An Evaluation and Comparison of Autonomous Orbit Transfer Controllers for the NASA Magnetospheric

MultiScale Satellite

Michael Borrelli1 and May-Win Thein2 University of New Hampshire, Durham, New Hampshire 03824

With the ever more demanding goals of space exploration and research comes the need

for more complex mission planning. Part of this complexity manifests itself in a satellite’s orbit specifications. An increasing number of explorer missions calls for the maneuvering and tight control of constellations of satellites. One such mission is the NASA Magnetospheric MultiScale (MMS) mission, which aims to investigate the temporal and special behavior of Earth’s magnetosphere. The primary focus of this work is to develop a controller for MMS spacecraft. The goal is to ensure high-accuracy performance while minimizing control effort, so as to optimize the use of on-board propellant. This paper shows through rigorous simulations that it is feasible to maintain strict control of spin-stabilized spacecraft during orbital maneuvers to within relatively small levels of accuracy (as dictacted by mission requirements). A PID controller is developed for this purpose and its performance is compared to that of an elementary Sliding Mode Controller (SMC). Simulation results show that the PID contoller meets all mission design requirements, including the limits on the orbital Semi-Major Axis (SMA) error for which the SMC is not as successful.

Nomenclature SMA = semi-major axis Ft = thruster force Iii = moment of inertia in the i-axis α = direction cosine matrix β = radial firing half-angle θ = yaw rotation angle θn = nutation angle φ = roll rotation angle ψ = pitch rotation angle ωi = rotation rate in the i-direction

I. Introduction HEsa

lliptic

Magnetospheric MultiScale mission (MMS), scheduled to launch in 2014, is to consist of a group of four tellites which will maintain a tetrahedron formation (the “constellation”) throughout high-altitude and highly

e orbits. These spacecraft are hosts to very large, thin antennae which extend from the positive and negative z-axis. In order to ensure optimum functionality of these antennae, mission requirements dictate that the spacecraft spin about their respective body z-axes at a fixed rate (“two-axis” or “spin” stabilized). Although spin-stabilized craft are less susceptible to torque disturbances, the benefit of stabilization is accompanied by the difficult task of orbital maneuvering. Since each satellite spins about its body z-axis, the system is considered to have only two degrees of freedom (radial and axial). In addition, any control thrust actuated in the radial direction (i.e. perpendicular to the axis of rotation) to correct for position and velocity errors is required to be aligned with the

T

1 Mechanical Engineer, AIAA Student Member 2 Professor of Mechanical Engineering, Kingsbury Hall, University of New Hampshire, AIAA Member

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AIAA/AAS Astrodynamics Specialist Conference and Exhibit18 - 21 August 2008, Honolulu, Hawaii

AIAA 2008-7084

Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

“true” direction of error, lest the control thrust propel the spacecraft in an unintended direction. The MMS mission success, with regards to the feedback control design, is measured by the level of Semi-Major Axis (SMA) error, along with the measured fuel efficiency. The objective of this paper is to show that an individual MMS spacecraft can be precisely controlled throughout an orbital transfer maneuver such that the errors in both the final orbit’s semi-major axis and in the spacecraft’s command Delta-V are within design specifications and that, at the same time, at least 90% fuel efficiency is maintained throughout the maneuver. This paper also investigates the use of two types of controllers for this application— a PID controller and an elementary Sliding Mode Controller (SMC)—and compares the results of both control methods.

II. Spacecraft System Model In this research, a high fidelity model of the spacecraft dynamics is required. To that end, the three major design

challenges to overcome developing this system model are: real-time coordinate transformations, accurate thruster modeling, and noise and disturbance models.

A. Coordinate Transformations Without loss of generality, this research uses Euler Angle transformations for all body frame rotations. All

rotation sequences are the standard 3-1-3 sequence and, as such, the coordinate transformation matrix is:

(1) ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−+−−−+−

=−−

)(cos)cossin()sin(sin)sin(cos)coscoscossinsin()sincoscoscossin()sin(sin)coscossinsin(cos)sincossincos(cos

313

θψθψθθφψθφψφψθφψφθφψθφψφψθφψφ

α

and the resulting attitude kinematic equations are:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

z

y

x

ωωω

θθφθφθφθφ

φφ

θφθψ

sincoscoscossin0sinsinsincos0cossin

sin1

&

&

& (2)

It is assumed that the MMS spacecraft experiences nutation error and, therefore, all body rates are assumed to be non-zero. All inputs to the spacecraft system are described in terms of a Delta-V vector, originally defined in the orbit-defined system (OCS, shown in Fig. 1 as xo, yo and zo), with a magnitude of 1m/s in all of the in-track, cross-track and orbit-normal directions. This vector, first converted to the Earth inertial frame and then subsequently converted into the spacecraft body frame, is derived from the corresponding spacecraft attitude. All error vectors used for feedback control, first converted from body Cartesian coordinates to body cylindrical coordinates, are represented by the final error vector shown in Eq. 3.

EarthXI

xo

xi

x

ZI

YI

z

y

zi

yi

yozo

EarthXI

xo

xi

x

ZI

YI

z

y

zi

yi

yozo

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⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

avel

rvel

apos

rpos

eeee

e

,

,

,

,

v

Figure 1. Satellite-, Earth-, and orbit-based coordinate systems. The coordinate system used primarily by this research is the orbit-defined coordinate system (xo, yo, zo) in which the x-axis is tangential to the orbit, the z-axis is orbit-normal, and the y-axis completes the orthogonal triad.

(3)

The subscripts r and a represent radial errors and axial errors, respectively. Once converted from Cartesian to cylindrical coordinates, the angle θ is used to synchronoize the radial thrusts appropriately, explained in Section II.B.

B. Thruster Model As stated, this research considers only the two-degree-of-freedom problem of orbital control. As such, there is no

consideration of attitude control or torque adjustments. The actual MMS spacecraft are expected to be equipped with twelve monopropellant hydrazine thrusters arranged along the ± z-axis and ± y-axis of the body frame. However, for ease of implementation and without loss of generality, it is assumed that the radial thrusters are, instead, aligned with the ± body x-axis (or, in cylindrical coordinates, aligned with the angles θ = 0° and 180°). Also, this research further simplifies the thrusters to assume a single thruster in each direction. Although this is a forgiving assumption in terms of potential thruster perturbations, it is not overly simplified as to be inaccurate3.

A crucial aspect of this type of thruster is that it is a binary (or “bang-bang”) type actuator which is either fully “on” or switched “off” and, as a result, is capable of only a single level of output. The resulting thruster model used in this study, therefore, is such that:

, (4) ⎩⎨⎧

<−>+

=0,0,

uFuF

Ft

tt

where Ft is the thruster output force and u is the desired control input value.

Because of the axial spin of the MMS spacecraft, radial thrusts must be accurately synchronized in order to ensure that the craft is propelled in the appropriate direction. To do this, the system model must actively track and calculate the difference (denoted as angle θ) between the error vector and the time-varying direction of the radial thrusters, as located with respect to the spacecraft body. Once θ decreases to within a desired burn-angle window (denoted as ±β), the radial thrusters are only then permitted to engage. To achieve this synchronization, a “tangent” function is used that repeats twice per cycle and is symmetric about θ = 180°. As Fig. 2 shows, the value tan(β) is user-defined and acts as a threshold such that all angles between ±β enable thruster firing.

C. Noise and Disturbance Models

2β

| |

firing threshold

resulting thrust profile

2β

| |

2β

| |

2β2β

| |

firing threshold

resulting thrust profile

firing threshold

resulting thrust profile

Figure 2. Radial thruster timing method - |tan(θ)| is the input to the synchronization algorithm and |tan(2β)| provides the user-defined threshold, below which the radial thruster is allowed to engage. The resulting thrust profile when using this firing mechanism is shown in blue.

Based upon current accelerometer models considered for implementation aboard the MMS spacecraft, a noise model is developed Dean C. Tsai (Aerospace Engineering at the Flight Dynamics Analysis Branch of the NASA Goddard Space Flight Center) using standard Butterworth filters. This noise model contains three frequency spectrums, the highest being between 500Hz and 10GHz. This presents a problem for large quantities of simulations since the simulated model must run at (ideally) ten times the largest frequency value. For the purposes of this research, a simplified accelerometer noise model is developed based upon Tsai’s model. The simplified model spans the same frequency ranges and is more conservative than that of Tsai’s model. A comparison of the two noise models can be seen in Fig. 3, where the noise

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power spectral density (based upon Welch’s method) is shown.

Figure 3. Comparison of accelerometer noise models. The Goddard

accelerometer noise model, shown in gray, is compared to a simplified noise model, shown in black, using Welch’s method.

The disturbances included in this research take into account spacecraft nutation, thruster bias and thruster

misalignment. Nutation is modeled as a secondary body spin. The MMS design requirements specify a 0.35° nutation angle, θn. Using this requirement, the secondary spin rate can be calculated by using the ratio of spin-axis angular momentum to non-spin-axis angular momentum:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⋅

⋅+⋅==

−

−−

zzz

yyyxxx

axisspin

axisspinnonn I

II

LL

ω

ωωθ

22 )()()tan( (5)

where Iii is the moment of inertia in the i-axis and ωi is the spin rate in the i-axis (for i=x, y, z). Using small angle approximations and assuming only one secondary spin rate, the above relationship is simplified to:

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅

≈zzz

xxxn I

Iωωθ (6)

With the given mission parameters of

Ixx = 2379 kg m2

Iyy= 4201 kg m2

θn = 0.35°= 0.061 rad,

calculations show that ωx is 1.11% of the nominal spin rate ωz. Thruster bias is included in the spacecraft model to take into account “hot” or “cold” thruster firing output. This

disturbance is included as a percentage increase/decrease with respect to the nominal thruster output. MMS design standards specify a 5% bias for simulations. For the purposes of this research, all biases are assumed to be negative (i.e. the thrusters are uniformly firing “cold”). With a more sophisticated thruster profile, there is a potential for thruster biases to cause undesired changes in spin rate or even nutation. The former is not considered in this research, and the latter has been included, thus making this method of bias modeling sufficient.

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The last disturbance included in this research is thruster misalignment. MMS design standards specify a 1°

misalignment angle between thrusters. For the purposes of this research, all misalignment angles are between radial and axial thrusters; no radial-radial or axial-axial misalignment angles are considered. It has been determined that the conservative scenario occurs when the radial thrusters are misaligned towards the axial direction. As such, this is the misalignment scenario incorporated in this research.

It should be noted that, while not truly a disturbance, there is one more consideration that must be taken into account when modeling the radial thrusters. This consideration is the fact that, since the spacecraft is spinning in the radial plane, any radial thrusts are almost never directly collinear with the intended thrust direction. Accordingly, the simulations in this work take into account the “sine” and “cosine” components of the radial thrust, ensuring that the “sine” component of the radial thrusts are implemented perpendicular to the intended thrust direction and the “cosine” component are in the direction of the intended control actuation.

III. PID Controller

A. Design A simple PID controller is first designed of for the MMS spacecraft system, using position and velocity errors in

both the radial and axial directions for control feedback such that KP, KI, and KD . As previously mentioned, the primary criterion for maneuver success is measured by the accuracy of the attained orbital semi-major axis (SMA) of each satellite. Controlling or monitoring the changes in SMA over time ensures that each satellite maintains its target orbital period. The initial gains are chosen based upon the relationship between the change in the orbit’s semi-major axis and the changes in the spacecraft body’s position and velocity errors. This relationship, shown in Eq. 7, is derived from the energy balance equation (Newton’s Vis Viva equation) and is with respect to changes in only the in-track direction, as defined by the orbit.

42×ℜ∈

(7) vrSMA Δ×+Δ×=Δ − )10115.9()10157.6( 31

It is important to note that Eq. 7 does not describe a general orbital relationship, but rather depicts an indicative orbit specific to the MMS mission. The above relationship suggests a basis for the order of magnitude of the gains on position and velocity errors. These initial values are further tuned until the final gain matrices are obtained, as shown in Table 1.

Upon finalizing the PID control design by

incorporating the accelerometer noise model and all disturbances (as discussed previously), simulations show a large amount of chatter occurring towards the end of the maneuver, causing the controller to continually overcompensate and thus compromise the intended fficiency goal of the controller design. This problem is result of the spacecraft nutation, causing the body-

frame position and rate errors to oscillate (even when there are no true inertial errors). The noise within the

accelerometer model also exacerbates this phenomenon. To avoid overcompensation, a relay model with a hysteresis band is included directly before the thruster model, allowing the controller to ignore all error signals within a certain band. This hysteresis band is then adjusted until the perceived errors (from nutation) and fluctuation (due to noise) are mitigated, while still allowing the controller to maintain overall accuracy.

ea

Table 1. Final PID gain values.

radial axial radial axial0 0 10000 00 0 0 100000 0 0.15 00 0 0 0.150 0 0 00 0 0 0

Velocity Errors

P

I

D

Position Errors

B. Results Two specific test scenarios are performed to determine the accuracy and robustness of the PID controller. For

each study, the results for these scenarios are shown by way of three figures and a summary table of results. The first figure shows plots of the inertial errors (position and velocity) and body-frame accumulated Delta-V (which can

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be thought of as fuel usage). The table following shows the major results from that case study. The next figure shows the behavior of the semi-major axis over the course of the maneuver. The final figure shows the thrust profiles for both the radial and axial thrusters.

The first study is referred to as the “ideal” case scenario. For this study there are no nutation errors, thruster biases, or thruster misalignments present. Accelerometer noise is included, however, along with the sine and cosine components of the radial thrusts, as mentioned in Section II.C. Figure 5 shows the X- Y- and Z-errors as seen from the Earth inertial frame. Note that although the velocity errors tend towards zero (upper right plot), there exists some lingering positional drift at the end of the orbital maneuver (upper left plot). The total accumulated Delta-V, shown in the bottom figure, is close to the targeted values. Table 2 shows that the velocity errors fall within 1% of total, the semi-major axis error is within 50 meters of zero, and the total accumulated Delta-V is within 10% of the intended value, thus all design criteria are successfully satisfied.

Figure 5. Position, velocity and Delta-V results for the ideal case using a PID controller. The plot in the upper

right shows rapid and accurate velocity corrections. The bottom figure shows little wasted Delta-V.

Note that in the bottom figure, depicting total accumulated Delta-V, the blue dash-dot line represents the total Delta-V accumulated by the radial thrusters, and the solid blue line shows the total Delta-V as seen in the direction of the error. The difference between the two is referred to as the “sine losses” and amounts to about 5% wasted fuel in the radial direction.

Table 2. Results from the PID ideal scenario.

Original SMA Error: 8667.2mFinal SMA Error: 9.6mFuel Efficiency:

Radial: 95.27%Axial: 93.58%Total: 95.22%

Resultant Velocity Error: 0.09%

Simulation Results:IDEAL

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The behavior of the semi-major axis over the course of this maneuver is shown in Fig 6. The thrust profile for the radial and axial thrusters is shown in Fig. 7. Note that in Fig. 7, there is a long pause in the axial control output. This pause is a design input to limit the amount of mid-transfer corrections that are performed in the axial direction. In this way, the axial thrusters are able to make final adjustments towards the end of the maneuver and thereby conserve fuel.

Figure 6: Semi-major axis error over time for the PID ideal case scenario.

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Figure 7: Radial (top) and axial (bottom) thrust profiles over time for the PID ideal case scenario.

The next results presented here are from the worst case scenario, in which all disturbances are included. Figure 8 shows the resulting errors and the Delta-V accumulated over the course of the maneuver. Table 3 summarizes the results. From these simulations, it can be seen that the PID controller shows good performance despite the presence of disturbances.

Figures 9 and 10 depict the behavior of the SMA error and the radial and axial thrust profiles, respectively, over the course of the maneuver. The thrust profile reveals that the disturbances demand more corrections towards the end of the maneuver. Otherwise, there is no significant difference between the two cases.

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Figure 8. Position, velocity and Delta-V results for the worst case using a PID controller. Although still accurate in terms of velocity correction, the worst case scenario demands more Delta-V from this linear

controller to account for the disturbances.

Table 3. Results from the PID worst case scenario

8667.2m49.88m

95.03%90.90%94.89%0.74%Resultant Velocity Error:

WORST CASESimulation Results:

Fuel Efficiency:Radial:Axial:Total:

Original SMA Error:Final SMA Error:

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Figure 9. Semi-major axis error over time for the PID worst case scenario.

Figure 10. Radial (top) and axial (bottom) thrust profiles over time for the PID worst case scenario.

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IV. Nonlinear Controller In an effort to compare the PID controller described earlier to a more complex design, a simple nonlinear

controller is implemented and tested in the same simulation environments as that of the PID controller to determine whether or not using a more complicated nonlinear controller could improve overall error resolution and possibly fuel efficiency for the Delta-V maneuver and without sacrificing the ease of implementation of the control design.

A. Sliding Mode Control Design The nonlinear controller used in this research is a elementary form of the Sliding Mode Controller (SMC). The

SMC is typically robust against bounded modeling uncertainties and disturbances. The development of a sliding mode control begins with the creation of a sliding surface. This surface, if stable and attractive, will force the error trajectories to zero. Weighting parameters λi are assigned to the state errors to create the sliding surface s:

)()( 21 arar eeees && +++= λλ (8) where er and ea are the errors of the radial and axial position, respectively, and and are the errors of the radial and axial velocity, respectively5. In an effort to make this SMC analogous to the previous PID controller (for appropriate comparison), the magnitudes of the weighting values are selected to be the same as that of the PID control gains. The magnitudes of these weighting values are shown in Table 4.

re & ae &

Table 4. SMC weighting parameters.

Weight Valueλ1 0.15λ2 10000

Once the sliding surface has been created using Eq. 8 and the gains from Table 4, the surface function s then

becomes the input to a switching function 1(s), which is multiplied by a “robustifier” term K. The purpose of the switching function and the gain is to nullify the effects of chattering that may occur using this type of control5. In keeping with the idea of using an SMC analog to the previous PID control, the switching function 1(s) is chosen to be the signum (relay) function with the addition of a hysteresis band. The gain K is selected to be simply 1. It should be noted here that a more sophisticated SMC could be used for comparative purposes. For this study, however, a simpler SMC is chosen for preliminary comparative purposes.

B. Results As in Section III.B, the results are shown by way of three figures, the first being subdivided into three subplots,

and a table of results. Also, as before, there are two tests performed: the first is the best-case scenario (no added disturbances) and the second is the worst case (all disturbances included).

Figure 11 shows the three subplots, detailing the inertial position and velocity errors and the body-frame Delta-V. From these results it can be seen that the overall error resolution is acceptable, with values falling under the desired thresholds, including that of fuel efficiency. However, comparing the results listed in Table 5 with those in Table 2 and weighing the importance of the SMA error over that of fuel efficiency, the PID controller has slightly better overall results. Note again that the SMC used in this comparison is designed to be analogous to the PID controller and is, by no means, that of an optimal design.

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Figure 11. Position, velocity and Delta-V results for the ideal case using a sliding mode controller. The position errors, velocity errors, and Delta-V used behave similarly to those shown in Fig. 5.

Table 5: Results from the SMC ideal case scenario.

Original SMA Error: 8667.2mFinal SMA Error: 20.3mFuel Efficiency:

Radial: 94.96%Axial: 100.00%Total: 95.11%

Resultant Velocity Error: 0.39%

Simulation Results:IDEAL

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Figures 12 and 13 show the behavior of the semi-major axis error and the thrust profiles, respectively, over the course of the maneuver.

Figure 12. Semi-major axis error over time for the SMC ideal case scenario

Figure 13. Radial (top) and axial (bottom) thrust profiles over time for the SMC ideal case scenario

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The next series of results are from the SMC in worst case scenario. Figure 14 shows the subplots that depict the inertial position and velocity errors over time, as well as the accumulated body-frame Delta-V. From Table 6, it can be seen that although the velocity error and fuel efficiency thresholds are satisfied, the controller (without further tuning) fails to bring the semi-major axis error to within 50 meters. Figure 15 shows the behavior of the SMA error over time. Figure 16 shows the thrust profiles of the radial and axial thrusters.

Figure 14. Position, velocity and Delta-V results for the SMC worst case. The velocity and Delta-V subplots reveal that, without further tuning,the result is a less accurate and less efficient controller than the PID.

Table 6. Results from the SMC worst case scenario.

Original SMA Error: 8667.2mFinal SMA Error: 96.37mFuel Efficiency:

Radial: 95.03%Axial: 90.90%Total: 94.89%

Resultant Velocity Error: 0.74%

WORST CASE SMASimulation Results:

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Figure 15. Semi-major axis error over time for the SMC worst case scenario

Figure 16. Radial (top) and axial (bottom) thrust profiles over time for the SMC worst case scenario.

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V. Conclusions and Future Work Although both controllers perform adequately, the results from the two tested scenarios indicate that the PID

controller is more capable of controlling the errors while mitigating the effects of various disturbances. It should be noted again that the SMC presented in this paper is not an optimal controller, but an analog to the PID controller to determine if a simple SMC would improve upon the results of the PID controller without compromising the ease of implementation that the PID is noted for. Because of the inclusion of the relay and hysteresis band in the PID controller design, the PID controller, ironically, mimics the action of a typical SMC. In fact, it is possible to argue that the PID controller is no longer a true linear PID controller. What is certain, however, is that without the presence of a hysteresis band, the errors due to nutation and the noise from the accelerometer model would corrupt the input to any controller, linear or nonlinear, such that all error resolution is poor and all fuel efficiency is well below the desire threshold.

Given the results on hand, a PID controller with the added relay shows to be a capable controller in terms of error resolution. Although the efficiency of the SMC is better in the worst case scenario than the PID controller, the SMA criterion, the measure of the success of the feedback control system, is not satisfied without further design of the SMC. As a result, the SMC’s improved fuel efficiency becomes irrelevant. It is, therefore, the authors’ opinion that the PID controller be further considered for use as a viable method of feedback orbit transfer control. The authors also suggest that further research be pursued regarding optimal nonlinear controllers (Sliding Mode or otherwise) to test the efficacy of other controllers before any further conclusions be drawn.

In addition, more simulations should be performed to qualify the controllers not only in terms of final results but also with regard to initial spacecraft attitude orientation 1. Future work also involves increasing the spacecraft model fidelity by incorporating additional sensor and actuator dynamics. Finally, the work in this research is to be augmentated with the work done by Mushaweh2, which investigates the potential for active attitude and nutation control aboard the MMS spacecraft.

Acknowledgments The authors would like to acknowledge the invaluable assistance of the aerospace engineers at the Flight

Dynamics Analysis Branch at the NASA Goddard Space Flight Center, especially Josephine K. San and Dean C. Tsai. Additional thanks go to both the NASA Goddard Space Flight Center and the New Hampshire NASA Space Grant Consortium for funding of this research.

References 1Borrelli, M. J., “An Autonomous Orbit Transfer Controller for the NASA Magnetospheric MultiScale Satellite,” Master’s Thesis, Mechanical Engineering Dept., Univ. of New Hampshire, Durham, NH, 2008. 2Mushaweh, N., “An Observer-based Attitude and Nutation Control and Flexible Dynamic Analysis for the NASA Magnetospheric MultiScale Mission,” Master’s Thesis, Mechanical Engineering Dept., Univ. of New Hampshire, Durham, NH, 2007. 3Noll, R. B. et al., “Spacecraft Attitude Control during Thrusting Maneuvers,” NASA, Washington, D.C., 1971.

4Vallado, D. A., Fundamentals of Astrodynamics and Applications, Microcosm Press, El Segundo, CA, 2001. 5Slotine, J.-J. E. and Li, W., Applied Nonlinear Dynamics, Prentice-Hall, Upper Saddle River, NJ, 1991.