SSC20-VII-02

Maneuver Planning for Demonstration of a Low-ThrustElectric Propulsion System

Madeleine Schroeder, Christopher Womack, Amelia GagnonMassachusetts Institute of Technology

Bldg. 37-335; (617) 253-7805msch1@mit.edu, cwomack@mit.edu, agagnon@mit.edu

ABSTRACT

As the testing and implementation of CubeSat technologies on-orbit becomes more prolific, the need forhigh-efficiency, low-mass propulsion systems continues to grow. Ionic propulsion systems have emerged asa potential technology to fill the niche of CubeSat propulsion. BeaverCube, a student-built 3U CubeSatfrom the Massachusetts Institute of Technology, will host an ionic propulsion system demonstration inlow Earth orbit. Slated to launch no earlier than October 2020, BeaverCube seeks to demonstrate theAccion Systems Inc. Tiled Ionic Liquid Electrospray propulsion system. This system utilizes an ionic liquidas propellant, giving BeaverCube the ability to make high-efficiency, low-thrust maneuvers. A successfulsystem demonstration will be able to detect a translational maneuver using the NovAtel OEM-719 GlobalPositioning System receiver onboard BeaverCube. Detectability requires the altitude change of a maneuverto be at least 9 meters, which is 3 standard deviations above the expected GPS altitude error. The goal ofthis work is to determine the duration of translational maneuver that will result in the highest probability ofdetection while producing the smallest error in thrust calculation. From simulations performed in SystemsTool Kit, a 3.5 hour maneuver was determined to be optimal, resulting in an altitude change of 280.6 meters.

INTRODUCTION

CubeSats are an increasingly important platform forearly stage, small-scale space technology develop-ment due to their small form factor and standardizedlaunch and deployment procedures. While CubeSatsprovide a low-complexity, low-cost way to test noveltechnologies in space, their use in larger missions,including multi-satellite constellations and space ex-ploration, is limited by their reliance on launch ve-hicles for delivery into their final orbit. The primarybarrier for the use of propulsion on CubeSats is thelack of high-efficiency, low-mass, low-power propul-sion systems. In response to the lack of CubeSatappropriate propulsion, engineers at Accion SystemsInc. (Accion) developed the novel Tiled Ionic Liq-uid Electrospray (TILE 2) thruster system, whichutilizes an ionic liquid as propellant.1 While someforms of electrospray propulsion have been exten-sively characterized in lab settings, most have notyet been demonstrated on orbit.2,3 As a result, thedemonstration of the TILE 2 is one of the missionobjectives of BeaverCube, a 3U CubeSat designedand built by the Massachusetts Institute of Technol-ogy (MIT).

BeaverCube provides an opportunity to test theTILE 2 propulsion system in space, validating per-formance of the thruster on orbit. The NovAtel

OEM-719 Global Positioning System (GPS) receiveronboard BeaverCube will be utilized to detect anychanges in its orbit resulting from maneuvers pro-duced by the TILE 2 thruster. These changes willbe compared to orbital simulations in Systems ToolKit (STK) to confirm that the system is function-ing as anticipated. In addition to demonstrating theTILE 2 system, the BeaverCube mission is requiredto prove that the thruster does not pose a risk tothe International Space Station (ISS), other satel-lites, or to its own payloads. This will be ensuredby continuously monitoring BeaverCube’s locationand preventing thruster firing until consent has beengiven by both the ISS and the Combined Space Op-erations Center (CSpOC). The system parametersof BeaverCube can be found in Table 1.

Maneuver planning must be completed to deter-mine the maneuver duration required to meet theprimary propulsion demonstration objective. Alti-tude change for a given maneuver duration is subjectto maneuver starting altitude along with uncertain-ties in the Attitude Determination and Control Sys-tem (ADCS). Errors in thrust calculation are subjectto errors in ADCS capabilities in addition to GPSerrors. The goal of this work is to determine the du-ration of translational maneuver that will result inthe highest probability of detection while producingthe smallest error in thrust calculation.

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BACKGROUND

BeaverCube Project Overview

BeaverCube, shown in Figure 1, is being built bystudents as part of the Space Systems Developmentcapstone course at MIT. Students are responsiblefor designing, assembling, testing, and integratingall components of BeaverCube. BeaverCube carriestwo payloads: a visible and long-wave infrared cam-era array to perform sea surface and cloud top tem-perature measurements, and the TILE 2 propulsiontechnology from Accion for orbital maneuvering.

Figure 1: Reproduction of BeaverCube.

BeaverCube is currently scheduled to launch onNG-14 as early as September 2020 and will be de-ployed into low Earth orbit (LEO) from the ISS.Following the activation of the imaging payload andapproval to fire, the propulsion demonstration willbe performed. This will most likely occur between 1and 3 months after deployment. The ADCS onboardBeaverCube is the iMTQ MagneTorQuer Board byInnovative Solutions in Space (ISISPACE). To powerall onboard systems, BeaverCube has 10 units (3x2U and 2x 2U) of solar panels fixed on the outersurfaces and a battery with a capacity of 40 Watthours.

Table 1: BeaverCube System Parameters

Parameter Quantity/Type

Mass 4 kg

Size 10 cm x 10 cm x 30 cm

Imaging Payload 1 Matrix Vision BlueFOX camera2 FLIR Boson IR cameras

Propulsion Payload Accion TILE 2 Thruster

GPS Receiver NovAtel OEM-719

ADCS ISISPACE MagneTorQuer

Battery Clyde Space 40 Wh

Solar Panels Clyde Space 10U

IMU TDK MPU-6000

Electrospray Propulsion

An electrospray propulsion system produces thrustby applying voltage between an extractor grid and aporous emitter chip of sharp tips wetted with propel-lant. The resulting electric field causes ions to evap-orate from the tips and accelerates them throughholes in the extractor. Electrospray propulsion isparticularly suited to small satellites due to the highthrust density and high specific impulse of the de-sign.2,3 Electrospray characteristics make it wellsuited for widely varying mission requirements in-cluding attitude control for imaging missions, sta-tion keeping for constellations, and large orbital ma-neuvers for space exploration.4,5

The TILE 2 system, shown in Figure 2, is suit-able for demonstration of the electrospray propulsionconcept on a 3U CubeSat. Table 2 lists the impor-tant characteristics of the TILE 2 system.

Figure 2: Image of Accion TILE 2 propulsionsystem.

Table 2: Accion TILE 2 System Parameters

Parameter Value

Thrust 50 µN

Specific Impulse 1800 s

Mass 0.48 kg

Size 0.5U

Low-Thrust Maneuvers

The demonstration of the Accion Systems TILE 2propulsion system will be considered successful if thefiring of the system results in a detectable maneuver.The two main types of maneuvers used to demon-strate the function of propulsion systems on orbitare rotational and translational maneuvers. Rota-tional maneuvers cause a rotation of the satelliteabout the center of mass by generating torque via

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off-axis thrusting. Although rotational maneuversmay require a lower maneuver duration to achieveresults detectable by most gyroscopes, spinning upthe satellite has drawbacks. In particular, it is pos-sible to spin the satellite to a rate at which the solarpanels are not able to supply enough energy for theADCS to detumble. Although the risk of this fail-ure mode is low, it could result in an end to themission. As such, rotational maneuvers were ruledout as a way to demonstrate the TILE 2 system onBeaverCube.

The propulsion demonstration will be achievedthrough the completion of multiple translational, ororbit-raising, maneuvers. In this type of maneuver,all four emitter chips on the TILE 2 system will firesimultaneously with the thrust vector antiparallelto the orbital velocity vector. The ∆V , a measureof the impulse per unit spacecraft mass, from thepropulsion system adds energy to the orbit of thesatellite. This energy addition results in an altitudeincrease.

The thrust produced by the TILE 2 system issmall enough to make the assumption that the satel-lite orbit will remain nearly circular throughout thecourse of any maneuvers.6 This allows for ∆V to beapproximated as the difference in velocities betweenthe initial orbit and the orbit after firing:

∆V ≈√

µ

R0−√µ

R(1)

where µ = Earth gravitational constant; R0 =initial altitude; R = altitude after maneuver.

The maximum ∆V achievable by the TILE 2 sys-tem is 8.8 m/s, which results from a burn of approx-imately 8 days in duration at the expected thrustof 50 µN. Thrust can be calculated by dividing Eq.(1) by the maneuver duration. Utilizing the low-thrust approximation, Eq. (1), this ∆V results in analtitude change of 14.2 km. However, due to the al-titude of the orbit of BeaverCube, atmospheric drageffects must be considered as well. Drag decreasesthe altitude change for a maneuver. The drag forceis given by:

FD =1

2CDAρv

2 (2)

where CD = drag coefficient; A = drag area; ρ= atmospheric density; and v = orbital velocity.

The drag coefficient and drag area of BeaverCubeare provided in Table 5. Atmospheric density varieswith altitude, leading to varying drag forces. Dragis not only dependent on the altitude of the satellite,but is also dependent on the velocity, drag area, andpointing error.

ADCS and Pointing

In order for the satellite to complete a translationalmaneuver, the propulsion system requires coordina-tion with the ADCS. The ADCS ensures the satelliteis in the correct pointing orientation to guaranteethrusting will occur along the intended vector. If asatellite were to orbit the Earth without force inputfrom the ADCS or elsewhere, the satellite would al-ways point in the same direction with reference tothe stars. In this example, the thruster would onlybe in the optimal position to maneuver for a singlemoment each orbit. To maneuver the satellite in anoptimal position for the entire orbit, the ADCS willslew the satellite to the desired, corrected positionto keep all thrust pointed in the anti-velocity direc-tion as the satellite orbits the Earth. Assuming allthrust is antiparallel to the direction of orbital ve-locity, Eq. (1) describes how the altitude will changefor a given ∆V .

BeaverCube utilizes the 3-axis iMTQ board forattitude control. Magnetorquers work by creating alocalized magnetic field that interacts with the mag-netic field surrounding the Earth. This interactioncauses the satellite to change orientation dependingon the direction and the strength of the field gener-ated by the magnetorquer.7

While the magnetorquer can adjust Beaver-Cube’s orientation as desired, there may still be er-rors in the pointing that result in thrusting in anoff-nominal direction. Pointing error is expected inboth azimuth (Az) and elevation (El). 0 degrees oferror is optimal, 5 degrees of error is assumed forrealistic off-nominal pointing, and 15 degrees is as-sumed for conservative worst-case off-nominal point-ing. Thrust in an off-nominal orientation will resultin a lower change in altitude, and therefore semi-major axis, than thrusting in the nominal orienta-tion. The proportion of on-axis thrust can be deter-mined by rotating the thrust vector by the pointingerror and projecting it onto the satellite velocity vec-tor. Additionally, because the CubeSat is pointedin an off-nominal direction, the ram-facing surfacearea with respect to orbital velocity direction mayincrease. Drag area is determined by projecting the3 dimensional shape of the rotated CubeSat onto theplane perpendicular to the orbital velocity.

Maneuver Detection and GPS Error

There are several different methods that can be im-plemented to verify that a translational maneuverhas been completed. One way is to utilize the ac-celerometers within the onboard inertial measure-ment unit (IMU) to detect any acceleration caused

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by the propulsion unit. However, the MPU-6000IMU onboard BeaverCube can only detect acceler-ations of magnitude 5.98 × 10−4 m/s2 and greater.Translational maneuvers produced by the TILE 2system have a maximum magnitude of 12.5 × 10−6

m/s2, rendering this type of maneuver undetectableto this IMU. Two-line element (TLE) propagationwill be used on BeaverCube for payload and othersatellite operations. However, it is unlikely thatTLE results will be accurate or precise enough todetermine the altitude change made by a maneuver.According to previous research, errors in TLE gen-erated altitudes are approximately 1 km for mostsatellites in LEO, with errors for satellites in ISS or-bit reaching 20 km in the worst case.8,9 Due to thelimitations of the onboard IMU and TLE propaga-tion, translational maneuvers will be detected usingthe GPS receiver onboard BeaverCube.

A GPS receiver collects information by receivingsignals from constellations of navigation satellitesin medium Earth orbit (MEO), such as the GPS,GLONASS, BeiDou, and Galileo constellations. Thereceiver determines its altitude and velocity by per-forming its own calculations based on the broadcastorbit and navigation message information from thenavigation satellites.

The accuracy of altitude determination by a GPSconstellation varies, and is dependent on several keyfactors. One of these factors is the geometric/po-sition dilution of precision (G/PDOP) of the GPSconstellation.10 GDOP is utilized to quantify howerrors in measurement will affect the final estimationof the position of the GPS receiver, and is definedas the ratio of the variance in the output location tothe variance in the input data.10 A low GDOP isdesirable, as it means that small errors in the inputdata will not result in larger errors in the outputlocation of the receiver. GDOP is most strongly in-fluenced by the relative positioning of GPS satelliteswith respect to the receiver; if visible satellites areclose together, it is more difficult to get an accurateposition of the receiver, which results in a higherGDOP value.11 Effects such as receiver clock driftand internal receiver noise also contribute to errorsin the accuracy of a GPS receiver on-orbit.12

METHODOLOGY

There are three major mission constraints thatare relevant to the success of the propulsion sys-tem demonstration: the maneuver starting altitude,pointing error, and GPS error. Maneuver startingaltitude and pointing error affect the altitude changemade by a maneuver of a given duration. GPS er-

ror affects the accuracy with which altitude changecan be measured on orbit. Table 3 lists each missionconstraint, along with details of their respective ef-fects on the propulsion demonstration. Additionallylisted are environmental factors and other missionparameters that determine this mission constraint.

Table 3: Summary of Key Factors

MissionCon-straint

Effect of Mis-sion Constrainton PropulsionDemonstration

What Determinesthis MissionConstraint

StartingAltitude

Starting altitudedetermines themagnitude of theeffect of drag dueto the variation ofatmospheric densityas well as the dis-tribution of orbitalvelocities.

Starting altitude isdetermined throughmission planning.The ISS programand CSpOC mustapprove the timingand parameters ofeach maneuver.

PointingError

Pointing error deter-mines theproportion of thrustthat is on-axis aswell as the dragarea of the satellite.

Pointing error isdetermined by theADCS and attitudecontrol softwareimplementation.

GPS Error GPS Errordetermines whataltitude changemust be made fora maneuver to bedetected.

GPS Error is de-termined by theproperties of theGPS receiver andthe position of thesatellite relative tothe constellations ofGPS satellites.

Expected values and uncertainties for each ofthese mission constraints were determined throughtheoretical and numerical analysis along with con-sideration of programmatic constraints. Altitudechange was determined for maneuvers of varying du-ration. Uncertainty in altitude change for a givenmaneuver duration was found by simulating boththe best- and worst-case mission constraint valuesgiven in Table 4.

Table 4: Best-, Expected-, and Worst-CaseMission Constraints

Parameter Best-Case Expected Worst-Case

StartingAltitude

428.65 km 392.32 km 361.9 km

Pointingerror

0◦ 5◦ 15◦

GPSError

2.8 m 2.9 m 3.0 m

Thrust determination error is the error in calcu-lating the system thrust from the results of a maneu-ver. It was determined by propagating uncertaintiesin mission constraints that are not controlled duringmaneuvers through Eq. (1). Optimal maneuver du-ration was chosen based on the uncertainty in ma-neuver detection error, the uncertainty in altitude

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change, and the expected thrust determination er-ror for a given maneuver duration.

Several assumptions, as listed below, were madeto simplify the following analysis. Thrust producedduring a maneuver was assumed to remain constant,and hardware was assumed to retain beginning of lifeperformance throughout the mission. Hardware wasalso assumed to remain at operational temperaturesduring maneuvers with no performance degradation.Additionally, power draw was assumed to remain atnominal levels for all hardware. Effects of orbitalparameters other than eccentricity were not investi-gated.

Table 5: Standard Mission Parameters

Mission Parameter Value

Earth Radius 6378 km

Semimajor axis 6798 km

Eccentricity (eccentric orbit) 0.0014898

Eccentricity (circular orbit) 0

RAAN 30.184◦

Inclination 51.702◦

Argument of Perigee 102.927◦

Maneuver Duration 4 hrs

Drag Area (for maneuvering) 0.01 m2

Drag Area (for propagating) 0.024 m2

Drag Coefficient 2.2

Ejection Velocity from ISS 1 m/s

Satellite Mass 4 kg

Pointing Error 0◦ in Az/El

GPS Velocity Error 0.03 m/s

All data was gathered from either theoretical ap-proximations which use Eq. (1) and (2) or from nu-merical simulations in STK. Altitude change in STKwas found by simulating the chosen orbit with andwithout a maneuver and finding the difference inthe semimajor axis (SMA) between these two simu-lations at the end of the maneuver. Changes in thealtitude of apogee and perigee were determined withthe same method. STK simulations used the EarthHPOP Default v10 (HPOP) force model, which ac-counts for drag effects in addition to variations inthe shape of the Earth. The Jacchia-Roberts modelwas used for the density of the atmosphere, which isconsidered accurate between altitudes of 90 and 2500km.13 The effects of drag were investigated by sim-ulating maneuvers both with and without drag. Allmission parameters were equal to the values given inTable 5 unless the parameters were explicitly statedto vary. The orbital parameters were identical to the

ISS orbit parameter prediction in STK for Novem-ber 1st, 2020. Eccentric orbits were modeled withthe eccentricity of the ISS orbit while circular orbitshad an eccentricity of 0.

Maneuver Starting Altitude

Demonstration of the propulsion system will com-mence after the imaging system of BeaverCube hasbeen commissioned. This is expected to occur afterapproximately one month on orbit, but this time pe-riod may be as long as three months. All maneuversperformed must be approved by the ISS and CSpOC.Expected-, best-, and worst-case starting altitudesfor maneuvers were found by modelling satellite de-ployment from the ISS and propagating the orbit for1 to 3 months.

To account for variations in the location of theISS, orbital position and altitude at deployment werevaried. ISS starting altitude range was determinedby looking at the hourly altitude of the ISS be-tween January 2015 and April 2020 as reported bythe National Aeronautics and Space AdministrationJet Propulsion Lab HORIZONS Web-Interface. As-suming the total altitude of all course correctionsremains under 5 km, the altitude of the ISS at thetime of BeaverCube release will be between 405 kmand 435 km. The separation velocity as a result ofdeployment was modeled as a 1 m/s ∆V impulsivemaneuver. The drag area for propagation was 0.024m2, which was found by estimating the average dragarea during imaging payload operations. It shouldbe noted that the orbit parameters for this model-ing are listed in Table 5, with the exception of semi-major axis and true anomaly. ISS orbital positionduring deployment was modeled both at perigee andapogee. Therefore the starting true anomaly was ei-ther 0 or 180 degrees while the semimajor axis wasvaried so that the modeled altitude at release wouldcoincide with the investigated altitude of release.

The effects of starting altitude were investigatedby comparing maneuvers starting between 350 and430 km, inclusive. Atmospheric density values fortheoretical calculations were found using a MAT-LAB implementation of the Jacchia reference atmo-spheric model.

Pointing Error

The effects of pointing error were investigated by si-multaneously varying the azimuth and elevation an-gle of the propulsion thrust vector from -20◦ to +20◦.The effects of pointing error on eccentricity and in-clination were investigated by varying azimuth andelevation angle independently. Altitude change for

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varying pointing error was determined by simulat-ing a maneuver with the chosen pointing error andcomparing it to a simulation of propagation with nopointing error. Drag area for a given pointing errorwas found using MATLAB to rotate the satellite bythe azimuth and elevation error and project it ontothe plane normal to the velocity vector. The effectsof drag area were found using Eq. (3):

dH =dH

dA

dA

dαdα (3)

where dH = altitude change for a given changein pointing error; dH

dA = slope of drag area vs. alti-

tude change; dAdα = slope of pointing error angle vs.

drag area; and dα = change in pointing error.dHdA was found by simulating the altitude change

for drag areas varying from 0.01 to 0.04 m2, findinga linear fit, and taking the derivative of the linearfit. dA

dα was found by calculating the drag area forpointing errors between -20◦ and +20◦, finding apiecewise linear fit, and taking the derivative of thepiecewise fit. The effects of off-axis thrust were de-termined by rotating the thrust vector by the point-ing error in azimuth and elevation and projectingthe resultant vector on the satellite velocity vector.This thrust was then used with Eq. (1) to determinealtitude change.

GPS Error

Variations in the accuracy of the GPS receiver on-board BeaverCube were investigated by calculatingthe estimated (1-σ) position error in STK. This wasdone using a dilution of precision (DOP) simulationfor a GPS receiver in LEO; the simulation utilizedJ2 propagation for a period of 30 days for estimatingthe GPS error in both circular and eccentric orbits.14

The effects of starting altitude on GPS error were de-termined by varying the initial altitude of the DOPsimulation from 300 to 430 km. The effects of the ini-tial date of firing were also determined through thesame DOP simulation. In this case, the start datewas varied between Nov. 1st, 2020 and Oct. 1st, 2021in one month increments. The initial conditions forBeaverCube each month were held constant, ensur-ing that any variations in GPS error were the resultof the positioning of the GPS constellation and notdue to irregularities in the orbit of BeaverCube.

Maneuver Duration

Orbit average power and maneuver power drawwere used to determine the time that the satellite

can perform propulsive maneuvers before propul-sion must be shut off for low-power-mode charg-ing. The relationship between maneuver durationand altitude change was determined by comparingaltitude changes while varying maneuver durationsbetween 15 min and the maximum maneuver dura-tion of approximately 9 hrs. The effect of startingthe maneuver at different locations on orbit was de-termined by comparing altitude changes for varyingmaneuver durations starting at perigee or apogee.Expected altitude changes for varying maneuver du-rations were calculated using expected starting alti-tude and pointing error values. Uncertainty in al-titude change was found by simulating varying ma-neuver durations with both the best- and worst-casemission constraints given in Table 4.

Thrust Determination Error

Thrust determination error is dependent on factorsthat are unknown when completing a maneuver,namely the detection and pointing error. These ac-curacies cannot be determined on orbit due to thelack of truth values for comparison. Errors from Eq.(1) must also be included. Thrust error for calcula-tions based on altitude change was found by takingthe derivative of the right side of Eq. (1) with re-spect to the altitude change for a maneuver, multi-plying by the uncertainty in altitude change to getthe velocity change uncertainty, and dividing by theduration of the maneuver to get thrust uncertainty.The result is shown in Eq. (4).

δTaltitude =

(1

2µt

(R0 + ∆H

µ

)−32

)δH (4)

where δT = thrust determination error using alti-tude change; ∆H = altitude change; t = maneuverduration in seconds; R0 = maneuver starting alti-tude; and δH = altitude change uncertainty result-ing from pointing error, GPS error, and error fromEq. (1).

Thrust error for calculations based on velocitywas found by taking the derivative of the left side ofEq. (1) with respect to velocity change, multiplyingby the uncertainty in velocity change, and dividingby the duration of the maneuver to get thrust un-certainty. The result is shown in Eq. (5).

δTvelocity =δH

t(5)

where δTvelocity = thrust determination error us-ing velocity change.

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Maneuvers were assumed to start at 392.32 kmand maneuver duration was assumed to have no er-rors. GPS velocity detection was assumed to havean error of 0.03 m/s.14

RESULTS

Starting Altitude

Table 6 summarizes the results of modeling Cube-Sat deployment from the ISS at varying ISS startingaltitudes and deployment orbit locations. The bestdeployment and maneuvering conditions from an al-titude perspective would be to deploy at apogee andthen start maneuvering at apogee one month afterdeployment. The worst condition from an altitudeperspective would be to deploy at perigee and thenstart maneuvering at perigee three months after de-ployment.

Table 6: Starting Altitude Results

Case ReleasePosition

ManeuverPosition

Best Apogee Apogee

Expected Apogee Perigee

Worst Perigee Perigee

ReleaseAltitude(km)

Time AfterRelease(months)

ManeuverAltitude(km)

435 1 428.65

420 2 392.32

405 3 361.9

Figure 3: Altitude changes for maneuversstarting at varying altitudes.

Figures 3 and 4 show the effects of varying start-ing altitude on altitude change due to drag and ve-locity variation. Altitude change for a given ∆V in-creases for higher altitudes due to the inverse square

root relationship between orbital velocity and alti-tude as well as the decrease in drag due to particledensity. The decrease in altitude change at loweraltitudes is smaller for the eccentric orbit as seen inFigure 3. It is possible that this is due to the alter-nation between higher and lower altitudes in the ec-centric orbit, which mitigates the effects of changingthe average altitude. Due to the effects of drag on al-titude change, the optimal procedure for propulsivemaneuvers is to initiate them as soon after deploy-ment as possible, maximizing starting altitude.

Figure 4: Difference between maneuvers sim-ulated with and without drag.

Pointing Error

Figure 5 shows the effect of varying azimuth and ele-vation simultaneously. The effect of varying azimuthalone was identical to the effect of varying elevationalone. A greater azimuth-elevation error combina-tion results in a lower altitude change because agreater proportion of the thrust vector is off-axis.As seen in Figure 6, an increase in off-axis thrustdue to increasing pointing error decreases the alti-tude change. The effects of drag also increase withpointing error due to the increased drag area. Theeffect of drag on altitude change is greater than theeffect of off-axis thrust until pointing error exceeds+/- 15◦. Exceeding +/- 15◦, the off-nominal thrustangle is the largest source of altitude change. As thedetrimental effects of drag area and off-axis thrustincrease with pointing error, it is clear that the opti-mal pointing error for making propulsive maneuversto increase altitude is 0◦ in azimuth and elevation.

The magnitude of inclination changes resultingfrom off-axis thrust in the azimuth and elevationdirections were lower than 5◦ × 10−5 with largerchanges for simulations with variations in azimuth.Eccentricity variations were smaller than 7 × 10−6

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for all simulations with little variation between theeffect of azimuth and elevation.

Figure 5: Altitude changes for maneuverswith varying pointing error.

Figure 6: Effect of drag area and off-axisthrust due to pointing error.

GPS Error

As seen in Figure 7, the dependence of GPS (1-σ) position error on the initial maneuver altitudeof BeaverCube is approximately linear. There is aclear decrease in the error calculated from the DOPsimulation as the initial altitude increases. This ismost likely the result of GPS signal propagation de-lay as the signal passes through different levels of theatmosphere. This results in greater errors at loweraltitudes with larger tangent paths through higheratmospheric density. Based on this relationship, thebest course of action to take in terms of maneuverdetectability would be to commence maneuveringwhile BeaverCube is at its highest altitude.

GPS position error is also dependent upon thedate when firing commences, as is illustrated by Fig-ure 8. While GPS error follows an overall linear

trend in relation to firing date, there are somewhatcyclic changes in the calculated error as time pro-gresses. These cycles see the error increasing as timeprogresses, but have a roughly three month periodbetween peaks. It is possible that the least favorableconfiguration of GPS satellites has a period of aboutthree months; further confirmation and analysis ofthis effect is planned in future work.

Figure 7: Expected GPS (1-σσσ) 3D positionerror with varying starting altitudes.

Figure 8: Expected GPS (1-σσσ) 3D positionerror with varying start dates.

Maneuver Duration

Maximum maneuver duration limited by poweravailability was calculated using the parameters inTable 7. Net power production over a 24 hour periodis found by subtracting the hourly power consump-tion from the hourly power production and multiply-ing by 24. Adding the net power production duringthe 24 hour period to the available battery capacitymultiplied by the allowed discharge percentage and

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dividing by the satellite power draw during maneu-vers yields the maximum maneuver time. As seenin Table 7, the BeaverCube power system is capa-ble of supporting a 9 hour maneuver. However, thisrequires 100% discharge, which could damage thebattery thereby causing an end to the mission. Toavoid this risk, battery discharge is limited to 75%,which results in a maximum possible maneuver du-ration of 7.7 hours.

Table 7: Power Budget Parameters

Mission Parameter Value

Orbit Average Power Consumption 5.09 W

Orbit Average Power Production 6.43 W

Battery Capacity 40 Wh

Power Draw During Maneuvers 8 W

Max. Maneuver Time - Full Discharge 9.0 hr

Max. Maneuver Time - 75 % Discharge 7.7 hr

Figure 9: Altitude increase for maneuvers ofvarying duration.

Figure 10: Effect of drag for maneuvers ofvarying duration.

Figures 9 and 10 show the effects of drag andmaneuver duration on altitude change. The depen-dence of altitude change on maneuver duration is lin-ear for theoretical approximations and sinusoidal fornumerical simulations. This sinusoidal behavior isdue to the spiral nature of low-thrust maneuvers notaccounted for in the derivation of Eq. (1).6 Longermaneuvers have a larger total drag effect as the dragforce is acting over a longer period of time. However,the ratio of total drag effect to total altitude changeremains approximately constant at 0.05. The rela-tionship between drag and maneuver duration is acomplex periodic function for eccentric orbits due tothe variation in density between apogee and perigee.

Uncertainty and Thrust Determination Error

Figure 11 shows the expected maneuver altitudechange with an uncertainty band. This band showsthe uncertainties in starting altitude and pointingerror. Table 4 shows the parameters used for theexpected, best, and worst cases used to determineuncertainty. Uncertainties vary between 4 and 9%of the expected altitude change.

Figure 11: Uncertainty in altitude change formaneuver of varying duration.

Figure 12 shows the error in thrust determinedusing the change in both altitude and velocity. Er-rors using altitude change are larger than errors us-ing velocity change because Eq. (1) assumes a con-stant eccentricity of 0. As a result, altitude changevaries between numerical simulations and theoret-ical calculations. The differences between numeri-cal and theoretical results vary between 0 and 10%,with larger errors for longer maneuvers. Errors us-ing velocity change decrease with increasing maneu-ver duration because velocity error is divided by themaneuver duration as shown in Eq. (5).

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Figure 12: Thrust determination error forvarying maneuver duration.

DISCUSSION

Maneuver Duration

Once the imaging system on BeaverCube has beencommissioned, the propulsion unit may only begranted permission to fire a limited number of times.As a result, the duration of each potential maneuvermust be chosen to maximize the likelihood and ac-curacy of maneuver detection, even in a worst-casescenario.

Worst-case scenario parameters are given in Ta-ble 4. The altitude change required for a maneuverto be detectable in this worst-case is 9 m, 3 timesthe standard deviation of the GPS error. For theTILE 2 propulsion system, the worst configurationin which thrust could be produced would occur ifonly 2 of the 4 thrusters were operational, resultingin half of the expected thrust and altitude increase.In this case, the altitude change would need to betwice the value previously calculated, or 18 m. Amaneuver with a duration greater than 15 minuteswould result in an altitude change greater than 19m, satisfying this requirement.

As thrust determination error decreases with in-creasing maneuver duration, the optimal duration tominimize thrust error is the longest maneuver thatthe system is capable of producing. Battery dis-charge must be limited to 75% to avoid damage tothe battery, which requires maneuvers to remain un-der 7.7 hours. Additionally, longer maneuvers havethe potential to increase the corrective force requiredfrom the ADCS to point the satellite, resulting inlarger pointing errors and possible loss of satellitecontrol.

A maneuver duration of 3.5 hours avoids the riskof damaging the battery while minimizing thrust er-

ror. This maneuver duration results in an altitudechange of 280.6 m. The best- and worst-case alti-tude changes are 261.7 m and 285.3 m, respectively.The lower bound of this uncertainty range is 252.7m above the 9 m, 3-σ uncertainty in the GPS error.For this maneuver duration, the expected thrust de-termination error is approximately 5.1% when us-ing altitude change and 25.6% when using velocitychange.

Operational Implications

The results of this maneuver planning maximize thelikelihood of detecting maneuvers subject to uncer-tainties in starting altitude, GPS error, and pointingerror. However, it is possible to plan the timing andexecution of the maneuver to improve the probabil-ity of detection and further reduce thrust determi-nation error.

One factor that will have a large impact uponthe propulsion demonstration detectability and thethrust determination error is the date on which fir-ing approval is granted. As the altitude of Beaver-Cube decays, any maneuver performed will result ina smaller change in altitude compared to one madeat a higher orbit. Additionally, GPS error increasesas time after deployment increases. To maximize thelikelihood of detection, maneuvers should be per-formed as early as possible. Time-dependent vari-ations in GPS error stem from GPS constellationorbital dynamics, which result in less-than-optimalconstellation configurations for the onboard receiver.Firing several months to a year after deployment, theexpected (1-σ) position error could increase from aslow as 2.8 m to more than 3 m. This indicates thatfiring should take place as soon as possible to mini-mize the GPS error, and thus, the thrust determina-tion error. However, it should be noted that, due tothe cyclic behavior of GPS error over time, it maybe beneficial to wait to fire until a local minimumerror value is reached.

While both GPS error and maneuver startingaltitude results indicate that maneuvers should beperformed as soon as possible, the time at which fir-ing approval is granted has significant implicationsupon the safety of the mission. Any CubeSat witha propulsion system could be considered a recon-tact hazard for the ISS. As such, there must be adelay between initial deployment from the ISS andthe initial firing of the TILE 2 system. During thisperiod, the primary focus of BeaverCube will be tosuccessfully commission the imaging system withoutthe additional risk incurred by firing the propulsionunit. Following the imaging payload demonstration,

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maneuvers will be initiated as soon as approval isgranted.

If the GPS receiver onboard BeaverCube wereto malfunction, translational maneuvers would diffi-cult to confirm with the IMU or TLEs. In this case,there remains the potential for the system to per-form a rotational maneuver. However, this shouldonly be utilized as a last resort, due to the risk ofspinning up the satellite.

Due to programmatic constraints, maneuverswill be individually commanded. Splitting up ma-neuvers may allow for a greater altitude change thaninitially estimated, as the system could fire for themaximum duration deemed safe, recharge the bat-tery, and fire again. Provided charging cycles areclose together, the altitude decay between subse-quent maneuvers could be small enough that thechanges in altitude would nearly sum. This ap-proach would be beneficial from a detectability pointof view, but requires more communication cycleswith BeaverCube, increasing the potential for delaysand transmission errors.

Limitations

All analysis assumed that hardware would retain be-ginning of life performance throughout the mission.Degradation of hardware over time due to exposureto the space environment could cause a significantdecrease in function. In particular, battery capacityand solar panel efficiency could decrease, shorteningthe maximum possible maneuver time. Additionally,there may be variations in hardware performancethat could affect the power budget given in Table 7.These variations have not yet been characterized forBeaverCube and as such were not included in thisanalysis. As the maneuver duration chosen to meetthe mission requirements is far from the maximumpossible maneuver duration of 9 hours, the authorsdo not believe that including these variations wouldsignificantly change the conclusions. As discussedpreviously, there is no data available for thrust vari-ation for electrospray thrusters over the lifetime ofthe system and as such this data was not includedin this work.

The magnitude of inclination and eccentricityvariations were less than 5◦ × 10−5 and 7 × 10−6,respectively. As seen in the starting altitude andmaneuver duration results sections, the eccentricityof the ISS orbit has a small effect on the altitudechange for a given maneuver. Thus, coupling be-tween the effects of pointing error on eccentricity andinclination and the effects of these orbit parameterson altitude change was assumed to be negligible.

Future Work

Looking forward, there are several topics whichcould be investigated further in order to fully char-acterize low-thrust maneuver planning. As the es-timated pointing error is one of the largest factorsinfluencing maneuver analysis, an exploration intoother potential ADCSs could be beneficial. Alongthe same lines, a military-grade GPS receiver maybe able to achieve a higher level of accuracy than thereceiver currently on BeaverCube, and should be in-vestigated. One scenario which must be consideredis where the GPS receiver fails completely. With-out a functioning, onboard receiver, TLE propaga-tion would be necessary for translational maneuverdetection. Continuing work in maneuver planningshould include a closer investigation of TLE propa-gation to mitigate the potential loss of maneuver de-tectability. Despite the lower magnitude of accuracyof TLE propagation compared to GPS, redundanciesin maneuver detection can only improve the odds ofmission success. In addition, the effects of the pre-cession of the ISS orbital plane relative to GPS mustbe examined to see if it coincides with the peaks inexpected GPS error seen in Figure 8. Finally, to fullycharacterize how the TILE 2 thruster will performover the course of its operation, the thrust variationof the system will need to be investigated.

CONCLUSIONS

This paper has demonstrated some of the techniquesused for low-thrust, translational maneuver plan-ning required to maximize the likelihood of successof a propulsion demonstration. Parameters that af-fect propulsive maneuvers such as starting altitude,pointing error, and GPS error were investigated todetermine the optimal maneuver duration for theBeaverCube mission. The proposed maneuver du-ration for the TILE 2 system is 3.5 hours. A ma-neuver of this duration satisfies the requirement ofexceeding the expected detection error of 3 m bythree standard deviations. This eliminates overlapbetween the uncertainty in detection error and al-titude change and produces the smallest error inthrust calculation.

Most low-thrust maneuver planning prior to thishas largely focused on station keeping for geosyn-chronous orbits along with Earth escape trajectoriesfor deep space exploration. However, the introduc-tion of high-efficiency propulsion systems, such asthe Accion TILE 2, could see a shift from past meth-ods. As ion electric propulsion systems become moreprolific, methods for maneuver planning in LEO,

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such as the methods discussed in this work, will be-come more relevant. While the mission parametersused in this work are specific to BeaverCube andthe TILE 2 system, the techniques used to investi-gate parameters relevant to the propulsion demon-stration can be generalized to other low-thrust sys-tems. In particular the dependence of maneuverson starting altitude, pointing error, altitude change,and GPS error can be utilized to guide future ma-neuver planning for low-thrust maneuvers in LEO.As low-thrust, high-efficiency systems become morewidespread for use in CubeSats, the need for maneu-ver planning for LEO missions will become a criticalpart of satellite mission planning.

ACKNOWLEDGEMENTS

The authors would like to thank Jeremy Wertheimer,Accion Systems, STAR Lab, WHOI, MIT MediaLab, Analytical Graphics Inc., and Northrop Grum-man for technical and financial support along withPeter Grenfell for his assistance with simulating GPSerror. Special thanks goes to the students and men-tors working on BeaverCube, in particular KateBrewer, Mary Dahl, and Gustavo Valez for theirpower budget calculations. This work was made pos-sible by the efforts of the many TAs, mentors, andstudents who have worked on BeaverCube, ProfessorKerri Cahoy for teaching the Space Systems Devel-opment class, and Robert Cato III for his work onthe BeaverCube propulsion team.

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