International Journal of Control, Automation , and Systems (2011) 9(4):805-813
http://www.5pringer.comlI2555
00110.1007/512555-Qll-0424-8
Modified-VSIMM Algorithm with an Application to the Naval Fire Control Technology Hui Qiang Zbuang, Chang Ho Yu, Hong Ping Gao, Jae Weon Cboi*, Young Bong Sea, and Tae II Sea Abstract: A modified variable structure interacting mUltiple model (M-VSIMM) estimator for complex hybrid maneuver target tracking is presented. The M-VSIMM could potentially be applied to fire control systems (FCS) used on warships. Target model groups were designed using 3D dimensional dynamic target models. Optimal model group selection logic was proposed, contrary to the activation and terminati on logic in the original VSIMM. The system will respond faster with optimal model group selection logic. After performing simulations, the tracking performances of the Kalman, a-p(-y), VDIE, IMM and M- VSIMM filters were compared under various maneuvering conditions. Keywords: Estimation, FCS, IMM, Kalman filter, M-VSIMM, target tracking, VSIMM.
1. INTRODUCTION A target tracking algorithm was designed for future applications to naval fire control systems (FCS). A FCS is a computer, often mechanical, designed to assist a weapon system in hitting its target. It perfonns the same task as a human gunner firing a weapon, but attempts to do so faster and more accurately. Naval firing control systems are more complex because they must simultaneously manage several firi ng guns. Weapons as well as targets move during naval engagements. Thus, the variables involved in naval engagements become more complex. The navel gun FCS comprises an infi:ared alarm system, a radar group, a guidance system, a combat management system, a fire control computer, an aim system and weapons. The infrared alann system is used to search and detect targets. Once a target has been detected, the combat management system sends an order to the radar group to track the target. The fire control computer estimates the target state via the tracking algorithm and combines the target state, ship data, and guidance data together to create a sequence of fire Manuscript received September 21, 2009; revised December II, 2010; accepted March 8, 20 11. Recommended by Editorial Board member Wen-Hua Chen under the direction of Editor Young lJ Lee. This study was supported by a grant (UD I I0007DD) from the basic research program of the Agency for Defense Development, Korea. Hui Qiang Zhuang, Chang Ho Yu, Hong Ping Gao, and Jae Wean Choi are with the School of Mechanical Engineering, Pusan National University, San 3D, Jangjeon-dong, Geumjeong·gu, Busan 609-735, Korea (e-mails: lovelyxiaoqiang@ 126.com,
[email protected], {changhoyu, choijw}@pusan.ac.kr). Youngbong Sea is with the Innovation Center fo r Engineering Education, Pusan National Univers ity, Korea (e-mail: ybseo@ pusan.ac.kr) Tae II Seo is with the Agency for Defense Development, JinHae, Korea (e-mail:
[email protected]). • Corresponding author.
o ICROS, KIEE and Springer 201 I
parameters that are used by the fi re controller. Finally, the fire controller and aim system gives the fire order to the weapons to fire. These processes above correspond to the combat management system. The structure of the naval FCS is shown by Fig. I. The larget tracking algorithm is the core technology of the naval FCS. This study aims to design an optimal tracking algorithm with broad applicability, fast response, small tracking error, and less computational burden. A modified variable structure interacting multiple model (M-VSIMM) estimator fo r complex mixed maneuver target tracking is presented. The IMM algorithm is a powerful estimator for target tracking because of its wide applicability, high precision, low computational load, and simple structure [I] . An IMM filter employs a target model witb a fixed structure. A fixed structure inadequately tracks hybrid maneuvering targets. Target models do not accurately describe hybrid maneuvers or motions. The variable structure IMM (VSIMM) was proposed following tbe improvements made to the IMM filter. The VSIMM filter is capable of handling adaptive variable mode sets tbat include many target models to track the targets in
Tar et Informat ion Combat Manilgement System
Infrared
Fig. 1. Naval fire control system structure .
.:Zl Springer
806
Hui Qiang Zhuang, Chang Ho Yu, Hong Ping Gao, Jae Weon Choi, Young Bong Seo, and Tae II Seo
I
Initialization
I
k::k+l
J
2. VSIMM ALGORITHM
~
The IMM estimation is powerful adaptive estimation approach . Most existing estimators possess fixed
.j,
I
I
IMMIM.1
t
. ActIVate a
i-
grou~
1 y
M.
structure IMMs (FSIMM); in particular, FSIMMs use a N
ou~ut
fixed set of models at all times. However, in many practical situations, especially with high-dimensional
=
M k• J Mk
systems, the FSIMM estimators need many target models to account for real system variations. However, the use
of many models in FSIMM considerably increases the computational burden. More importantly, the performance deteriorates if too many target models are used due
=newly activated group
Mo= M.,.
M,, : M. · MI;
IMM[M"I
to the excessive unlikely models. In other words, an excessive amount of models inadequately tracks perfonnance and increases computational burden.
Fusion of M" and Mk
Mj;:=MI;U M.
~
y
Terminate group M.?
~
The VSIMM was proposed in order to overcome the limitations of the FSIMM [4,5, 14]. The VSIMM algorithm does not require all of the models at the sarne time, thus reducing computational costs. The VSIMM algorithm includes an IMM cycle, a model-conditional filter (Kalman filter), and a target model group switching (MGS) logic. The MGS logic includes activation logic
Output
M k• 1 =M
N
y Terminate group Mo?
M Ia1 =M.
1
and termination logic . The activation logic determines
N
the new target model group that will tracks targets with
MIaJ = MI;. k:-k+l
better perfonnance. The termination logic prepares the new target model group and terminates the group exhibiting the worst tracking performance.
~
I
I
IMM[MI:+11
I Fig. 2. Flowchart of the VSIMM.
2. I. Dynamic target model System Equations:
complex and hybrid maneuver conditions. A VSIMM filter flowchart is shown in Fig. 2. However, VSIMM algorithms are designed with target model groups based only on constant acceleration models with varying initial acceleration vectors in 2-
State equation: X k = Fk _1 X' _1 + Gk-iW'_1 , Measurement equation: Z, = H,Xk + V"
(I) (2)
for 3-dimensional space is difficult because a large number of target models will be created in 3 directions.
where Xk is the state vector, Fk is the state transition matrix, Ok is the system noise input matrix, Zk is the measurement vector, Hk is the measurement matrix, Wk_\ and Vk are the white noises whose covariance are Qk and
Thus, the model group design logic will no longer be
R..respectively.
suitable. Furthennore, it poorly tracks perfonnance when the accelerations of the maneuvers are out of bound.
2.2. IMM algorithm
dimensional space [5] . Designing target model groups
Step 1: Model-conditional initialization.
We proposed a modified VSIMM filter in order to compensate
for
the
aforementioned
I) Predicted mode probability:
shortcomings,
producing a new algorithm called the M-VSIMM filter. (3)
The target model groups were redesigned using a new
method in 3 dimensional space through the Constant Velocity (CV) Model [8], Constant Acceleration (CA) Model [8], Abrupt Acceleration (AA) Model, 3D Constant Turn (CT3) Model [9] and Singer Models with different maneuver times [10]. Furthermore, we proposed an optimal model group selection logic instead of the activation and termination logic in the VSIMM
given by Li [4]. The VSIMM filter will be able to track the target in 3D space based on the new model group switching logic. The simulation results were then compared to the tracking performance of Kalman, a-p(-y), VDIE, IMM and M-VSIMM filters during non-maneuver conditions,
hybrid maneuvers, respectively.
and
highly hybrid
maneuvers,
where mj
7(ij
is the probability of switching from model mj to
belonging to model group Mk• Its value is given by the
model transition matrix 7(;
probability ofmj and 2) Mixing weight: J..i
,uLI
;Ii _ Pi ; I Mj M mk_\ k' k -\
fii'lk-l
is the predicted mode
is the mode probability ofm;.
,Z
k- I}_
"ij,uk-I
- - 'J- . - '
(4)
J..iklk - I
where I./ i is the mixing probability of switching from model m j to mi. 3) Mixing estimate and mixing covariance
Modifred-VSIMM Algorithm with an Application to the Naval Fire Control Technology
xO i =£[xk-1 \ m£, M '_I ,z'- I ] = .
I
xLI, _ullli , (S)
m/e Mk. _]
i M pOJ-- £[(xk -xOJ)(xk _xOi )' \ mk' z'- I] k-I'
"[p,' (' Oi L.... k- Ilk- I + X
=
,I
- Xk-Jlk-J
)
(6)
(XO i -xLI'_I)']f'~i,
where iO) is the mixing estimate state of mj and is the mixing covariance of mj. Step 2: Model-conditional fil ter (Kalman Filter).
pO}
2.3. VSIMM algorithm loop The main steps ofVSIMM algorithm are as follows: Step 1: IMM Algorithm Cycle. Increase the time counter k by 1. Run the IMM [M,] cycle. Step 2: Activation Logic. Check if a candidate model-group is activated. Activate group f0 while M,=M, where ; '" j, jfboth of the following two conditions are satisfied: .
Probability condition:
xi =E[xk I m1, Mk_ .z,t_d= F/_I xo J
+GLlwk'_I'
(7)
= FLI P Oi CF!_,)'+ GL,Qi_I(GL)' ·
(10)
S) Filter gain: l
Here la is the design parameter known as the probabili-
X,\" p,\"
and {f'ilm,EM, which can be obtained from the IMM
4) Residual covariance:
K i = pi (H ; )'(Si r
(19)
(8)
,i =z,-E[z,\m;,Mk-I,z'-I]=z,-Hfxi-'ii{ (9)
= cov[,i \ m; , M'_I,z'-I ] = H{pi (H; )'+ R{
,4 > la '
Step 3: IMM Cycle with New Mode l Group If no model group is activated, then output
3) Measurement residual :
Si
(18)
ty threshold.
2) Predicted covariance:
p i = E[(x, - x J)(x, - xi)' \ m£, M'_I ,z'-I]
I
Likelihood condition: L{ = maxm/ EM,t Lk •
1) Predicted state: j
807
[M,] cycle. Let MHI = M, and go to Step 1. Here M, becomes the new target model group. If a model group M. is activated, then let ko=k, Mo= M, and run the IMM [M,] cycle, where M,= M. Mo is the set of new and only new models. Let M , = M, U M o = M. U M o. Calculate the estimates, error covariance, and model probabilities for the union set M, :
(11)
(20) 6) Updated state: x'iklk
= E[ xk \ rn ki , M k_ l'z '] =x-i + Ki-i z.
(12)
(21 )
7) Update covariance:
p,klA: i = E[( 'i)( ' Xk - Xk lk Xk
I
Pk\k = 'iklk ) ' \ rn ki , M k-J, Z '] - X
(22)
(13)
where the estimations {x~lk}' error covariance {pl lk}' likelihoods {L',} and predicted probabilities {,u~IH}
= pi _ K i Si( K i)' . Step 3: Mode probability update. Likelihood function:
Li = P[-z \ rn ki , M k-I' Z ' - I] = N[-I z ;0. Si ] •
[(xi\, - £,\, )(5:;\, - Xk\k ) + P:\, J.uk,
m;EMJ:
were obtained in the above IMM [Md and IMM[M,] (14)
cycles. Output xklk' Pklk , and {J..i~}mi EM,t' Step 4: Termination Logic
where i is measurement vector. Mode probability:
For model groups ~= Mo, Ma , compute
L
J.l;f/ =
(IS)
J.l~ .
(23)
MjEM/
ft:(LJ =
Step 4: Combination.
L
ft~lk-l'
(24)
IIIjEAf/
Overall estimate: (2S)
(16)
If
Overall covariance: P'I' = E[(x, - x'I' le x, - x'I' )' \ M, ,M'_I ,z'] =
I mJE M,t
[P4, + (X' I' - xil')(X'I' -x;I')'J.uf.
(26) ( 17)
then terminate group M.: let M, ., = Mo and go to Step 1.
808
Hui Qiang Zhuang, Chang Ho Yu, Hong Ping Gao, Jae Wean Chai, Young Bong Sea, and Tae II Sea
If Ma
~> t# Mo 2
k
LMa
TI -'->
or
'="
fl,
Mo
(27)
IL 2
L,
then tenninate group Mo: let M.. , = M, and go to Step I. Here Ii and If is the group probability ratio threshold; 1/ and 1/ is the group likelihood ratio threshold. They are all design parameters. Step 5: IMM Cycle with Un ite Model Group If there is no tennination, increase the time counter k by I and let = M,. Run the IMM [Mk+d cycle. Go to Step 4.
M,.,
2.4. Model group design In the original VSIMM algorithm, target model groups were designed using constant acceleration models with different initial acceleration vectors in 2-dimensional space. The system dynamic equitation of the CA model is presented in (28).
T'
T
-
2
T
0
=
xk+!
0
0
0
0
0 0
0
0
0
T
T' T
0
0
0
0
I
0
0
0
0
0
0
0
0
0
0
T
0
0
0
-T'
0
0
z, =[~
0
0
2
0 0 0
x, + 0
2
0
T'
m, :
0
= [0
~]Xk + v"
2
T
3. MODIFIED-VSIMM ALGORITHM
20]'
m, :a =[-20 0]'
m, :0 = [- 20 m9 : a
= [20
mll : 0 = [0
20]'
(30)
(28)
m, :0 = [20 m6 : 0
M\ = {mJ, m2, m), m." ms}, M2= {m2, mJO, m6, mJ, m9}, M)= {m ), mJl , m7, mJ, m6}, M4 = {m4' m12, mg. m], m7}, Ms= {m s, ml3, m9, m" mg}.
0
0]'
ms : 0 = [0 - 20]
3) Each model group should have at least one common model for switching between the common model group and other groups. 4) The number of models in each model group should not be excessive. Otherwise, it will increase the computational quantity for each IMM cycle. Equation (30) provides the model groups for modelgroup switching,
w"
where Wk is the process noise with variance Q; Vir: is the measurement noise with variance R; T is time interval of sampling. Fig. 3 represents the total model-set, and each model is characterized by the expected acceleration vector as given in (29).
m, :0 = [0
Fig, 3. Graph-representation of the total model-set.
= [20
0]'
20]
m,:0 =[-20 -20]
-20]
mlO :0=[40
40]
mll :o =[-40
3.1. Model group design Many target tracking models have been proposed and implemented in different applications. For example, previously developed models include the constant velocity (CV) model, constant acceleration (CA) model, abrupt acceleration (AA) model, singer model, jerk model [II], Song model [12], Berg model [13], constant tum model(2D) [8]. curvilinear model [8], constant tum model (3D), variable turn model (3D) [8]. Our work utilized the CV, CA, CT (3D), AA, and Singer models operating under different maneuver time points as the total model set.
(29)
0] 0]'
3.1.1 3D dynamic target models I) Constant Velocity Model In continuous-time, the state equation is:
mil : 0 = [0 -40]'. The model-group design should consider the following aspects: I) The initial model group should include the common target models that usually appear in practical conditions . Such as m], tn2, m) , tn4, lTIs. 2) Every model group should include a most common model, such as m i . It can reduce the frequency of switching between the model groups.
X(I) = Ax(t)+Bw(t),
A=[~
a
B=[~J
(31)
From the discretization, the discrete-time equation is :
X,+, = k+I.'X, +W"
,.,., =[~ ~J
(32)
The covariance of the discrete-time process noise WIr: is:
Modified·VSIMM Algorithm with an Application to the Naval Fire Control Technology
I [T - r fT[r -r]
Q=cov(W,)=
1)qdr
0
(33) =[T'/3 T2/2 q=
E[W(I)WT(I)).
B09
if: perturbing acceleration and angular velocity covariance. 4) Singer Model The Singer model assumes that the target acceleration aCt) is a zero-mean stationary first·order Markov process with autocorrelation
(34) (41)
2) Constant Acceleration Model In continuous·time, the state equation is: 0
x(t) = Ax(l) + Bw(t),
A=
[
The state vector is x(l) = [x(l) x(l) x(I)]'.
I
°° °°
Power spectrum:
so(w) = 2aa'/(w' +a').
In discrete· time, the equation is:
X'+I = ¢>k+I" X,
T'/20
+ W"
T'/8
[~
X(I) = Ax(l) + Bw(t),
A=[~ ~ ~], B=[~]. ° ° -a
T'/6 2
Q= T'/8
T'/3
r' / 6
T2 / 2
I
¢>k+I,k =
In continuous-time, the state equation is:
r / 2 Sw, T
(37)
(42)
I
In discrete-time, the equation is:
J
Power spectral density: Sw=2aa'. 3) 3D Constant Turn Model The state vector is:
(43)
a=llr: is the reciprocal of the maneuver time constant r.
For continuous-time, the state equation is:
T'/20
=[~: ~ ~:]X(t)+[~:]W(I)' ° 0 1 = [-my
Q = cov(W,) =2aa'
x(l)
0,
0,
0,
-Ill,
M
(38)
-Olx'
0
t1Jx
a = !'lxv, (39)
v = [x
y iJ':
a = {i
y
0. =
[{J)x
velocity vector,
(j)z ( :
angular velocity vector.
= xk _1 + [vk_I '
[ I,
B
= 0, I, +A 0,] 0, ,r= [0,] 0, ,Q= [0, 0, 03 I, I, 0, 0,
(44)
T
3.1.2 Target Model Define We defined the target models as: m ,: Constant Velocity Model(CV), m,: Constant Acceleration Model(CA), nl3: Constant Turn 3D(CT), m,: Abrupt Acceleration Model(AA), m, : Singer I r=60s (lazy turn), m6: Singer2 r=60s (evasive maneuver). In,: Singer3 r=60s (atmospheric turbulence), where r is maneuver time.
Mo: [mh M l : [ml' M,: [m" M3: [m" M4 : [m I.
For discrete·time, the equation is: xk
T' / 6
;:
~:;~ ;~l'
T2 / 2
3.1.3 Model group definitions
it: acceleration vector, (j)y
I
13
Illy
Wz
T' 18
0, 0,
0,] 0) 0- 2 ,
0,
I,
(40)
m2. m3. m4. m~" m6. m7] m2. m4]
m" m,l
m4, m,l m2. m7]
Here, Mo is total target model group used in the model group switching logic; Ml is the common model group set as the initial model group; M2 is used to track the target during turns; M, is used to track the target during
810
Hui Qiang Zhuang, Chang Ho Yu, Hong Ping Gao, Jae Weon Choi, Young Bong Sea, and Tae II Seo
high maneuvers; M4 is used to track the target during low maneuvers. The total target model group transition probability matrix given as:
I
I
Initial ization
,lk:=k+l
I
{-
I IMM 1M.! ZJ I
7rMO ::::
0.45
0.05
0.05
0
0.05
0.1
OJ
0.05
0.42
0.1
0.2
0.1
0.08
0.05
0.05 0. 1
0.63
0
0.1
0.07 0.05
0
0.2
0
0.5 0. 1
0.2
0
0.05
0.1
0.1
0.1 0.5
0.1
0.05
0.1
0.08
0.07
0.2
0.1
0.25
0.2
0.3
0.05
0.05
0
0.05
0.2
0.35
-1 M1
