The simulation time of the sixParameter orbits of the𝐶𝐶 eclipse fraction Valueat 408 km altitude is depicted in Figure 2 6. In each orbit period, the eclipseSolar fraction flux (S) lasts about 2180 s,1367 whereas Wm− the remaining daylight period 2 lasts around 3352 s. Accordingly,Earth the radiation fraction of solar flux 400follows Wm− suit (i.e., occurs only during the daylight period). As shown Albedoin Figure factor 6, the (aF )Earth-emitted radiation 0.33 is constant throughout the orbit, 12 3 Atmospheric density (ρatm) 3.725 10 kg/m since we previously assumed that it is uniform over the surface× − of the earth, whereas the reflected Drag coefficient (CD) 2 radiation due to albedo varies with the Sun vector and satellite vector from the Earth.

(a) 1.5 Daylight Eclipse 1

0.5 Eclipse Fraction

0 0 1 2 3 4 5 6 7 8 9

(b) 356

2) 354 -

352 (Wm Earth Radiaton 0 1 2 3 4 5 6 7 8 9

Time (hours)

Figure 6. (a) Eclipse fractions, (b) earth radiation, and (c) radiation due to albedo at a 408 km altitude. Figure 6. (a) Eclipse fractions, (b) earth radiation, and (c) radiation due to albedo at a 408 km altitude.

5.1. Solar Radiation Torque on Model 1

The simulation results of the solar radiation torque in model 1 in the x-axis (Tx) and y-axis (Ty) are presented in Figure 7. The results show the one-year orbit period with the start of seasons indicated, whereas sub-figures provide more detailed disturbance characteristic of the six orbits’ simulation time. At the inclination set, the RAAN varies over 360° in a one-year simulation. There is no net torque in the z-axis (Tz), since the deployable solar panels form a symmetric satellite configuration with respect to the solar forces acting on it. Differences in the disturbance torque level due to seasonal effects can be seen as the Earth’s axis tilts with respect to the Sun’s rays changing. These effects are more apparent on Tx than Ty. During the summer solstice and winter solstice, the Tx is at a maximum and drops to almost zero during the autumnal equinox and vernal equinox. Ty registers its maximum and minimum in opposite seasons. Figure 7a shows the level of Tx during the summer solstice. The disturbance is mainly caused by the solar flux pressure on the deployable solar panels as they face the Sun throughout the daylight period. The level slowly increases when the

Aerospace 2019, 6, 50 12 of 22

5.1. Solar Radiation Torque on Model 1

The simulation results of the solar radiation torque in model 1 in the x-axis (Tx) and y-axis (Ty) are Aerospace 2019, 6, x FOR PEER REVIEW 12 of 23 presented in Figure7. The results show the one-year orbit period with the start of seasons indicated, whereassatellite sub-figuresleaves the eclipse provide, and more reaches detailed its disturbancepeak when the characteristic satellite passes of the through six orbits’ the simulation equatorial time. line Atand the subsequently inclination set, declining the RAAN untilvaries it enters over the360 eclipse.◦ in a one-year However, simulation. the more There concern is noing net characteristic torque in the is zthe-axis sudden (Tz), since torque the increase deployable and solar decrease panels at form the aedge symmetric of the eclipse satellite, since configuration the rapid with change respect in the to thedisturbance solar forces torque acting might on it. cause Differences sudden in rotational the disturbance motion torqueof the satellite. level due For to this seasonal particular effects case, can the be seencondition as the occurs Earth’s because axis tilts when with the respect satellite to theis in Sun’s the transition rays changing. from eclipse These to effects daylight are moreand vice apparent versa, ontwoT xdifferentthan Ty. sides During excha thenge summer positions solstice to andface winterthe solar solstice, flux from the T thex is Sun. at a maximumFor example, and when drops the to almostsatellite zero emerges during from the autumnal eclipse, the equinox solar andfluxvernal first hit equinox.s the solarTy registers cell surface its maximum on its body and for minimum a while inbefore opposite the deployable seasons. Figure solar7 apanels shows face the levelthe Sun. of TFigurex during 7b theshows summer the level solstice. of T They during disturbance autumnal is mainlyequinox, caused at which by the the solar Ty is flux at maximum. pressure on As the in deployable Tx, sudden solar torque panels changes as they can face be theseen Sun during throughout eclipse thetransitions. daylight period.During Thethe day, level T slowlyy has two increases peaks due when to thedifferences satellite leavesin the theposition eclipse, vector and of reaches the Sun its when peak whenthe satellite the satellite is in passesthe southern through and the northern equatorial hemispheres line and subsequently of the Earth. declining until it enters the eclipse. However,Next, the the more shapes concerning of solar characteristicradiation torques is the with sudden inclination torque increase angles andvarying decrease from at 0° the to edge90° are of thedepicted eclipse, in since the three the rapid-dimensional change in ( the3D) disturbance plots of Figure torque 8. mightEach plot cause is suddenan orbit rotational long simulation motion oftaken the satellite.during Forthe thissummer particular solstice. case, The the conditionsurface plots occurs show because the when time the and satellite angle is at in thewhich transition the torque from eclipsemagnitudes to daylight are among and vice the versa, maximum two different at their sidesrespective exchange axes. positions In Figure to 8a, face the the Tx solar change fluxs fromdirection the Sun.at the For inclination example, of when 23.4° the due satellite to the Earth’s emerges obliquity. from eclipse, The maximum the solar flux Tx occurs first hits in the the solar region cell when surface the oninclination its body foris around a while 70 before°. The the shape deployable of Ty with solar varying panels inclination face the Sun. angles Figure is 7shownb shows in theFigure level 8b. of TheTy duringoverall autumnal shape of equinox,the plot at is which basically the T they is atsame maximum. regardless As inofT thex, sudden inclination, torque but changes the levels can be differ seen duringaccordingly. eclipse From transitions. the 0° Duringinclination, the day, its Tmaximumy has two peakslevel dueincreases to differences until the in highest the position, which vector occurs of thewhen Sun whenthe inclination the satellite is 23.4°, is in the then southern subsequently and northern declining hemispheres almost linearly of the with Earth. increased inclination.

(a) (b)

Spring Autumn 0.04 Summer Winter 0.04

0.02 0.02

Nm) 0 Nm) 0 ( ( x y T T -0.02 -0.02

-0.04 -0.04

0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350

Time (days) Time (days)

0.02 0.04 Eclipse Transition Eclipse 0.03 0.01 transition 0.02

0.01 Nm) 0 (

Nm) 0 x T

-0.01 ( y -0.01 T

-0.02 -0.02 -0.03

-0.03 -0.04 92.8 92.9 93 93.1 93.2 189.8 189.9 190 190.1 190.2 Time (days) Time (days)

Figure 7. Solar radiation torques in model 1 in the International Space Station (ISS) orbit for one-year Figure 7. Solar radiation torques in model 1 in the International Space Station (ISS) orbit for one-year simulation and six orbit periods at seasonal start time: (a) Tx and (b) Ty. simulation and six orbit periods at seasonal start time: (a) Tx and (b) Ty.

Aerospace 2019, 6, 50 13 of 22

Next, the shapes of solar radiation torques with inclination angles varying from 0◦ to 90◦ are depicted in the three-dimensional (3D) plots of Figure8. Each plot is an orbit long simulation taken during the summer solstice. The surface plots show the time and angle at which the torque magnitudes are among the maximum at their respective axes. In Figure8a, the Tx changes direction at the inclination of 23.4◦ due to the Earth’s obliquity. The maximum Tx occurs in the region when the inclination is around 70◦. The shape of Ty with varying inclination angles is shown in Figure8b. The overall shape of the plot is basically the same regardless of the inclination, but the levels diﬀer accordingly. From the 0◦ inclination, its maximum level increases until the highest, which occurs when the inclination is 23.4Aerospace◦, then 2019 subsequently, 6, x FOR PEER declining REVIEW almost linearly with increased inclination. 13 of 23

(a)

23.4°

(b)

23.4°

FigureFigure 8. One8. One orbit orbit period period of of solar solar radiation radiation torques torques in modelin model 1 in 1 ISSin ISS orbit orbit with with varying varying inclination inclination

angles.angles. (a )(aT)x Tandx and (b ()bT)y T. y. 5.2. Solar Radiation Torque in Model 2 5.2. Solar Radiation Torque in Model 2 The simulation results of the solar radiation torque in model 2 are presented in Figure9. The shape The simulation results of the solar radiation torque in model 2 are presented in Figure 9. The of the one-year long simulation, Tx, is the opposite of model 1, whereas Ty, similar to model 1, shape of the one-year long simulation, Tx, is the opposite of model 1, whereas Ty, similar to model 1, always fluctuates between positive and negative torques throughout the orbit. Compared to model 1, always fluctuates between positive and negative torques throughout the orbit. Compared to model the fluctuation range of Tx and Ty increased by 134% and 108%, respectively. The sub-figures detailing 1, the fluctuation range of Tx and Ty increased by 134% and 108%, respectively. The sub-figures the shape of the torques in the six orbit periods further reveal the differences in comparison to model 1. detailing the shape of the torques in the six orbit periods further reveal the differences in comparison The torques produced during the eclipse transitions look rather smooth compared to those in model 1, to model 1. The torques produced during the eclipse transitions look rather smooth compared to which occurred due to differences in the solar panel configurations and the satellite’s velocity direction, those in model 1, which occurred due to differences in the solar panel configurations and the although the vector direction of the Sun to the model 2 satellite is different, resulting in the occurrence satellite’s velocity direction, although the vector direction of the Sun to the model 2 satellite is of Tz as well, where the order of magnitude is 10 times lower than Tx. different, resulting in the occurrence of Tz as well, where the order of magnitude is 10 times lower than Tx. From the 3D torque plots during the summer solstice in Figure 10, the inclination angles would be a factor of the level of Tx. The higher the inclination, the higher the magnitude of Tx. For Ty, regardless of the inclination angle, maximum levels always occur during eclipse transitions. The effects of the inclination angle change can also be observed in the pattern of Tz. Four peaks occur in high inclination, but only two peaks in low and mid inclinations.

Aerospace 2019, 6, 50 14 of 22 Aerospace 2019, 6, x FOR PEER REVIEW 14 of 23

(a) (b) 0.06 0.1 Summer Winter Autumn

0.04 0.05

0.02

Nm) 0

0 ( Nm) y T

( -0.05 x

T -0.02

-0.1 -0.04

-0.06 -0.15 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 Time (days) Time(days)

0.05 0.1

0.04 0.05

0.03

Nm) 0 Nm) ( x

0.02 ( y T

T -0.05 0.01

-0.1 0

Eclipse Transition -0.01 -0.15 92.8 92.9 93 93.1 93.2 189.8 189.9 190 190.1 190.2 Time (days) Time(days)

-3 -3 10 10 (c) 2 2

1 1 Nm) 0 Nm) 0 ( ( z z T T

-1 -1

-2 -2 0 50 100 150 200 250 300 350 341.2 341.3 341.4 341.5 341.6 Time (days) Time (days)

FigureFigure 9. 9. SolarSolar radiation radiation torques torques in in m modelodel 2 2 in in ISS ISS orbit orbit for one one-year-year simulation andand sixsixorbit orbit periods periods at atseasonal seasonal start start time: time: ( a(a))T Txx,(, (bb))T Tyy,, andand ((cc)) TTzz.

From the 3D torque plots during the summer solstice in Figure 10, the inclination angles would be a(a factor) of the level of Tx. The higher the inclination, the higher the magnitude of Tx. For Ty, regardless of the inclination angle, maximum levels always occur during eclipse transitions. The eﬀects of the inclination angle change can also be observed in the pattern of Tz. Four peaks occur in high inclination, but only two peaks in low and mid inclinations.

Aerospace 2019, 6, x FOR PEER REVIEW 14 of 23

(a) (b) 0.06 0.1 Summer Winter Autumn 0.04 0.05

0.02

Nm) 0 ( Nm) 0 y T (

x -0.05

T -0.02

-0.1 -0.04

-0.06 -0.15 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 Time (days) Time(days)

0.05 0.1

0.04 0.05

0.03 Nm)

Nm) 0 ( x 0.02 ( y T T -0.05 0.01

-0.1 0 Eclipse Transition -0.01 -0.15 92.8 92.9 93 93.1 93.2 189.8 189.9 190 190.1 190.2 Time (days) Time(days)

1 1 Nm) 0 Nm) 0 ( ( z z T T

-1 -1

-2 -2 0 50 100 150 200 250 300 350 341.2 341.3 341.4 341.5 341.6 Time (days) Time (days) Figure 9. Solar radiation torques in model 2 in ISS orbit for one-year simulation and six orbit periods Aerospace 2019, 6, 50 15 of 22 at seasonal start time: (a) Tx, (b) Ty, and (c) Tz.

(a)

Aerospace 2019, 6, x FOR PEER REVIEW 15 of 23

(b)

(c)

FigureFigure 10. One10. One orbit orbit period period of solarof solar radiation radiation torques torques in in model model 2 in2 in ISS ISS orbit orbit with with varying varying inclination inclination angles: (a) T ,(b) T , and (c) T angles: (ax) Tx, (by) Ty, and (c)z. Tz. 5.3. Solar Radiation Torque in Model 3 5.3. Solar Radiation Torque in Model 3 The simulation results of the solar radiation torque in model 3 are presented in Figure 11. The simulation results of the solar radiation torque in model 3 are presented in Figure 11. The The shapes of solar radiation torques, taken during the summer solstice with the inclination varying shapes of solar radiation torques, taken during the summer solstice with the inclination varying from from 0 to 90 are depicted in 3D plots in Figure 12. As expected, results of model 3 are similar to those 0° to◦ 90° are◦ depicted in 3D plots in Figure 12. As expected, results of model 3 are similar to those of of modelmodel 11 because because the the vector vector direction direction ofof the the Sun Sun to to the the deployable deployable solar solar panels panelsand and the the pointing pointing mission are similar to model 1. The main diﬀerence is the lower peak torques in both Tx and Ty. This mission are similar to model 1. The main difference is the lower peak torques in both Tx and Ty. This waswas expected expected since since the the distances distances between between solar solar panels panels and and the the satellite satellite body body of of model model 3 3 are are shorter shorter than in model 1. Compared to model 1, the ﬂuctuation ranges of T and T decreased by 62% and than in model 1. Compared to model 1, the fluctuation ranges of xTx and yTy decreased by 62% and 70%,70%, respectively. respectively.

Aerospace 2019, 6, 50 16 of 22 Aerospace 2019, 6, x FOR PEER REVIEW 16 of 23 Aerospace 2019, 6, x FOR PEER REVIEW 16 of 23

(a) (b) 0.01 0.015 (a) Summer Winter (b) 0.01 Spring Autumn Summer Winter 0.015 Spring Autumn 0.01 0.005 0.01 0.005 0.005 0.005 Nm) Nm) 0 Nm) 0 Nm) ( 0 x 0 ( ( x y T ( T y T -0.005 T -0.005 -0.005 -0.005 -0.01 -0.01

-0.01 -0.015 -0.01 -0.015 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 Time (days) Time (days) Time (days)Time (days)

0.01 0.01 0.015 0.015 EclipseEclipse Transition Transition

0.01 0.01 Eclipse Eclipse 0.005 0.005 transition transition 0.005 0.005 Nm) Nm) Nm) 0 0 Nm) 0 0 ( ( ( x ( y x y T T T T -0.005 -0.005 -0.005 -0.005 -0.01 -0.01

-0.01 -0.01 -0.015 -0.015 92.8 92.9 93 93.1 93.2 189.8 189.9 190 190.1 190.2 92.8 92.9 93 93.1 93.2 189.8 189.9 190 190.1 190.2 Time (days) Time (days) Time (days) Time (days)

Figure 11. Solar radiation torques in model 3 in ISS orbit for one-year simulation and six orbit periods Figure 11. Solar radiation torques in model 3 in ISS orbit for one-year simulation and six orbit periods Figure 11.at seasonalSolar radiation start time: torques (a) Tx and in model(b) Ty. 3 in ISS orbit for one-year simulation and six orbit periods at seasonal start time: (a) Tx and (b) Ty. at seasonal start time: (a) Tx and (b) Ty.

(a) (a)

23.4° 23.4°

Aerospace 2019, 6, x FOR PEER REVIEW 17 of 23

(b)

Figure 12.FigureOne 12. orbit One periodorbit period of solar of solar radiation radiation torques torques in in model model 3 3 in in ISS ISS orbit orbit withwith varying inclination inclination

angles:angles: (a) Tx and(a) T (x band) T y(.b) Ty.

5.4. Solar Power Generation The simulation results of the solar power generation in models 1, 2, and 3 over a one-year orbit period are presented in Figure 13. The sub-figures for the six orbits’ simulation times, taken during the summer solstice, provide a more detailed plot of the generated power. Seasonal effects can be seen affecting the peak level of the power generated. As expected, models 1 and 3 have similar shapes since the solar radiation torques over the one year period are similar. For model 1, the peak power is 25.2 W and the average power is 10.4 W. Model 2 has the lowest peak power and average power at 16.26 W and 6.1 W, respectively. Model 3 has the highest peak power at 40.3 W. However, its average power, at 9.9 W, is lower than that of model 1. The use of deployable solar panels has significantly improved power generation, especially for models 1 and 3. Considering the same nadir-pointing missions specified, when compared with their respective configurations without deployable solar panels, model 1 performs 201% better, whereas model 3 achieves a 237% increase in average power. Model 2 produces around a 107.4% improvement.

(a) (b)

Aerospace 2019, 6, x FOR PEER REVIEW 17 of 23

(b)

Figure 12. One orbit period of solar radiation torques in model 3 in ISS orbit with varying inclination

angles: (a) Tx and (b) Ty.

Aerospace 2019 6 5.4., Solar, 50 Power Generation 17 of 22 The simulation results of the solar power generation in models 1, 2, and 3 over a one-year orbit 5.4. Solarperiod Power are Generation presented in Figure 13. The sub-figures for the six orbits’ simulation times, taken during the summer solstice, provide a more detailed plot of the generated power. Seasonal effects can be Theseen simulation affecting resultsthe peakof level the of solar the power power generate generationd. As expected, in models models 1, 2,1 and and 3 have 3 over similar a one-year shapes orbit period aresince presented the solarin radiation Figure torques13. The over sub-figures the one year for pe theriod six are orbits’ similar. simulation For model 1, times, the peak taken power during is the summer25.2 solstice, W and provide the average a more power detailed is 10.4 W. plot Model of the2 has generated the lowest power.peak power Seasonal and average effects power can at be seen affecting16.26 the peak W and level 6.1 W, of respectively. the power generated.Model 3 has Asthe highest expected, peak models power at 1 40.3 and W. 3 haveHowever, similar its average shapes since power, at 9.9 W, is lower than that of model 1. the solar radiationThe use torques of deployable over the solar one panels year has period signific areantly similar. improved For power model gene 1, theration, peak especially power isfor 25.2 W and the averagemodels 1 powerand 3. Considering is 10.4 W. Modelthe same 2 nadir-pointing has the lowest missions peak specified, power and when average compared power with attheir 16.26 W and 6.1 W,respective respectively. configurations Model without 3 has the deployable highest sola peakr panels, power model at 40.3 1 performs W. However, 201% better, its average whereas power, at 9.9 W,model is lower 3 thanachieves that a of237% model increase 1. in average power. Model 2 produces around a 107.4% improvement.

(a) (b)

Aerospace 2019, 6, x FOR PEER REVIEW 18 of 23

(c)

Figure 13. Solar power generation over one year period of (a) model 1, (b) model 2, and (c) model 3. Figure 13. Solar power generation over one year period of (a) model 1, (b) model 2, and (c) model 3.

5.5. Discussion—Effects of Adding Deployable Solar Panels on the Overall Disturbances From the initial results above, the characteristics of the solar radiation torque of each CubeSat configurations were obtained. By including other external disturbances to those results, the effects of adding the deployable solar panels on the total external torques can be better observed. While the total resultant torques presented are by no means the maximum torques that will be experienced by the satellites, especially since the initial condition after deployment is not considered, forecasts on the maximum resultant torques to maintain the nadir-pointing mission proposed would be very useful for design engineers to plan the overall power usage. The one-year long period results of total torque in ISS orbit for models 1, 2, and 3 are plotted in Figures 14–16, respectively. In each figure, a comparison between the total torque with deployable solar panels and with none are made. Although not shown separately, the type of disturbance which dominates the total torque will be explained. Firstly, the resultant Tz for each model is not presented since the value is negligible compared to Tx and Ty. For model 1, the total torques are mainly contributed by the satellite’s residual dipoles. Due to the nadir-pointing mission chosen and the model’s configuration, both the gravity gradient torque and the aerodynamic torque are negligible. However, the solar radiation torque contribution is still obvious, especially in the x-axis, as shown in Figure 14a. At the maximum, for example, during the summer solstice, the increase of disturbance torque is about 13.3% for Tx and 12.7% for Ty when compared to not using the solar panels.

Aerospace 2019, 6, 50 18 of 22

The use of deployable solar panels has significantly improved power generation, especially for models 1 and 3. Considering the same nadir-pointing missions specified, when compared with their respective configurations without deployable solar panels, model 1 performs 201% better, whereas model 3 achieves a 237% increase in average power. Model 2 produces around a 107.4% improvement.

5.5. Discussion—Effects of Adding Deployable Solar Panels on the Overall Disturbances From the initial results above, the characteristics of the solar radiation torque of each CubeSat configurations were obtained. By including other external disturbances to those results, the effects of adding the deployable solar panels on the total external torques can be better observed. While the total resultant torques presented are by no means the maximum torques that will be experienced by the satellites, especially since the initial condition after deployment is not considered, forecasts on the maximum resultant torques to maintain the nadir-pointing mission proposed would be very useful for design engineers to plan the overall power usage. The one-year long period results of total torque in ISS orbit for models 1, 2, and 3 are plotted in Figures 14–16, respectively. In each figure, a comparison between the total torque with deployable solar panels and with none are made. Although not shown separately, the type of disturbance which dominates the total torque will be explained. Firstly, the resultant Tz for each model is not presented since the value is negligible compared to Tx and Ty. For model 1, the total torques are mainly contributed by the satellite’s residual dipoles. Due to the nadir-pointing mission chosen and the model’s configuration, both the gravity gradient torque and the aerodynamic torque are negligible. However, the solar radiation torque contribution is still obvious, especially in the x-axis, as shown in Figure 14a. At the maximum, for example, during the summer solstice, the increase of disturbance torque is about 13.3% for Tx and 12.7% for Ty when compared to not using the solar panels. Aerospace 2019, 6, x FOR PEER REVIEW 19 of 23

Model 1 Tx with solar panels Model 1 Tx without solar panels (a) (b) 0 0

Summer Winter -0.05 -0.05

-0.1 -0.1 Nm) Nm) ( ( x -0.15 x T -0.15 T

-0.2 -0.2

0 100 200 300 0 100 200 300

Time (days) Time (days)

Model 1 Ty without solar panels (c) Model 1 Ty with solar panels (d) 0.4 0.4

0.2 0.2 Nm) Nm) 0 0 ( ( y y T T -0.2 -0.2

-0.4 -0.4 0 100 200 300 0 100 200 300

Time (days) Time (days)

Figure 14. Total disturbance torque in model 1 in ISS orbit over a one year simulation period. (a) Tx Figure 14. Total disturbance torque in model 1 in ISS orbit over a one year simulation period. (a)Tx with solar panels, (b) Tx without solar panels, (c) Ty with solar panels, and (d) Ty without solar panels. with solar panels, (b) Tx without solar panels, (c) Ty with solar panels, and (d) Ty without solar panels.

Next, for model 2, a significant increase of total torque in the x-axis can be seen in Figure 15a when the deployable solar panels are mounted. This is mainly contributed by the aerodynamics drag as the space dart configuration contributes to a higher projected area with respect to the velocity direction. At the maximum, the increase of disturbance torque is about 322.6% for Tx and 6.5% for Ty when compared to not using the solar panels. Meanwhile, referring to the plot in Figure 9a, the solar radiation torque is less than 6% of the total disturbance torque.

Model 2 Tx with solar panels Model 2 Tx without solar panels (a) (b) 1 1

0.5 0.5 Nm) 0 Nm) 0 ( ( x x T T -0.5 -0.5

-1 -1 0 100 200 300 0 100 200 300

Time (days) Time (days)

Aerospace 2019, 6, x FOR PEER REVIEW 19 of 23

Model 1 Tx with solar panels Model 1 Tx without solar panels (a) (b) 0 0

Summer Winter -0.05 -0.05

-0.1 -0.1 Nm) Nm) ( ( x -0.15 x T -0.15 T

-0.2 -0.2

0 100 200 300 0 100 200 300

Time (days) Time (days)

Model 1 Ty without solar panels (c) Model 1 Ty with solar panels (d) 0.4 0.4

0.2 0.2 Nm) Nm) 0 0 ( ( y y T T -0.2 -0.2

-0.4 -0.4 0 100 200 300 0 100 200 300

Time (days) Time (days)

AerospaceFigure2019 ,146,. 50 Total disturbance torque in model 1 in ISS orbit over a one year simulation period. (a) T19x of 22 with solar panels, (b) Tx without solar panels, (c) Ty with solar panels, and (d) Ty without solar panels.

Next,Next, for modelmodel 2,2, a signiﬁcantsignificant increase of total torque in the x-axis-axis can be seen in FigureFigure 1515aa whenwhen thethe deployable solarsolar panelspanels areare mounted.mounted. This is mainly contributed by the aerodynamics drag as thethe spacespace dart conﬁgurationconfiguration contributes to a higher projected area with respect to the velocity direction. At the maximum, the increase of disturbance torque is about 322.6% for T and 6.5% for T direction. At the maximum, the increase of disturbance torque is about 322.6% for Txx and 6.5% for Tyy whenwhen comparedcompared toto notnot usingusing thethe solarsolar panels.panels. Meanwhile, referringreferring to the plot in Figure9 9a,a, thethe solarsolar radiation torquetorque isis lessless thanthan 6%6% ofof thethe totaltotal disturbancedisturbance torque.torque.

Model 2 Tx with solar panels Model 2 Tx without solar panels (a) (b) 1 1

0.5 0.5 Nm) 0 Nm) 0 ( ( x x T T -0.5 -0.5

-1 -1 0 100 200 300 0 100 200 300

Aerospace 2019, 6, x FOR PEERTime REVIEW (days) Time (days) 20 of 23

Model 2 Ty with solar panels Model 2 Ty without solar panels (c) (d) 1 1

0.5 0.5 Nm) Nm) 0 0 ( ( y y T T -0.5 -0.5

-1 -1 0 100 200 300 0 100 200 300

Time (days) Time (days)

Figure 15. Total disturbance torque in model 2 in ISS orbit over a one year simulation period. (a) T Figure 15. Total disturbance torque in model 2 in ISS orbit over a one year simulation period. (a) Txx with solar panels, (b) Tx without solar panels, (c) Ty with solar panels, (d) Ty without solar panels. with solar panels, (b) Tx without solar panels, (c) Ty with solar panels, (d) Ty without solar panels,. Lastly, for model 3, the characteristics are similar to model 1 albeit with lower total torques in Lastly, for model 3, the characteristics are similar to model 1 albeit with lower total torques in both the x-axis and y-axis. As shown in Figure 16, at the maximum, the increase of disturbance torque both the x-axis and y-axis. As shown in Figure 16, at the maximum, the increase of disturbance torque is about 3.4% for Tx and 3.2% for Ty when compared to not using the solar panels. is about 3.4% for Tx and 3.2% for Ty when compared to not using the solar panels. The data obtained above show that the solar radiation torques produced due to the usage of The data obtained above show that the solar radiation torques produced due to the usage of deployable solar panels are still much smaller compared to other disturbances. The proportion should deployable solar panels are still much smaller compared to other disturbances. The proportion be much lower if the nadir pointing mission is not used and the satellite’s center of mass is not exactly should be much lower if the nadir pointing mission is not used and the satellite’s center of mass is at the center of the body. On the other hand, the solar radiation torque proportion will increase when a not exactly at the center of the body. On the other hand, the solar radiation torque proportion will higher altitude is used. Therefore, future work may include these variables to enable better selection of increase when a higher altitude is used. Therefore, future work may include these variables to enable panels’ conﬁguration. better selection of panels’ configuration.

Model 3 Tx with solar panels Model 3 Tx without solar panels (a) (b) 0 0

-0.05 -0.05

-0.1 -0.1 Nm) Nm) ( ( x x -0.15 -0.15 T T

-0.2 -0.2

0 100 200 300 0 100 200 300

Time (days) Time (days)

Model 3 Ty with solar panels Model 3 Ty without solar panels (d) (c) 0.4 0.4

0.2 0.2

Nm) 0 Nm) 0 ( ( y y T T -0.2 -0.2

-0.4 -0.4 0 100 200 300 0 100 200 300

Time (days) Time (days)

Figure 16. Total disturbance torque in model 3 in ISS orbit over a one year simulation period. (a) Tx with solar panels, (b) Tx without solar panels, (c) Ty with solar panels, (d) Ty without solar panels.

Aerospace 2019, 6, x FOR PEER REVIEW 20 of 23

Model 2 Ty with solar panels Model 2 Ty without solar panels (c) (d) 1 1

0.5 0.5 Nm) Nm) 0 0 ( ( y y T T -0.5 -0.5

-1 -1 0 100 200 300 0 100 200 300

Time (days) Time (days)

Figure 15. Total disturbance torque in model 2 in ISS orbit over a one year simulation period. (a) Tx with solar panels, (b) Tx without solar panels, (c) Ty with solar panels, (d) Ty without solar panels,.

Lastly, for model 3, the characteristics are similar to model 1 albeit with lower total torques in both the x-axis and y-axis. As shown in Figure 16, at the maximum, the increase of disturbance torque is about 3.4% for Tx and 3.2% for Ty when compared to not using the solar panels. The data obtained above show that the solar radiation torques produced due to the usage of deployable solar panels are still much smaller compared to other disturbances. The proportion should be much lower if the nadir pointing mission is not used and the satellite’s center of mass is not exactly at the center of the body. On the other hand, the solar radiation torque proportion will Aerospace 2019, 6, 50 20 of 22 increase when a higher altitude is used. Therefore, future work may include these variables to enable better selection of panels’ configuration.

Model 3 Tx with solar panels Model 3 Tx without solar panels (a) (b) 0 0

-0.05 -0.05

-0.1 -0.1 Nm) Nm) ( ( x x -0.15 -0.15 T T

-0.2 -0.2

0 100 200 300 0 100 200 300

Time (days) Time (days)

Model 3 Ty with solar panels Model 3 Ty without solar panels (d) (c) 0.4 0.4

0.2 0.2

Nm) 0 Nm) 0 ( ( y y T T -0.2 -0.2

-0.4 -0.4 0 100 200 300 0 100 200 300

Time (days) Time (days)

Figure 16.16. Total disturbancedisturbance torquetorque inin modelmodel 33 inin ISSISS orbitorbit overover aa oneone yearyear simulationsimulation period.period. ((aa)) TTxx with solar panels,panels, ((bb)) TTxx without solar panels, (c) Ty with solar panels, ( d) Ty withoutwithout solar panels panels..

6. Conclusions In this study, we examined the solar radiation torques encountered by low earth orbiting nanosatellites with deployable solar panels. The solar power generated when using different configurations of deployable solar panels was presented. The software algorithm used to simulate the disturbance torque and solar power generation was described in detail. The optical properties of different surfaces were distinguished and pointing missions from previous literature were considered when deciding the satellite pointing missions. For demonstration and comparison purposes, three common configurations of 3U-sized CubeSat with deployable solar panels were tested in the orbit occupied by the International Space Station. The first CubeSat has four deployable solar panels attached 90 degrees at the short edges (model 1), the second CubeSat resembles the space-dart configuration (model 2), and the third CubeSat has two-doubled solar panels deployed along the long edge of its body (model 3). With respect to the nadir-pointing missions defined, models 1 and 3 would be affected by the eclipse effect in the form of sudden torque increases and decreases. However, model 1 experiences higher peak radiation torques than model 3. For model 2, although not affected by the eclipse transition effect, the overall torque magnitudes are relatively higher compared to those in models 1 and 3. In terms of the average solar power generation, model 1 leads, followed by model 3 with just 5% less than model 1, and model 3 lags far behind, generating just 59% of the power of model 1. When compared with other external disturbances, the solar radiation torques produced are within 3.2% to 13.3% of the total torques. Model 1 and model 3 experience the residual dipole torque the most while model 2 experiences aerodynamic drag the most. Overall, the solar radiation torque and solar power generation data presented would help CubeSats developers to select the right deployable solar panel configuration for their respective missions. Based on two factors, solar radiation torque and solar power generation, the model 2 configuration would be Aerospace 2019, 6, 50 21 of 22 the best option for nadir-pointing missions. Future work will extend the impact of the solar radiation torque on the dynamics of the respective CubeSat models used in the study, as well as the required control torque to maintain the proposed nadir pointing mission.

Author Contributions: Analysis, software, and writing—original draft preparation, S.A.I.; supervision and writing—review and editing, E.Y. Funding: This research was funded by Jabatan Perkhidmatan Awam Malaysia (HLP2015). Acknowledgments: The authors gratefully acknowledge the support from the National Space Agency of Malaysia (ANGKASA) in conducting this research. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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