Adaptive Control System Design for Solar Electric PropulsionSystems used during Near-Earth Asteroid Capture Missions

Prepared by:Benjamin Dykstra

Faculty Advisors:Dr. Randy Hoover

Director, SDSM&T Computer Vision and Motion Capture Laboratory

Dr. Thomas MontoyaREU Site Director, Department of Electrical and Computer Engineering

Dr. Alfred BoysenProfessor, SDSM&T Department of Humanities

Program Information:National Science Foundation

NSF Grant #1359476

South Dakota School of Mines and Technology501 E Saint Joseph Street

Rapid City, SD 57701

Contents

Abstract 2

1 Introduction 3

2 Broader Impact 4

3 Procedure 53.1 Simulation Environment . . . . . . . . . . . . . . . . . . . . . 53.2 NEA/SEP Experimental System Characterization . . . . . . . 53.3 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Results 94.1 System Characterization . . . . . . . . . . . . . . . . . . . . . 94.2 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Discussion 12

6 Conclusion 136.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Reference 15

Acknowledgments 15

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Abstract

NASA’s Asteroid Redirect Mission (ARM) is a robotic mission concept thataims to return near-Earth asteroids (NEAs) to a safe and stable orbit incislunar space (situated between the Earth and the Moon) for the purposesof asteroid mining, scientific study, and planetary defense. Once captured,NASA proposes to employ a solar-electric propulsion (SEP) system to directNEA’s from solar orbit to cislunar space.

This study investigates and develops a control system structure that maybe used for the NEA/SEP system. Additionally, due to the difficulties as-sociated with accurately characterizing the mass and density distributionof target near-Earth asteroids, this research looks primarily to investigatea means by which the SEP/NEA system dynamics may be experimentallycharacterized once a target NEA is captured by the SEP system.

After system characterization, a predictive control scheme is employed todirect a captured NEA to a stable cislunar orbit.

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1 Introduction

The robotic Asteroid Redirect Mission (ARM) concept is centered around useof a solar-electric propulsion (SEP) system that, in conjunction with an aster-oid capture mechanism, proposes to redirect near-Earth asteroids (NEAs) toa stable cislunar orbit [2]. As with any robotic mission concept such as this,many obstacles are presented to the mission planners as well as the engineersof the spacecraft which must achieve the proposed mission goals. Perhapsamong the greatest obstacles presented to engineers of the ARM/SEP space-craft pertains to difficulties in accurately identifying the mass and densitydistribution of possible NEA targets before sending a spacecraft to performretrieval. As such, the SEP spacecraft’s propulsion control system must havethe capability to adapt to a range of possible NEA characteristics in order todirect a captured NEA along the desired trajectory. The currently proposedARM/SEP spacecraft configuration (depicted in Figure 1) consists of a cap-ture mechanism, two solar arrays, and and four 10-kW dual-axis gimballedHall thrusters [2].

Figure 1: ARM spacecraft configuration

This research investigated a means by which the NEA/SEP dynamics maybe characterized experimentally once an NEA is captured as to provide thepropulsion control system with an accurate model from which to operate.Additionally, only a single-input single-output (SISO) version of the system

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is considered here for sake of simplifying the discussion, however a more accu-rate multiple-input multiple-output (MIMO) model of the system is alreadyplanned for future work.

Once system identification is accomplished, a predictive controller is appliedto regulate the NEA/SEP position along a single inertial frame axis. Theregulative control presented here is crude and meant only to demonstratethat the model developed during system characterization may be used forcontrol. Much of the focus of this work was centered around developmentof an expandable simulation environment and investigation of the systemcharacterization method as a valid approach to identifying SEP/NEA dy-namics. What is presented here represents merely a preface work to a futuregraduate-level thesis to be investigated by the same author.

2 Broader Impact

Near-Earth asteroid retrieval would provide asteroidal resources to a widerange of application areas to include scientific study pertaining to the gene-sis of the solar system as well as asteroidal mining for industrial material use.Another, perhaps less obvious, use for captured NEAs is known by NASA asPlanetary Defense - the disruption or destruction of potential Earth-collidingobjects (mostly other asteroids). More specifically, a captured NEA wouldserve to disrupt the trajectory of a potential Earth-collider by placing theNEA somewhere along the intruder’s collision course [2].

NASA and other agencies around the world are looking to expand humanspace exploration, and eventually realize human settlement in space and onother astronomical bodies. The retrieval and processing of raw materialsin space will become increasingly vital to expansion efforts of our in-spaceinfrastructure. As such, one of the most noted benefits to pursuing NEAcapture as a source of raw materials in space comes down to a simple energybudget analysis. More specifically, to retreive an NEA, bring it into cislunarorbit, and extract desirable materials would cost less energy (and therefore,cost less money) than mining resources from the Moon and subsequentlyferrying those resources back off of the Moon [2].

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3 Procedure

Investigating an adaptive control scheme for the NEA/SEP system consistedof three main tasks - creation of a simulation environment, system character-ization of randomly generated NEA/SEP characteristics, and finally, controlof the NEA/SEP sytem. In order to satisfy the requirement that the SEPthrust control be capable of adapting to a range of NEA characteristics, aneural network (NN) model was used to identify system dynamics. The net-work was trained by iteratively introducing random inputs to the systemand subsequently comparing the NEA/SEP response to the NN predictedresponse.

The NEA/SEP system characterization and control were intimately relateddue to the fact that a nueral network predictive control approach was em-ployed. MATLAB/SimulinkTM software was used to create the simulationenvironment as well as used to assist in the development of the neural networkand control.

3.1 Simulation Environment

In order to both simulate and visualize NEA/SEP dynamics, a SimMechanicsTM

environment was created that features the SEP system attached to an NEAobject for which random characteristics may be generated (mass, center ofmass, moments of inertia, etc.). The randomly generated NEA sizes arebounded by the proposed target NEA sizes as indicated within the NASAAsteroid Retrieval Mission concept parameters (about 5-10 meters in diam-eter). Other NEA characteristics (mass, moments of inertia, etc.) were thenderived by using an average asteroid density of 2g/cm3 [1].

Figures 2 and 3 depict the NEA/SEP dynamic model block diagrams, whileFigure 4 shows a screenshot of the simulation visualization environment.

3.2 NEA/SEP Experimental System Characterization

As with any control system, in order to establish control of the NEA/SEPsystem, a mathematical model of the (nonlinear) system dynamics must bedeveloped. A nueral network model of the NEA/SEP plant was created andtrained in order to experimentally characterize any given NEA/SEP system

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Figure 2: System Dynamics and Measurement Output Block Diagram

Figure 3: NEA/SEP Dynamics Configuration

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Figure 4: NEA/SEP Simulation Visualization

dynamics by iteratively providing both the plant and the NN model with arandom input value. The simplified SISO version of the system uses thrustdirection as the input (± 30 degrees with respect to the SEP base) and they-axis (inertial frame) position as the output. Each test iteration uses arandomly determined input thrust angle and a 5-30 second thrust pulse du-ration. Each thrust pulse utilizes the same constant thrust magnitude.

Figure 5: Nueral Network Training Process(image: MathWorks Inc.)

Once the random input is ap-plied, the response of the plant iscompared to the NN response inorder to develop an error. Thiserror is then fed to a dual-layerbackpropagation training algo-rithm which subsequently up-dates the NN model in order todevelop a more accurate repre-sentation of the true NEA/SEPsystem dynamics. Figure 5 de-picts a block diagram repre-sentation of the nueral networkmodel training process.

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The backpropagation algorithm trains the nueral network by utilizing pre-vious plant inputs and outputs to predict future outputs of the NEA/SEPsystem [4].

3.3 Control

Figure 6: Nueral Network PredictiveController (Image: Math-Works Inc.)

After the nueral network is trained,a predictive controller is employedwhich uses the previously estab-lished nueral network plant model tocontrol the plant output about somedesired regulated input yr. The pre-dictive control method used by thenueral network controller is based onthe receding horizon technique [3].More specifically, the previously es-tablished nueral network model pre-dicts the plant output over somegiven time horizon. These predic-tions are then used by a numericaloptimization algorithm (embedded in MATLAB software) that determinesthe control signal which minimizes a performance criterion over a specifiedtime horizon using the following:

Where N1, N2, andNu define the horizons over which the tracking error andcontrol increments are evaluated. The u′ variable is the tentative controlsignal, yr is the desired response, and ym is the NN model response. Theρ value determines the contribution that the least squares sum of the con-trol increments has on the performance index [4]. Figure 6 illustrates thepredictive controller block diagram.

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4 Results

As mentioned previously, the simulation environment and system charac-terization method were the primary focus of this work, and are thereforescrutinized more heavily here.

4.1 System Characterization

After the training process was complete, validation of the NN model was ac-complished by stimulating both the plant and the NN model with a series ofrandom inputs (thrust angles) and subsequently comparing the response ofthe plant to the predicted response of the NN model. Figures 7a-7d highlightthe NN prediction error (difference between plant response and NN response)for various numbers of NN training iterations. The NN model was able topredict the NEA/SEP SISO dynamics with minimal error after, on average,100 training iterations. More specifically, as is illustrated in figure 7c, theposition prediction error of the neural network was on the order of millime-ters after about 100 training iterations.

For training sessions with around 75 or less iterations, however, the NN pre-diction error was on the order of meters as shown in figures 7a and 7b. Lastly,for training sessions of 150 iterations (figure 7d) and higher, a negligible dif-ference was noted in the NN prediction error as compared to 100 trainingiterations.

4.2 Control

As mentioned before, the predictive control method employed was rathercrude, and as such, the regulation about the inertial frame y-axis was notoptimized. Figure 8 illustrates the result of the NN predictive controllerregulating the NEA/SEP position about the inertial frame y-axis. Note thelarge steady-state error as well as the highly oscilliatory behavior of thesystem.

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(a) 50 training iterations

(b) 75 training iterations

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(c) 100 training iterations

(d) 150 training iterations

Figure 7: Neural Network Model Validations

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Figure 8: NEA/SEP Control Step Response

5 Discussion

As stated previously, the main focus of this stage of the research was thedevelopment of the simulation environment and investigation of whether ornot a neural network is an appropriate way of handling the inherently non-linear dynamics of the NEA/SEP system. As to the former, the simulationenvironment developed here will serve as an expandable base from which amore complex version of the system may be analyzed in the future.

As it pertains to the NN model and NN training process, it would seem thatapproximately 100 training iterations are appropriate to establish a mini-mal NN model prediction error. Further investigation in future stages of theresearch will reveal whether or not a position prediction error on the scaleof millimeters is sufficient for planning trajectories millions of kilometers inlength.

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The application of the NN training method outside of the simulation envi-ronment perscribes another consideration pertaining to the number of NNtraining iterations (as well as the length of each iteration) required to estab-lish an acceptable NN prediction error. More specifically, the SEP spacecraftwill have a limited fuel supply, and as such, not be able to provide inputthrust iterations indefinitely to train the neural network. Future work willlook to understand and analyze this cost-benefit further - to optimize thetraining process around minimal fuel usage while still developing an accurateNN model.

Altogther, due to the unknown nonlinear dynamics of any given NEA/SEPsystem, nueral network adaptive control seems a viable option to handleproblems pertaining to nonlinearity and unknown system dynamics simulta-neously. Additionally, due to the nature of the ARM mission concept, systemidentification would not be rushed and would therefore allow for the approx-imate 100 training iterations to properly establish a model of the nonlineardynamics.

6 Conclusion

6.1 Summary

In this study, a potential method for adaptively controlling Solar-ElectricPropulsion systems designed to fly near-Earth asteroids from solar orbit intocislunar space was proposed and investigated. The proposed system identi-fication and control method was that of a nueral network predictive controlsystem that could, in theory, act as a universal propulsion control systemdesign for a wide range of NEA/SEP combinations. It was found that theneural network (utilizing a backpropagation algorithm) was able to identifyand predict the nonlinear NEA/SEP dynamics with minimal error with ap-proximately 100 training iterations.

6.2 Future Work

Future work under the same author (in pursuit of a Thesis on the subject) isalready planned, and will look advance the NEA/SEP control investigation

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in a number of areas. First, the neural network system identification methodwill be extended to include a more accurate, MIMO version of the NEA/SEPsystem. Additionally, the simulation environment will be expanded such thatthe NEA/SEP objects may be placed in a physics analogue of the solar sys-tem to test control along realistic trajectories.

The control structure which utilized the developed NN model will be greatlyexpanded to include fuel and transit time optimizations as well as considermany more phsical limitations of the NEA/SEP system (perhaps solar panelflexing and vibration, for example).

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References

[1] Britt et al. Ear-a-5-ddr-asteroid-densities-v1.1. NASA Planetary DataSystem, 2002.

[2] Daniel D. Mazanek. Asteroid retrieval mission concept – trailblazingour future in space and helping to protect us from earth impactors. InPlanetary Defense Conference, 2013.

[3] D. Soloway and P.J. Haley. Neural generalized predictive control. InProceedings of the 1996 IEEE International Symposium on IntelligentControl, pages 277–281, 1996.

[4] C. T. Sun. Neuro-Fuzzy and Soft Computing. Prentice-Hall Inc., 1997.

Acknowledgments

The Author would first like to extend a sincere thank you to the NationalScience Foundation for funding this research opportunity as provided bythe Research Experience for Undergraduates (REU) Program. Additionally,the Author would like to thank the SDSM&T faculty that supported theresearch - Dr. Randy Hoover for his technical guidance and perspective, Dr.Alfred Boysen for his simultaneously comical and effective documentationand presentation direction, and Dr. Thomas Montoya for his organizationand support as the REU Site Director.

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