Analysis of the Pointing Accuracy of a 6U CubeSat Mission for ProximityOperations and Resident Space Object Imaging

B. Udrea1, M. Nayak2, F. Ankersen3†

1Embry-Riddle Aeronautical University, FL, USA;2Red Sky Research, NM, US;

3European Space Agency, The Netherlands

This paper describes the results of the initial analysis of the pointing accuracyof a nanosatellite of the 6U (34×22×10cm) CubeSat class. The Applicationfor RSO Proximity Analysis and IMAging (ARAPAIMA) mission is designedto perform in-orbit demonstrations of autonomous proximity operations forvisible, infrared, and three-dimensional imaging of resident space objects(RSOs). ARAPAIMA will fly a payload consisting of a commercial off-the-shelf (COTS) infrared (IR) camera and miniature laser rangefinder. The startracker (STR) from the attitude determination and control system (ADCS) isalso used as a payload instrument for imaging in the visible spectrum. Thesatellite’s actuators include a reaction control system (RCS) comprised of athree-axis RCS thruster and reaction wheel array together with a single or-bital maneuvering thruster. The star tracker is complemented by an InertialMeasurement Unit (IMU) and a GPS unit. The satellite’s ability to reject ex-ternal and internal disturbance torques while maintaining a pointing positionrelative to a stationary object is tested utilizing only the reaction wheel triad.

1. Introduction

The goal of the ARAPAIMA mission is to perform the in-orbit demonstration of au-tonomous proximity operations for visible, infrared, and three dimensional imaging ofresident space objects (RSOs). The payload consists of a commercially available IRcamera and a miniature laser rangefinder with a range of a few km. The star trackerfrom the attitude determination and control system is used as a visible light camera.The three instruments are installed on the nanosat so that their optical axes are point-ing in the same direction. The nanosat is equipped with a warm gas propulsion systemwhich enables it to perform orbital maneuvers and reaction attitude control. Furtherdetails on the satellite subsystems are presented in [11]

The goal of the mission is achieved in five steps of increasing complexity. During thefirst two steps the nanosat uses pre-loaded and ground initiated commands to ma-neuver within rangefinder range of the RSO and acquire a relative circular orbit withrespect to it. The third step consists of the nanosat using constraint-based algorithms

†The views expressed in this article are those of the author in the capacity of a private person and donot necessarily express the views of and should not be attributed to ESA.

1

and visual navigation to maneuver autonomously to reduce the size of the relative orbitto a few hundred meters. The fourth step involves detailed visible and IR imaging ofthe RSO from multiple relative orbit profiles [7]. During the fifth step a combination ofchaser attitude motion and relative motion between the nanosat and the RSO is em-ployed to perform 3D imaging by combining rangefinder measurements and knowledgeof the nanosat inertial attitude and position [6].

The approach to the design of the ARAPAIMA mission closely follows that of a fullyfledged science mission. The basic concept has been developed into the design of theexperiment and initial simulations of the mission have shown encouraging results. Thesubsequent steps are 1 ) implementing the experiment; 2 ) executing it; 3 ) extractingthe data products; 4 ) and interpreting the results. However, unlike a fully fledged spacemission only a few of the mission level requirements have been flown down to systemand subsystem level. The reason for the partial flow down of the requirements comesfrom the practical aspects of developing a space mission as a student project and withinthe budget and physical size constraints of a university built and operated mission. Asa consequence , parts of the mission are developed in a bottom-up approach incorpo-rating simulation, design, integration and testing of components and subsystems, withchecks of their performance against an ideal mission model.

A CubeSat class satellite with this level of expected complexity and maneuverability isunprecedented in the field. This mission’s success will expand the expected range ofcapabilities of small satellites and establish a standard of performance hitherto unex-pected without compromising the initial idea of universal feasibility. As a pioneer on thisfront, ARAPAIMA’s ADCS utilizes COTS components combined with an experiencedknowledge of mission management and design, and complex system dynamics whichwill result in capabilities limited only by the satellite’s lifespan.

2. Mission Design

The ARAPAIMA mission concept consists of the previously referenced five steps lead-ing to the successful orbit and three-dimensional imaging of an RSO, using passivevisual-only navigation and real-time near-optimal guidance. The mission design fo-cuses on a 6U (34×22×10cm) CubeSat class nanosatellite of approximately 12kgmass operating in low earth orbit (LEO). ARAPAIMA’s payload consists of an MLR2Klaser rangefinder from FLIR Systems Inc. (maximum range of approximately 2km) anda short wave GA640C IR camera from Goodrich Aerospace (640x512 pixel resolutionand 15µ pixel depth) with a 40◦ full angle FoV. An S3R star tracker from the ADCS(discussed in Section 4) for visible spectrum imaging complements this setup. Due tothe heavy computational loads associated with image processing, a dedicated ARMprocessor will form part of the payload unit.

To perform its maneuvers, ARAPAIMA is equipped with a warm gas propulsion systemthat is calculated to last for a mission length of approximately one year, including theamount required for a deorbit and re-entry maneuver. This propulsion system drivesthe main orbital maneuver thruster (OMT) as well as eight RCS thrusters installed such

as to produce control torques in both directions about each body axis (Figure 1). Yellowrepresents the geocentric inertial frame; blue represents the Earth-centered-Earth-fixedframe; the red represents the local-vertical local-horizontal frame; and the green is thebody fixed frame.

Figure 1: RSC thrusters and theirplacement relative to the satellite’sbody fixed frame.

ARAPAIMA will fly as secondary payload on aroutine host mission that is yet to be determined.The RSO of choice is the spent upper stageof the host rocket; its last action will be to de-ploy ARAPAIMA from a canisterized satellite dis-penser (CSD)specially designed to house 3U,6U, 12U, and 27U CubeSats. After separationthe satellite will enter a detumble mode, duringwhich solar panels will be deployed and angu-lar rates canceled. The satellite then assumesa nadir pointing position and begins a full sys-tems, guidance, propulsion and proximity navi-gation checkout while in communication with theMission Operations Center (MOC).

During this process, the attitude and position ofthe satellite is described using four different ref-erence frames shown in Figure 2. The body fixedframe (BFF) describes the orientation of the RCS

thrusters. The traditional geostationary frame (ECI) acts as the inertial frame of refer-ence. The Earth-centered-Earth-fixed (ECEF) frame rotates with the Earth; the z-axisoverlaps the Earth’s rotation axis and the x-axis crosses the 0◦ latitude and 0◦ longitudemark. The fourth and final reference frame is a local-vertical local-horizontal (LVLH)type frame in the north-east-down (NED) notation. This definition is dependent on itscenter’s location in the latitude-longitude grid, which changes over time as it tracks thebody’s orbital path. By definition, the z-axis of the frame will always point radially downtowards the Earth.

Figure 2: Reference frames used forattitude determination and control.

During the 3D imaging operations and maneu-vers, the relative motion between the CubeSatand the RSO are modeled by choosing a se-ries of waypoints around the RSO and lettingthe chaser fly freely between them, modeled bya combination of Clohessy-Wiltshire-Hill chasingmaneuvers and relative orbit maintenance pat-terns. The modeling of disturbance torques andtheir effects on these relative trajectories are ofextreme interest for proximity operations missionplanning. The following section will describe the

setup of environmental and internal disturbance torques, and the couples driving ef-

fects of reaction wheel torque. This will then be used to draw conclusions regarding thepointing accuracy and execution of ARAPAIMA mission objectives.

3. Disturbance Torques

The disturbance torques acting on the CubeSat have been assigned to two categories.The first is environmental disturbance torques; due to aerodynamic forces, gravity gra-dient, residual magnetic moment, and solar radiation pressure. The second category isthat of internal disurbance torques; due to reaction wheel imbalance, propellant slosh,flexible modes of the deployable solar panels, and misalignment of the orbital maneuverthruster.

Methods employed for the evaluation of each disturbance torque are described in thissection.All disturbance torques are applied along the axes of the body-fixed frames ofthe CubeSat.

3.1. Aerodynamic Torques

The aerodynamic torques are produced by the upper atmospheric particles collidingwith the CubeSat. The worst case scenario of solar max activity is modeled for the cal-culation of the aerodynamic torques in this study. The Naval Research Laboratory MassSpectrometer and Incoherent Scatter Radar (NRLMSISE-00) model of the atmosphereat the nominal altitude of 500km gives a composition of 94% Oxygen and 6% Nitrogen.The number and mass densities are n=3.769×1014m−3 and ρ=1.02×10−11kg/m−3 re-spectively. The mean free path is λ=27.33km; when compared with a reference lengthlref=0.1m, this results in a Knudsen number, Kn= λ/lref= 273,300, indicating that theCubeSat flies in the free molecular flow regime.

Table 1: Component properties used in the DSMC calculations.

Comp. Mass Rot. Ref. dia. Ref. temp. Visc. temp.(kg) DoF (m) (K) pwr.

O 2.656×10-26 0 3.000×10-10 273.0 0.80

N 2.326×10-26 0 4.200×10-10 273.0 0.79

The aerodynamic torqueson the CubeSat havebeen calculated witha freely available di-rect simulation MonteCarlo (DSMC) pro-gram called DS3V.The program em-ploys algorithms basedon the methods described by Bird [2]. A variable hard sphere model (VHS) has beenassumed for the simulation of the collisions between atomic oxygen, atomic nitrogen,and the CubeSat. Parameters of the model are shown in Table 1. The free streamspeed V∞ assumed was 7.612 km/s, the orbital speed in a 500km altitude circularorbit.

A total of 91 runs of the DS3V code were performed for 13 angles of attack and 7sideslip angles. The angle of attack α and the sideslip angle β are illustrated in Figure 3.α was varied between -90◦ and 90◦ in steps of 15◦ and β between 0 and 180◦ in stepsof 30◦.

Figure 3: Sketch of the ARA-PAIMA geometric fixed frameand the wind axes.

The maximum cross-sectional surface area of theCubeSat, with all solar panels deployed, is used asthe reference area, Saero=0.2276m2. The smallestsize of the CubeSat is used as the reference length,laero=lref=0.1m. The aerodynamic coefficients for thetorques about each body axis, calculated about thegeometric center of the CubeSat bus, are obtainedby calculating Taero:

Taero =1

2ρV 2∞CmSaerolaero,

The aerodynamic torque coefficients are loaded intwo-dimensional look-up tables called by a Simulinkmodel of the ARAPAIMA attitude dynamics. Theseare plotted in Figure 4, bottom row. The maximumvalue of the aerodynamic disturbance torque aboutany of the body axes is calculated to be 10-6Nm.

Figure 4: Aerodynamic force and torque coefficients for theARAPAIMA.

The validity of the DSMCmethod is confirmed bythe results obtained forthe aerodynamic forcecoefficients. As seen inFigure 4, the maximumaerodynamic force co-efficient isCy,max=2.11,within 5% of the dragcoefficient Cd = 2.2widely used to calcu-late drag for satellitesin LEO.

3.2. Gravity GradientTorque

The 1/r2 dependencyof the gravitational forceproduces a torque thatacts in such a way that,

in the absence of any other torques, one of the principal axes of the CubeSat alignsitself with the local gravity field vector. The gravity gradient torque is calculated with:

Tg = 3µr× (Ir) /r3

where µ=398,600km3/s2 is the gravitational parameter of the Earth, I is the momentof inertia tensor about the body axes, r is the unit vector along the position vector of

the CubeSat, and r is the magnitude of the position vector. The mass of ARAPAIMAwith a full propellant tank is 7 kg. To calculate the moment of inertia, it is assumed thatthe principal axes coincide with the geometric fixed frame (GFF) and the body-fixedframe (BFF) axes. This is calculated to be I = diag(0.0541, 0.0914, 0.1054) kgm2. Themaximum value of the gravity gradient torque is calculated to be 10-7Nm.

3.3. Magnetic Torque

The magnetic disturbance torque has two major sources 1) the force produced on acharge q by the magnetic component of the Lorentz force, qv × b and 2) the torqueexperienced by an aspheric paramagnetic body which, in the absence of other torques,aligns its long axis with the local magentic field field [9]. All the electrically conduc-tive parts of the CubeSat contribute charges (electrons) to the magnetic Lorentz forceand, as a consequence, the electric currents in all the energized CubeSat subsystemsproduce a Lorentz-type torque. In addition, Eddy (Foucault) currents induced by timevarying magnetic fields, both external and internal, produce disturbance torques. Di-rect evaluation of the magnetic disturbance torques on a low Earth orbit satellite is anextremely complex endeavor and researchers in the field often resort to empirical datato estimate them. For example, Inamori [5] discusses magnetic disturbance torquesfor small satellite systems and, based on empirical data, lumps the effects describedabove into a disturbance residual magnetic moment (RMM). Using Inamori’s data it hasbeen determined that a RMM of 0.1Am2 is typical for the uncompensated 6U CubeSatanalyzed here.

For implementation of in the Simulink model, the worst case is assumed, i.e., the RMMof ARAPAIMA is parallel to the Oy body-fixed axis. The maximum magnetic disturbancetorque can be determined assuming that the magnetic field and the RMM in Trmm =mrmm ×B are perpendicular. With an average value of the magnetic field of 2.5×10-5Tthe expected magnetic disturbance torque is 2.5×10-6Nm.

3.4. Solar Radiation Pressure Torque

Solar radiation pressure (SRP) is created by solar photons exchanging momentumwith the CubeSat. If the point of application of the SRP force is shifted from the centerof mass of the CubeSat, an SRP disturbance torque is created. The magnitudes ofthe SRP force and moment are complex functions of the shape of the CubeSat, theshading, and the optical properties of the materials exposed to the sunlight. The worstcase solar radiation torque [12] can be calculated with Tsrp = Fsrp lsrp, with Fsrp =ΦAsrp,max(1 + qr) cos(φ)/c, where Φ=1,366W/m2 is the solar constant, c=3×108m/s isthe speed of light in vacuum, Asrp,max=0.2276m2 is the maximum exposed surface area,the moment arm is lsrp=0.1m, and qr is the reflectance factor. Since about 85% of theCubeSat surface exposed to the Sun is covered by solar cells, a reflectance factor ofq=0.6 typical of solar panels [12] is used. For worst-case modeling, the incidence angleof the Sun is assumed as φ=0◦ which gives the maximum SRP disturbance torque. Themaximum solar radiation torque is calculated to be 1.7×10-7Nm.

For Simulink modeling, the SRP force is assumed to point in the direction of the Sunvector. An average surface area of Asrp,avg=0.11m2 is used instead of the maximumvalue, while the SRP moment arm remains at 0.1m. The normalized (unit) Sun vectoris calculated with Satellite Tool Kit (STK) in the Earth-centered Earth-fixed (ECF) co-ordinate system, loaded to Simulink and transformed to the BFF for calculation of theSRP disturbance torque.

Aerodynamic, gravity gradient, magnetic and Solar Radiation Pressure torques com-prise the environmental disturbance torque model. The initial attitude of the CubeSatis chosen such that the body axes are parallel with the Earth-centered inertial (ECI)axes, making the initial body angular rates are null. Figure 5 shows the variation of theenvironmental torques with respect to time across one orbit.

Figure 5: Evolution of the environmental disturbance torques, expressed in the BFF,during the first orbit of the CubeSat.

Next, we address internal torques, produced as a result of control inputs. A robustcontroller will be able to minimize the external torques on the system while coping withthe internal torques discussed in the following subsections.

3.5. Reaction Wheel Imbalance Torque

Since reaction wheels do not have completely uniform mass distributions, mass lumpscan be considered at the ends of the disk. These lumps cause the reaction wheelsto present with two primary imbalances: Static and Dynamic. Static imbalance is alsoknown as the “hopping” effect, observed when a mass lump exists on the edge of the

rotating disk, causing a radially outward force. The force produced is sinusoidal andis heavily related to the spin rate of the reaction wheel. Static imbalance is calculatedusing the following vector expression[1]

~Fxyz = {0, USω2 sin(ωt, USω

2 cos(ωt)}

US is the coefficient of static imbalance of the wheel, ω is the wheel’s angular velocity,and the wheel spins along the x-axis. This imbalance becomes a disturbance torqueafter converting Fxyz into the BFF with ~rRW × vecFxyz, where ~rRW is the position of thereaction wheel from the center of mass with respect to the BFF.

Figure 6: Reaction Wheel Torque Imbal-ances: Dynamic Imbalance (Left) andStatic Imbalance (Right). The “wob-bling” from the dynamic imbalance pro-duces torques perpendicular to the axisof rotation. The hopping from the staticimbalance produces a radially outwardforce disturbance perpendicular to theaxis of rotation

Similarly, the dynamic imbalance results frommass lumps changing the principal momentof inertia axis from the spin axis. A torqueis produced from the slight precession of thereaction wheel, which is calculated using:

~τxyz = {0, UDω2 sin(ωt, UDω

2 cos(ωt)}

UD is the coefficient of dynamic imbalance ofthe wheel, ω is the wheel’s angular velocityand the wheel spins along the x-axis. ~τxyz issimilarly transformed into into the BFF.

US and UD are coefficients that are inheritedfrom the manufacturer and cannot be con-trolled. Our analysis uses US and UD valuesof 5×10−10 Ns2 and 10−10 Nms2 respectively.

3.6. Propellant Slosh Torques

While complex propellant slosh is an activeresearch area, simple linear lateral slosh-ing models give good approximations to theslosh of propellant in a zero-g environment[8]. The model used consists of a solidmass which is“attached” to the tank. A second mass is used to represent the free sur-face motion. This mass is attached to the wall by a spring-damper system with thefollowing equation[10]

mx+ bx+ kx = FTank,

where m is the portion of propellant mass attached to the spring, b is the dampingcoefficient of the system emulating the propellant’s viscous tendencies (or other damp-ing elements like the tank’s baffle design), and k is the spring coefficient representingthe springiness of the dominating force (capillary forces dominate in the simulation’sregime).

The disturbance force is represented by the spring force. Tank shape, fill fraction, andthe tank’s acceleration drive the parameters of the mass-spring-damper model. It is

assumed that a sloshing frequency of νslosh=0.03Hz and a fill fraction of 0.032, and thefollowing parameters are used: m=0.9534kg, b=0.3577 Ns/m, and k=0.0126N/m, withthe FTank forced by the attitude motion of the satellite.

3.7. Solar Panel Flexible Mode Torque

The joints of the solar panel are modeled after a torsional spring-damper system. Thepanel itself is considered to be a rigid body attached to the CubeSat by a flexible joint.The spring and damping coefficients are approximated from by a video model of thedeployment of a 3U solar panel provided by Clyde-Space, similar to the hardware pro-posed for ARAPAIMA[11].

Jθ + bθ + kθ = τPanel

Figure 7: System response to the of the initial de-ployment of the solar panels

J is the moment of inertia of thepanel with respect to the joint axis,b is the damping coefficient of thejoint, k is the torsional spring co-efficient of the joint, and τPanel isthe torque acting on the hinge,modeled as being in the oppo-site direction of the reaction wheeltorque response. J is calculatedusing measurements made avail-able by Clyde-Space. The topand bottom (3U Panel) are calcu-lated to have a moment of inertia0.0350kgm2 and the side panels(6U Panel) have a moment of in-ertia of 0.0193kgm2. The springand damper coefficients are ap-proximated to be 0.45Nm/rad and0.09Nms/rad respectively. The initial deployment response is shown in Figure 7.

3.8. Disturbance Torque Results

Simulink results of the of each internal disturbance torques are shown. Because thereaction wheel torque drives some of the disturbances, the reaction wheel torque re-sponse is also shown. The total internal disturbance plot is shown in Figure 8 togetherwith individual plots for each internal disturbance.

Figure 8: Sum all internal imbalances throughout one orbit with no control torquesapplied (top right) and individual contribution of body torques due to solar panel flexing,propellant slosh, and reaction wheels respectively.

Figure 9: Body torques due to the static imbalance of the reaction wheels

Figure 10: Body torques due to the dynamic imbalance of the reaction wheels.

4. Attitude Determination and Control System

During normal operations (scientific mode) the CubeSat uses an STC-CLC202A monochrome camera from SenTech Inc. and a S3S star tracker (STR) from Sinclair Interplan-etary for attitude determination. [11] The S3S star tracker has a rated accuracy of 7asabout axes normal to its optical axis and 70as about the optical axis. The monochromecamera has a resolution of 1628x1236 with an element size of 4.4/mum. Three RW-0.03-4 reaction wheels are used to reject internal and external perturbations. The RW’sare mounted so that their axes are closely aligned with those of the satellite; each canproduce a nominal torque of 2mNm and store an angular momentum of 30mNms at5600RPM. They are also used to perform minor changes in the satellite’s attitude suchas slewing on different axes. The RCS thrusters are used to periodically off-load theangular momentum accumulated in the RW’s.

During detumbling operations, information gathered form the STR and the monochromecamera is unusable due to the large angular rates of the satellite. In this situation ananoIMU from MEMSense [11] is used to measure these rates which are then coun-teracted by the RCS thrusters and RW’s for detumble. These two sensors, togetherwith the Novatel OEMV-1 GPS receiver and patch antenna, complete the guidancenavigation and control suite.

During proximity operations, the main OMT is used to get within range of the RSOand supply the small ∆V needed to transfer to and maintain the relative orbit. RCSthrusters are used to spin stabilize the satellite during these maneuvers. During RSO

rendezvous, the thrusters are also used to set the satellite in its relative circular orbit.

While the ADCS of the ARAPAIMA mission is very well defined in terms of require-ments for successful execution, the state of the system is limited by unknown budgetsand constraints arising from other aspects of the mission still in development. The de-sign team has not advanced to hardware-in-the-loop testing; at present time, the onlyhardware tested with the ADCS is the laser rangefinder, during testing of selected 3Dimaging algorithms. [4] However, significant progress has been made in the softwaredesign of a control system. A plant faithful to the encountered disturbances discussedpreviously has been designed from scratch and implemented with the attitude dynam-ics of a satellite in orbit by applying the torques directly to the body axis. The celestialmechanics portion of the model has been incorporated by means of an oblate Earthmodel in which the orbital altitude fluctuates with time. It also has a great impact in de-termining the attitude and position of the NED frame; this ensures that the disturbancesmodeled in this paper have a solid foundation on position with respect to the Earth’ssurface.

The current model utilizes RW’s alone as actuators and neglects inherent sensor inac-curacies. These inaccuracies are of future interest, particularly those of the STR andmonochrome camera. Due to visual-based navigation and image processing, thesecould result in significant orbital path error if not handled appropriately.

The model of the system takes advantage of the small time step nature of numericalintegrators such as Simulink to perform single-input single-output (SISO) control on thesystem. Three Proportional-Derivative (PD) controllers control the angular position ofone body axis relative to the NED frame. The angular positions being monitored are thex, y, and z components of the rotation vector formed by the product of the unit Euler axisand the Euler angle. The PD controllers send a control signal consisting of a torquecommand into the RW actuator model which is then further filtered by a second layerof Proportional-Integral (PI) controllers that manage the RW DC motors using voltagesignals.

5. Simulation and Results

As mentioned previously, the simulation runs on MATLAB/SIMULINK. The solver usedis ODE14x, an extrapolation method that is very accurate when using fixed step simu-lation. The fixed step size is 0.01 s. The simulation is run for one orbit period (5676sfor a 500km circular orbit). Figure 11 show the output of ECEF position and velocity asthe orbit time lapses.

With only RWs as actuators, keeping good pointing accuracy in the presence of allmodeled disturbances requires a well-tuned controller. The initial direction was to useZiegler-Nichols tuning methods, but after in-depth simulation it was determined that thesystem was too sensitive for integral action. This, along with no classical method forPD control left manual tuning. The controller is structured as a three-axis controllerwith a zero radian reference signal. The controller response is interpreted as a torque

Figure 11: Changes in position (left) and velocity (right) vectors of the body framerelative to the ECEF over one orbit.

command and sent directly to the plant (disturbances) and into the reaction wheel blockinput. The RW torque command and the 6DOF outputs are fed into all the other internaland external disturbance blocks. Resulting disturbances are fed into the 6DOF block toproduce a quaternion from the BFF to the NED frame.

With good tuning, the perfect quaternion response produced by the model of the atti-tude dynamics should result in a constant [1 0 0 0] quaternion as time lapses.

Without control, due to all the disturbance torques present, the satellite cannot maintainalignment between the BFF and NED frames. Figure 12 shows the quaternion and the3-2-1 Euler angles from NED to BFF without any control. Clearly, the quaternion is farfrom identity, i.e., the two frames are not aligned. The angles between BFF and NEDalso shown in Figure 12 further supports this observation.

Figure 12: Quaternion (left) and Euler angles (right) from BFF to NED (target frame)without any active (closed loop) control

Next, the PD controller is implemented. Figure 13 shows results of the satellite’s Eulerrotation vector[3] components from the BFF to NED frame after manual tuning. Con-troller gains pertaining to the classic PID error equation used to produce the results

are: Proportional Kp= 0.001 and derivative Kd = 0.022. As it is evident in Figure 13 theangles are in the range of 0.002radians or approximately 7arcmin.

Figure 13: Euler angles from BFF to NED (target frame) with active PD control.

This falls short of our ideal 1arcmin mark. However, as a preliminary analysis bench-mark, the results are promising. As the manual tuning process accuracy improvesmore accurate results are expected. Of note in Figure 13 is a pseudo-steady state er-ror. Properly introduced integral action should reduce this error significantly. Furtheranalysis is ongoing to find a proper integral coefficient that does not excite the systemdynamics excessively.

6. Conclusions

The main purpose of the current work was to develop and simulate the pointing accu-racy of a nadir-pointing CubeSat in the presence of external and internal disturbances.Aerodynamic drag, magnetic dipole moment, gravity gradient, solar radiation pressure,propellant slosh, reaction wheel imbalance, and solar panel array flex were studied. Al-though a preliminary analysis, results were found to be very encouraging. A very highaccuracy goalhas been set for an ideal control scenario; positive results at a pre-PDRdesign point indicate the likelihood of a favorable final response, even when sensorinaccuracies are implemented. Further, the high frequency oscillatory motion in Figure15 is expected to decrease significantly in magnitude with the introduction of the RCSthrusters into the simulation. Finally, modeling will be streamlined by the acquisition ofa D-Space satellite test bed which will allow the ARAPAIMA team to test flight hardwarealongside software simulations. Future implementations planned include complex ce-lestial mechanics analysis involving orbital maneuvers towards and relative to the RSO.

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