Comparative Observer-Based Nutation Control Techniques

for NASA Magnetospheric Multiscale (MMS) Mission

Spacecraft

Neil Mushaweh∗

Raytheon Corporation

May-Win L. Thein†

University of New Hampshire

Current research with the NASA Goddard Space Flight Center (GSFC) involves thedynamic modeling and control of the spacecraft of the NASA Magnetosphereic Multiscale(MMS) Mission, a Solar-Terrestrial Probe mission to study Earth’s magnetosphere. Fourobserver-based attitude and nutation controllers are designed and evaluated to determinethe comparative performance of each of the feedback control systems as it applies to theMMS mission.

Linear (not shown in this paper) and nonlinear observers are developed through sim-ulations to estimate satellite attitude and angular body rates without the use of ratesensors. After observing that linear estimation techniques do not satisfy mission designrequirements, linear and nonlinear (Sliding Mode Control) techniques are implementedin conjunction with only the nonlinear observers to complete the observer-based controlsystem.

The results of this research show that, of the methods analyzed, both the ExtendedKalman Filter and Sliding Mode Observer implemented with Sliding Mode Control yieldthe most satisfactory performance out of the systems studied. Both observer-based controlsystems satisfy NASA design requirements while reducing thruster control effort and re-ducing the effects of measurement noise and spacecraft uncertainties/disturbances. Moreinvestigation, however, is needed to verify performance of the proposed observer-basedcontrol system over all possible ranges of mission operation.

I. Introduction

The NASA Magnetospheric Multiscale satellite constellation is scheduled for launch in 2014. Eachsatellite is composed of six instrumentation booms reaching up to 50 meters in length that will be used tocollect astrophysical data. Once operational, the four-satellite, tetrahedron formation will provide a threedimensional understanding of the Earths magnetosphere, small scale plasma processes and other astro-physical phenomena. The mission will occur in three stages, each which require large orbital maneuvers.These large orbital transfers, coupled with the high sensitivity of the instruments require the satellite tomaintain a constant spin about its z-axis only. Mass imbalances, external torques and other unknowndisturbances will cause the satellite to nutate about its x and y axis undesirably. In order to effectively

∗Engineer, neil.mushaweh@gmail.com†Professor of Mechanical Engineering, mthein@cisunix.unh.edu, AIAA member

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reject all satellite nutation, while maintaining a constant spin, it is essential that the satellite angular bodyrates are accurately known.

This paper investigates the feasibility of an estimation algorithm that is capable of determining space-craft angular rates with only star tracker attitude measurements. The application of such estimationcapabilities has implications on designs that avoid costly angular rate measurement systems while main-taining highly accurate state estimates. These body rate estimates will then be used in a control algorithmthat will utilize thrusters to reject satellite spin nutation. Several estimation and control techniques areexplored and compared to deliver the most effective results given NASA design requirements. A SlidingMode Observer (SMO) is compared with an Extended Kalman Filter (EKF) for estimation of spacecraftbody rates, while an optimal controller and Sliding Mode Controller are investigated for nutation rejec-tion. A significant amounts of research has been performed in the area of state reconstruction withoutthe aide of angular rate measurements. The goal of this paper is not to introduce new observer-basedcontrol algorithms. On the contrary, the purpose is to compare the performances of specific estimationand observer-based control techniques for use in NASA MMS spin-stabilized spacecraft.

The SMO is noted for robustness to modeling uncertainty and unknown disturbances. The SMO is anextension of the Sliding Mode Controller, where the estimation error trajectory, rather then the controlerror trajectory, is made to converge to zero. Applications of variable structure estimation and controlinclude Luk’yanov 1 who used an SMO to control a spacecraft while excluding rate estimates. Misawa hasalso performed extensive research in the field of nonlinear estimation and sliding mode observers ,2 .3

The Extended Kalman Filter (EKF), a nonlinear extension of the Kalman filter, has been used bymany researchers to develop algorithms that use one set of sensor data to obtain full state and inputknowledge. Psiaki 4 used magnetometor vector measurements to determine attitude, angular rates, andexternal torques while Gai 5 used star sensor measurements to determine attitude and angular rates. Inanother application of nonlinear filters, Markley and Crassidis 6 used a feed-forward predictive estimatorto determine attitude, rate and model error trajectories without gyroscopic sensors.

Work done by Grasshoff 7 and Lin 8 explore nutation control through accelerometer measurements andthruster actuation. However, neither offer a comprehensive comparison of nonlinear and linear observerbased control systems. Wilson 9 also presents a bang-bang control method for spin reduction and stabi-lization of space vehicles using gas jets. This prior research serves as a reference in the development andcomparison of an observer based controller for the NASA MMS mission.

This paper presents a comparative analysis between several observer-based nutation controllers designedand analyzed specifically for the purposes of the NASA MMS mission. For the full state estimation method,a feasibility analysis is performed to determine whether nutation control can be achieved without the use ofrate gyros for body angular rate measurements. In addition, a nutation controller is developed, comparinga linear optimal control technique to that of the Sliding Mode Controller (SMC).

In this paper the Sliding Mode Observer (SMO) and the Extended Kalman Filter (EKF) are briefly pre-sented. Next, the simulation parameters are presented followed by results for each of the state estimationtechniques that are analyzed for performance and robustness. This analysis is followed by the compara-tive analysis of the implementation of each of the techniques into an observer-based nutation controllerdeveloped for MMS spacecraft. The observer-based control simulation results and analysis are followed byconclusions and future work.

II. Spacecraft Attitude Kinematics and Rigid Body Dynamics

A rigid spacecraft is assumed in this work. Euler angle angle representation is used to define attitude.A 3-2-1 rotation sequence with rotation angles ψ, θ, and φ, respectively, is applied. The resulting Euler

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angle rates (ψ, θ, φ) for a rigid spacecraft, then, are as follows:

ψ = ωx + (ωz cosψ + ωy sinψ) tan θ

θ = ωy cosψ − ωz sinψ

φ = (ωz cosψ + ωy sinψ) secθ (1)

The spacecraft dynamics are described by Euler’s moment equations, succinctly written as:

ω(t) = I−1 [ M(t) − ω(t) × I ω(t) ] (2)

where

I =

Ix −Ixy −Ixz

−Iyx Iy −Iyz

−Izx −Izy Iz

(3)

Here, M(t) represents the input thruster torque as determined by the observer-based (full state feedback)nutation controller. In addition, the spacecraft body misalignments are taken to account by the presenceof non-zero cross products of inertia. The true sixth order spacecraft dynamic model, therefore, is that ofEquation (1) augmented with Equation (2).

For this study, the observers (both the SMO and EKF) incorrectly assume that the spacecraft bodyframe coincides with the principle axes of inertia. That is, the observer model for spacecraft inertia is givenas I such that

I =

Ix 0 0

0 Iy 0

0 0 Iz

(4)

Therefore, from Equations (1) and (2) and the observer model, the true spacecraft dynamics may berepresented as

x(t) = f(x(t), t) +B [ u(t) + ∆u(t) ] + ∆f(x2(t), t)[

x1(t)

x2(t)

]

=

[

f1(x(t), t)

f2(x2(t), t)

]

+

[

0

B2 + ∆B2

]

[ u(t) + ∆u(t) ] +

[

0

∆f2(x2(t), t)

]

y(t) = x1(t) + ν(t) (5)

where x1(t) = [ ψ, θ, φ ]T , x2(t) = [ ωx, ωy, ωz ]T , u(t) = M(t), and

f1(x(t), t) =

ωx + (ωz cosψ + ωy sinψ) tan θ

(ωz cosψ + ωy sinψ) secθ

ωy cosψ − ωz sinψ

f2(x2(t), t) = − I−1[ ω(t) × I ω(t) ]

B2 = I−1 (6)

and

∆f2(x2(t), t) = − I−1[ ω(t) × I ω(t) ] + I−1[ ω(t) × I ω(t) ]

∆B2 = I−1 − I−1 (7)

In addition, ν(t) represents bounded sensor noise, and ∆u(t) represents added disturbance torques.

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Here, body frame misalignment (non-zero cross products of inertia), thruster error, and external dis-turbance torques represent matched uncertainties (via u(t)) and unmatched uncertainties (i.e., parametricuncertainties and lumped modeling errors). These modeling errors/uncertainties may be lumped into asingle term Γ(t) such that

x(t) =

[

f1(x(t), t)

f2(x2(t), t)

]

+

[

0

B2

]

u(t) +

[

0

Γ(t)

]

(8)

whereΓ(t) = ∆f(x2(t), t) + ∆B2[ u(t) + ∆u(t) ] + B2∆u(t) (9)

III. Nonlinear Estimation Methods

In many applications, system dynamics can be highly nonlinear and in most cases, linear observers areinsufficient in estimating nonlinear states, as is the case in this study (not shown here).

A. Extended Kalman Filter

When using the Kalman filter, nonlinear system equations are linearized off-line about a predeterminedstate vector and the Luenberger gain is then calculated. The Extended Kalman Filter (EKF) involvesupdating the linearized system and linearized measurement model at each estimation step. These lin-earizations, or Jacobian matrices, are updated with each state estimate vector, rather then off-line withpredetermined equilibrium points as with the linear Kalman filter. This linearized update makes the EKFmore effective for systems with highly nonlinear dynamics (for the formulation below, the measurementmodel is assumed linear). For a given nonlinear system the state estimation model is defined as (Thereader is referred to 10 for further mathematical background on the EKF):

˙x(t) = f(x(t), t) +K(t)[y(t) − h(x(t), t)] (10)

where y(t) = h(x(t), t) + z(t) is sensor outputs (h(x(t), t)) and measurement noise (v(t)).The gain matrix (K(t)) calculation begins with propagating the estimation error covariance matrix

equation given by:

P(t) = F (x(t), t)P (t) + P (t)F T (x(t), t) +Q(t) − P (t)HTR−1(t)HP (t) (11)

where the linearization of the nonlinear system model about each estimate is given by:

F (x(t), t) =∂f(x(t), t)

∂x(t)|x(t)=x,(t) (12)

and Q(t) and R(t) are weight matrices selected based on process noise and measurement noise respectively.The gain matrix is calculated by:

K(t) = P (t)HTR−1(t) (13)

The error covariance equation, Eq. (11), and state matrix are then updated using the gain matrix andmeasurement error.

The EKF offers exceptional results when it comes to measurement noise rejection for highly nonlinearsystem models. However, the inability to guarantee closed loop system stability, as with the linear Kalmanfilter, is one of the drawbacks of the EKF. Also, since the calculation of the gain matrix is dependentupon the Jacobian of the nonlinear system equations, a highly accurate system model is necessary and,

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thus, makes the EKF less robust to parametric or modeling uncertainties (unless the uncertainties can bewell-characterized as process noise). Finally, the evaluation of complex matrix operations such as inversesin Eq. (11) that must be calculated at each time step, make the EKF less desirable for applications withstrict computational constraints.

B. Sliding Mode Observer

The Sliding Mode Observer is another nonlinear state estimation technique. Unlike the EKF, the SMOrequires no linearization about any operating points during estimation. This makes the technique desirablein that the state equations are not being simplified in order to estimate states.

The SMO is very similar to the SMC. Like the SMC, the SMO also utilizes a switching function andsliding surfaces to force the error trajectory to zero. For SMO development the observer dynamics is afunction of estimated states, x, and is defined as:

˙x = f(x(t), t) + Bu(t) +Hz +K1s (14a)

z = Cx (14b)

where the estimation error signal is defined as z = z− z. Gain matrices Hǫℜn×m and Kǫℜn×p are selectedthrough design iteration where n, m and p represent the number of states, measurements and slidingsurfaces, respectively. The switching function, as defined in the specific SMC used in this application, is asaturation function such that:

1s = sat

(

s

ρ

)

(15)

As with the SMC, the number of sliding surfaces and how they are defined are both part of the observerdesign process. The boundary layer and gain matrices are also defined through design and iteration.

For a more complete development of Sliding Mode Observers and Nonlinear Systems the reader isreferred to2 and.3

IV. Attitude and Body Rate Estimation

In this section of the paper, the Extended Kalman Filter (EKF) and Sliding Mode Observer (SMO)are both implemented in the MMS spacecraft attitude dynamics to compare the relative performancesof the two estimation techniques for providing full state estimates for nutation control. Both estimationtechniques use the dynamic model provided in Equations (1) and (2), except that the modeling errors anduncertainties discussed in the previous section are not taken into account. That is, the observer modeldynamics assumes that the body frame lies along the principal axes of inertia (i.e., all cross product inertialterms are zero) and that there are no parametric uncertainties or input disturbance torques, such that:

x(t) = f(x(t), t) +B u(t)

y(t) = x1(t) (16)

where, using Equation (4), f2(x(t), t) simplifies to:

f2(x(t), t) =

((Iy − Iz)/Ix)ωyωz

((Iz − Ix)/Iy)ωxωz

((Ix − Iy)/Iz)ωxωy

(17)

and the only thing assumed known about measurement noise ν are the sensor noise bounds.

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Simulation parameters for the Sliding Mode Observer and EKF are listed in Table 1. An externalmoment of ∆u(t) = 0.001 sin(.3t) N-m is added to all three axes to simulate unknown disturbances as wellas a 10 percent uncertainty in each of the principle moments of inertia.

Actual System Observer

I.C.: ψ, θ, φ (rad) 0, 0, 0 0, 0, 0

I.C.: ωx, ωy, ωz (rad/s) 0.01, 0.01, 0.3 0,0,0

Inertia Matrix (kg ·m2)

8402.64 −58.8 −44.6

−58.8 8411.97 −100

−44.66 −100 16414.66

8000 0 0

0 8600 0

0 0 17500

Noise bounds (rad) ±0.03 0

Table 1. Simulation Parameters

A. Extended Kalman Filter Results

EKF matrices are selected through partitioning. It is because of the unknown disturbances and para-metric uncertainty that the Euler moment equations describing ωx, ωy and ωz are most negatively affectedand, therefore, require Qβ to be several orders of magnitude greater then Qα. The relative magnitude ofthe two partitioned matrices that make up Q are first determined through tuning, followed by the indi-vidual tuning of the diagonal elements. Iteration and analysis of each simulation’s filtering and trackingeffectiveness determine the final matrices. The resulting Q and R weight matrices are:

Q =

0.1 0 0 0 0 0

0 0.1 0 0 0 0

0 0 0.1 0 0 0

0 0 0 120000 0 0

0 0 0 0 100000 0

0 0 0 0 0 130000

(18a)

R =

106 0 0

0 106 0

0 0 106

(18b)

To further improve estimation results, body rate state estimates in particular, the overall influence ofthe observer loop on body rate correction terms is investigated. It can be seen through inspection of theclosed-loop observer simulation diagram that the EKF is updated via Euler angle errors only, since thesestates are the only ones being measured. Body rate state estimate errors, for example, are not directlyobserved by the closed loop EKF, and therefore, cannot guarantee convergence to zero. By investigatingthe correction terms that affect the satellite body rates, and independently tuning those, the percentageerror and steady state error can be greatly improved. The gains are tuned through iterations starting withthe ωx correction term and then independently tuning for the ωy and ωz correction terms. Steady-stateerror is drastically reduced while maintaining effective noise filtering as gains increase. Slowly, as the gainsincrease, so does the noise that is being amplified by the gains. The final gain selections of 3.75, 1.9 and1 for ωx, ωy and ωz, respectively, are used to tune body rate correction terms. Figure 1 illustrates anexample of the improvement in body rate tracking for ωx with the additions of correction term tuning.Improvements are most relevant after 10 seconds of simulation when the estimator reaches steady state

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0 5 10 15 20 25 30 35

−0.02

−0.01

0

0.01

0.02

0.03

0.04

time(s)

omeg

a x

Body Rate (rad/s)

ActualEstimated tuningEstimated no tuning

Figure 1. EKF Correction Input Tuning Influence on Body Rates (ωx)

and begins to track the body rates. Since this tuning occurs about selected initial conditions, furtherinvestigation into the bounds of spacecraft operating ranges are needed to determine when the proposedcorrection input tuning is effective.

The Extended Kalman Filter offers a much more comprehensive and effective estimation technique,as compared to its linear counterpart. From inspection of the simulation results, it can be seen thatnoise filtering and body rate estimation are greatly improved by using the EKD. Figure 2 shows theEKF’s effectiveness at filtering measurement noise, and Figure3 shows that the EKF estimates body rateseffectively, especially given unknown disturbances and modeling uncertainties which typically affect thesestates the most. Although the error magnitude is small, a percent error is evaluated from the steady-stateamplitude of oscillations of all six states and their respective amplitude of stead-state error.

B. Sliding Mode Observer Results

The Sliding Mode Observer dynamics, using the dynamic model defined previously are:

˙x = f(x(t), t) +Bu(t) + L(Cx− Cx) +K1s

y = Cx (19)

where C = [ I3×3 03×3 ]. Observer gain matrices are represented by L and K in which L,K ∈ ℜ6×3. TheLuenberger gain, L, is an optimal Kalman gain determined from a linearzied system model and remainsconstant throughout the simulation. The Sliding Mode correction term ensures that the error trajectoryremains on the sliding surface and is selected through design iteration.

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0 10 20 30 40 50−0.01

0

0.01

0.02Euler Angle Estimation Error (rad)

psi

0 10 20 30 40 50−0.01

0

0.01

0.02

thet

a

0 10 20 30 40 50−0.5

0

0.5

1

phi

time (s)

Figure 2. Extended Kalman Filter Euler Angle Estimation Error

0 10 20 30 40 50−0.02

0

0.02Body Rate Estimation Error (rad/s)

omeg

a x

0 10 20 30 40 50−0.02

0

0.02

omeg

a y

0 10 20 30 40 50−0.5

0

0.5

time (s)

omeg

a z

Figure 3. Extended Kalman Filter Body Rate Estimation Error

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The sliding surface s is defined such that

s =

3∑

i=1

λi[ yi(t) − (Cx(t))i ]

= λ1zψ + λ1zθ + λ1zφ (20)

where z denotes the error between measured state and the corresponding estimated state, λi are slidingsurface weighting terms, and the switching function 1s is defined as:

1s = sat

(

s

ρ

)

(21)

Here, ρ is an additional SMO design parameter referred to as the sliding surface boundary layer.The Luenberger gain is adopted from a Kalman filter analysis for the linearized spacecraft dynamics

while the switching gain K is chosen as:

L =

1.6697 0 0

0 1.6697 0

0 0 1.6818

1.3939 −0.2388 0

0.2388 1.3939 0

0 0 1.4142

, K =

1

1

1

1

1

1

∗ 10−3 (22)

Figure 4. Sliding Mode Observer Correction Input Tuning

with the boundary layer chosen as ρ = .0006. As with the EKF, the correction terms for ωx, ωy, and ωzare tuned to improve estimates when gain tuning is exhausted as with the EKF design. Figure 4 illustratesthe gains that are used for correction input tuning.

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0 10 20 30 40 50 60 70

−2

0

2

4x 10

−3 Euler Angle Estimation Error (rad)

psi

0 10 20 30 40 50 60 70

−2

0

2

4x 10

−3

thet

a

0 10 20 30 40 50 60 70−0.05

0

0.05

0.1

time (s)

phi

Figure 5. Sliding Mode Observer Euler Angle Estimation Error

0 10 20 30 40 50−5

0

5

10x 10

−3 Body Rate Estimation Error (rad/s)

omeg

a x

0 20 40 60 80 100−5

0

5

10x 10

−3

omeg

a y

0 20 40 60 80 100

0

0.1

0.2

time (s)

omeg

a z

Figure 6. Sliding Mode Observer Body Rate Estimation Error

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Once again, it is important in recognizing overall percentage error between estimated and actual statesto determine the effectiveness of the observer and usefulness of the observed states. States with largelevels of error are ineffective when used to update a control algorithm. In the case of MMS spacecraft,the observer-based controller is less likely to be able to maintain constant spin and desired attitude whilerejecting nutation. Compared to the EKF, it can be seen that the SMO steady-state estimates have asmaller levels of error than that of the EKF. However, the SMO estimates appear slighly more susceptibleto measurement noise as seen in Figure 5. Euler angle estimation error results are shown in Figures 5and 6. Estimation errors are small for φ and ωz in comparison to the other states because the satelliteis rotating about its z axis at a constant rate, thus, making it easier for the estimator to estimate thesestates. Initial errors for both body rates and Euler angles are small for the SMO compared to the EKFestimates. It can also be seen from steady-state error analysis that the SMO is much more effective atestimating satellite states in the presence of unknown disturbance torques.

V. Observer Based Control Results

The first control method tested uses linear feedback to control the satellite body rate and orientation.The desired states are defined as follows:

xdes =

ψdes

θdes

ωxdes

ωydes

ωzdes

=

0.052

−0.052

0

0

0.3

(rad, rad/s) (23)

The final feedback gains and configuration involve a PD controller. Many combinations of proportional,integral, and derivative control presented in12 are tested for application. Below are the final gain selectionsbased on simulation iteration and tuning:

Kp =

300 50 20 1 0

50 300 1 20 0

0 0 0 0 20

, Kd = Kp ∗ 0.1 (24)

Results for linear control of satellite attitude and nutation using SMO and EKF estimates are shownin Figures 7 and 9 while body-rate control results are illustrated in Figure 8 and 10. The EKF, althoughoptimal in measurement noise filtering, does not offer acceptable results when implemented with a linearfeedback controller. The derivative feedback error signal results in excessive control noise that can causeactuator failure. Also, since the EKF is ineffective at estimating states with unknown disturbance torques,the control system is incapable of recognizing them in the error signal and acceptable control compensationdoes not occur.

Linear feedback control coupled with SMO state estimate feedback again results in control signals withsignificant noise. In addition, body rate control errors are at unacceptably high levels.

The second control technique investigated is the Sliding Mode Controller (SMC). As with linearfeedback control, SMC involves multiple design iterations and simulations to tune feedback gains andsliding surfaces to acquire an effective feedback control system.

In designing a SMC, as many as five and as few as one sliding surfaces are investigated for controlpurposes. These sliding surfaces include combinations of error summations, derivatives and integrals todetermine the most effective form of variable structure control. The selection of the sliding surfacesinfluenced the performance of the observer more then other design parameters. Five sliding surfaces are

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0 20 40 60 80 100 120−0.08

−0.06

−0.04

−0.02

0

Euler Angle Control Error (rad)

psi

0 20 40 60 80 100 120−0.02

0

0.02

0.04

0.06

0.08

time (s)

thet

a

Figure 7. SMO-Based Linear Controller: Euler Angle Control Error

0 20 40 60 80 100 120−0.1

−0.05

0

0.05Body Rate Control Error (rad/s)

omeg

a x

0 20 40 60 80 100 120−0.02

0

0.02

omeg

a y

0 20 40 60 80 100 120−0.5

0

0.5

time (s)

omeg

a z

Figure 8. SMO-Based Linear Controller: Body Rate Control Error

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0 50 100 150 200−0.5

0

0.5EKF Body Rate Control Error (rad/s)

omeg

a x

0 50 100 150 200−0.5

0

0.5

omeg

a y

0 50 100 150 200−0.5

0

0.5

time (s)

omeg

a z

Figure 9. EKF-Based Linear Controller: Euler Angle Control Error

0 50 100 150 200−0.15

−0.1

−0.05

0

0.05EKF Euler Angle Control Error (rad)

psi

0 50 100 150 200−0.05

0

0.05

0.1

0.15

time (s)

thet

a

Figure 10. EKF-Based Linear Controller: Body-Rate Control Error

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selected and defined as:

si = λ1zi + λ2

∫

zidt, i = 1, 2, 3, 4, 5 (25)

where zi is the error between actual and desired states. During the sliding surface selection process,weighting factors λ1 = 1 and λ2 = 0.5 are also determined to affect the overall influence of these errorsignals on the control signal. The two gain matrices and boundary layer are defined as:

Hsmc =

10 0 2 0 0

0 10 0 2 0

0 0 0 0 2

, Ksmc =

0.05 0 0.001 0 0

0 0.05 0 0.001 0

0 0 0 0 0.001

, ρ = 0.01 (26)

Results for Sliding Mode Control of satellite attitude and nutation using SMO and EKF estimates areshown in Figures 11 and 13 while body-rate control results are illustrated in Figure 12 and 14. The SMCappears to be more effective at rejecting nutation and maintaining attitude while filtering measurementnoise. Feedback control implementation with SMO also offers acceptable results based on NASA designcriteria.

Results indicate that although thrusters are not saturated from a linear control configuration, there is alarge transient demand placed on the thrusters. Sliding Mode Control effort, however, places a significantlyless demand on thrusters during control sequences.

0 20 40 60 80 100 120−0.06

−0.04

−0.02

0

0.02Euler Angle Control Error (rad)

psi

0 20 40 60 80 100 120−0.02

−0.01

0

0.01

0.02

thet

a

time (s)

Figure 11. SMO-Based SMC: Euler Angle Control Error

VI. Conclusions and Future Work

For attitude and body rate estimation with only star tracker attitude measurements available, twodifferent observers are explored. The Sliding Mode Observer (SMO) and Extended Kalman Filter (EKF)

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0 20 40 60 80 100 120−0.05

0

0.05Body Rate Control Error (rad/s)

omeg

a x

0 20 40 60 80 100 120−0.02

−0.01

0

0.01

omeg

a y

0 20 40 60 80 100 120−1

−0.5

0

0.5

time (s)

omeg

a z

Figure 12. SMO-Based SMC: Body Rate Control Error

0 20 40 60 80 100 120 140−0.2

0

0.2Body Rate Control Error (rad/s)

omeg

a x

0 20 40 60 80 100 120 140−0.2

0

0.2

omeg

a y

0 20 40 60 80 100 120 140−0.5

0

0.5

time (s)

omeg

a z

Figure 13. EKF-Based SMC: Euler Angle Control Error

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0 20 40 60 80 100 120 140−0.1

−0.05

0

0.05

0.1Euler Angle Control Error (rad)

psi

0 20 40 60 80 100 120 140−0.1

−0.05

0

0.05

0.1

thet

a

time (s)

Figure 14. EKF-Based SMC: Body Rate Control Error

offered acceptable results for attitude and body rate estimation given measurement noise, parametricuncertainties and disturbance torques. The results presented clearly illustrate the strengths and weaknessesof the EKF and SMO. The EKF does an exceptional job at filtering measurement noise for all six states.However, its weakness is in estimating satellite states with unknown disturbance torques and parametricuncertainties. Results illustrate a larger magnitude of oscillation error as compared to the SMO, dueto the inability of the EKF to effectively estimate satellite states, especially body rates which are moredirectly influenced by external torques. Conversely, the SMO has slightly more noise in the estimation errorsignals. However, the magnitude of error oscillation is reduced through effective estimation in the presenceof unknown disturbances. In either design, acceptable estimates of all six states, while only measuringsatellite orientation, allowed for progression to the control development.

Not only are feedback gains, sliding surfaces, and weight matrices tuned for the development of the SMOand EKF, but also observer correction inputs are investigated for their influence on observer performance.It is determined that by re-tuning the observer correction inputs, estimation results are improved for bodyrate estimation. Noise amplification is another design parameter that is monitored in tuning estimationcorrection inputs. Although there is a limit to these fixed, tuned gains because of their amplification ofnoise in the signals, they do vastly improve results. Further analysis of this type of tuning is necessarysince the gains are acquired for given initial conditions, noise characteristics, and unknown disturbances.Ranges of satellite operating conditions need to be determined for the given set of correction terms andtheir effectiveness for a wide range of system characteristics.

A PD controller and Sliding Mode Controller are both tested using state estimate feedback to determinethe overall most effective system for observer-based attitude and nutation control. The EKF, althoughoptimal in measurement noise filtering, does not offer acceptable results when implemented with a linearfeedback controller. The derivative feedback error signal results in excessive control noise that can causeactuator failure. Also, since the EKF is ineffective at estimating states with unknown disturbance torques,

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the control system is incapable of recognizing them in the error signal and acceptable control compensationdoes not occur.

The SMC is far more effective at rejecting nutation and maintaining attitude while filtering measure-ment noise. Feedback control implementation with SMO also offers acceptable results based on NASAdesign criteria. Linear feedback control again results in control signals with significant noise. However,when used for feedback into the SMC, the SMO’s robustness allows for effective rejection of unknowntorques and disturbances compared to the EKF.

Future work includes more rigorous simulations of all observer-based control techniques for all rangesof MMS operation. In addition, various other SMC and SMO designs are to be further developed andanalyzed. Ultimately, what ever observer-based nutation controller is deemed the most appropriate forthe MMS mission, this method is to be augmented with an orbital feedback controller for use in thespin-stablized MMS spacecraft.

Acknowledgments

The funding for this work was granted by NASA Goddard Space Flight Center’s MagnetosphericMultiScale (MMS) Mission through the Flight Dynamics Analysis Branch of the NASA Goddard SpaceFlight Center. Special acknowledgments go to Josephine San, Attitude and Control Systems Lead, andDean Tsai, Aerospace Engineer, both who played a key role in the progress of this work.

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