Efficient In-flight Transfer Alignment Using
Evolutionary Strategy Based Particle Filter Algorithm
Suvendu Chattaraj and Abhik Mukherjee
Department of Computer Science and Technology, Bengal Engineering and Science University, Shibpur, West Bengal, India
Keywords:
Transfer Alignment, Nonlinearity, Particle Filter, Evolutionary Strategy, Target Tracking.
Abstract:
Large initial misalignment between mother and daughter munitions make transfer alignment system nonlin-
ear, because small angle approximation applicable to the system dynamics does not hold. Further, when the
parameters of state transition matrix are based on current measurements, the system becomes time varying. A
conventional Kalman filter fails to estimate misalignment in such situations. A particle filter performs satis-
factorily, but, the performance suffers when the knowledge about the system is not accurate. Out of particles
that get propagated through such improper system dynamics, only a few are retained and used for estimation
purpose, due to sample impoverishment problem. In this work, it is claimed that better result can be obtained
by employing an evolutionary strategy. Set of support points are generated for each particle by propagating the
particle through an array of perturbed system dynamics, and, then by choosing best weight support point as
apriori estimate from that set. The current work considers design of such evolutionary strategy based particle
filter. For the purpose of proving robustness of proposed algorithm, simulation is first carried out on target
tracking problem. Then it is applied to in-flight transfer alignment problem and its performance is found to be
satisfactory.
1 INTRODUCTION
In the context of delivering guided munitions from
a moving platform, Transfer Alignment (TA) refers
to the process of determining the orientation of in-
ertial reference axes of daughter munitions with re-
spect to that of the mother platform. Mere copying of
mother data to daughter is not enough, as it does not
consider the misalignment between the two. Hence,
an algorithm is needed to estimate the misalignment.
Through some TA algorithm, initial velocity, position
and attitude of daughter, imparted by mother vehicle,
is determined as accurately as possible prior to ejec-
tion, to facilitate further navigation. Various TA tech-
niques have been reviewed in (Chattaraj et al., 2013;
Ali and Jiancheng, 2004).
An estimator (like Kalman Filter (KF)) is used in
TA problem for estimating system states, which uses
the dynamics of the system (based on the Newton`s
laws of motion) and some noise contaminated mea-
surements. KF can provide optimal estimate for linear
systems perturbed by white Gaussian noise, but, in re-
ality, such assumption does not hold, and necessitates
use of some variants of KF. Factors such as large ini-
tial misalignment angles (> 5
o
), non white-Gaussian
noise models etc., makes TA problem nonlinear and
has been discussed and presented in (Dmitriyev et al.,
1997).
For a non-linear system, variant of KF, like Ex-
tended KF (EKF), Unscented KF (UKF) etc. are used.
These approaches handle non-linearity based on the
principle of piecewise linear approximation of sys-
tem model, which, may not give accurate estimation
for poorly designed system models, un-modelled non-
Gaussian system or measurement noise etc (Wan and
Van Der Merwe, 2000; Julier and Uhlmann, 1997).
Another approach to handle non-linearity is Par-
ticle Filter (PF) in which, randomly chosen sample
points (particles) with associated weights are used
to compute posterior density of estimates of system
states. PF does not require rigorous noise modelling
(as in KF) for system and measurement noise to pro-
duce optimal estimate for the system states. However,
performance of PF is heavily depended on system
modelling and suffers due to loss of samples which
represent the solution space (sample impoverishment
problem) (Uosaki et al., 2003). Real time system
behaviour is unpredictable and cannot be modelled
properly by means of system dynamics. For a PF
based estimator, if initially generated particles are
5
Chattaraj S. and Mukherjee A..
Efficient In-flight Transfer Alignment Using Evolutionary Strategy Based Particle Filter Algorithm.
DOI: 10.5220/0005006000050013
In Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2014), pages 5-13
ISBN: 978-989-758-039-0
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)
propagated through any such improper system dy-
namics, it results in retention of a few imperfect sam-
ples which may affect overall performance.
One solution to the above problem is to employ
larger number of particles at the beginning, but, that
increases computational complexity of the entire pro-
cess and thus avoided. Better approach may be to gen-
erate few support points for each particle in each iter-
ation representing few more system dynamics, rather
than increasing number of particles arbitrarily. This
approach may be regarded as an evolutionary strat-
egy (ES) and has been recommended in literature for
handling sample impoverishment problem of conven-
tional PF (CPF) (Uosaki et al., 2003; Uosaki et al.,
2004). Residual (difference between predicted and
measured system states) plays an important role in the
estimation. The closer this value is to zero, the closer
the system is to being perfectly modelled and hence
reliable. Generating support points for particles is es-
sentially strengthening system models by bringing the
value of residual close to zero. Such approach is thus
termed as Residual Evolutionary Strategy Based Par-
ticle Filter (REPF) algorithm in this work.
In the current work, following the design of the
proposed algorithm, performance of CPF with 1000
particles (CPF) and REPF are compared for track-
ing problem. Target tracking is a classical problem
in which, some estimator is used to predict the navi-
gation information of manoeuvring target in the next
time step. Manoeuvring target emits signal which is
received by some sensors placed in the tracker, and
the tracker process those signals by applying some
estimation algorithm, to estimate navigation informa-
tion of manoeuvring target. Trackers may use vari-
ous measurements for this purpose such as position,
velocity acceleration or any combination of these.
This particular problem has number of applications
in real world such as mobile robot localization, mo-
bile (phone) tracking etc. Non-linearity in tracking
problem arises mainly due to unpredictable depen-
dencies of measurement noise and system states (Li
and Jilkov, 2001) and also on the non-uniform avail-
ability of measurements (Li and Jilkov, 2001; Li and
Jilkov, 2003).
Application of the proposed algorithm to tracking
problem exercises the robustness of the algorithm. It
has been shown that, proposed algorithm works at par
with that of CPF in situation when the system dy-
namics is considered perfect and works better in ad-
verse situation. Then similar algorithm is applied to
in-flight TA problem. One simulation exercise is con-
ducted with assumption of proper system knowledge.
Performance of REPF is found to be at par with other
two filters. In the second simulation scenario, a more
practical one, the system knowledge is perturbed and
performance of CPF with 1000 particles and REPF is
compared. The REPF that uses the concept of support
points for each particle has been found most suitable
to meet accuracy requirement for multiple delivery of
munitions, when knowledge about the system is poor.
The presentation is organized as follows. First the
TA requirement is discussed in Section 2 where the
problems of Kalman filter and conventional particle
filter convergence due to presence of nonlinearities in
the system matrix is highlighted. Following this, a
particle filter design, capable of handling the nonlin-
earities in system matrix, is proposed in Section 3.
The preformance of the designed filter is first tested
on tracking problems and the result is given in Sec-
tion 4. Following satisfactory performance of the fil-
ter in case of tracking problem, it is tuned to tackle the
nonlinearities of the TA problem and the algorithm is
detailed in Section 5. Peformance of the algorithm
for TA is analyzed in Section 5.2. Complexity being
a major issue in any particle filter design, the com-
plexity analysis of the designed filter is presented in
Section 5.3. Scope of further work in this direction is
provided in Section 6.
2 BACKGROUND
Misalignment angle results in components of veloc-
ity, attitude and position errors between mother and
daughter measurements, otherwise, measurements of
mother and daughter would have matched. In naviga-
tion, these error propagation equations form the basis
of the system equations used in the state space formu-
lation of the TA problem (Groves, 1999). Deliberate
manoeuvre is used to get appropriate differences in
velocity measurement of mother and daughter inertial
navigation system (INS). Assumption of small
misalignment angle makes the TA problem fit into a
KF framework, which is best described by the fol-
lowing state propagation and measurement equations:
x
k+1
= Fx
k
+ Bu
k
+ w (1)
z
k
= Hx
k
+ v (2)
where, x
k
is the system state vector, F and H are linear
functions of system states, w and v are white Gaus-
sian system and measurement noise with covariance
Q
k
and R
k
respectively, and, B is the control matrix.
The term Bu
k
in this context is the regulator to the sys-
tem(Bemporad et al., 2002), which computationally
corrects the misalignment to zero, without involving
any actuator for physical correction of misalignment.
The TA algorithm used in the simulation and real
time environment is described in Figure 1. State
ICINCO2014-11thInternationalConferenceonInformaticsinControl,AutomationandRobotics
6
transition matrix is computed in every iteration from
measurement provided by mother INS, which has
low noise contamination, as mother INS is costly
and more accurate. TA filter uses this externally
supplied state transition matrix to predict apriori
estimate, in the time update phase, which is corrected
using measurement to form aposteriori estimate once
measurement is available for that cycle. This system
functions well when mother INS is aided with more
accurate measurement, like GPS (Groves, 1999)
but suffers in absence of those. Transient nature of
external aiding makes the state transition matrix time
variant and erroneous, which in turn, affect overall
filter performance. Also, factors like large initial
misalignment, non white, non Gaussian system /
measurement noise, makes system non-linear and a
conventional KF fails to provide correct estimation
of misalignment in such cases (Julier and Uhlmann,
1997). Such a non-linear, time varying as well
as erroneous system necessitate the use of some
non-linear filter.
Figure 1: TA in Simulation and real time environment.
For a non-linear system, the state and measurement
equations can be expressed as follows:
x
k+1
= f
k
(x
k
,w
k
,u
k
) (3)
z
k
= h
k
(x
k
,v
k
) (4)
where, f
k
and h
k
are non-linear functions of system
states, with v
k
and w
k
being noises described as be-
fore.
PF is used extensively in non-linear TA prob-
lem (Ding et al., 2009), but suffers from sample im-
poverishment problem which can be solved by re-
sampling. Resampling is costly and its practical im-
plementation is complex, and thus avoided in real
time situation like TA problem.
3 PROPOSED FILTER DESIGN
3.1 CPF Algorithm
CPF essentially solves the recursive Bayesian estima-
tion problem by using Monte Carlo approach for a
non-linear system. A very good description of PF
may be found in (Arulampalam et al., 2002). A CPF
tries to estimate the posterior distribution p(x
0:k
|z
1:k
),
i.e. the estimate of x
k
based on the available measure-
ments and using a set of randomly chosen samples
and associated weights.
A sample importance re-sampling PF (SIR-PF) is
considered similar to a genetic algorithm because the
steps stated above are similar to the primitive steps
of genetic algorithms (GAs) (Kwok et al., 2005) (i.e.
initialize population, calculate fitness of individual el-
ements of population and evolve candidate popula-
tion by applying mutation and crossover). Moreover,
the weight calculation phase in SIR-PF bears some
similarities with the selection process used in GA as
both are probabilistic in nature. Such probabilistic se-
lection is the main cause of sample impoverishment
problem just described.
3.2 Proposed REPF Algorithm
Following (Uosaki et al., 2004), in this approach, ’n’
support points x
(i, j)
k
{i=1,2,...N
s
, j=1,2,...n}, are gen-
erated per particle x
(i)
k−1
{i=1,2,...N
s
)}, sampled from
the importance density function q(x
k−1
|y
1:k−1
), based
on the importance density function q(x
k
|x
(i)
k−1
: y
1:k
).
Weights corresponding to each particle are generated
following:
w
(i, j)
k
= w
(i)
k−1
p(y
k
|x
(i, j)
k
)p(x
(i, j)
k−1
|x
(i)
k−1
)
q(x
(i, j)
k
|x
(i)
(k−1)
,y
1:k
)
(5)
i = 1,2,...N
s
, j = 1,2,...,n The system now has
(n × N
s
) particle-weight combination x
(i, j)
k
;w
(i, j)
k
,
and one out of every n support points of a particle is
chosen to select N
s
most likely particle-weight com-
binations, from this set. In the context of the TA algo-
rithm design, N
s
distinct particles, in the form of state
vector x, are considered. Altogether n different state
transition matrices (obtained by perturbing elements
of F matrix) are used for apriori propagation of each
such state vector. Choice of most likely system dy-
namics for the propagation is based on closeness of
the corresponding apriori estimate state vector to the
current measurement vector, and the mean of these
most likely ones is considered as aposteriori estimate.
EfficientIn-flightTransferAlignmentUsingEvolutionaryStrategyBasedParticleFilterAlgorithm
7
The main difference between the proposed algo-
rithm and the CPF algorithm is the selection process
used by both the methods. Contrary to CPF algo-
rithm, proposed algorithm uses some deterministic
method for selection of particles for next time step.
The selection is (µ + λ) selection of ES, which se-
lects best µ points out of the total of (µ + λ) sam-
ples and supports rather than a (µ,λ) selection in
which selection is made from λ supports, excluding
µ points(Beyer and Schwefel, 2002). Determinism
is due to the use of known probability distribution
function (pdf) to create new population used in next
time step. This approach, is analogous to multiple
model adaptive estimation (MMAE) concept(Hanlon
and Maybeck, 2000) and is particularly useful in real
time system simulation.
4 TRACKING PROBLEM
4.1 System Used
Performances of two estimation algorithms, CPF and
REPF is compared by designing these algorithms for
tracking problem. A point moving with constant ve-
locity in two dimensional plane is considered, where
its distance and elevation angle are the measurements
(as in case of radar measurement). Relating to Equa-
tion 3 and 4 above, x
k
= [x1x2
˙
x1
˙
x2], the discretized
F matrix and measurement z are as follows:
F =
1 0 ∆k 0
0 1 0 ∆k
0 0 1 0
0 0 0 1
and
z =
tan
−1
((x(2)/x(1))
sqrt(x(1)
2
+ x(2)
2
)
+ v
where, x and ˙x denotes the x position and velocity re-
spectively. Clearly, this measurement model is non-
linear (Li and Jilkov, 2003; Li and Jilkov, 2001) and
thus, conventional KF is not applied. To simulate im-
proper system dynamics, ∆k in F is altered by incor-
porating normrnd(0,0.1) error, which in turn, pro-
duces 8% error in X-position and Y-position. Prac-
tical significance of this may be explained by the fact
that data may arrive at irregular interval which makes
the system non-linear. For the purpose of simulation,
it is assumed that, the object is initialized with the
position [10 m,10 m] in X and Y axis respectively,
and, initial velocity is assumed to be 1 m /sec along
both the axes. Acceleration is assumed normally dis-
tributed and is modelled as white Gaussian noise, w
k
.
Parameters corresponding to variables used in algo-
rithms used in this simulation are listed in Table 1.
Table 1: Parameters used in simulation of tracking problem.
Symbol Value
Q (1e− 6)/2. ∗ eye(4)
w N([0000]
′
,sqrt(Q))
R 0.05/2. ∗ eye(2)
v N([0 0]
′
,sqrt(R))
X
0
[10 10 1 1]
′
dx [0.1 0.1 0.1 0.1]
′
x
0
X
0
+ dx. ∗ randn(4, 1000)
10 12 14 16 18 20 22 24
10
12
14
16
18
20
22
24
26
28
30
Distance (X) in Meter
Distance (Y) in Meter
Trajectory Tracking plots − System Perfectly Modeled
True
CPF
REPF
Figure 2: Tracking when system knowledge is proper.
10 15 20 25
10
12
14
16
18
20
22
24
26
28
30
Distance (X) in Meter
Distance (Y) in Meter
Trajectory Tracking plots − System NOT Perfectly Modeled
True
CPF
REPF
Figure 3: Tracking fails when system knowledge is not
proper.
4.2 Result Analysis
Performance of CPF is compared with REPF. From
Figure 2 it can be inferred that, the proposed algo-
rithm works at par with CPF when the system is prop-
erly modelled. It is shown by simulation that, pro-
posed algorithm manages to track satisfactorily in sit-
uation when the system is not perfectly modelled,
while the CPF fails (see Figure 3). Proposed algo-
rithm simulates multiple models by generating sup-
port points for every particles, and thus, manages to
track the system when it is improperly modelled. Fig-
ICINCO2014-11thInternationalConferenceonInformaticsinControl,AutomationandRobotics
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Figure 4: Mean of Means of Position Errors (in Meter) in
100 Monte Carlo Runs for both filters
ure 4 shows the result of 100 Monte Carlo runs. Mean
of Means of position estimation has been considered
for relative comparison of performances of two fil-
ters. Closer the value of this quantity to zero, better
the performance of the filter. As described before, the
value of ∆k has been contaminated with around 8%
error, which results in wrong estimation of X and Y
position for CPF in wrongly modelled system, where
REPF performs satisfactorily.
5 SOLUTION FOR TA PROBLEM
5.1 TA System Model
To capture complete non-linearity, complex navi-
gation error equations can be used. Inclusion of
gyro, accelerometer error components as system state
in the state vector increases the overall complex-
ity of computation. For practical implementation
purpose, less complex error equations can be used
to estimate misalignment angle accurately. Current
work assumes the system state vector to be x
k
=
[δα δβ δγ δv
n
δv
e
δv
d
δL δh], where, [δα δβ δγ] are
attitude errors along N-E-D axes, [δv
n
δv
e
δv
d
] are
velocity errors and [δL δh] are errors in latitude and
heights as described in (Titterton and Weston, 2004).
Longitude as state variable has been omitted in cur-
rent formulation because of its negligible contribution
in estimation equations. State transition matrix is for-
mulated based on the error propagation equations as
described below.
˙
ψ = −ω
n
in
× ψ−C
n
b
δω
b
ib
+ δω
n
in
(6)
where, ψ =
δα δβ δγ
]
T
the alignment error
vector,
δω
b
ib
= (
˜
ω
b
ib
−ω
b
ib
) the gyroscopicmeasurement errors
in the slave system,
δω
n
in
= (
˜
ω
n
in
− ω
n
in
) the errors in the reference frame
rate estimates.
˙v
n
= C
n
b
f
b
− g (7)
where, v
n
is the velocity of the mother in navigation
frame, f
b
is the output of accelerometers of daughter
in body frame, along body axes and g is the gravity
vector. C
n
b
is the rotation matrix from body frame to
navigation frame. Then the estimates of velocity error
in the launcher (δv) will be –
˙
δv = f
n
× ψ+C
n
b
δf
b
(8)
f
n
is the accelerometer measurement of the master in
body frame resolved in the navigation frame, and δf
b
is the accelerometer noise.
In Equation 6 and Equation 8, C
n
b
represents the
Direction Cosine Matrix (DCM) which express the
orientation of the daughter instrument cluster with
respect to navigation frame (Titterton and Weston,
2004). For large initial misalignment, small angle ap-
proximation in constituent elements of this DCM is
not possible, which, in turn, makes the system nonlin-
ear. Referring to Figure 1, in absence of external aid-
ing (or otherwise as well), mother measurement be-
comes erroneous, which in turn, corrupt the externally
supplied state transition matrix. In such situation, par-
ticles of a CPF propagates through the erroneous state
transition matrix and thus fails to perform satisfacto-
rily. Better approach is to propagate particles through
various state transition matrices and generate support
points for each particle, which is essentially the prac-
tice followed in ES based PF.
Generation of multiple support points avoids re-
sampling technique and still produces good estimate.
Since the particles cover a larger space, the surviving
points cover a larger space than CPF where the closest
points are retained. Performance of REPF does not
noticeably improve with respect to the performance
of conventional PFs, when the knowledge of the sys-
tem dynamics is accurate. However, the REPF can
cover more non-linearity associated with the system
and yields more realistic convergence when knowl-
edge about the system is poor due to the ability of
covering wider range of solution space in terms of
generated support vectors for each particle which is
not possible in case of CPF.
5.2 Performance Analysis
The current work implements and evaluates perfor-
mances of CPF (with N
s
= 1000 particles) and REPF
(with N
s
= 1000 particles and n = 15 support points
for each particle). A regulator has been designed
to correct the misalignment computationally. Perfor-
mance of these filters have been evaluated in two dif-
ferent simulations by using Monte Carlo technique.
EfficientIn-flightTransferAlignmentUsingEvolutionaryStrategyBasedParticleFilterAlgorithm
9
Table 2: Parameters used in simulation runs.
Parameters Value
Initial Mis-
alignment
δα = 5
o
δβ = 8
o
δγ = 10
o
Gyro Drift 3
o
/hr
Accelerometer
Bias
500µg
System noise
covariance
Q(8× 8)
diag([(4.7596e − 7) (1.2185e−
7) (1.9039e − 7) (0)
1×5
])
Measurement
noise covari-
ance R(5× 5)
diag([(2.25e − 4)
3×1
(6.4e −
4) (7.284e− 4)]
One simulation portrays the situation where system
dynamics is known. In this simulation, state transi-
tion matrix has been kept unperturbed, and, all three
algorithms are executed. In another simulation, state
transition matrix has been made erroneous by inject-
ing normally distributed noise to constituent elements
of F matrix (zero mean with 10% variation), to cap-
ture the situation of improper system dynamics. Par-
ticles of REPF are propagated apriori through fifteen
such matrices, and its performance is evaluated with
that of CPF with 1000 particles in which particles are
propagated through one state transition matrix. Pa-
rameters used in the simulations are given in Table 2.
For proof of concept, trajectory data has been gen-
erated from a simulated manoeuvre. Figure 5 shows
profile of acceleration in North East Down (NED) di-
rections used to generate the manoeuvre, based on the
kinematic approach (Adhikari et al., 2002). It may be
observed that, small amount of acceleration has been
used in all three directions, which is desirable to avoid
easy detection. Typical plots of misalignment angles
(δα,δβ, δγ) of CPF with 1000 particles and CPF with
3000 particles and REPF are given in Figures 6, 7, 8.
All three filters take approximately same time to con-
verge and the accuracy is also comparable. Figure 9
and 10 depicts the RMSE plots (Mean and STDs)
0 50 100 150 200
−20
0
20
Time in Seconds
Meter / Sec
2
Acceleration in North
0 50 100 150 200
−0.02
0
0.02
Time in Seconds
Meter / Sec
2
Acceleration in East
0 50 100 150 200
−20
0
20
Time in Seconds
Meter / Sec
2
Acceleration in Down
Figure 5: Acceleration profile used.
Figure 6: Typical Plot of δα (Zoomed).
Figure 7: Typical Plot of δβ (Zoomed).
Figure 8: Typical Plot of δγ (zoomed).
of misalignment angles for 100 runs of Monte Carlo
simulation (MCS) which shows the similar nature of
convergence of all three algorithms. Comparisons of
mean of means of the RMSE values of the misalign-
ment angle estimates computed over 100 MCS runs
are tabulated in Table 3. Due to presence of regula-
tor, the expected value is zero. The values depicted
is a clear indicator of the comparable performance of
above three filters.
Next, MCS is performed for the scenario where
ICINCO2014-11thInternationalConferenceonInformaticsinControl,AutomationandRobotics
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Figure 9: Mean of δα,δβ,δγ for 100 MC runs.
Figure 10: Variance of δα,δβ,δγ for 100 MC runs.
Table 3: Performance of three filters for 100 MC runs when
the system knowledge is correct.
Mean
of
means
(in arc
min)
CPF with
1000 par-
ticles
CPF with
3000 par-
ticles
REPF
δα -3.372067 -3.542256 -3.463119
δβ -2.749540 -2.190724 -2.577882
δγ -0.022702 0.011495 0.001139
knowledge of the system is improper. While the con-
ventional filters fail to converge, REPF delivers good
convergence. Values of mean of means of the CPF
and REPF algorithm for 100 MCS runs in this sce-
nario has been tabulated in Table 4. It clearly indi-
cates the inefficiency of CPF algorithm in overcom-
ing lack of knowledge of system dynamics. In this
respect, the proposed REPF algorithm performs well.
5.3 Complexity Analysis
Total complexity of running the TA algorithm in each
iteration can be expressed as the sum of:
Table 4: Performance of CPF and REPF over 100 MC runs
when system knowledge is incorrect.
Mean of
means (in
arc min)
CPF with 1000
particles
REPF
δα 36.709314 -3.384981
δβ -100.142260 -2.591800
δγ 27.838801 0.024971
• Time to generate apriori estimates (uses common
mother INS data)
• Time to select the appropriate support vector (uses
daughter INS measurement)
• Time to arrive at aposteriori estimate (daughter
specific computation)
In the first step, the generation of apriori estimate
of the support particles is linear in number of parti-
cles but would need amortized analysis that depends
on the size of state vector. Generation of each apri-
ori estimate (support points for all particles) requires
multiplication of matrices having size of the state vec-
tor. The size of state vector is also responsible for
the space complexity to be handled by the algorithm
during execution. Any reduction in run-time memory
requirement in turn can increase the processor effi-
ciency in handling floating point operations.
The current work restricts the number of state
variables to 8. There are TA algorithms which ad-
ditionally estimate state variables like gyro and ac-
celerometer bias, which increases the number of state
variables twofold or even more (Kong and Nebot,
1999), so that the amortized value increases eight-
fold or more. The bias estimation as state vari-
ables may result in more accurate state estimation
of misalignment angles. But the present work tries
to improve performance even with inaccurate system
model, where estimation of bias cannot improve the
accuracy. Hence the overhead of these extra states
has been eliminated in this work, resulting in better
amortized complexity.
The second step involves particle sampling. A
SIR-PF with N number of particles, employs a sys-
tematic re-sampling technique which runs in O(N)
time, which dominate the overall time complexity of
the algorithm (Arulampalam et al., 2002; Carpenter
et al., 1999). REPF does not require any such re-
sampling, and, performs a sorting algorithm to choose
the best of n support points for each N
s
particles. Such
sorting requires O(n
2
) time in the worst case, yielding
an overall time complexity for REPF algorithm T(N)
as described below:
T(N) = N
s
O(n
2
)
∼
=
c O(N
s
) (9)
EfficientIn-flightTransferAlignmentUsingEvolutionaryStrategyBasedParticleFilterAlgorithm
11
where, c = n
2
. As n ≪ N
s
, it can be conferred that,
the execution time of REPF is comparable to that of
a SIR-PF. The computation of aposteriori estimate is
linear in N
s
, so that the overall complexity also stays
linear.
6 CONCLUDING REMARKS
The work addresses the issues arising out of the lack
of knowledge about time varying state transition ma-
trix that is used for system modelling. A particle filter
based algorithm based on evolutionary strategy has
been designed to tackle such scenario. Performance
of the designed filter algorithm has been compared
with CPF by employing Monte Carlo simulation. Ro-
bustness of filter performance has been studied by ap-
plying to tracking problem. Then the similar algo-
rithm is applied to handle the problem of non-linearity
in TA problem in presence of large initial misalign-
ment angle as well as poorly modelled system.
Their performance is comparable when the non-
linearity of the system is well configured. But in sit-
uations where knowledge about the system is poor,
REPF performs better than that of conventional PF,
as evident from large number of Monte Carlo runs.
Time and space complexity associated with the real
time implementation of such filter is discussed in de-
tails. It is shown that the complexity is comparable
and amortized analysis shows improvement in overall
complexity.
In the case of multiple daughter ejection, scope
of multi-threading of TA algorithms and faster and
concurrent convergence of TA algorithms needed
for multiple daughter ejection have been discussed
in (Das et al., 2009). It was identified that, some
task involving mother INS data, that is common for
all daughters, can be assigned to the mother On
Board Computer (OBC), thereby reducing computa-
tional overhead of daughter OBC. This concept will
be useful in the REPF algorithm implementation. The
computation of state transition matrix and propaga-
tion of particles for each daughter needs only mother
INS data which may be run as different threads in the
mother OBC and can be passed to daughter as and
when required. This reduces daughter processor time
overhead, which may be utilized solely for daughter
specific tasks as discussed above. This can help in si-
multaneous convergence of TA algorithms of all the
daughters.
ACKNOWLEDGEMENTS
The work is partially supported from DRDO project.
Inputs from the scientists M. Kumar, S. Das and Dr.
S.K Chaudhuri were valuable. Shrutilekha Santra
contributed as project fellow.
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