Compensation of Data-Loss in Attitude Control of Spacecraft Systems

International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.1, No.2, July 2014
1
COMPENSATION OF DATA-LOSS IN ATTITUDE
CONTROL OF SPACECRAFT SYSTEMS
Naeem Khan1
, and Dawei Gu2
1
Assistant Professors in Electrical Engineering Department., UET Peshawar, Pakistan
2
Professor at Department of Engineering, University of Leicester, UK
ABSTRACT
In this paper, a comprehensive comparison of two robust estimation techniques namely, compensated
closed-loop Kalman filtering and open-loop Kalman filtering is presented. A common problem of data loss
in a real-time control system is investigated through these two schemes. The open-loop scheme, dealing
with the data-loss, suffers from several shortcomings. These shortcomings are overcome using
compensated scheme, where an accommodating observation signal is obtained through linear prediction
technique -- a closed-loop setting and is adopted at a posteriori update step. The calculation and
employment of accommodating observation signal causes computational complexity. For simulation
purpose, a linear time invariant spacecraft model is however, obtained from the nonlinear spacecraft
attitude dynamics through linearization at nonzero equilibrium points -- achieved off-line through
Levenberg-Marguardt iterative scheme. Attempt has been made to analyze the selected example from most
of the perspectives in order to display the performance of the two techniques.
KEYWORDS
State Estimation, Kalman filtering, Open-loop Kalman filtering, Intermittent Observations, Compensated
Closed-loop KF
1. INTRODUCTION
State estimation in uncertain environments [15, 9] or noisy information [29] is a broad field of
communication and control theory. This is because the problem of state or parameter estimation is
of paramount importance in the analysis and design of control systems [14]. The most celebrated
techniques for state estimation are Kalman filtering and its adaptive forms, particle filtering, and
H∞ filtering [8] etc. For a linear system, Kalman filter is an optimal approach where state of an
LTI system is estimated based on an optimal Kalman filter gain.
For various reasons including understanding of system behaviour, designing and implementation
of an optimal control scheme, state (attitude) estimation has remained an important research topic
for spacecraft control. Spacecraft systems in particular, mainly depend on data achieved and
processed from the ground that ultimately results in time delay [25]. However, perfect
communication is a valuable and the most desired asset in the event of fault and failure. To
handle such unfavourable conditions, several techniques like hardware redundancy, including
duplicate, triplicate and voting schemes, has remained consistently adopted [24]. But issues like
weight, complexity and cost of the supplementary elements in these hardware-based techniques
have diverted the attention toward software based approaches (Model Based FDI) to overwhelm
the aforementioned limitations [31].

2
A precise communication is a fundamental factor in achieving fruitful results in any
control system. However, scenarios including finite gateways of networks, bounded
spaces and overpopulated networks might cause the data packets to be lost. As a
consequence, the spacecraft performance may be significantly degraded specially in
terms of delay and failure due to any of these diverse conditions [26]. Hence the loss of
observation or output data plays a vital role in spacecraft attitude control and remedies
need to be explored in order to provide reliable state (attitude etc.) estimation.
Open-loop Kalman filtering (OLKF), also known as Open-loop estimation is, perhaps, the
predominant scheme in the literature since it is frequently utilized for data-loss cases. There are
numerous research articles which elaborate this technique in details such as [28, 4, 30, 8] and the
references therein, to name a few. Some of these literatures have demonstrated the associated
limitations of this technique too. It is an intense need to propose some novel techniques that could
handle data-loss situations more efficiently and to overcome those limitations. For such reasons, a
compensated closed-loop Kalman filtering algorithm is proposed in [17, 18]. In the compensated
closed-loop scheme, an accommodating observation signal is reconstructed using linear
prediction coding, for which one parameter is crucial to decides i.e. the order of linear prediction
filter.
The present study is aimed to elaborate two objectives: (a) to provide handsome details of this
recently proposed scheme and (b) the extensive study and comparison of these two state
estimation schemes (Open-loop Kalman filtering and compensated closed-loop Kalman filtering)
for a rigid body spacecraft system which is subjected to an induced data-loss. A minimum mean
square error based algorithm is proposed to decide the computation of linear prediction filter
order. In order to provide the true and complete picture, these two schemes are compared with
conventional estimation scheme (normal Kalman filtering without any data-loss). In fact it
provides a common base for the comparison. Both rotational dynamic and kinematic equations
are used to derive the state-space equations for the spacecraft system [1, 20], contrary to the
normal trend of using `Kinematic equation' as discussed by [10, 22, 26, 6, 13] and the references
therein.
The remaining paper is organized as follows: Section 2 presents the nonlinear model of a rigid
body spacecraft system using Modified Rodrigues Parameterizations i.e. MRP representation.
Section 3 is devoted to a brief discussion of an existing control system design. A detailed
discussion of the accommodating closed-loop Kalman filtering scheme is described in Section 4.
The performance of open-loop estimation and the recently introduced accommodating estimation
schemes is analyzed through a numerical case study in Section 5. The paper is concluded with
suggestions for the future work in Section 6.
2. SPACECRAFT RIGID BODY
It is common to observe spacecraft analysis while employing kinematic equations and/or dynamic
equations in Euler angles and quaternion parameterizations. These two parameterizations have
certain limitations; nonlinear trigonometric functions and singularity issues are linked with Euler
angles while a redundant element and unit constraint are associated with quaternion
parameterizations. To overcome these shortcomings, Modified Rodrigues Parameters (MRPs) are
utilized in this work which is found advancement to the parameterization's family.

3
2.1 Spacecraft Dynamics
As mentioned before spacecraft dynamics are usually specified by its 'Kinematic' system only
[10, 22, 6, 13] In this manuscript, however, both Euler equations of rotational dynamics and the
Kinematic equations in order to examine the complete behaviour of the spacecraft system.
2.1.1 Kinematic Equations
In the compact form, Kinematic equations are
)(σσ T=& (1)
where σ is the Modified Rodrigues Parameter and )(σT is defined as






++




 −
= ×
T
T
SIT σσσ
σσ
σ )(
2
1
5.0)( 33 (2)
wherein )(σS denotes the skew symmetric matrix defined as
(3)










−
−
−
=
0
0
0
)(
12
13
23
σσ
σσ
σσ
σS
The attitude vector σ and noisy angular vector ω are of dimension 3 x 1 with










−
−
−
=










=
33
22
11
3
2
1
:
n
n
n
ω
ω
ω
ω
ω
ω
ω (4)
The gyroscope output model is )(ty j is selected as
}3,2,1{)()( =∀+= jtntcy jjjj θ& (5)
where jjj nandc ,θ represent the scale coefficient, the angular position and gyroscope noise
respectively. The noise is assumed to be Gaussian white noise with zero mean, i.e.
)(0,~ ΠΝnj (6)
where Π is the bias variance.
2.1.2 Dynamic Equations
The dynamics are defined using Euler's equations as

4
τωωω +−= JSJ )(& (7)
where J is the spacecraft’s inertia, τ is control input and “ ωωω ×=)(S ” is the skew
symmetric matrix which shows the cross-product operation as










−
−
−
=
0
0
0
)(
12
13
23
ωω
ωω
ωω
ωS (8)
Equations (1) and (7), collectively constitute the full state vector for the spacecraft plant model,
i.e. [ ]T
x ωσ= . The linearized attitude dynamics of the plant are represented as
)()()()( tGtButAxtx ξ++=& (9)
where Gand, BA are Jacobian matrices of the linear spacecraft dynamics which are derived in a
straightforward manner. Equations (1) and (2) can be considered six coupled equations, from
which the linearized plant model can be achieved through Jacobian linearizations. Some related
theory can be found in [11].
3. CONTROL SYSTEM DESIGN
It is important to mention that in this paper special attention is paid to obtain efficient state
estimation in case of loss of observation and not the control issues related to the spacecraft
system. Hence, an already established and employed control scheme [1] is briefly recalled for the
sake of complete view. This control technique consists of two loops. An inner loop comprises a
simple transfer function while an outer loop is merely a unity feedback gain. The recalled control
system design is:
)]()(~[)( *
ttST P
T
σσστ −= (10)
where τ is the control input and






+++
=
=
−=
=
3
3
3
2
2
2
1
1
1
*
321
,,
),()(
),(ˆ)(~
),,,(
α
α
α
α
α
α
σσ
σσσ
s
s
s
s
s
s
s
s
sdiagN
tNt
tt
sssdiagS
ddd
d
pppP
(11)
The positive definite matrices ),( NSP are the tuning elements need to be tuned. The
candidate Lyapunov function is
)}{~~(5.0)~,,( *1***
σασσσσσσσσ −
++= d
T
p
TT
SSHV &&& (12)
with d
*
SandH are defined as

5
),,(
&)())((
321
11*
dddd
T
sssdiagS
JTTH
=
= −−
σσ
(13)
The derivative of Lyapunov function w.r.t time is
*1*
*1**
)(}{
σσ
ασσαασσσ
−
−
−=
−+−=
d
T
dd
TT
S
SSV &&&
(14)
Figure 1. Schematic of Feedback Controller
The choice of gain parameters ),( idipi andss α to stabilize the plant model could be any suitable
positive values as long as convergence is associated. Such detailed discussion can be found in [1].
In the following section, the two robust algorithms are discussed along with the normal Kalman
filtering (without any data-loss) scheme.
4. ROBUST KALMAN FILTERING TECHNIQUES
Loss of observations or data-loss might produce adverse scenarios for state estimation in Kalman
filtering, as it heavily depends on measured data [7] and [3]. Such conditions are sometimes
unavoidable, result in poor estimation and could lead Kalman filter to diverge very swiftly. A
usual technique to avoid such shortcomings is the so-called Open-Loop Estimation (OLE)
algorithm when observations are subjected to random loss -- see e.g. [27, 23, 33, 28]. In these
references, the authors have presented the Kalman filter running in an open-loop method, when
the plant is subjected to data-loss. Simply the predicted quantities (state and covariance) are
processed without any measurement update to the next step. It is considered helpful to present a
brief discussion of this OLE along with its related drawbacks.
4.1 Open-Loop Estimation
Open-Loop Estimation (OLE) or Open-Loop Kalman Filtering (OLKF) scheme is an effective,
simpler and computationally efficient method in practice to accommodate data-loss [12]. In this
scheme Kalman filter gain Kk is forced to zero if, a data-loss is observed at instant ‘k’ and hence

6
no update step is performed. Therefore, estimation is performed with zero sensitivity matrix [12].
Open-loop estimation is briefly introduced in the following lines with necessary comments.
4.1.1 Time update step
The a priori step or time update quantities (state and covariance) are
kkkkko BuAxx +=+ ||1 (15)
k
T
kkkko QAAPP +=+ ||1 (16)
4.1.2 Accommodating measurement vector
The pseudo-observational vector in Open-loop estimation is
kkoko xCz |11 ++ =
where the leading subscript ‘o’ shows Open-loop Kalman filtering approach. It causes zero
residual vector (2-norm) and hence, the Kalman gain would be
01 =+ko K (17)
4.1.3 Measurement update step
Since no data is available for measurement update, hence the a posteriori state and covariance
quantities will be
kkkko xx |1o1|1 +++ ← (18)
kkkko PP |1o1|1 +++ ← (19)
Therefore, in this approach the a posteriori parameters strictly follow the a priori quantities
respectively.
4.2 Shortcomings of the OLE
Although Open-loop scheme is a fast remedy to accommodate data-loss in state estimation due to
skipping of measurement update step, it suffers from the following shortcomings:
1. OLE diverges swiftly in the presence of adequate data-loss [33],
2. Spikes and/or oscillations (particularly when the output data is recovered),
3. It is harder to attain steady state values completely when data-loss is resumed [19].
4.3 Closed-loop Estimation Scheme
Due to the aforementioned disadvantages of OLE approach, an improved estimation scheme,
based on linear prediction concept, presented in [18, 19], is utilized. This scheme is known as
"Compensated Closed-loop Kalman Filtering (CCLKF)" wherein the lost observation signal is
reconstructed through linear prediction subsystem. There are various methods to predict a lost

7
sample or signal such as Particle Swarm Optimization [2], Maximum-a-posteriori or MAP [34],
Linear Prediction Coefficients/Coding (LPC) [5], etc. For several advantages such as close
resemblance to FFT, based on source-filter model, easy calculations, LPC technique is adopted to
reconstruct missing data signal.
For efficient linear prediction, as commonly adopted, it is speculated that the measurement signal
has correlation of some extend. In addition, the statistical properties of the plant model vary
slowly with time. According to this scheme, the missing signal is assumed to be
∑=
−=
p
i
ikik zz
1
α (20)
where the Linear Prediction Coefficients (LPCs) '' iα represent weights assigned to the previous
observations which is decided according to their correlation-degrees and '' p is the order of the
linear prediction filter (LPFO). The optimal value(s) of LPC and order of the Linear Prediction
Filter (LPFO) in Equation (20) are important elements in achieving an efficient filter [18] in term
of optimal results. For this reason, special attention has been considered to compute these two
critical elements. For this reason, a simple strategy is adopted using Algorithms 1, which can
assure the optimal value of LPFO leading to the optimal values of LPCs.
Algorithm 1: For LPFO
1: Initialization j = 1, Compute γγ randR {Equations (24-26)}.
2: Recursion j =2, 3, . . . M ( LPFO)
Calculate LPC {Equation (27)}
• Calculate compensated observation signal J
Rkz
γ
| {Equation (28)}
• Calculate compensated state estimation j
kc xˆ based on this signal {Equation (29)}
• Calculate compensated state estimation error 2||ˆ||: j
kckj xxe −=
3: Trace ϵth },...,{),min( 32 Mjj eeeewherease ∈= ,
4: Select jz which results in ϵth
5: Decide jp ← i.e. LPFO.
The detailed implementation of compensating scheme which will help in the understanding of the
above algorithm can be described as follows. Consider the discrete LTI plant dynamics are
described by the following equations:
kkkk
kkkk
vCxy
BuAxx
+=
++=+
)(
1
γ
ξ
(21)
where ,,,,,,....},2,1,0{ mn nnt
RRRRR ×
∈∈∈∈=∈ Azuxk ξ
nmln
RR ××
∈∈ CandB are state transition matrix, the input matrix, and output matrix with
)).,,(),0,0,((~),,(( 000 kkkk RQPxNvx ξ The random variable kγ is characterized as
follows:

8


 −
=
otherwise
ectedislossdataif
k
;1
det;0
γ (22)
Assume data-loss is observed at time instant ''k , the employed Kalman filtering technique
(CCLKF) is outlined as follows:
• Prediction cycle
At time step )(k-1 ,
K
T
kkckkc
kkkckkc
QAPAP
BuxAx
+=
+=
−−−
−−−−
1|11|
11|11|
(23)
where kk Q=]E[ T
kξξ is the process error covariance matrix, with ''E denoting expectation.
• Check loss of observation:
If 1=kγ i.e. no loss has occurred.
⇒Run conventional or normal Kalman filter (see e.g. [7], [3], [32]).
if 0=kγ , it means a data-loss condition has been detected
⇒No measurement (data) is available, hence at a posteriori step, accommodating-
measurement
update procedure is carried out as follows:
• Chose a nominal frame size of the previously stored measurement data (say
M ) modelled through the constraint ks tfM *≤ [17], where sf is the
sampling frequency and tk is starting instant of data-loss.
• Compute the autocorrelation matrix γR as
















−−−
−
−
−
=
]0[]3[]2[]1[
]3[]0[]1[]2[
]2[]1[]0[]1[
]1[]2[]1[]0[
RpRpRpR
pRRRR
pRRRR
pRRRR
R
L
MOMMM
L
L
L
γ (24)
and the auxiliary autocorrelation array γr is
[ ]T
prrrrr ][]3[]2[]1[ L=γ (25)
where
][]
ji|],[|
ji],0[
jrzz
ifjiR
ifR
zz
jk
T
k
jk
T
ik
=



≠−
=
=
−
−−
E[
]E[
(26)
• Compute the Linear Prediction Coefficients (LPC) as

9
γα γα rRA T
j .][ 1−
== (27)
• Compute the accommodating observation vector as
kk
p
j
jkjk vxCzz +≡= ∑=
−
1
α (28)
• Compute the compensated residual vector as kk zz ˆ− .
• Compute compensated Kalman gain as 1
32 )( −
+= k
T
kc
T
kckc RCPCCPK
• Perform a posterior step as
)( 1|1|| −− −+= kkckkckkckkc xCzKxx
kck
T
kc
T
kckckkc PCRCPCCPPP 2
1
321| )( −
+−= (29)
• Return to step 1 (prediction cycle).
Three covariance matrices that appear in above Equation are defined as:
]))([(]))([(
]))([(]))([(
]))([(]))([(
1|1|1|1|1|33
1|1|1|1|1|22
1|1|1|1|1|11
T
kkckkkck
T
kkkk
def
kkckc
T
kkckkkck
T
kkkk
def
kkckc
T
kkckkkck
T
kkkk
def
kkckc
xxxxeePP
xxxxeePP
xxxxeePP
−−−−−
−−−−−
−−−−−
−−===
−−===
−−===
EE
EE
EE
(30)
where 1|1 −= kkkc PP is the normal predicted error covariance matrix. The compensated closed-
loop scheme along with the controller strategy is shown in the block diagram fashion in Fig
(2). The switching mechanism between conventional Kalman filter (when there is no data-loss)
and compensated closed-loop scheme (when there is a data-loss) is shown in Fig (3).
Figure 2. Complete Diagram of Compensated scheme and spacecraft system

10
Figure 3. Switching mechanism between loss of information and fault free systems
5. NUMERICAL SIMULATION
As mentioned before, a rigid spacecraft model, subjected to intermittent output data is
employed here to test the performance of two schemes. These two schemes are thoroughly
discussed in the simulation results with their respective advantages and disadvantages.
Emphasis has been made on simulation results based on a data-loss at a specific location (30
- 45 sec). However, to show the flexibility and efficiency of the compensated scheme, a few
simulation results based on a data-loss at another location (15 - 25 sec) are also shown.
5.1 Spacecraft model
The non-zero equilibrium points in Equations (1) and (7) are computed off-line through
Levenberg- Marguardt iterative least-square scheme in Matlab-Simulink environment. The
plant model is linearized using Jacobian linearization at these operating points to conclude the
state-space model. The mathematical description of the linearized spacecraft attitude model is
given by
tttt
tttt
DuCxz
GBuAxx
θ
ξ
++=
++=&
(31)
where zux ,,, ξ are state vectors, deterministic system input, plant disturbance and measured
output. The linear time invariant system matrices DandCBA ,, computed through Jacobian
linearization at non-zero operating points are as follows:

11




















−−
−−
−−−−
−−−−
−−
=
0628.00274.01921.0000
1275.00163.00435.0000
0234.00661.00465.0000
2247.00779.00580.00160.01462.03567.0
0962.02010.02010.01462.00160.00621.0
0133.02087.02151.03567.00621.00160.0
A










−−
−−
−−
=
0673.00054.00027.0000
0054.00595.00033.0000
0027.00033.00503.0000
T
B










−−
−−−−−−
−−
=
0628.01275.00234.02247.00962.00133.0
0274.00163.00661.00779.02010.02087.0
1921.00435.00465.00580.02010.02151.0
T
G










−
−
−
=
0006252.22864.01156.1
0001132.15571.26196.2
0003026.07731.27069.2
C
and
330 ×=D
For simulation purpose, the initial state vector is selected as
T
x ]2.04.03.02.02.02.0[0 −−= ,the operating points computed through Levenberg-
Marguardt methods are ;]9365.05767.01284.0[ T
u ==τ 0.04471741.0[=σ
T
]0.4097- and T
]0.2280.47794779.0[ −=ω . The output and process noise covariance
matricesarechosen as 6633 *01.0and*05.0 ×× == IQIR where I is an identity matrix.
5.2 Spacecraft model
Certain typical simulation results are shown in this section for the system discussed above,
subjected to an induced data-loss. The simulation results obtained for CCLKF are compared
with that of OLKF and loss-free (Normal) Kalman filter results. This data-loss is assumed to
commence at time instant t = 30 sec and remains for 15 sec. Various studies are investigated
and addressed in the subsection below.
5.2.1 MRP Attitude
Figures 4 and 5 show the estimation results of MRP-σ1 for the said two techniques. Although
all the three attitude parameters σ1, σ2 and σ3 could be shown, however to avoid repetitions,
they are not included in this paper. It is obvious that the open-loop estimation diverges heavily
and instantly. Contrary to this, in the accommodating CCLKF approach, the estimation during
data-loss time is significantly stable as the highlighted picture shows in Figure 5. No doubt, the
deviation is directly related to the duration of loss of observation. From Figure 5, it can be

12
observed that less deviation caused by CCLKF scheme results in minor moment of inertia
(juggling) just after the data-loss is resumed, and hence the steady state is captured sooner than
OLKF approach. This is a significant achievement compared to OLKF scheme.
0 20 40 60 80 100 120 70 80 90 100
0.2
0.4
0.6
0.8
1
Time (sec)
σσσσ1
Normal KF
Open Loop KF
Comp. Closed-loop KF
Figure 4: Estimated result of First MRP (σ1)
25 30 35 40 45 50 55 60
0.94
0.96
0.98
1
1.02
1.04
Time (sec)
σσσσ1
NormalKF
Open-LoopKF
Comp.Closed-LoopKF
Figure 5: Data-loss period is highlighted
5.2.2 Angular Velocity
Similar to MRPs, three angular velocities related to the spacecraft model are also analyzed.
Figures 6 – 8 show the comparison of the two schemes (OLKF and CCLKF) during the data-loss
time for the angular velocities ω1, ω2 and ω3 along with the base comparison of normal
Kalman filtering. These figures illustrate, that the unavailability of observation has made
OLKF a poor estimating tool for nominal data-loss. Abrupt changes ("spikes and oscillations")
can be frequently realized in the estimation of angular velocities through OLKF scheme. On the
other side, the accommodating CCLKF scheme provides smaller chattering compared to OLKF
and hence outperforms.

13
0 10 20 30 40 50 60 70 80 90 100
-2
-1
0
1
2
3
4
Time(sec)
rad/sec
NormalKF
Open-LoopKF
Comp.Closed-LoopKF
Figure 6: Estimated result of (ω1)
0 10 20 30 40 50 60 70 80 90 100
-0.4
-0.2
0
0.2
0.4
0.6
Time(Sec)
rad/sec
NormalKF
Open-LoopKF
Comp.Closed-LoopKF

14
0 10 20 30 40 50 60 70 80 90 100
0.5
0.75
1
1.25
Time(Sec)
rad/sec
NormalKF
Open-LoopKF
Comp.Closed-LoopKF
5.2.3 Control Effort
MRPs another important aspect which delivers significant performance impact is the control
input signal. The data-loss at output terminal transverses its influence to the input parameters as
well due the recursive behaviour of Kalman filter and feedback control system. During the
data-loss time, significant overshoots can always be observed using the OLKF scheme than the
accommodating CCLKF scheme as shown in Figures 9-11. To observe the effectiveness and
flexibility of compensated scheme, Figure 13 shows the performance of OLKF and CCLKF
when data-loss has occurred at another location (i.e. from 15 – 25 sec). This figure too shows
the efficiency of CCLKF over the OLKF scheme.
0 10 20 30 40 50 60 70 80 90 100
-10
-5
0
5
10
Time(sec)
ττττ1
NormalKF
Open-LoopKF
Comp.Closed-LoopKF

15
Figure 9: Input Signal 1τ
0 10 20 30 40 50 60 70 80 90 100
-20
-10
0
10
20
30
40
Time (sec)
ττττ2
Normal KF
Open-Loop KF
Comp. Closed-Loop KF
0 10 20 30 40 50 60 70 80 90 100
-10
0
10
20
30
40
Time (sec)
ττττ3
Normal KF
Open-Loop KF
0 50 100 150 80 100
-12.5
0
12.5
ττττ1
(a)
0 20 40 60 80 100
-25
0
25
ττττ2
(b)
Normal KF
Open-Loop KF
0 20 40 60 80 100
-2
0
2
4
6
(c)
Time (sec)
ττττ3
10 15 20 25 30
-15
-10
-5
0
5
(d)
Time (sec)
ττττ3
Figure 12: Simulation results of Control inputs (a) 1τ (b) 2τ (c) 3τ and (d) Enlarge view of 3τ when
data loss occurs from 15-25 sec.

16
5.2.4 Error Analysis
Error analysis is the another investigated characteristic which reveals the efficient performance of
the accommodating CCLKF scheme i.e. )(ˆ)()( ∀−= iiie krk σσ . }3,2,1{=i . Similar to the
other parameters, less disruption can be seen utilizing CCLKF scheme than that of OLKF
approach as shown in Figures 13-15. In other words, state estimation using CCLKF approach is
less influenced by data-loss than the OLKF scheme.
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
Time (sec)
e1
Normal KF
Open-Loop KF
Figure 13: Estimated error in σ1 by the three schemes
0 10 20 30 40 50 60 70 80 90 100
-0.5
0
0.5
1
Time (sec)
e2
Normal KF
Open-Loop KF

17
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
Time (sec)
e3
Normal KF
Open-Loop KF
6. CONCLUSIONS AND FUTURE WORKS
6.1 Conclusion
In this work, theoretical study of two accommodating estimation schemes are discussed for
a linearized rigid body spacecraft model subjected to loss of observations. Linearized output
dynamics of the spacecraft model are derived by examining the two Direction Cosine
Matrices of the same sequences for the Euler angels and MRP. The conventional Kalman filter
fails to provide bounded error estimation during loss of measurement, open-loop Kalman
filtering (OLKF) is frequently utilized to overcome loss of output data. However, the
compensated closed-loop Kalman filtering (CCLKF) scheme has been found impressive
compared to OLKF scheme to the linearized spacecraft model. Simulation results of the two
approaches are compared to normal Kalman filter estimation approach under no loss of output
data. A comprehensive analysis of OLKF and CCLKF approaches is presented by demonstrating
various characteristics through a numerical example.
6.2 Future Work
“In this paper a stationary process of spacecraft dynamics has been considered. However, non-
stationary processes are intended to be tested in future. Stability and convergence issues related
to accommodating CCLKF approach also need to be explored. In this paper, usual linear
prediction technique (Normal Equation) is employed to compute linear prediction coefficients.
In future, faster techniques including Levinson-Durbin and Leroux-Gueguen algorithms, to
handle computational burdensome of the CCLKF approaches are intended to entertain for the
discussed case study.”
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