A Comparative Analsysis of Body-Rate Estimation

Techniques for the NASA Magnetospheric Multiscale

(MMS) Mission Spacecraft

Neil Mushaweh∗

Raytheon Corporation

Benjamin Jenkins†, Daniel A. Castelli‡, and May-Win L. Thein§

University of New Hampshire

The NASA Magnetosphereic Multiscale (MMS) Mission is a constellation mission, con-sisting of four spin-stablized satellites flying in a tetrahedron-shaped formation, to belaunched in 2014. The MMS design specifications dictate strict pointing and body ratecontrol requirements without the use of onboard gyroscopic instruments. The authorshave previously developed an observer-based attitude and nutation control scheme for theNASA MMS Mission. The Extended Kalman Filter (EKF) and Sliding Mode Observer(SMO) were used to provide full state feedback to PID and Variable Structure controllers.These observer-based control algorithms were proven to meet MMS Mission requirementswithout the need for gyros for state feedback.

In this paper, the authors extend their previous work and further analyze variousestimation techniques for use in the NASA MMS Mission Attitude Control System (ACS).In addition to the EKF and SMO, the performances of other estimation techniques willbe compared under the MMS Mission scenario. Additionally, the EKF and SMO will beexplored further to determine a mission optimal system for attitude estimation and control.Performance criteria of transient response, steady state accuracy, speed of convergence,robustness to input and output uncertainties and modeling errors, noise sensitivity, fuelconsumption and system stability will be analyzed for each observer technique.

As it has already been shown that an observer-based control algorithm without the useof gyros is a feasible option for the NASA MMS Mission, this continuing work will serveto further shed light on the advantages and disadvantages of each estimation algorithm, aswell as uncovering the most appropriate observer technique(s) for use on MMS spacecraftfor gyroless attitude and nutation control.

I. Introduction

Current research with the NASA Goddard Space Flight Center (GSFC) involves the dynamic modelingand control of the NASA Magnetosphereic Multiscale (MMS) Mission, a Solar-Terrestrial Probe missionto study Earth’s magnetosphere. The mission will progress in three stages, each requiring large orbital

∗Engineer, neil.mushaweh@gmail.com†Graduate Student of Mechanical Engineering, bmb5@cisunix.unh.edu, AIAA member‡Graduate Student of Mechanical Engineering,daz6@cisunix.unh.edu, AIAA member§Associate Professor of Mechanical Engineering, mthein@cisunix.unh.edu, AIAA member

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maneuvers. These large orbital transfers, coupled with the high sensitivity of the instruments, require thesatellite to maintain a constant pure rotation about its local z-axis. Mass imbalances, external torques andother unknown disturbances cause the satellite to tend to nutate about its x and y axis undesirably. Inorder to effectively reject all satellite nutation, while maintaining a constant spin, it is essential that thesatellite angular body rates are accurately known.

The MMS satellite has 6 instrumentation booms (2 Axial Double Probe, 4 Spin Double Probe) that ex-tend up to 50 meters in length at just under two millimeters of thickness. These highly sensitive instrumentsrequire that the satellite spin at a constant rate about its z-axis at 0.3 rad/s while rejecting nutation (asa result of mass imbalances and/or disturbance torques) about the x and y axis (±0.02 rad/s 3σ) whichmay occur from mass imbalances, or external torques. While maintaining these satellite body rates, thespacecraft must also have an orientation that is 0.052 radians (± 0.009 rad 3σ) about its x and y body-fixedreference frame.

This research investigates the feasibility of various nonlinear estimation algorithms that are capableof determining spacecraft angular rates with only star tracker attitude measurements. Additionally, thispaper will build upon past research to offer a comparative analysis of nonlinear estimation techniques forspacecraft attitude determination. Estimation techniques, unlike in past research, will be evaluated forcomputational efficiency and steady-state fuel consumption, while still investigating state estimation effec-tiveness as in previous research. The focus of this research is on the rotational (attitude) dynamics of thespacecraft rather then the translational motion (orbit). Although these dynamics are highly coupled, thework in this thesis will decouple attitude and orbital dynamics (the control of an integrated attitude/orbitalsystem is left for future work). Each estimation algothrim considered will be integrated with a sliding modecontroller as presented in1 and2 for attitude control and nutation rejection. It is within the control loopthat the estimation techniques will be evaluated for effectiveness and robustness. The application of suchestimation capabilities has implications on designs that avoid costly angular rate measurement systemswhile maintaining highly accurate state representations.

Similar research was conducted as indicated in3 and,4 however, each fail to consider alternative esti-mation techniques to determine the most effective system for a given application. The Extended KalmanFilter (EKF) and Sliding Mode Observer (SMO) are revisited from 1 as part of the comparative analy-sis. Additionally, the H-infinity, also referred to as the minimax filter, is explored as presented in.5 Thehigher-order SMO as presented in67 was not considered in this study because of excessive complexitiesand computational demand associated with higher order SMOs compared to that of the SMO used in thiswork.

As presented in1 the SMO and EKF offer effective results for full attitude estimation. The EKF wasnoted to be superior in noise filtering for body rate estimation while having difficulties rejecting unknowndisturbances and modeling uncertainties. At steady-state, the EKF is shown to have excessive erroroscillations of the attitude states on the same order of magnitude of the disturbance torques, indicating itsineffectiveness to unknown disturbances. Conversely, the SMO was found to be more effective in rejectingmodeling uncertainties and unknown disturbances while ineffective in rejecting zero-mean gaussian whitenoise.

In this paper the H-infinity filter is introduced to explore its potential in improving attitude stateestimation for attitude and nutation control as compared to previously presented EKF and SMO results.Additionally, changes in the observer design of the SMO and EKF are implemented to build upon pastresearch to determine the optimal estimator for the MMS mission. First, all nonlinear state estimationtechniques are introduced for their application in the MMS mission. Next, the MMS mission profile andsimulation parameters as well as disturbance torque, modeling uncertainties and white noise bounds of thesimulation are presented. Estimation results of the three techniques are then presented in a comparativefashion where steady state performance is more throughly inspected unlike in prior papers.

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II. MMS Model and Mission Profile

For this reasearch, a rigid body, euler angle approach for attitude kinematics is used for MMS modeldevelopment. The time dependency of Euler angles of the 3-2-1 sequence with angles ψ, θ, and φ is definedas:

φ = (ωb/o3cosψ + ωb/o2

sinψ) secθ (1a)

θ = ωb/o2cosψ − ωb/o3

sinψ (1b)

ψ = ωb/o1+ (ωb/o3

cosψ + ωb/o2sinψ) tan θ (1c)

The Euler moment equations are based on the law of conservation of angular momentum and used inthe formulation of attitude motion. Eq.(2) may be expanded as follows:

∑

Mb =

[

dH

dt

]

b

+ ω × H (2)

where the subscript b refers to the body fixed axes of the spacecraft. If there are no momentum exchangedevices, then the angular momentum of a spacecraft can be defined as:

H = Iω =

Ixx −Ixy −Ixz

−Iyx Iyy −Iyz

−Izx −Izy Izz

ωx

ωy

ωz

(3)

where I is the symmetric inertia matrix of the spacecraft about the body-fixed reference frame.A 3-2-1 rotation sequence with angles ψ, θ, and φ coupled with Euler moment equations, assuming no

cross products of inertia is used as a MMS model of attitude dynamics and defined as:

f(x(t), t) =

ωx + (ωz cosψ + ωy sinψ) tan θ

ωy cosψ − ωz sinψ

(ωz cosψ + ωy sinψ) sec θ

((Iy − Iz)/Ix)ωyωz

((Iz − Ix)/Iy)ωxωz

((Ix − Iy)/Iz)ωxωy

(4)

where x = f(x(t), t) and the vector of states is x(t) = [ψ, θ, φ, ωx, ωy, ωz]T . Eq.(4) is used for nonlinear

estimation models and do not take into account cross products of inertia, unknown disturbances, or externaltorques. For dynamic equations of motion in which external torques and cross products of inertia areconsidered, the following decoupled equations are used to find ωx, ωy, ωz in terms of ωx, ωy, ωz and hx, hy, hz :

Ixωx − Ixyωy − Ixzωz = Mx − ωyhz + ωzhy (5a)

−Ixyωx + Iyωy − Iyzωz = My − ωzhx + ωxhz (5b)

−Ixzωx − Iyzωy + Izωz = Mz − ωxhy + ωyhx (5c)

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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)

2

6

4

8402.64 −58.8 −44.6

−58.8 8411.97 −100

−44.66 −100 16414.66

3

7

5

8000 0 0

0 8600 0

0 0 17500

Noise bounds (rad) ±0.03

Table 1. Simulation Parameters for the Extended Kalman Filter

where:

hx

hy

hz

=

Ixx −Ixy −Ixz

−Iyx Iyy −Iyz

−Izx −Izy Izz

ωx

ωy

ωz

(6)

It should be noted that for consideration of cross products of inertia and external torques, the first threestate equations of Eq. (4) remain the same since Eqs. (5) are not dependent on states ψ, θ, φ.

II.A. MMS Simulation Parameters

Simulation parameters are listed in Table 1. An external moment of M = 0.001 sin(.3t) N-m is addedto all three axes to simulate unknown uncertanties/disturbances acting on the satellite that oscillate atthe same frequency as the satellite rotation. This is to emulate disturbances that could be acting on thespacecraft while spinning at 3 rotations per minute (0.3 rad/s). These disturbance torques act about thex, y and z axes of the spacecraft body-fixed reference frame and are consistent in magnitude with externaltorques acting on orbiting spacecraft at a given altitude of approximately 105 km. Also, a 10 percent errorin moment of inertia values are in the observer dynamics to simulate further parametric uncertainty. Allestimation techniques’ robustness to such common inconsistencies are analyzed.

III. Nonlinear Estimation Methods

This research considers three nonlinear estimation techniques for comparative analysis in the use forMMS application. The following sections will outline the Extended Kalman Filter (EKF), Sliding ModeObserver (SMO) and H-infinity.

III.A. Extended Kalman Filter

For a given nonlinear, continuous-time system defined as:

˙x(t) = f(x(t), t) +K(t)[y(t) − h(x(t), t)] (7)

where y(t) = h(x(t), t) + z(t) is sensor outputs (h(x(t), t)) and measurement noise (v(t)).The continuous-time Kalman filter equations are given as:

P(t) = F (x(t), t)P (t) + P (t)F T (x(t), t) +Q(t) − P (t)HTR−1(t)HP (t) (8a)

F (x(t), t) =∂f(x(t), t)

∂x(t)|x(t)=x,(t) (8b)

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K(t) = P (t)HTR−1(t) (8c)

where Q(t) and R(t) are weight matrices selected based on process noise and measurement noise respec-tively. The error covariance equation, Eq. (8a), and state matrix are then updated using the gain matrixand measurement error.

III.B. Sliding Mode Observer

As previously introduced in8 and9 the Sliding Mode Observer (SMO) utilizes a switching function andsliding surfaces to force the error trajectory to zero. The number of sliding surfaces and how they aredefined is part of the SMO trade space for development and application. For nonlinear observer dynamicsthat are a function of the estimated states, x, the SMO is defined as:

˙x = f(x(t), t) +Bu(t) +Hz +K1s (9a)

z = Cx (9b)

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 SMC, is a saturation function such that:

1s = sat

(

s

ρ

)

(10)

For a more complete development of Sliding Mode Observers and Nonlinear Systems the reader isreferred to8 and.10

III.C. H-infinity Filter

The H-infinity, or sometimes referred to as the minimax,11 filter works to minimize the worst case sce-nario estimation error of the desired states while, unlike the EKF, need no prior knowledge of the noisecharacteristics. Similarly to the EKF, however, is the optimal nature of the H-infinity filter through theutilization of the Riccati equation. Additionally, the H-infinity filter has been shown to have improvedrobustness to modeling uncertainties when compared to the EKF.12 As with the formulation of the EKF,the H-infinity filter is derived from the nonlinear state estimation model of Eq. (7). A detailed formulationof the H-infinity filter can be found in,11 where it can be shown that the covariance matrix equation basedon a specific cost function is defined as:

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

where the matrix S is a symmetric, positive semi-definite time, invariant performance matrix and n andm are the number and states and number of measurements respectively.

IV. Attitude Control with Perfect State Measurements

For baseline comparative purposes, the sliding mode controller (SMC) will be presented assumingperfect state measurements. This will provide a baseline to which the nonlinear estimation techniques canbe measured against. The SMC presented in 1 will be used as the attitude control technique for comparisonof the estimation techniques. The Sliding Mode Controller introduced in 1 is again defined as:

x = f(x, t) +Bu(t) +Hsmcz +Ksmc1s (12)

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where the nonlinear equations of motion f(x, t) are defined in Eq. (4) and the switching function isdependent upon the boundary layer ρ and is defined as 1s = sat(s/ρ).

Five sliding surfaces are selected and defined as:

si = λ1zi + λ2

∫

zidt, i = 1, 2, 3, 4, 5 (13)

where zi is the error between actual and desired states.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 (14)

Euler angle control results are presented in Figures 1 and 2 and it can be seen that the steady stateerror is essentially zero. Control effort shown in Figure 3 is of reasonable values and is a major criteriathat will be used to evaluate the estimation techniques.

0 20 40 60 80 100−0.02

0

0.02

0.04

0.06Euler Angles (rad)

psi

0 20 40 60 80 100−0.06

−0.04

−0.02

0

0.02

thet

a

time (s)

Figure 1. Sliding Mode Control Euler Angles

V. Comparative Analysis of Attitude Estimation Methods

The following sections show results of the considered estimation techniques augmented with the SMC.Particular interest is taken in the estimation technique that has the least amount of control effort as tonot induce unnecessary thruster commands that will consume valuable fuel supplies.

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0 20 40 60 80 100−0.04

−0.02

0

0.02Euler Angle Control Error (rad)

psi

0 20 40 60 80 100−0.015

−0.01

−0.005

0

0.005

0.01

thet

a

time (s)

Figure 2. Euler Angle Error Using Sliding Mode Control

0 10 20 30 40 50−0.5

0

0.5

M1

Control Effort (N)

0 10 20 30 40 50−0.1

0

0.1

M2

0 10 20 30 40 50−2

−1

0

1

time(s)

M3

Figure 3. Sliding Mode Control Effort for Perfect Measurements

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V.A. EKF based SMC

Past research 1 indicated that the SMO is a much more effective estimator for tracking satellite attitudestates in the presence of unknown disturbances when compared to the EKF. Further EKF tuning hasindicated that there is a trade in thrust optimization and effectiveness in tracking satellite states. Incomparison to past research as presented in 1 it was determined that a marginal increase in control effortresulted in improved state estimation. The improved EKF control effort can be seen in figure 4. Addi-tionally, Figures 5 and 6 show new results for EKF Euler angle and body-rate control errors respectivelywhere:

Q =

0.0001 0 0 0 0 0

0 0.0001 0 0 0 0

0 0 0.0001 0 0 0

0 0 0 150000 0 0

0 0 0 0 250000 0

0 0 0 0 0 100000

(15a)

R =

106 0 0

0 106 0

0 0 106

(15b)

are the new EKF weight matrices for Q and R respectively.

0 20 40 60 80 100 120 140 160 180 200−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Control Effort for Extended Kalman Filter

M1

Time(s)

Mom

ent (

N/m

)

0 20 40 60 80 100 120 140 160 180 200−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25M2

Time(s)

Mom

ent (

N/m

)

0 20 40 60 80 100 120 140 160 180 200−0.8

−0.6

−0.4

−0.2

0

0.2M3

Time(s)

Mom

ent (

N/m

)

Figure 4. EKF Control Effort

V.B. H-infinity results

The H-infinity observer was introduced with the sliding mode control investigate the improved robustnessof the estimator while still effectively rejecting measurement noise as with the EKF. H-infinity tuning wasconducted in the same manner as EKF tuning, in that gains were selected through iterative processes. The

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0 20 40 60 80 100 120 140 160 180 200−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

Euler Angle Control Errors for Extended Kalman Filter

Psi Error

Time(s)

Err

or (

rad)

0 20 40 60 80 100 120 140 160 180 200−0.03

−0.02

−0.01

0

0.01

0.02

0.03Theta Error

Time(s)

Err

or (

rad)

Figure 5. EKF Euler Angle Control Errors

0 20 40 60 80 100 120 140 160 180 200−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

Body−Rate Control Errors for Extended Kalman Filter

Wx Error

Time(s)

Err

or (

rad/

s)

0 20 40 60 80 100 120 140 160 180 200−0.04

−0.03

−0.02

−0.01

0

0.01

0.02Wy Error

Time(s)

Err

or (

rad/

s)

0 20 40 60 80 100 120 140 160 180 200−0.4

−0.3

−0.2

−0.1

0

0.1Wz Error

Time(s)

Err

or (

rad/

s)

Figure 6. EKF Euler Angle Body-Rate Control Errors

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Q and R matricies were equivalent to the EKF and S is defined as:

S =

2 0 0 0 0 0

0 4 0 0 0 0

0 0 2 0 0 0

0 0 0 2 0 0

0 0 0 0 2 0

0 0 0 0 0 2

(16)

and the θ scalar term is 1e− 7.H-infinity Euler angle control errors can be seen in Figures 7, while body-rate control errors and control

effort are shown in Figures 8 and 9 respectively.

0 20 40 60 80 100 120 140 160 180 200−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

Euler Angle Control Error for H−Infinity

Psi Error

Time(s)

Err

or(r

ad)

0 20 40 60 80 100 120 140 160 180 200−0.03

−0.02

−0.01

0

0.01

0.02

0.03Theta Error

Time(s)

Err

or(r

ad)

Figure 7. H-infinity Euler Angle Control Errors

V.C. SMO extensions

In expanding on previous research, the integration of the observer error was performed to improve SMOperformance. Additionally, the sliding surfaces were divided so that each measurement feedback had itsown sliding surfaces. There was a saturation placed on integration feedback to eliminate overpowering theproportional term before the controller commenced. Figure 10 shows the new sliding mode observer where:

s3

s2

s2

=

8.901e − 3 0 0 9.2577 0 0

0 0.8901 0 0 9.2577 0

0 0 0.8901 0 0 9.2577

(17a)

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0 20 40 60 80 100 120 140 160 180 200−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

Body Rate Control Error for H−Infinity

Wx Error

Time(s)

Err

or(r

ad/s

)

0 20 40 60 80 100 120 140 160 180 200−0.04

−0.03

−0.02

−0.01

0

0.01

0.02Wy Error

Time(s)

Err

or(r

ad/s

)

0 20 40 60 80 100 120 140 160 180 200−0.4

−0.3

−0.2

−0.1

0

0.1Wz Error

Time(s)

Err

or(r

ad/s

)

Figure 8. H-infinity Body-Rate Control Errors

0 20 40 60 80 100 120 140 160 180 200−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Control Effort for H−Infinity

M1

Time(s)

Mom

ent(

N−

m)

0 20 40 60 80 100 120 140 160 180 200−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25M2

Time(s)

Mom

ent(

N−

m)

0 20 40 60 80 100 120 140 160 180 200−0.8

−0.6

−0.4

−0.2

0

0.2M3

Time(s)

Mom

ent(

N−

m)

Figure 9. H-infinity Control Effort

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[

ks1 ks2 ks3]

=

4.1e − 3 0 0

0 4.1e − 3 0

0 0 14.1e − 3

1.55e − 4 0 0

0 1.55e − 4 0

0 0 1.55e − 4

(17b)

Figure 10. SMO Block Diagram

Code was written to optimize SMO gains about MMS flight requirements by using weighted standarddeviations and thrust as performance requirements. This improvement in gain selection improved SMOperformance and ensured that for the given sliding surface combination, and optimal gain structure wasselected.

Figures 11, 12, and 13 show the Euler angle control error, body-rate control error and control effort fora SMO based SMC system.

VI. Conclusions and Future Work

The results of this research show that both the Extended Kalman Filter and Sliding Mode Observer im-plemented with Sliding Mode Control yield satisfactory performance. Additionally, the H-infinity observeroffered improved results for tracking error, while sacrificing thrust performance. Conversely, it was foundthrough the tuning techniques that the H-infinity observer had stability issues at certain gain selectionswhich need to be further investigated in future work. These observer-based control systems both meetNASA design requirements while reducing thruster control effort and reducing the effects of measurementnoise and spacecraft uncertainties/disburtances over past research. Although all observer-based control

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0 20 40 60 80 100 120 140 160 180 200−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

Euler Angle Control Errors for Sliding Mode Observer

Psi Error

Time(s)

Err

or(r

ad)

0 20 40 60 80 100 120 140 160 180 200−0.03

−0.02

−0.01

0

0.01

0.02

0.03Theta Error

Time(s)

Err

or(r

ad)

Figure 11. SMO Euler Angle Control Errors

0 20 40 60 80 100 120 140 160 180 200−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

Body Rate Control Errors for Sliding Mode Observer

Wx Error

Time(s)

Err

or(r

ad/s

)

0 20 40 60 80 100 120 140 160 180 200−0.04

−0.03

−0.02

−0.01

0

0.01

0.02Wy Error

Time(s)

Err

or(r

ad/s

)

0 20 40 60 80 100 120 140 160 180 200−0.4

−0.3

−0.2

−0.1

0

0.1Wz Error

Time(s)

Err

or(r

ad/s

)

Figure 12. SMO Body-Rate Control Errors

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0 50 100 150 200 250−0.02

−0.01

0

0.01

0.02

0.03

Control Effort for Sliding Mode Observer

M1

Time(s)M

omen

t(N

−m

)

0 50 100 150 200 250−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25M2

Time(s)

Mom

ent(

N−

m)

0 50 100 150 200 250−0.8

−0.6

−0.4

−0.2

0

0.2M3

Time(s)

Mom

ent(

N−

m)

Figure 13. EKF Control Effort

systems are satisfactory for meeting NASA requirements, it can be seen that the SMO-based control sys-tem is more robust to un-modeled disturbance torques, while the EKF and H-infinity control system aremore effective at rejecting measurement noise.

As seen in Table 2 the SMO was able to achieve lower errors on the euler angles, while only marginallyincreasing maximum control effort during transient responses. During steady-state operations, however,the SMO used 40 percent more effort over a 100 second interval, which would have serious impacts on along term mission. It was determined that the H-infinity could get more efficient than the EKF, however,and with less impact to fuel consumption when compared to the SMO.

H-infinity SMO EKF

Control Error ψ (rad) 0.0025 0.0019 0.0033

Control Error θ (rad) 0.0029 0.0020 0.0037

Control Error ωx (rad/s) 0.0176 0.0190 0.0176

Control Error ωy (rad/s) 0.0190 0.0191 0.0182

Control Error ωz (rad/s) 0.0050 0.0059 0.0049

Max Control Effort (N ·m) 0.618 0.610 0.624

Total Steady-State Effort for 100s (N ·m) 0.902 1.17 0.839

Table 2. Estimator Performance Measures

Computer demand is another design parameter that was considered in this comparison of each estimatorbased control system. The accelerator mode in simulink, which compiles the model into C code, wasutilized to allow for faster computation. The H-infinity took approximately 28 seconds to run through asimulation while the SMO only took 6.4 seconds and the EKF took 22 seconds. It should be noted, thatfor simulations that involved perfect state feedback simulation times took 4.5 seconds. For computationalefficiency purposes, the SMO is far superior while the EKF and the H-infinity took about the same time torun. It should be noted that some of the processing was for the estimator, some was for the controller andsome was to perform the simulated system response. Therefore, it is concluded that the SMO was likelymore than 5 times more efficient when compared to the H-infinity and EKF based controllers. Computations

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in the estimator loop that require inverse matrix operations are a larger hinderance on simulation timeand computational efficiency, thus making the SMO more desirable in this aspect.

Ultimately, the SMO offers the best option for an observer based attitude controller because of itscomputational efficiency and its improvement in state tracking over other considered estimation techniques.The impacts of fuel consumption will need to be better understood and MMS fuel availability requirementscould impact this conclusion.

The authors plan to develop stability proofs for these observer methods, as permissible, and will incor-porate Monte Carlo analysis methods as appropriate to take into account all initial conditions, unknowndisturbance and large orbital maneuvers that will be necessary. The authors also plan to add other estima-tion techniques to this study. Additionally, augmentation of the attitude estimation and control algorithmwith orbital state estimation and control will be required to ensure MMS mission success. The SMO hada number of terms that were found to be very interrelated, and because of this, future work should includeexploration of these interdependencies by performing a more iterative process in the tuning.

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. Additionally,special acknowledgments are extended to Raytheon Company for their financial support of the primaryauthor in attending and presenting at the 2009 AIAA GNC Conference.

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