AAS 20-560

GUIDANCE, NAVIGATION, AND CONTROL FOR THE DWARFFORMATION-FLYING MISSION

Vincent Giralo∗, Michelle Chernick†, Simone D’Amico‡

The Demonstration with Nanosatellites of Autonomous Rendezvous and Formation-Flying (DWARF) mission consists of a pair of identical 3U CubeSats which willact as a spaceborne testbed to further advance, rigorously validate, and embed newrelative navigation and control technologies in order to meet the needs of futuredistributed space systems. This paper focuses on the design and implementationof the DWARF on-board Guidance, Navigation, and Control (GNC) system. TheDWARF mission will demonstrate unprecedented real-time centimeter-level nav-igation accuracy on board and new safe, robust, and autonomous relative orbitcontrol algorithms.

INTRODUCTION

Distributed space systems (DSS) employ multiple co-orbiting units to enable advanced missionsin space science, technology, and infrastructure. As mission concepts become more complex, fu-ture space architectures must address new challenges in relative navigation and control accuracy,robustness, and safety. The Demonstration with Nanosatellites of Autonomous Rendezvous andFormation-Flying (DWARF) mission, developed by the Stanford Space Rendezvous Laboratory(SLAB) in collaboration with Gauss S.R.L. and King Abdulaziz City for Science and Technology(KACST), seeks to demonstrate advancements in relative navigation and control to meet the needsof future ambitious distributed space systems. The on-board dedicated Guidance, Navigation, andControl (GNC) payload consists of software developed by SLAB and is integrated on two identi-cal and autonomous 3U CubeSats using commercial-off-the-shelf (COTS) hardware and a cold-gaspropulsion system. The satellites, which are scheduled to launch in the first quarter of 2022, will flyin a LEO sun-synchronous orbit. Both spacecraft can assume the role of chief and deputy in order tobalance fuel consumption and double the mission lifetime. The GNC software is split into two inte-grated modules, navigation and control. The navigation module employs carrier-phase differentialGlobal Navigation Satellite System (GNSS) techniques to demonstrate real-time centimeter-levelprecision through the use of Integer Ambiguity Resolution (IAR) on board in the presence of fre-quent control maneuvers. The control module will perform safe, robust, and autonomous formationacquisition, keeping, and reconfiguration at separations between 100 meters and 100 kilometers.These capabilities will be demonstrated in flight for the first time, enabling future missions such as

∗Doctoral Candidate, Stanford University, Department of Aeronautics and Astronautics, Space Rendezvous Laboratory,Durand Building, 496 Lomita Mall, Stanford, CA 94305-4035†Doctoral Candidate, Stanford University, Department of Aeronautics and Astronautics, Space Rendezvous Laboratory,Durand Building, 496 Lomita Mall, Stanford, CA 94305-4035‡Assistant Professor, Stanford University, Department of Aeronautics and Astronautics, Space Rendezvous Laboratory,Durand Building, 496 Lomita Mall, Stanford, CA 94305-4035

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virtual telescopes,1, 2 optical interferometers,3 on-orbit servicers for lifetime prolongation4 and as-sembly of larger structures in space.5 Specifically, these technologies have been identified by NASAand NSF as enablers for missions such as Starling,6–8 a swarm of four CubeSats demonstrating thecapability to navigate in deep space without Global Positioning System (GPS), the MiniaturizedDistributed Occulter/Telescope (mDOT),9 a precise formation of two SmallSats intending to di-rectly image exoplanets and exozodii, the Space Weather Atmospheric Reconfigurable MultiscaleExperiment (SWARM-EX),10 a swarm of three CubeSats that will study Earth’s upper atmosphere,and the Virtual Super-resolution Optics with Reconfigurable Swarms (VISORS),11, 12 a precise for-mation of two CubeSats that will perform solar observations of unprecedented resolution. Thesemissions are all under study and development at SLAB and will directly benefit from the algorithmsdescribed in this paper.

The concept of multiple spacecraft operating together was first demonstrated in the 1960s dur-ing the Gemini program, with advancements continuing through the Apollo program and the SpaceShuttle.13 While these early demonstrations were rudimentary by current standards, they paved theway for recent spacecraft formation-flying and rendezvous missions that require more advancedautonomous navigation and control algorithms. Numerous science missions such as the GravityRecovery and Climate Experiment (GRACE, 2002),14 the TerraSAR-X add-on for Digital Ele-vation Measurement (TanDEM-X, 2010),15 and the Magnetospheric Multiscale Mission (MMS,2015)16, 17 used multiple spacecraft to conduct unprecedented experiments for gravity field recov-ery, synthetic aperture radar interferometry, and magnetospheric observation, respectively. Anotherclass of formation-flying and rendezvous missions is technology demonstrations, such as PRISMA(2010),18, 19 CanX-4/5 (2014),20 and CPOD (2020).21 Advancements in DSS missions have demon-strated state-of-the-art navigation and control algorithms on board, while also moving from large,expensive spacecraft (GRACE, TanDEM-X) to smaller, cheaper nanosatellites (CanX-4/5, CPOD).

Precision navigation for DSS has largely been enabled through the use of GNSS, such as GPS.GNSS-based absolute positioning accuracies of 1m have been demonstrated for a single spacecraftin real time, while relative navigation using differential GNSS (dGNSS) techniques provide greatlyhigher accuracy.13 By exploiting synchronous measurements from two receivers, common errorscan be cancelled out, resulting in low-noise relative measurements. This technique was validated onboard in real time on missions such as PRISMA, where an Extended Kalman Filter (EKF) showedprecise relative positioning results of less than 10cm (3D, RMS) between two small spacecraftthroughout most mission scenarios.18, 19, 22, 23 Similar results were obtained during the CanX-4/5mission,20, 24, 25 demonstrating dGNSS for the first time on CubeSat avionics in flight.

Navigation accuracies have been further improved by fixing the carrier-phase ambiguities to theirinteger values, a technique referred to as Integer Ambiguity Resolution. However, due to computa-tional overhead and a lack a guarantee of correctly fixing the integers, IAR has never been performedin flight. GRACE used IAR in post-processing to demonstrate 1mm (1D range-only, 1σ) relativepositioning accuracy at a separation of 200km when compared with the high-precision on-boardK/Ka-band ranging system.14, 26, 27 More recently, advances in spacecraft avionics have allowed formore complex algorithms such as IAR to be run on board in real time. The Distributed Multi-GNSSTiming and Localization system (DiGiTaL) is a navigation payload for nanosatellites that achievescentimeter-level positioning accuracy and nanosecond-level time synchronization throughout ar-bitrarily sized swarms.28 A reduced-dynamics estimation architecture on board each individualsatellite processes low-noise measurements from multiple GNSS constellations and frequencies toreconstruct the full formation state with high accuracy. DiGiTaL demonstrated successful IAR to

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provide less than 1cm (1D, RMS) of relative positioning accuracy in real time for a swarm of fourspacecraft over short baselines using full CubeSat avionics in the loop.29 The system was developedby SLAB in cooperation with Goddard Space Flight Center and Tyvak Nanosatellite Systems as partof the NASA Small Spacecraft Technology Program (SSTP). The DWARF mission will be the firstin-flight demonstration of both DiGiTaL and online IAR.

Advancements have also been made to push towards robust and autonomous control of spacecraftrelative motion. An on-board relative control software aims to minimize the delta-v cost of a set ofimpulsive control actions while achieving a desired relative state in fixed time. Recent flight demon-strations of control software such as the Spaceborne Autonomous Formation Flying Experiment(SAFE) on PRISMA30 and the TanDEM-X Autonomous Formation Flying (TAFF) system15 castthe relative orbit reconfiguration problem into quasi-nonsingular relative orbit element (ROE) space.Use of the ROE state representation overcomes the limitations of the Cartesian-based linear modelsof relative dynamics like the Hill-Clohessy-Wiltshire equations (HCW) and Yamanaka-Ankerson(YA) equations, which are accurate only for small spacecraft separations. The ROE state allows forlinearization of the equations that govern relative motion with minimal loss of accuracy.

In addition, use of the ROE state provides a geometric intuition that can be exploited to derivesimple geometric conditions for collision avoidance and safety, called relative eccentricity/inclina-tion (E/I) vector separation.22 Typically, mission designers maintain safe separation using a relativespacecraft configuration called the safety ellipse,31 which is defined such that the relative motionin the radial-tangential/radial-normal (RT/RN) planes is bounded and periodic. The OSAM-1 (for-merly Restore-L) mission, set to launch in 2020, plans to use safety ellipses in their approach tra-jectory.4 However, D’Amico et al. showed that the minimum separation between the spacecraftover several orbits can be expressed analytically as a function of the ROE, and derived simple ex-pressions for the relationship between the relative eccentricity vector and relative inclination vectorthat ensure that the satellites will maintain a prescribed relative distance in the radial/cross-trackplane. The method was implemented successfully in several missions such as GRACE,32 TanDEM-X/TerraSAR-X,33 and PRISMA.34

Though recent missions have employed the use of the ROE state representation, formation controlimplementation has been restricted to near-circular orbits. In particular, TAFF used pairs of along-track maneuvers separated by half an orbital period, which is optimal for in-plane control in near-circular orbits, but is infeasible in eccentric chief orbits.15 SAFE used additional radial and cross-track maneuvers, but radial control was only used in certain modes, and not optimally.18

To improve upon the state of the art in autonomous, on-board spacecraft relative orbit control,Chernick et al. developed globally energy-optimal closed-form impulsive control algorithms fororbits of arbitrary eccentricity.35 The solutions are derived using the domain-specific benefits ofthe ROE state representation and leveraging the geometric advantages of reachable set theory, a toolcommonly used to assess cost-reachability and safety,36, 37 and only recently used in control itself.38

The maneuver planning algorithms do not assume the form of the maneuvers; unlike the aforemen-tioned missions that are limited to near-circular orbits, Chernick’s relative control algorithms arerobust to orbit regime and therefore employ radial, tangential, and normal (RTN) maneuvers toachieve an optimal reconfiguration solution, in-plane and out-of-plane alike. The DWARF missionis the first in-flight demonstration of these globally optimal control algorithms. In addition, DWARFwill employ a newly developed collision avoidance method inspired by D’Amico’s method whichguarantees passive safety throughout a reconfiguration, even if a maneuver fails. Finally, the newclosed-form solutions are implemented in a model-predictive-control fashion to provide robustness

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to dynamics, navigation, and control errors.

This paper presents the architecture of the GNC payload for the DWARF mission. Following thisintroduction, an overview of the system is presented, with detailed description of each the navigationand control modules. A test case is examined, demonstrating the state-of-the-art capabilities in ascenario representative of the upcoming DWARF mission.

GNC SYSTEM OVERVIEW

This work presents the full architecture for a dedicated GNC payload that implements novel al-gorithms in flight for the first time. Figure 1 details the DWARF payload, including communicationwith external subsystems. In addition to the software, the payload also consists of a GNSS receiver,two GNSS antennas, an Inter-Satellite Link (ISL), and a cold-gas propulsion system. The layout ofthe exterior hardware components on the 3U CubeSat is shown in Figure 2.

DWARF GNC PAYLOADTC

ADCSTM

ADCS

TIME

DWARF GNC SOFTWARE

NAV CTRL

GNSS Receiver

ISL

Prop

PropTo OtherSpacecraft

Figure 1: High-level Architecture of DWARF GNC Payload

GNSS antenna

LEOP TM/TC antenna

Deployable solar panel

Omni-directional ISL antenna

(a)

Propulsion

Secondary GNSS antenna

LEOP TM/TC antenna

HSL antenna

(b)

Figure 2: Rendering of DWARF Satellite Detailing Hardware Component Layout

To perform these navigation and control tasks, DWARF requires inputs from various sources fromthe spacecraft. The system receives messages from the GNSS receiver that includes measurementsand GNSS broadcast ephemeris data. It is also necessary to have attitude and maneuver informa-tion, the latter of which is either generated internally within the control subsystem of the DWARFpayload or provided via telecommand. In addition, the GNC payload requires all of the above in-formation from the other spacecraft which is received via the ISL. Finally, telecommands are sentto the payload from ground to set parameters and to trigger different modes of operation. As anoutput, DWARF provides absolute and relative orbit information, including both the orbit state and

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associated covariance, and control maneuver commands for formation keeping and reconfiguration.As described above, information necessary for ISL communication is sent to the other spacecraft,and telemetry data is generated to monitor the system and output to ground. Further details on boththe navigation and control modules are provided in the following sections.

Navigation System

Navigation for DWARF is accomplished using DiGiTaL, which integrates a multi-GNSS receiverand antenna system with an ISL to provide centimeter-level relative positioning accuracy in realtime. The DiGiTaL payload has a 0.5U CubeSat form factor, with a maximum power requirementof 3W and a mass of 225g.29 DiGiTaL shares synchronous low-noise measurements from receiverson each spacecraft through the ISL and forms powerful error-cancelling combinations in an EKF toprecisely estimate absolute and relative orbits of the satellites.

Figure 3 shows the hardware architecture of DiGiTaL, as implemented in the DWARF payload.Two Tallysman TW3972E Embedded Triple Band GNSS antennas39 collect radiofrequency (RF)signals, pointing in anti-parallel directions to allow for near-omnidirectional coverage (see Figure2). The RF switch is designed to select the signals from the most zenith-pointing antenna basedon the available attitude information, maximizing the visibility of GNSS satellites. A NovAtelOEM628 High-Performance GNSS receiver,40 capable of receiving signals from multiple GNSSfrequencies and constellations, processes the signals and provides raw measurements to the localflight computer as well as to the remote spacecraft via the ISL.

Figure 3: Hardware Architecture of DiGiTaL System29

The software architecture of DiGiTaL is shown in Figure 4. The system receives GNSS dataand attitude information directly from the local (user) spacecraft, as well as the same informationfrom the remote spacecraft via the ISL. This information is processed in the DiGiTaL Data Inter-face (DDIF) module, where measurements from the local and remote spacecraft are checked forvalidity and synchronicity. DDIF also uses the navigation solution provided from the on-boardGNSS receiver to initialize its own navigation filter. All of this information is passed to the DiG-iTaL Orbit Determination (DOD) module, where the EKF estimates the absolute inertial state ofeach spacecraft, subject to the tight relative constraints provided from dGNSS. In addition, DODprovides estimates of the on-board receiver’s clock offset with respect to GNSS system time whichis used to measure the time synchronicity of the two spacecraft. In general, the DiGiTaL SwarmDetermination (DSD) module fuses the output from the local and remote instances of DOD to forma state estimate of the entire swarm. While the DWARF mission consists of only two spacecraft,DSD will still be demonstrated in-flight by fusing the estimates from both instances of DOD torecreate the ”swarm” state. Due to their complexity, DOD and DSD are run with a sample time of30 seconds. However, navigation solutions can be provided at a faster rate via the DiGiTaL OrbitPrediction (DOP) module, which interpolates the spacecraft states through the use of orbit polyno-mials formed within the numerical integration of DOD. Further details on the orbit determination

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process are provided below.

DataInterface

(DDIF)

Orbit Determination

(DOD)

Swarm Determination

(DSD)

OrbitPredicton

(DOP)

User S/C

Subset S/C Swarm S/C

Intersatellite Link

Telemetry & Telecommands

GNSS Data GNSS

Data

Attitude & Maneuvers

Attitude & Maneuvers

GNSS Data

GNSS Time

GNSS Time

Attitude & Maneuvers GNSS

Data from Subset

Subset State & Covaiance

Swarm State & Orbit Coefficients

GNSS Time

Subset State from Swarm

DOD

Subset State

Swarm State for User Subsystem

GNSS

AOCS

GNSS

GNSS Time for Synchronization

GNSS

AOCS

S/C

Figure 4: High-level Software Architecture of DiGiTaL System29

The EKF in DOD estimates the absolute inertial position, r, and velocity, v, of each spacecraftin the J2000 reference frame, eliminating both the need for an explicit relative state and the need todeal with rotating reference frames during the time update. To propagate each state, DOD uses areduced-dynamics architecture, where only a 20×20 spherical harmonic gravity model is used14 inorder to reduce computational load. To account for unmodelled dynamics, empirical accelerations,aemp, are estimated as a first-order Gauss-Markov process. The reduced-dynamical equations arenumerically integrated with a fourth-order Runge-Kutta scheme, including a Richardson step toprovide orbit polynomials that are then are used to interpolate the state during the time update of thefilter. The variational equations are also integrated, allowing for computation of a state transitionmatrix (STM) to linearly update the covariance matrix.22

In the presence of maneuvers, the state is propagated to the time of the maneuver, where the com-manded delta-v is added to the state and additional process noise is added to the velocity entries ofthe covariance matrix. Note that maneuvers are treated as true impulses, represented as an instanta-neous change in velocity. The orbit polynomial coefficients used in DOP are also updated to reflectthe maneuver.22 Following this, the time update proceeds as normal.

The full estimation state for each spacecraft is given as

x =[rc vc acemp cδtc Nc rd vd ademp cδtd Ndc

], (1)

where superscripts c and d denote chief and deputy, or local and remote, respectively. In addition tothe position, velocity, and empirical accelerations, the state includes the GNSS receiver clock offsetfrom GNSS time, cδt and the carrier-phase biases, N. The clock offset is denoted as as vectorbecause each GNSS constellation operates on a separate time scale, requiring a different offset withrespect to each constellation. Note that the state is identical for each spacecraft, with the exceptionof the carrier-phase bias in the secondary state. These are estimated as the single-difference between

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chief and deputy, or Ndc = Nd − N c, described in more detail below. This allows for the single-difference bias to be fixed from IAR, while still allowing for the un-differenced ambiguities to beestimated as float values.

The key to success of orbit determination in DiGiTaL is the use of dGNSS in combination withIAR. The raw code and phase measurements are modelled by

ρpr(t) = ||r(t)− r(t− τ)GNSS ||+ c (δt− δtGNSS) + I + εpr (2)

ρcp(t) = λΦ = ||r(t)− r(t− τ)GNSS ||+ c (δt− δtGNSS)− I + λN + εcp. (3)

Here, r represents the phase center of the receiving antenna and rGNSS is the phase center of theGNSS satellite’s broadcasting antenna. Because GNSS systems operate in an Earth-fixed frame andthe estimated position is an inertial quantity, reference frame transformations between inertial andEarth-fixed frames are required, taking into account precession and nutation of the Earth. The clockoffsets of the receiver clock and the GNSS satellite clock are given by δt and δtGNSS , respectively,and the ionospheric path delay is given as I . The carrier-phase bias is included as N , multiplied bythe wavelength λ. The signal time-of-flight is modelled as τ , and the measurement noise is givenby ε, which is on the order of 10cm for code and 1mm for phase.

Measurement combinations are formed from the code and phase measurements from each re-ceiver into the coarse Group And Phase Ionospheric Correction (GRAPHIC) data type, given by

ρgr =ρpr + ρcp

2= ||r(t)− r(t− τ)GNSS ||+ c (δt− δtGNSS) + λ

N

2+ εgr, (4)

and the precise single-difference carrier-phase (SDCP) data type, given by

ρsdcp = ρdcp − ρccp = ||r(t)− r(t− τ)GNSS ||dc + cδtdc + λNdc + εsdcp. (5)

The SDCP measurments are formed as differences of carrier-phase measurements from two re-ceivers, represented by c and d, tracking the same GNSS service vehicle, creating a relative mea-surement between the receiving antennae’s phase centers with millimeter-level noise. In this formu-lation, (·)dc = (·)d − (·)c.

Once the measurements are processed, the filter performs the additional step of IAR using theModified Least-Squares Ambiguity Decorrelation Adjustment (mLAMBDA) method.41 To fix theambiguities, float-valued double-differenced carrier-phase (DDCP) ambiguities are formed by dif-ferencing two SDCP ambiguities from two GNSS satellites, or

Nddcp = Nksdcp −N

jsdcp, (6)

where superscripts j and k represent different GNSS satellites. The DDCP ambiguities are thenfixed to their integer values by mLAMBDA. By selecting a single reference GNSS satellite j todifference all others with, the SDCP ambiguities can be resolved and inserted back into the state.At this time, the ambiguities are treated as deterministic by zeroing out the corresponding rowsand columns in the covariance matrix. As discussed above, the un-differenced ambiguities in theestimation state are left as stochastic quantities.

To increase the robustness of the IAR process, a series of checks are performed including theSuccess Rate Test42 and the Discrimination Test,43 as well as the novel Residual Test.29 This finaltest checks the measurement residuals before and after the ambiguities are fixed to ensure that IARdid not incorrectly set the ambiguities, which would be seen as a bias in the post-fit residual of thecorresponding measurement.

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Control System

The DWARF control system is implemented as a regularly called finite state machine within theframework of the GNC payload and is defined as a set of states, transition conditions, and actions.Figure 5 represents the states as blue boxes with their associated actions highlighted in blue directlyto the right.

0NO_CONTROL

I NI T_CONTROL 1

PLAN_CONTROL 2

Flag to turn on control received

I N_CONTROL_WI NDOW

Queue of upcoming control events (event _queue) is not empty

3

MANEUVER_I N_PROGRESS 5

Maneuver occurs within next timestep

Thr ust er s

Empty event _queue?

- Find optimal maneuver scheme(s)- Choose maneuver scheme that satisfies constraints - Translate to thruster board commands

Maneuver completed, no errors, all thrusters off

- Send command to thrusters- Remove maneuver information from queue

ATTI TUDE_CONTROL 4 Satellite must rotate to align thruster with maneuver direction

Thrusters aligned with maneuver direction, no errors

ATTI TUDE_TO_NOM 6 Satellite must rotate back to nominal attitude

Attitude control end

If mode is KEEP,and guidance command receivedswitch to RECONFI G

If mode is RECONFI G,Switch mode to KEEP

If mode is KEEP,Stay in mode

If mode is KEEP,internal command received

Waiting reconfiguration

commands?

Control initialized from ground

Yes

GOTO 3

No

GOTO 1

No

Yes GOTO 2

In control window, but no maneuver in progress

Figure 5: Compact state machine for on-board maneuver-planning and control. Thereconfig_list holds all received guidance commands, and the event_queue holds all up-coming attitude and relative orbit control events including timing information.

There are three modes, NONE, RECONFIG, and KEEP, associated with no control, reconfigura-tion control, and formation keeping, respectively. KEEP is the default mode for the controller, whichis only changed if a reconfiguration command is received from ground. The state machine is thesame for formation keeping and reconfiguration/acquisition, but the source of guidance commandschanges: external for reconfiguration/acquisition, internal for keeping. Otherwise, the controllerplans and executes formation keeping maneuvers according to a set formation keeping window andnominal relative (orbit element) state. In addition, there are seven states, labeled 0 to 6, associatedwith actions that are briefly described in the compact state machine in Figure 5. The action is per-formed whenever the state machine is called and until the transition condition, denoted to the rightof the black horizontal lines in Figure 5, is satisfied.

Much of the maneuver planning within the control software occurs in the PLAN_CONTROL state.As discussed previously, the software utilizes the in-plane and out-of-plane globally-optimal closed-form control solutions developed by Chernick et al.35 leveraging reachable set theory and the ROE

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state representation. The planner is robust to orbit regime and produces maneuver schemes that areprovably optimal.35 For in-plane control, the maneuvers include both radial and tangential compo-nents, which makes them robust to all orbit regimes, while for out-of-plane, the maneuvers occurin the normal direction. The maneuver schemes use the minimum required number of impulses,which is three for in-plane control, and one or two for out-of-plane control. The locations of themaneuvers are dependent on which ROE change dominates the required delta-v usage for a givenreconfiguration. For example, in a near-circular orbit where the relative eccentricity vector domi-nates, the maneuvers occur at half orbit multiples from the mean argument of latitude that alignswith the phase of the desired change. In reconfiguration cases that cannot be reached optimally, thesame algorithms can be used to generate quantifiably sub-optimal solutions.

The PLAN_CONTROL state also handles performance bounds, collision avoidance, and failuremitigation. Using relative state covariance data from the navigation module, the controller usesa new method to assess maneuver scheme performance. Given a desired confidence level and acovariance matrix, an n-dimensional bounding box is fitted to an n-dimensional error ellipsoid todetermine if performing a reconfiguration is viable (if a given maneuver scheme will achieve thedesired end state within reasonable bounds).35 From there, the controller must also ensure that theproposed maneuver scheme does not cause a collision between the spacecraft. The controller uses aset of conditions based on relative E/I vector separation to check that the deputy (controlled) space-craft does not enter a configured “keep-out-area” (KOA) around the chief (reference) spacecraft,which assures that they do not collide, even in the case of a maneuver failure. If the maneuverplanner fails to meet those conditions, it attempts to re-plan the reconfiguration for a configurednumber of ‘pings’ to the state machine. If it cannot successfully re-plan that reconfiguration in time,the controller returns automatically to autonomous formation keeping. Maneuver replanning is alsoused as a failure mitigation method to counter the effects of perturbations, maneuver execution er-rors, and navigation uncertainty. After the first in-plane maneuver, using the difference betweenthe achieved ROE and expected ROE, the planner modifies the remaining maneuver vectors whilekeeping the maneuver times constant. This replanning process occurs continuously on a diminishingtime horizon until all maneuvers have been executed.

It should be noted that the navigation module uses a Cartesian-based inertial state for ease of usewith GNSS, in particular GPS, which operates in the World Geodetic System (WGS) 84 Earth-fixedframe. However, the control module operates using an orbital element state to take advantage ofgeometric properties and slowly varying dynamics. Therefore, it is necessary for a transformationto take place between the navigation and control modules for both the absolute and relative states.This is done using well-known Cartesian to osculating Keplerian orbital element transformations,followed by a transformation to mean orbital elements, including J2 effects, which uses an itera-tive approach with a Newton-Raphson solver using the Jacobian of the nonlinear transformation.44

Then, the quasi-nonsingular ROE are constructed by their definition in Eq. (3) in Chernick et al.’spaper35 for use in maneuver planning.

VALIDATION

To validate the on-board software, extensive testing is conducted at Stanford University’s GNSSand Radiofrequency Testbed for Autonomous Navigation of DSS (GRAND),45 shown in Figure 6.This system was designed to enable testing of GNSS-based navigation systems in both a hardware-in-the-loop (HIL) environment, including receivers, ISLs, and flight computers, and a software-in-the-loop (SIL) environment. For SIL testing of DWARF, a high-fidelity orbit propagator generates

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translational and rotational trajectories through numerical integration of the equations of motion.These trajectories are input in a GNSS Receiver Emulator, which outputs raw measurements andnavigation solutions representative of the NovAtel OEM628 receiver40 in the DWARF payload.Effects of broadcast ephemeris errors and atmospheric delays, as well as sensor noise, are accountedfor in the emulator. The DWARF payload then receives the GNSS data for processing. Based on thenavigation solution and telecommanded reconfiguration messages, control maneuvers are plannedand incorporated in the navigation filter, in addition to being looped back into the truth propagator.The navigation and control algorithms are developed in C/C++ and incorporated into the simulationenvironment through Simulink s-functions, allowing for a seamless transition from a simulationenvironment to an on-board microprocessor.

Space Rendezvous Lab Satellite Software

(S3)IFEN GNSS

Signal Simulator

Septentrio GNSS Live Antenna

GNSS Software Receiver Emulator

GNSS Receiver

Flight Computer Microprocessor

Workstation MATLAB/Simulink

Software GNSS Receiver Emulator

GNSS RF Signal Generation

GNSS Measurement Generation

GNC Software and Algorithms

Orbit/Attitude Dynamics Simulation

State Vector

RF Signal

Control CommandsGNSS

Data

Cold-GasPropulsionEmulator

ActuatorSimulation

Control Acceleration

Figure 6: GRAND testbed configuration. Red line indicates data flow for DWARF testing45

Maneuvers are included in the orbit propagator as perturbing accelerations spread over multipletimesteps and centered on the commanded maneuver time. Performance degrades if the lengthof the maneuver application increases because the maneuver no longer approximates an impulse.However, with a 4 mN thruster as is typical in many SmallSat applications, a burn time of 1 minuteis more than enough to meet the maneuver requirements of the DWARF mission, and for that controlapplication time, performance does not degrade significantly. In fact, performance does not degradesignificantly for maneuvers that span a longer time, such as 5 minutes, though long maneuverswould not be necessary on DWARF. In addition, the aforementioned maneuver re-planning schemewill help to mitigate any small control performance issues.

Test Scenario

The capabilities of the DWARF GNC payload will be demonstrated in a representative missionexample in a LEO, sun-synchronous orbit. The system will perform formation keeping around a setnominal relative state until a ground-commanded reconfiguration is received. The goals of the navi-gation module for this demonstration are to show convergence of the navigation solution within oneorbit and position errors less than 1cm (3D RMS) in the presence of reconfiguration and formationkeeping maneuvers. The goal of the control module is to demonstrate that the control maneuversachieve the desired ROE configurations with meter-level accuracy while maintaining a minimumseparation of at least 50 meters in the RN-plane and satisfying delta-v maxima and minima restric-tions imposed on the system. The test scenario will also show that the controller is able to mitigatethe aforementioned maneuver execution error and navigation uncertainty using maneuver replan-ning as compared to the control error when maneuvers are not replanned. In the Results section, thecontrol accuracy will be measured by comparing the achieved ROE to the desired ROE for each of

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the keeping and reconfiguration phases.

The ground truth simulation of this scenario propagates two identical spacecraft as defined inTable 1 using the forces listed in Table 2. To increase fidelity, the scenario simulates noisy sen-sors and imperfect actuators. The NovAtel OEM628 GNSS Receiver Emulator provides code andphase measurements to the system with white noise of 20cm and 1mm, respectively. Precise GNSSephemerides from archived International GNSS Service (IGS) data are used for measurement gen-eration, while broadcast ephemerides with meter-level error are provided to the flight software.46

Additionally, imperfect maneuver execution is simulated by applying a pointing error chosen froma normal distribution with a standard deviation of 1.5 degrees.

Table 1: Spacecraft parameters used in ground truth simulation

Parameter Value

Mass 4 kgEffective Drag Area 0.09 m2

Coefficient of Drag 2Effective SRP Area 0.06 m2

Coefficient of Reflectivity 1.8

Table 2: Force models used in ground truth simulation

Force Model

Gravity field GGM01S (120x120)14

Atmospheric drag Harris-Preister47

Cannonball spacecraft model47

Solar radiation pressure Flat plate model47

Conical Earth shadow model47

Geomagnetic and solar flux data NOA daily KP AP indicesThird-body perturbation Analytical Sun and Moon47

DE340 (all planets)48

Relativistic corrections First-order corrections for specialand general relativistic effects47

A 24-hour experiment takes place beginning March 1, 2018 00:00:00.0 UTC. For the entirety ofthe experiment, the chief orbit is passive, with initial mean orbital elements given by

ααα =[a e i Ω ω M

]=[6969km 0.003 97.98 25.85 239.25 343.59

]T. (7)

The scenario definition is divided into four phases which represent different modes of the deputyorbit with respect to the chief. During Phase 0, the navigation system will initialize its state fromthe GNSS receiver and begin to filter measurements, followed by the activation of IAR. The actualabsolute and relative states are not yet known. The desired relative state at the end of Phase 0 is

δαααnom,0 =[aδa aδλ aδex aδey aδix aδiy

]=[0 −1000 0 300 0 600

]T m. (8)

Phase 1 begins when the simulation has progressed for approximately one half orbit, when thenavigation system has had sufficient time to converge. The ROE at this point are found to be

δααα0 =[−0.9481 −994.8811 0.7364 301.8681 0.1557 599.1719

]T m. (9)

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To demonstrate the system’s formation keeping capabilities, maneuvers are internally planned andexecuted to maintain the ROE given in Eq. (8) on a scheduled 2.5 orbit cycle.

Phase 2 begins when a reconfiguration command is received from ground. As long as there areno errors and it is safe to change state, formation keeping is halted and the state is immediatelychanged to RECONFIG. The desired relative orbit elements are given by

δαααf,des =[0 −500 185 501 4.5 626

]T m. (10)

This reconfiguration is to be completed within 2.5 orbits.

When the new desired state is reached, Phase 3 begins. The new nominal state becomes δαααf,des

in Eq. (10), and the controller returns to KEEP mode about this state. This formation is kept for theremainder of the experiment. Table 3 outlines the timing of the experiment.

Table 3: Experiment timing for demonstration of formation keeping and reconfiguration

Phase Time since sim.start (hours)

Current relative state(m)

Desired relative state(m)

Duration(orbits)

Mode

0 0 - - - OFF1 0.75 Initial (Eq. (9)) Nominal (Eq. (8)) 2.5 KEEP2 2.25 Nominal (Eq. (8)) Eq. (10) 2.5 RECONFIG3 6.3-24 New nominal (Eq. (10)) New nominal (Eq. (10)) 2.5 KEEP

The relative motion of the deputy spacecraft in the RT- and RN-planes of the chief are shown inFigure 7. The black dot represents the chief spacecraft. The dashed line circle represents a RN-plane keep-out region of 50 meters in radius. As shown, the controller logic chose only keeping andreconfiguration schemes that satisfied the given collision avoidance constraint.

(a) RT (b) RN

Figure 7: Planar Relative Motion

Results

Figure 8 shows the relative position and velocity estimate errors. Covariance bounds (3σ) areincluded, as well as statistics calculated directly from the steady-state estimate error. After a briefinitialization period, IAR is activated and the estimate reaches steady state. This is shown in theplots just before the one-hour mark, demonstrating a marked decrease in both the error variance

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and uncertainty bounds. At this point, the navigation system shows sub-centimeter level position-ing error (3D RMS), consistent with HIL testing by Giralo et al.29 After steady-state is achieved,97.8% of ambiguities are fixed. This high value is due to strict verification checks and data editingprocedures. It is also important to note the effect of maneuvers on the state estimate, denoted byvertical red lines. The velocity uncertainty bounds increase at these times due to the addition of pro-cess noise to account for maneuver execution errors. The relative velocity error stays within theseinflated bounds after the maneuver before settling back to steady-state. The effect on the relativeposition is only present for large reconfiguration maneuvers.

(a) Relative Position Error (b) Relative Velocity Error

Figure 8: Relative state estimate errors, with vertical red lines indicating execution of maneuvers.

The associated measurement residuals can be seen in Figure 9. It’s interesting to note the dif-ference in performance before and after IAR is activated. Due to the loss of the free parameter ineach measurement (the float ambiguity), the filter cannot freely adjust to minimize the residualsbelow the noise level. Therefore, the appearance of inflated post-fit residuals is actually the filterbeing more consistent with the expected noise of each measurement, indicating a healthy filter. Inparticular, the SDCP residuals show millimeter-level noise.

(a) GRAPHIC (b) Single-Difference Carrier-Phase

Figure 9: Measurement Residuals

To measure reconfiguration accuracy, the ROE achieved at the end of the commanded reconfig-uration (Phase 2) are compared to the desired ROE. It is shown that reconfiguration accuracy is

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high even in the presence of perturbations (given in Table 2), navigation uncertainty, and maneu-ver execution errors. The analysis is repeated for the same simulation twice: including maneuverreplanning on a diminishing horizon and excluding replanning. This comparison is used to demon-strate the effectiveness of maneuver replanning on achieving high reconfiguration accuracy.

Figure 10 shows the evolution of the four in-plane ROE throughout the experiment. The blue linesrepresent the path of each of the ROE when maneuvers are replanned on a diminishing horizon,while the orange lines represent the ROE without any maneuver replanning. The vertical dashedlines denote times at which maneuvers are executed (for the case where maneuvers are replannedonly). Notice that the out-of-plane ROE, the relative inclination vector, are not shown. This isbecause relative inclination vector changes require only a single maneuver, and are therefore notreplanned.

(a) Relative semi-major axis (b) Relative mean longitude

(c) Relative eccentricity vector, x-component (d) Relative eccentricity vector, y-component

Figure 10: Evolution of ROE during simulation.

As shown in Figure 10, the paths diverge, meaning there is a significant difference between theROE when replanning is active vs. when it is not. The difference is most obvious in Figure 10b,which shows the evolution of the relative mean longitude, aδλ. The relative mean longitude driftsat a rate proportional to the value of the relative semi-major axis. As given in Eq. (10), the desiredrelative semi-major axis at the end of the reconfiguration was zero, in which case the relative meanlongitude would not drift at all. When the maneuvers are replanned, the relative semi-major axisachieved at the end of the reconfiguration is closer to zero, and the corresponding aδλ drift rate is

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much lower than in the no-replanning case. This significant reduction of error is indicated in Table4.

Table 4: Reconfiguration accuracy in simulation

Metric Relative orbit element

aδa aδλ aδex aδey aδix aδiyDesired ROE (m) 0 -500 185 501 4.5 626

Maneuver replanning onAchieved ROE (m) 0.09 -499.12 183.87 500.71 3.37 625.14Absolute Error (m) 0.09 0.88 1.13 0.29 1.13 0.86

Maneuver replanning offAchieved ROE (m) 0.96 -520.58 183.56 502.34 3.52 625.32Absolute Error (m) 0.96 20.58 1.44 1.34 0.98 0.68

As in Figure 10, the largest error in Table 4 when maneuvers are not replanned is in the relativemean longitude. With replanning off, the achieved state in aδλ is 20 meters away from the desiredstate. With replanning on, it is less than 1 meter away. In general, the absolute error with maneuversreplanned is less than 1.5 meters for all ROE. Notice that in Table 4, the components of the relativeinclination vector have almost the same error in both cases. As discussed earlier, this is because ingeneral only one maneuver occurs to correct the relative inclination vector, so the same replanningtechnique cannot be used.

In addition to reconfiguration accuracy, it is also important to measure the accuracy during thekeeping phase. In Phase 3, the simulation proceeded through three full phases of formation keeping,aiming to maintain the ROE achieved at the end of Phase 2, which are given in the first row of Table4. Table 5 lists the ROE achieved at the end of each keeping window and the associated absoluteerror in meters.

Table 5: Formation keeping accuracy in simulation

Metric Relative orbit element

aδa aδλ aδex aδey aδix aδiyAchieved ROE (m) 0.05 -499.80 183.11 501.73 4.33 626.14Absolute Error (m) 0.05 0.20 1.89 0.73 0.17 0.14Achieved ROE (m) -0.03 -499.85 183.04 501.81 4.57 626.14Absolute Error (m) 0.03 0.15 1.96 0.81 0.07 0.02Achieved ROE (m) 0.16 -501.19 183.31 501.59 4.46 626.14Absolute Error (m) 0.16 1.19 1.69 0.59 0.04 0.18

As demonstrated in Table 5, the absolute error is less than two meters for all ROE.

Data Gap

A contingency scenario is also considered, demonstrating the capabilities of the system to con-tinue operations even in the case where GNSS measurements are unavailable. Without measure-ments, the navigation filter cannot refine the state but continues to perform time updates by nu-merically integrating the equations of motion. The goal of this test is to demonstrate continuedfunctionality, orbit prediction performance, and re-convergence of the filter upon reception of newmeasurements.

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To demonstrate these goals, the previous test scenario is augmented by introducing two periodsof measurement outages. The first is a short data gap, lasting 100 seconds (three calls to the filter).The second is a longer gap, lasting 45 minutes, which is roughly half of an orbit period. Duringthese periods, both receivers are experiencing a loss of measurements. The test is only run for 4hours to focus on the performance during the data gaps.

Figure 11 shows the relative position and velocity navigation errors throughout this test. Duringeach data gap, the covariance is inflated due to the addition of process noise. This inflation isindicative of a filter performing only the time update. During the short data gap, position errorsremain small, with 2.5cm error in both the radial and tangential components at the end of the outage.Since this period is short, there is no noticeable effect on the relative velocity error. During the largedata gap, the errors are more prominent, with the largest position error component being 0.2m inthe tangential direction. At the end of this longer period, the maximum velocity component erroris 0.25mm/s in the tangential direction. When measurements return, the filter re-initializes and theperformance is on par with the filter before IAR is activated. However, successful IAR occurs againwithin 5 calls to the filter, resulting in a quick reconvergence back to steady state.

(a) Relative Position Error (b) Relative Velocity Error

Figure 11: Relative state estimate errors in the presence of data gaps, with vertical red lines indicat-ing execution of maneuvers and gray boxes indicating periods of measurement outages.

CONCLUSIONS

This work presents the Guidance, Navigation, and Control system for the DWARF mission, dueto launch in 2022. Developed by the Stanford Space Rendezvous Lab in collaboration with GaussS.R.L and KACST, DWARF will advance the state of the art for distributed space systems throughon-board demonstrations of novel navigation and control algorithms. The GNC payload will beintegrated on two identical 3U CubeSats, where commercial-off-the-shelf hardware is used to reducepower and cost.

The navigation subsystem will be the first in-flight demonstration of the Distributed Multi-GNSSTiming and Localization system. Differential GNSS techniques use synchronous low-noise mea-surements to form powerful error-cancelling combinations that enable precise relative state esti-mates. The precision provided by DiGiTaL allows for integer ambiguity resolution to be performedin real time on CubeSat avionics, producing centimeter-level relative positioning estimates. DWARFwill also demonstrate this capability on orbit for the first time.

The control subsystem on-board DWARF will demonstrate the accuracy and robustness of the

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globally optimal impulsive control algorithms developed by Chernick.35 Maneuvers are plannedsuch that they achieve a desired reconfiguration optimally and maintain a safe separation accordingto user-set parameters. The planner is integrated into a state machine that continuously replansmaneuver magnitudes, which improves accuracy and mitigates failures.

A software-in-the-loop experiment using a high-fidelity orbit propagator simulated two space-craft, with prescribed phases of acquisition, formation keeping, and reconfiguration. A GNSSreceiver emulator used this orbit information to generate realistic measurements used for naviga-tion. The control system generated actuation commands based on the navigation solution and thephase of the scenario that were used in both the ground propagation and the navigation filter. Or-bit determination results demonstrated less than 1cm of relative positioning error (3D RMS). Theexperiment also demonstrated that the system attains the desired relative state within 1.5 meters ofthe commanded state in both formation keeping and reconfiguration, and that maneuver replanningprovides a significant increase in reconfiguration accuracy in the presence of perturbations, error,and uncertainty.

Further evaluation and stress-testing of the GNC system is ongoing in an effort to ensure flightreadiness, especially in corner cases and contingency scenarios. In particular, an extensive hardware-in-the-loop testing campaign is planned, where the GRAND testbed will use an IFEN GNSS SignalSimulator to stimulate the Novatel OEM628 receiver, and the DWARF software will execute on an800MHz Endeavour flight computer. DWARF is scheduled for launch in 2022. When available,flight data will be analyzed to show the in-flight demonstration of the work described in this paper.

ACKNOWLEDGEMENTS

The authors would like to thank Taqnia International for sponsoring this work and the King Ab-dulaziz City for Science and Technology (KACST) for the DWARF project execution and support.

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