precise formation flying control of multiple spacecraft using carrier ...

AAS 00-109

PRECISE FORMATION FLYING CONTROL OF MULTIPLE SPACECRAFT USING CARRIER-PHASE DIFFERENTIAL GPS1 Gokhan Inalhan 2 , Franz D. Busse

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and Jonathan P. How

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Formation flying is a key technology for deep space and orbital applications that involve multiple spacecraft operations. Imaging and remote sensing systems based on radio interferometry and SAR require very precise (subwavelength) aperture knowledge and control for accurate relative data collection and processing. Closely tied with the Orion and TechSat21 projects, this work describes the ongoing research to investigate precise relative sensing and control via differential GPS for multiple spacecraft formation flying. Specifically, we present an autonomous control architecture for formation flying that integrates low-level satellite control algorithms (formation keeping and relative error correction) with high-level fuel/time optimal formation coordination and planning. The basic features of this architecture are implemented on a nonlinear orbital simulation of Orion vehicles with disturbances and a realistic differential GPS measurement model. Also, we generalize closed-form solutions of passive apertures for constellations with mean formation eccentricity.

INTRODUCTION Formation flying is a key technology for deep space and orbital applications that involve multi-spacecraft operations [1, 2, 3]. Imaging and remote sensing systems based on radio interferometry and SAR involve relative data collection and processing over an aperture [4, 5, 6, 7] where the resolution of the observations is inversely proportional to the baseline lengths. For this reason, orbital and deep space distributed apertures formed by formation flying spacecraft provide desirable characteristics compared to earth based or centralized apertures located on one structure. Also, the ability to easily form and reconfigure very long baselines for uniform and dense u-v (observing) plane coverage [8, 9] provides a clear advantage over traditional systems. However, there are major technical challenges in achieving the aperture coordination, control, and monitoring of these distributed vehicles that will be necessary to achieve the stringent payload pointing requirements [6, 7]. In addition, the fleet control design must be done with careful consideration of the onboard computation, inter-vehicle communication, and power requirements for the formation flying spacecraft. Thus a systems-level approach is essential, allowing explicit inclusion of the hardware limitations (power, mass, computation) in the theoretical analysis of the various multi-level control architectures [14, 16, 19]. For 1 This work is funded in part under Air Force grant # F49620-99-1-0095 and NASA GSFC grant #NAG56233-0005 2 Research Assistant, Dept. of Aeronautics and Astronautics, [email protected] 3 Research Assistant, Dept. of Aeronautics and Astronautics, [email protected] 4 Assistant Professor, Dept. of Aeronautics and Astronautics, [email protected]

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Fig. 1: ORION: On-orbit demonstration of formation flying. Nonlinear simulation with realistic disturbances and actuator limitations.

Fig. 2: Schematic of the Orion Satellite (0.45m cube with cold gas propulsion and GPS sensing)

this reason, the following system level challenges for such distributed systems have been identified under the overall formation flying problem: 1. 2. 3. 4. 5.

Sensing and metrology (relative/absolute sensing, sensor fusion) Aperture optimization (orbit and formation selection) Fleet and vehicle autonomy (control architecture) Control computation (formation planning, maneuvering and data collection) Spacecraft bus design (RCS, crosslink communications and CDH)

This paper describes ongoing research at Stanford University to address these systems-level issues for two future on-orbit demonstrations of formation flying: Orion and TechSat21 [13, 18]. The particular emphasis of this work is to demonstrate precise relative sensing and formation flying control of multiple spacecraft via carrier-phase differential GPS.

FORMATION FLYING ON-ORBIT: ORION & TECHSAT21 TechSat21 is an AFRL/DoD program focused on the development and on-orbit demonstration of various formation flying technologies [18]. Orion is a high-risk, low-cost NASAfunded project dedicated to the research and development of microsatellites that are capable of performing distributed relative sensing and formation flying. Orion and TechSat21 represent major stepping-stones/technology demonstrators for many of the basic control and sensing elements of future formation flying missions. The specific goals of Orion include: • Ability to organize a group of small satellites into a pre-determined formation on-orbit. In particular, this will include the ability to exchange GPS, position and scientific data between the satellites, as well as execute pre-planned, organized maneuvers [17]. The execution of the maneuvers will be governed by on-board real-time autonomous control software (typical simulation results shown in Fig. 1) 2

• Use and operation of a low-power, low-cost, multi-channel GPS receiver for real-time attitude and position determination of the satellites (antenna layout in Fig. 2). • On-orbit autonomy using various control architectures. Demonstrate autonomous mode switching from coarse to fine (and back) formation flying. • Ability to perform formation flying and station keeping with various baselines. With these main objectives for the Orion and TechSat21 projects, we have developed, simulated, and experimentally demonstrated basic features of an autonomous control architecture for formation flying. Low-level satellite control algorithms (formation keeping and relative error correction) are integrated with high-level, operations-driven, fuel/time optimal formation coordination and planning [22] within a multi-layered architecture. Fig. 3 shows the basic elements and the information flow for this control architecture. The formation planning part of this architecture was implemented on the Formation Flying Test Bed (FFTB - free flying vehicles on a granite table) to perform a nonlinear “rigid body” formation retargeting maneuver requiring a coordinated translation and rotation while keeping the relative distances fixed [21, 22]. This paper provides further analysis of the three key components of the autonomous control architecture in Fig. 3 and extends it to orbital operations. In particular, we illustrate the design and analysis of the low-level regulator for relative error correction and formation keeping, present results from the CDGPS estimator, and discuss the implementation of a coordinator for passive aperture formation. All of these components are integrated on an commercial high-fidelity nonlinear orbit propagation tool [31]. The simulations include realistic disturbance models, measurement errors, and typical propulsion system nonlinearities such as finite thrust and minimum impulse bit. Within the context of this autonomous formation control architecture, the following sections present detailed discussions of the tools developed to address the system level issues (e.g. sensing & metrology, vehicle & fleet autonomy) associated with the coordination and control of the multiple vehicles in the fleet. Specifically, we present the algorithms and methods used in control input generation for high-level coordination and low-level satellite control based on CDGPS measurements (absolute and relative).

FORMATION CONTROL ARCHITECTURE Within the architecture shown in Fig. 3, autonomy is based on intelligent decisions provided by the High-Level Coordination in two separate levels. The first level involves vehicle autonomy where system-level decisions, checks, and updates are conducted under the GNC Housekeeping algorithm. In addition, this process provides an operations and flight status link between the GNC subsystem and satellite command and data handling unit, which explicitly controls the satellite bus and hardware activities. The second level includes fleet autonomy handled by the Multiple Vehicle Formation Coordinator. In a centralized application, the coordinator essentially acts as a fuel efficient “distilling” algorithm that uses information such as the status and location of each spacecraft to determine the desired reference orbits (mean element update) and desired individual locations for each satellite. This complex decision process is tied to the fleet objectives and operations. The next section discusses tools developed to perform this coordination. Note that the architecture is sufficiently flexible that distributed coordination could be used for large fleets of vehicles. 3

Fig. 3: Autonomous Control Architecture As shown, the major hub of the external GNC information flow is handled via the crosslink unit with two different data rates. The high data rate intersatellite communication is needed for the raw GPS data used by the relative navigation process that is part of the differential GPS (CDGPS) unit. The CDGPS unit consists of 3 primary algorithms that solve for the absolute, relative, and attitude measurement information. The low data rate intersatellite communication is used to handle the system-level information regarding formation flight modes, desired reference orbits and locations, and updates of the fleet parametric models. With high fidelity on-orbit propagators, these estimators provide state and parametric model estimates of the current satellite and environment models. All of this information and the flight mode (based on the operations plan) are processed by the highlevel coordinator to determine the actual low-level satellite attitude and position control. The algorithms that have been developed for the low-level satellite control are discussed in the following sections.

High-Level Coordination In this modular architecture, the coordinator explicitly handles the task of coordination and scheduling of the operations of the formation based on mission defined objectives. The majority of the mission operations involve task distribution, formation selection/planning 4

Precise Formation Flying in LEO Orion 1 Orion 2 Orion 3

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Fig. 4: Typical Nonlinear Formation Initialization Simulation With 3 Orion Spacecraft. [12], collision avoidance and operation mode selection (e.g. coarse formation flying, precise formation flying, parking orbit) while performing imaging and remote sensing experiments. How the computational aspects of the formation planning are handled and implemented will strongly depend on the number of vehicles in the fleet and their computational capabilities. For a small fleet (typically 3-4 vehicles ), all duties of the coordinator can be implemented as a centralized process. For such a case, the multiple vehicle fuel/time-optimal formation planning can be solved as a convex Linear Programming (LP) optimization problem using the linearized relative dynamics [21, 22, 28]. Refs. [21, 22] have previously shown that fuel-optimal trajectories [11] and thrusting sequences of the vehicles can be generated for typical cooperative maneuvers such as initialization, resizing, and retargeting of passive apertures. Note that this LP design process used by the coordinator explicitly accounts for the bounded thrust capabilities and the limited fuel capacity of each vehicle [20]. Fig. 4 shows the results from a typical formation initialization and precise formation keeping maneuver in a nonlinear simulation with standard LEO disturbances and DGPSlevel measurement errors. In this implementation, two Orion spacecraft are moved from their initial positions to predefined formation locations with a ±50m intrack separation with respect to the central spacecraft. The LP algorithm was used by the coordinator to design the fuel optimal trajectories to move these two vehicles. After the initialization is complete, the coordinator selected the precise operation mode to keep the formation within a 0.5×2.5×2.5m radial/intrack/cross-track error box. In this operation mode, linear quadratic regulators with deadband are used for stationkeeping. The LP optimization can be used to answer many interesting aspects of optimal coop5

Time Optimal Formation Flight via Differential Drag

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Fig. 6: Drag Panel Sequence for Cooperative Control

erative control of multi-vehicle formation flight. For example, although differential disturbances can result in excessive fuel usage, such effects can also be used to the advantage of the formation flight [29]. Fig. 5 shows a cooperative time optimal formation flying control of two vehicles using drag panels as the actuation. This problem was solved using the LP technique. With an initial radial separation of 300m and 750m in the intrack direction, the vehicles operate their drag panels to come into close proximity. Fig. 6 shows the opening and closing sequence for the drag panels. The positive and negative effect is obtained by cooperation between the two vehicles. For close proximity formations on the order of a few hundred meters, the linearized dynamics provide useful and precise models for formation flight design. However for larger or longer maneuvers, which can take more than a few orbits, the effects of measurement noise, nonlinear orbital effects, and differential disturbances will cause deviations in the final relative states. For these cases, the following standard techniques could be utilized to handle the errors in the linearized dynamics: • Iterative Procedure: By using updates on the relative states of the vehicles, the coordinator could iterate and replan the maneuver to account for any large formation deviations and anomalies encountered during the execution of the initial plan. • Inclusion Method: Based on initial plan, all of the unaccounted effects can be traced along the initial trajectory design. These calculated effects could then be included in the replan to capture the approximate magnitude of the neglected effects. Note that a similar idea is used in the low-level satellite control subsection to design a long-term formation keeping trajectory with differential disturbances (see Fig. 10). • Trajectory Morphing: Ref. [27] presents an approach that explores the trajectory space of a nonlinear system by starting from simplified models. With the addition of linear time-varying feedback, feedforward, and homotopy, the linear trajectories can then be morphed into nonlinear ones, with the result that stable trajectory tracking is obtained. • Nonlinear Optimization: Can also use the linearized trajectory to initialize a nonlinear optimization technique. A similar approach was successfully implemented on the FFTB for the experimental nonlinear “rigid-body” retargeting maneuver [22]. 6

FORMATION INITIALIZATION OF 8 SPACECRAFT SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8

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Fig. 7: Formation Initialization Maneuver Selected by the Coordinator For fleets with a large number of vehicles, the computational aspects of the formation planning can become very difficult given the amount of information flow and the processing required. In such a case, the formation coordination can be distributed over the whole fleet with a distributed coordination scheme which involves an individual bidding process [12]. Fig. 7 shows the result of such an implementation where eight vehicles initialize to a closedform ellipse (e = 0) after a bidding and selection process for their locations. Initially all of the vehicles are on a intrack formation separated by 100m from each other. The only hard constraint on the planning process is that the vehicles should be placed with equal phasing along a closed-form ellipse with a semi-major axis of 600m. Given each vehicles fuel state and initial location, the main issue is to determine which vehicle should move to each location on the target ellipse. The key parts of the distributed solution to this problem are the bidding and selection process carried out by each of the satellites (shown in Fig. 8). Each satellite analyzes the alternative final locations (over a discrete grid of 1◦ resolution) and associates a cost with each of them. Fig. 9 shows a typical fuel usage vs. aperture location (phase angle) map created by one of the spacecraft. The ∆V calculations of the individual spacecraft are solved using the standard LP method discussed previously. These calculations are very simple because they involve only the vehicle itself. For this example, the individual bidding decisions of the satellites are based on their fuel usage vs. aperture location map. With all the bids obtained, the coordinator starts

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DV MAP vs APERTURE LOCATION FOR SC # 1 0.95

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identifying possible optimal aperture locations with the highest bidder for any starting aperture location. Having set that location and the vehicle, all other possible aperture locations are filled one-by-one using the best remaining highest bidders. This same process is run again for every phase angle over the discrete grid of 1◦ resolution. After this step, the coordinator selects the best 3 out of the 360 possible configurations (each consists of all 8 vehicles with equal phasing around the ellipse). Collision avoidance checks are then run on these 3 cases to decide on the best final configuration. The selection and collision avoidance process is not computationally intensive and can be run on any single vehicle (or be distributed). Note that it would also be possible to include rebidding and learning for each vehicle under an iterative scheme that uses knowledge of the previous bids made and the trajectories obtained (would be used to modify the spacecraft bidding strategy). Although this process is not guaranteed to be globally optimal, it provide some key benefits because it 1) distributes the computational intensity, 2) removes the possible threat of single point failure, and 3) allows the vehicles to develop individual decision models that will be reflected in their bids. For example, a vehicle can decide on a maneuver which requires minimum amount of orientation requirements due to a reaction wheel failure. Although the selected location in the aperture might be not the best fuel efficient choice, the bidding process allows the individual vehicles to include such complex decision parameters, which is very difficult to incorporate in a generalized optimization due to discrete logical nature and case dependence.

Low-Level Satellite Control The low-level satellite control combines linear-quadratic regulators with a formation keeping algorithm [15]. The linear-quadratic regulators are stored in a gain-schedule table based on different operational characteristics that are determined by the control effort and relative error weights. These regulators are used for trajectory tracking and relative error correction to place the spacecraft in the desired relative error boxes. Once the spacecraft are placed within the desired error boxes, the controller switches to a fuel optimal formation keeping algorithm. This algorithm is based on the solution of 8

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Fig. 11: Fuel Usage and Error Box Tradeoff Under Differential Disturbances

an extension of the LP formation planning problem with the addition of differential disturbances. Note that the very accurate CDGPS measurements could be used to produce a parametric estimation of the constant and periodic components of the disturbance environment. Fig. 10 shows the results of a fuel-optimal formation keeping maneuver for 10 orbits under a constant differential disturbance (drag) of 10−8 m/s2 . Given a disturbance model (e.g. differential drag, differential J2 ), the fuel-optimal thrusting sequence of a vehicle can be calculated to keep the vehicle within a desired error box around a reference point or another vehicle. This is a complicated process for differential J2 disturbances which are a function of the vehicle separations. However, if it is initially assumed that the low-level control keeps the vehicles within the prescribed error boxes, then the size of these boxes can be used to bound the maximum separation of the vehicles and thus also the disturbances of the fleet. The low-level control design process can then be iterated to develop new thrusting sequences that account for the updated differential disturbance models. As might be expected, the fuel cost for a zero tolerance error box is directly equal to counter-acting the differential disturbance at all times. However, an interesting problem is the trade-off analysis of fuel savings associated with larger error boxes. For this case we can use the LP method to generate the data required to map the error box size against fuel usage for a set of typical differential accelerations observed on on-orbit. Such a map is shown in Fig. 11 for differential drag ranging from 4.02×10−7 m/s2 to 0.05×10−7 m/s2 . For example, for a constant differential acceleration of 1.5×10−7 m/s2 , we could obtain almost a 30% fuel savings by changing from a 1 unit size error box to a 3 unit size error box (which is ±20m in the intrack and ±4m in the radial direction). With analysis of this type, the methods developed for formation keeping with disturbances can also be used to develop insights on the overall mission design. Both of the low-level schemes directly use the operation mode information from the coordinator for selection of error box sizes. Table 1 shows the expected operation modes for Orion with the corresponding separations and relative error boxes. In addition, a phase-plane based approach is currently being extended to control in-

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Experimental Modes Coarse Parking Fine Parking Precision Formation Flying

In-track Separation (m) 300 100 100

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Table 1: Separations and Tolerances for Orion Experiment Modes

dividual vehicles to desired formation locations. The approach divides the vehicle motion into two parts that involve cyclic and secular motion. Simple control strategies can then be developed to utilize differential energy correction and mean motion sizing to perform formation keeping with a desired error box. The approach is similar in concept to the eccentricity minimizing control in Ref. [29]. For both high-level coordination and low-level satellite algorithms, on-orbit estimation plays a crucial role in the flight mode selection, system and environment parametric modeling, and the control input generation. The next section describes the measurement models (absolute, relative and attitude) of the Carrier-phase Differential GPS (CDGPS) unit and the estimation schemes for high precision formation flying.

Carrier-Phase Differential GPS The sensing component of the control architecture is performed by the Carrier-phase Differential GPS sensing for precise absolute and relative navigation for formation flying [23, 24, 25]. The complete state of the formation of user satellites can be determined using GPS. This includes the absolute position and velocity of each vehicle (generally determined in Earth Centered Inertial frame), the relative position and velocity (the user vehicles with respect to each other), and the attitude of the user vehicles that have sufficient antenna and RF processing capability. This three-in-one capability of GPS sensing makes it very attractive option, often able to replace three separate sensor systems, at a fraction of the cost and weight. The estimator produces three solutions: the attitude, the relative state between users, and the absolute state of the (current) vehicle. The absolute state is determined by standard GPS pseudo-ranging, which measures the transmission time for the RF signals from at least 4 NAVSTAR satellites. Using raw GPS data sent by direct crosslink from the other satellites in the formation, and its own measurements, the relative state is determined using carrier-phase differential GPS, where the phase within the 19.2cm carrier wave is measured by two antennas, subtracted, and from that the relative distance can be measured. Attitude is also determined by carrier-phase differential GPS, but uses multiple antennas at fixed locations on a single vehicle frame. The method for determining relative position is inherently about three orders of magnitude more accurate than the method for determining absolute position. It is possible to use the known orbital dynamics in combination with the much higher precision of the relative solution to improve the absolute position estimate. Improved absolute position knowledge, especially in the radial direction, leads directly to improved control performance. As shown in Fig. 3, these solutions are transferred to the high-level coordinator. The coordinator uses this information for both the high-level constellation management as well

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as forwarding it down to the low-level satellite control. The next section discusses the relative navigation estimator and the accuracy levels predicted.

Estimation Note that both a Kalman filter and Weighted Least Squares approach are used for the relative state estimation. For the Kalman filter, the model for the relative motion of the vehicles in inertial space is used to propagate the current state and covariance matrix of the vehicle to the next measurement time 1 Hz rate. This propagated state is then updated with the measurements, where the model and measurements are combined using an optimal weighting matrix. For this particular problem, the state was propagated with a non-linear model, and the covariance matrix propagated with a linearized model. The Weighted least squares is based entirely on the current measurements from a single time step. It weights the measurements according to the expected noise levels on the signals. Simulations were run to test the accuracy of these two estimators. The state of the GPS NAVSTAR constellation was propagated over time, and measurements simulated from the constellation. The simulation results shown assumed an orbit with 35 degree inclination, 400km altitude, and eccentricity 10−3 . Figs. 12, 13 show the position and velocity error for Kalman filter relative state estimation between 2 vehicles with only an in-track initial separation of 1000 meters. The noise model for the measurements was based on previous actual GPS receiver performance. The absolute position ranging noise was assumed 25m, the differential code phase noise (where common error sources, like S/A, are eliminated) was assumed 2m, carrier noise 2cm, and Doppler noise 5mm/sec. Fig. 14 compares the errors of the weighted least squares and Kalman filter solutions. Multiple simulations were run with random noise included, and the final results from these are in Table 2. Note that these are the averages over all simulations of the mean and standard deviation on the absolute and relative errors for the second half of the orbit shown. Note the units for each quantity. These results show the expected 2-5cm levels of accuracy that should be achievable on-orbit in the relative position estimates at a 1 Hz update rate.

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PASSIVE APERTURES WITH MEAN FORMATION ECCENTRICITY Many earth imaging and remote sensing applications would benefit from long periods of operation over particular regions of interest, as is typical of Molniya-type orbits. In addition, highly eccentric orbits can be used to increase the percentage of useful observation time by decreasing the occlusion time of the Earth [26]. Thus there is significant interest in extending the design of “passive apertures” to the case of eccentric orbits, thereby providing a natural extension of the classic closed-form solutions to Hill’s equations. Using the linearized relative equations of motion with respect to any Keplerian orbit (see [28] for details), a generalization of the closed-form solutions [30] can be made for constellations with a non-zero mean formation eccentricity. Ref. [28] also provides the necessary conditions for obtaining T -periodic solutions for passive formations in eccentric orbits, where T corresponds to orbital period. Figs. 15 and 16 show a typical periodic closed-form solution for a reference orbit of a=46000km, e=0.67, i =62.8◦ . The in-plane and out-of-plane motion correspond to incremental changes in eccentricity (δ e = 0.0001) and inclination (δ i = 0.005◦ ). An interesting feature of these eccentric orbits is the “figure 8” shaped out-of-plane motion. For circular

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Error Absolute Position (m) Relative Radial Position (cm) Relative In-track Position (cm) Relative Cross-track Position (cm) Relative Radial Velocity (mm/s) Relative In-track Velocity (mm/s) Relative Cross-track Velocity (mm/s)

Mean 25.31 2.73 1.31 1.28 0.095 0.018 0.014

St. Dev. 148.83 2.34 1.69 0.82 0.066 0.051 0.027

Table 2: Error Statistics for Estimation using a Kalman Filter

Fig. 15: In-Plane Relative Motion For a Fig. 16: Out-of-Plane Relative Motion For High Eccentricity Reference Orbit, Nonlin- a High Eccentricity Reference Orbit, Nonear Simulation linear Simulation reference orbits, small inclination differences only result in a one-dimensional out-of-plane motion. This feature of highly eccentric orbits could be used to provide higher u-v plane coverage per orbit for aperture filling observations [3]. Current efforts are focused on modifying the control design tools to account for mean formation eccentricities.

CONCLUSION Formation flying is a key technology for deep space and orbital applications that involve multi-spacecraft operations such as remote sensing and imaging applications. However, under the overall formation flying problem, there are system-level challenges for such distributed systems due to limited resources and hardware constraints of space systems. This paper describes ongoing research at Stanford University to address these systems-level issues for two future on-orbit demonstrations of formation flying: Orion and TechSat21. A particular emphasis of this work is on precise relative sensing and formation flying control of multiple spacecraft via carrier-phase differential GPS. The paper discusses a multi-layer autonomous formation flying control architecture. This GNC tool combines low-level satellite control algorithms (formation keeping and rela-

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tive error correction) with high-level fuel/time optimal formation coordination and planning for multiple vehicles. The paper also discusses algorithms that can be used to perform the high-level coordination, trajectory design, low-level satellite control, and CDGPS estimation (absolute, relative and attitude). The basic features of this autonomous control architecture was implemented on an independent nonlinear orbital simulation of 3 Orion vehicles with realistic disturbances and a differential GPS measurement model. Finally, we demonstrate that it is possible to obtain closed-form solutions for constellations with mean formation eccentricity, which generalizes previous work on circular reference orbits.

ACKNOWLEDGMENTS The authors would like to thank Dr. Marc Jacobs (AFOSR), Rich Burns (AFRL), John Bristow (NASA GSFC), and Dr. Frank Bauer (NASA GSFC) for their guidance and support on the TechSat21 and Orion flight programs at Stanford University.

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