A Decision-Based Model and Algorithm for Maneuvering Target Tracking

Transcription

1 WSEAS RANSACIONS on SYSEMS A Decision-Based Model and Algorith for Maneuvering arget racking JIAHONG CHEN ZHONGHUA ZHANG ZHENDONG XI YONGXING MAO China Satellite Maritie racking and Control Departent, Jiangyin, Jiangsu, 4 43 stone_cjh@sina.co; zhangzh_stc@sina.co; dong369@hotail.co; yx88@o.co Abstract: he decision-based odels for aneuvering target tracking were studied in this paper. Focusing on the proble of dissatisfied with single odel tracking and the optial odel-set is difficult to design of the ultiple-odel (MM) algorith, we odified the current odel, and proposed an adaptive single odel (ASM) to track angular otion. he united velocity-acceleration estiation-based and direct easureentbased decision algoriths for odel-switching were provided and discussed. At the sae tie, we discussed the question of the thresholds setup, and gave a significance test ethod based on hypothesis testing theory. Siulation results show that the tracking perforance of this new adaptive single odel is stationary, and is iproved uch ore than the current odel, also better than the interacting ultiple-odel (IMM) for strong aneuvering target tracking. Furtherore the coputational load of the new odel can be less than that of IMM. Key-Words: adaptive single odel, ultiple-odel, aneuvering target tracking, current odel, decisionbased algorith, odel-switching Introduction he key to successful target tracking lies in the effective extraction of useful inforation about the target s state fro observations. A good odel of the target will certainly facilitate this inforation extraction to a great extent. [-6] gave a coprehensive and up-to-date survey of the techniques for tracking aneuvering target, especially of the tracking odels. Interrelationships aong odels and insight to the pros and cons of odels were provided by []. In the history of the developent of aneuvering target tracking (M) techniques, single odel based adaptive Kalan filter free of decision cae into existence first. As one single odel can only atch one otion ode best, and the perforance of tracking tie-varying or strong aneuvering target is dissatisfied, so the decisionbased algorith was proposed, by [4], which belonged to the switching-odel approach in a broad sense. he followed approaches of ultipleodel (MM) algoriths have becoe quite popular for M, aong which the interacting ultipleodel (IMM) is ore preferable [5-0]. he universal aneuvering target tracking techniques based on single odel are adaptive Kalan filtering algoriths [,, 3], which faced the difficulties to find a suitable bank of paraeters to atch the syste otion ode precisely. he algoriths, which use statistical ethods with ulti-banks of paraeters to describe the otion ode approxiately, were called MM algoriths. o iprove the tracking perforance of MM will face to coplex odel-set designing and burdened calculating. It was shown that the syste is optial only when there is one odel in the odel-set and the odel also ust atch the otion ode copletely by [8]. Here we will study the decision-based algoriths and try to avoid the coplex designing of the odel-set and to find a ore suitable single odel for M. As surveyed by [4], for M, there are three ain classes of decision-based techniques: equivalent noise, input detection and estiation, and switching odel, which are based on decisions regarding to target aneuver. In a broad sense, algoriths in the equivalent noise or input detection and estiation approaches also belong to the switching odel approach since following different decisions these algoriths taking different actions, which ay be construed as filtering based on different odels. he fundaental proble is the detection of aneuver onset and terination. wo popular choices for the detection ter are the ISSN: Issue 8, Volue 8, August 009

2 WSEAS RANSACIONS on SYSEMS easureent residual z and the input estiate u, others are based of the or of there cobined functions. In the open literature available to us, one decision-based algorith usually uses only one detection ter, which can not be ore adaptive to the tie-varying aneuver generally. Moreover there are only two single odels in essence which are switched each other, e.g., between CV (constantvelocity) and CA (constant-acceleration) [4] or between CV and Singer (Singer Acceleration) Models [5], etc. he ethods can generally be called as estiation-based ethods where u and z or v (velocity) are all estiation-based. he precision of estiation is relative to the syste noise (process noise and easureent noise) and odel reliability. Generally the infection of noise can be decreased by odel filtering. We can also take the direct easureent as detection ter which will be siple. Furtherore the validity of odel is irrelative to the decision, whereas the noise infection will be increased. In this paper we intend to propose a united velocity-acceleration estiation-based decision algorith (EBDA) and a direct easureent-based decision algorith (MBDA) with a new single united aneuvering and nonaneuvering odel. he rest of the paper is organized as follows. Section presents a new adaptive odel. he united velocity-acceleration estiation-based decision algorith and thresholds setup are presented in Section 3 and 4. Section 5 gives the easureentbased decision algorith with the new odel. Coparison of the new algoriths and IMM for siulated different trajectories is given in section 6. Concluding rearks are provided in Section 7. he New Adaptive Single Model. he Singer Model and Current Model he CV odel is ore proper for non-aneuvering target, while the CA odel is ore suitable for the otion whose acceleration derivative (i.e., jerk) a(t) is an independent process (white noise) w(t) : a ( t) = w( t). he Singer odel assues that the target acceleration a(t) is a zero-ean first-order stationary Markov process with autocorrelation α τ R a ( τ ) = E[ a( t + τ ) a( t)] = σ e. Such a process a(t) is the state process of a linear tie-invariant syste: at () = αat () + wt () () where a = x, is the acceleration; σ = E[ a( t) ] is the instantaneous variance of the acceleration; α = / τ is the reciprocal of the aneuver tie constant τ and thus depends on how long the aneuver lasts. For exaple for an aircraft, τ 60s for a lazy turn and τ 0 0s for an evasive aneuver, for the issile τ aybe a little longer []; w( t ) is zero-ean white noise with constant power spectral density S w = ασ. As the aneuver tie constant τ increases or decreases the Singer odel corresponds to a otion in between of (nearly) CA and (nearly) CV. It should thus be clear that the Singer odel has wider coverage than CV and CA odels [, 6]. Another acceleration odel, called the current odel by its authors, proposed by [6], is in essence a Singer odel with an adaptive ean of acceleration that is ean-adaptive acceleration (MAA) odel: at () = αat () + αat () + wt () () With state vector x = [ x x x], where the superscript denotes the transpose of atrix, the state-space representation of this odel is: x () t = 0 0 () t 0 a() t 0 x + + w () t (3) 0 0 α α excluding the tie instants at which saples are taken since a ( t ) is assued piecewise constant. he discrete-tie equivalent is: α ( + α + e )/ α α xk+ = 0 ( e )/ α xk + α 0 0 e α ( /) ( + α + e )/ α α ( e ) / α ak + wk α e = Fαxk + Bαak + wk (4) where is the sapling interval. As α <<, the covariance of the noise is given by: w k /0 /8 /6 4 3 Qk = ασ /8 /3 / (5) 3 /6 / A key underlying assuption of the current odel given by [6] (but not so stated explicitly) is that ak+ = aˆ k k, that is to use the last estiation as the current ean. his is questionable and can be actually avoided [, 6]. Getting the expectation of the last ite of (4), we have: ISSN: Issue 8, Volue 8, August 009

3 WSEAS RANSACIONS on SYSEMS α α ak+ = E[ ak+ zk] = e E[ ak zk] + ( e ) ak α α = e aˆkk + ( e ) ak = aˆk + k where z k is the easureent vector. (6). Modification to the Current Model Application of the current odel can be found in the literatures. A weighted current odel adaptive Kalan filtering algorith is suggested by [7], while an iproved adaptive filtering algorith of noise variances self-adaptation was given by [8] based on the current statistical odel. Here we take into account the atching proble of the odel. he process of target tracking is the servo syste of radar driving antenna to point to the target with tie-varying angular velocity and acceleration controlling instruction. By the sense of physics, it is viable to use ean-adaptive acceleration odel to atch the relative angular otion of the target. Actually, the angular velocity and acceleration are often tie-varying for radar target tracking, especially for rectilineal otion with constant velocity the angular otion is not constant. As τ 0s and the sapling interval is sall ( 0.5s, it s possible), expand e α of the current odel (4) with aylor series: α 3 e = α + ( α) ( α) + (7)! 3! Substituting (7) into (4), ignoring the high order ite, we can obtain nearly current odel: 3 3 / α /6 α /6 xk+ = 0 α / x k + α /ak + wk 0 0 α α (8) It s not difficult to find that the result of (6) is also correct for (8). As the new odel should be universal and adaptive, we odified the odel of (8) by iporting the adaptive factor λ : 3 3 λ( / α /6) α /6 xk+ = 0 λ( α /) xk + α / ak + wk 0 0 λ( α) α (9) Denotes (9) as: x k+ = Fk x k + Bkak + w k (0) where λ only can be 0 or, α is also a fixed value (in this paper we choose 0.) or zero. Analyzing the odel of (9), we can see that: ) When α and a k are not equal to zero, and λ =, the syste is nearly equivalent to MAA odel; ) When α 0 and λ = but a k = 0, the syste is nearly equivalent to Singer odel; 3) When α = 0 and λ =, the syste is nearly equivalent to CA odel but for rando noise ite; 4) When α = 0 and λ = 0, the syste is nearly equivalent to CV odel but for rando noise ite. We call the universal odel of (9) as iproved adaptive single odel (ASM). It s not difficult to find that if the paraeter α and λ varying in real tie, this new odel can atch constant-velocity, constant-acceleration and tie-varying acceleration otion, so it has strong adaptability. 3 United Estiation-Based Decision Algorith 3. Adaptive Rules o atch the target otion exactly, ASM should be switched in real tie between MAA, CA and CV according to the easureents and states estiation. We will design the adaptive rules with united velocity-acceleration estiations by distinguishing three kinds of situations. he ean (average) and odified standard deviation (MSD) of acceleration and velocity in continuous steps are: µ a, = aˆ k ik i () i= / σ a, = ( ( aˆ k ik i µ a, ) ) i= µ v, vˆ k ik i i= / σv, = ( ( vˆ k ik i v ) ), i= k () = (3) µ (4) where denotes current step of filtering and estiating. Here can not be too large because of aneuver will take place at any tie, so we use the odified standard deviation instead of the standard deviation by statistical theory. he adaptive rules for odel-switching are followed as: ) If the syste is under MAA odel currently ( λ =, α 0 ) (a). Meeting the condition σ a, < δ, i.e., when the MSD of acceleration of the final steps is sall enough, the odel should be switched to CA odel, which eans λ =, α = 0. ISSN: Issue 8, Volue 8, August 009

4 WSEAS RANSACIONS on SYSEMS (b). Meeting the condition µ a, < δ and σ v, < δ 3, i.e., when the MSD of velocity of the final steps is sall enough and the ean of acceleration is near to zero, the odel should be switched to CV odel, which eans λ = 0, α = 0. ) If the syste is under CA odel currently ( λ =, α = 0 ) (a). he (b) of () is still applicable. (b). Meeting the condition σ a, > δ 4, i.e., when the MSD of acceleration of the final steps is large enough, the odel should be switched to MAA odel, which eans λ =, α 0. 3) If the syste is under CV odel currently ( λ = 0, α = 0 ) Meeting the condition σ v, > δ 5, i.e., when the MSD of velocity of the final steps is large enough, the odel should be switched to CA odel, which eans λ =, α = 0. If the conditions are not et, the odel should be hold. 3. State Estiation in Spherical Coordinate-Syste Many studies of target tracking are focused on describing the otion ode in inertial Cartesian coordinate-syste (CS). But radar syste can only provide the bearing (or aziuth) b and elevation e easureents, and possibly soe radar can also provide range r or range rate (Doppler) r (in spherical CS). o study target otion in the Cartesian CS is the direct ethod, but the coplex nonlinear odel ust be introduced because of no x, y and z easureents in Cartesian CS. All kinds of approxiation and transforation will bring on bias of angular controlling instruction. We consider that studying independently of radar tracking in spherical CS fro state estiation in Cartesian CS is doable, because in spite of any way the radar and target oving, the essence of tracking is to ascertain the relative bearing and elevation angular otion of the target proptly and exactly. An angular error tracking filter and odel was researched by Ekstrand B. in [9], where only the nearly constant-velocity (CV) odel was used to estiate the inertial angular rate of the line-of-sight (LOS), whereas no optial angle estiation directly. Here we will estiate the angle directly. Because the range r, bearing b and elevation e in spherical coordinate-syste of the radar syste can be considered as independently, and the state and easureent equations of the three channels are the sae, so we can filter and estiate the three channels independently with the sae algorith. he state equation of the syste is just like (9), whereas the easureent equation is: zk = Hxk + vk (5) = [ 0 0] xk + vk where z k denotes the easureent and v k is the easureent noise, the covariance of vk is Rk. We can see that the odel is linear, despite it is tievarying, so the optial Kalan filter can be used here. o take the bearing b channel for exaple, denotes x = b b b, than the EBDA for state estiation ainly consists of two steps: value F k, B k by the adaptive rules and estiate the state with Kalan filter. he coplete steps for the state estiation algorith are concluded as in able. We proposed the new odel based on angular otion of radar target tracking, which can atch the unifor otion and variable otion together. It is suitable for range channel at the sae tie too, which is just the exhibition of the strong adaptability of the new odel algorith. able he EBDA of ASM for State Estiation. value Fk and Bk by the adaptive rules. odel-conditioned filtering (if switching filter fro k ) Predicted state: xˆk+ k = Fkx ˆkk + Ba k k Predicted easureent: z ˆk + ˆ k = Hx k + k Predicted covariance: Pk+ k = FP k k kfk + Qk Measureent residual: z k+ = zk+ z k+ k Residual covariance: Sk+ = HPk+ khk + Rk+ Filter gain: Kk+ = Pk+ kh Sk + Updated state: xˆk+ k+ = xˆk+ k + Kk+ z k+ Updated covariance: Pk+ k+ = Pk+ k Kk+ HPk+ k 4 hresholds Setup he key to successful perforance iproving of ASM tracking lies in the exact setup of the thresholds δ, δ, δ 3, δ 4 and δ 5. Besides experient and experience, we will analyze in view of probability and statistics theory. 4. hresholds of Variable Motion ake the statistic of acceleration for exaple. If the syste oves with constant acceleration, we can ISSN: Issue 8, Volue 8, August 009

5 WSEAS RANSACIONS on SYSEMS assue the acceleration has Gaussian distribution N( µ a, σ a ). he first oent (ean) µ a varies by the setup, whereas the second oent (covariance) σ a is relatively fixed, and depends on the perforance of the syste, such as sapling interval, easureent precision, etc.. We also can say that σ a is deterined by the syste, and can be obtained approxiately by experient sapling. If σ a, σ a obviously under MAA odel, we can consider the acceleration is nearly constant reasonably; on the other hand, if σ a, > σ a obviously under CA odel, odify the odel to MAA is also believable. Definition : assue F(y) is the distillation function of rando variable Y, p is a real nuber, 0 p, if: P{ Y yp} = F( yp) = p (6) where P{ } stands for probability. We call y p the p percentile of the distribution. o ascertain δ and δ 4 is a proble of hypothesis testing. ) Inferior (lower) liit δ We establish the original hypothesis H 0 : σ a, σ a against the alternative hypothesis H: σ a, < σ a. Definition : ype error is the error ade when rejecting H 0 while H 0 is true. ype error is the error ade when accepting H 0 while H is true. o control the risk of istaking σ a, < σ a when σ a, σ a actually, i.e., the ype error is less than β (significance level). According to the Gaussian distribution of a, the rando variable ( a ˆk ik i µ a, ) σ a i= is chisquare distributed ( χ ( ) ), so the rejection region W (rejecting H 0 and accepting H ) is: { ( aˆk k,, aˆk k ) : ( ˆ ak ik i µ a, ) σ a i= σ a, χ ( β ) } a ( ) = σ (7) χ β ( ) is the β percentile of χ ( ) distribution. If β is sall, rejecting H 0 and accepting H is credible by event with sall probability should not take place in one experient. χ Here we take β = 0. and = 5, query the distribution table, χ 064, so when 0. ( 4 ) =. σ a, 0. 56σ a, (7) coes into existence. hat is to say we can set δ = σ a /. ) Superior (upper) liit δ 4 Here we should control the risk of istaking σ a, > σ a when σ a, σ a actually, so we establish the original hypothesis H0: σ a, σ a against the alternative hypothesis H: σ a, > σ a. he rejection region W of the rando variable ( a ˆk ik i µ a, ) is: σ a i= aˆk k aˆk k ˆ ak ik i µ a, σ a i= ( ) σ a, = > χ β ( ) } σ a { (,, ) : ( ) (8) β, then χ 0 = 7.779, so when If = 0..9(4) σ a, >. 395σ a, (8) coes into existence. hat is to say we can set δ4 =. 395σ a. If we set δ 4 = σ a, then β < , so the reliability of H : σ a, > σ a is very high ( > ). In brief, we can set δ near to σ a / and δ 4 near to σ a fro the above analysis. In fact the setup of δ and δ 4 should also accord with the experient results. wo questions ust be paid attention to. he first is that σ a / and σ a are only reference values because can not be too large by the realtie requireent. he deterinistic value ust be experienced typical experients. Second, σ a, is relative to the sapling interval, also sae with σ a, so when is varied the thresholds should be varied accordingly. 4. hresholds of Unifor Motion he ethods of setup of δ 3 and δ 5 reseble to the above δ and δ 4. Assuing the syste is under CV odel, and the distribution of velocity is N( µ v, σ v ), so δ 3 can be near to σ v /, and δ 5 can be near to σ v. he atching perforance of CV odel is inferior to that of CA and MAA odel except that the syste is actually oving with constant velocity. Switching the CA and MAA odel to CV odel ISSN: Issue 8, Volue 8, August 009

6 WSEAS RANSACIONS on SYSEMS should be cautious, so we appended the constraint of acceleration ean: µ a, < δ. he physical eaning of this condition is obvious. If the acceleration noise is N(0, σ a ), we establish the original hypothesis H0: µ a, = 0 against the alternative hypothesis H: µ a, 0. he rando variable µ a, 0 σ a is standard Gaussian distribution N(0,), so the confidence interval W0 should eet: µ a, σ a < u β / (9) where u β / is the percentile of Gaussian distribution, β = 0. as the above tests, so we have: u β / u σ a µ a, < σa = σa = σa = (0) δ is near to σ a /. 36. Different fro the above tests, accepting H 0 should be credible here. Here we should also control the ype error, i.e., control the risk of istaking µ a, = 0 when µ a, 0 actually. As can not be too large, according to the hypothesis testing theory we can increase β. In order to be ore credible of H 0 : µ a, = 0, δ can be set between ( ~ )σ a 0 conservatively by experient. By the way, the short sapling interval and suitably large can iprove the stability of statistical data, and is of advantage to atch otion ode in real-tie. 5 Direct Measureent-Based Decision Algorith and Discussion 5. Adaptive Rules he first-order difference of position ( ) is z,k relative to the target velocity. he second-order difference of position ( z,k ) is relative to the target acceleration, and the third-order difference of position ( z 3,k ) is relative to the target acceleration derivative. Denote: z, k = zk+ z k () z, k = z, k+ z, k () z3, k = z, k+ z, k (3) Regarding of the randoicity of easureent, the average (ean) of the n th-order difference in steps is used here (denote as z,k, z,k and z 3,k ). We give the thresholds of velocity η and η, and of acceleration η3 and η4. ) If the syste is under MAA odel currently ( λ =, α 0 ) (a). Meeting the condition z 3, k < η 3 and z, k < η, i.e., when the acceleration and its derivative of the final steps are sall enough siultaneously, the odel should be switched to CV odel, which eans λ = 0, α = 0. (b). Meeting the only condition z 3, k < η 3, i.e., when the acceleration derivative of the final steps is sall enough, the odel should be switched to CA odel, which eans λ =, α = 0. ) If the syste is under CA odel currently ( λ =, α = 0 ) (a). Meeting the condition z 3, k > η 4, i.e., when the acceleration derivative of the final steps is large enough, the odel should be switched to MAA odel, which eans λ =, α 0. (b). Meeting the condition z 3, k < η 3 and z, k < η, i.e., when the acceleration and its derivative of the final steps are sall enough siultaneously, the odel should be switched to CV odel, which eans λ = 0, α = 0. 3) If the syste is under CV odel currently ( λ = 0, α = 0 ) (a). Meeting the condition z, k > η and z 3, k > η 4, i.e., when the acceleration and its derivative of the final steps are large enough siultaneously, the odel should be switched to MAA odel, which eans λ =, α 0. (b). Meeting the only condition z, k > η, i.e., when the acceleration of the final steps is large enough, the odel should be switched to CA odel, which eans λ =, α = 0. he estiation steps and thresholds setup of this MBDA are siilar to that of EBDA in able. 5. Discussion he new algoriths of ASM can switch to the better atching odel based on the analysis of the state estiation or the direct easureents. If the real target trajectory is taken as an unknown coplex curve, than the new ASM will fit the ode coprehensively with first degree, quadratic and cubic curves. It is ore adaptive than the traditional single odels or switching-odel approaches only with first degree and quadratic curves. If the atching odel of ASM is appropriate, than the tracking perforance will be iproved especially ISSN: Issue 8, Volue 8, August 009

7 WSEAS RANSACIONS on SYSEMS for the strong aneuvering targets. he siulation results have shown this well. We give a decision-based switching-odel approach for M, which is different fro the traditional switching-odel approaches. First, the nearly n th-order ( n =,, 3 ) odels were used to fit the otion ode, the fitting perforance of the new algorith should be better. Second, the switched odels have a unifor expression, odelswitching are realized by revaluing the paraeters. hird, the union of velocity, acceleration and jerk is taken as the detection ite rather than that only one of the is used in the traditional switching-odel approaches. Forth, the physical sense of the detection ter and the thresholds are obvious and understandable. One iportant assuption is that if the syste oves with constant acceleration (or constant velocity), the acceleration (or velocity) has Gaussian distribution. In the new algoriths, if the ode variation was detected at tie k, the odel should be switched fro k. When is sall enough (< τ ), switching fro k or k has no obvious difference. Switching fro k can lighten the coputational load. On the other hand if τ is saller, the aneuver acceleration can be considered as noise. For EBDA, considering the statistical reliability of the detection ters, the step can not be too sall. With the acceleration to be estiated, > 3. In view of real-tie coputation, can not be too large. Generally when <= 0, the coputational load is nearly approxiate to the IMM algorith. For the coputational syste with high perforance, if is very sall, than can be larger. For MBDA, the third-order difference should be calculated, then to copute the average. he covariance of the n th-order difference is in (direct) n n σ σ z proportion to (it can be proved about z, is the variance of easureent z ). should be large enough to reduce the infection of noise, > 5 generally. Sae as the EBDA also can not be too large ( < τ ). he coplexity of the two algoriths corresponds to each other. Fro the siulation results we can see that the tracking perforance of MBDA+ASM is also satisfied, which shows the strong adaptability of ASM. 6 Siulation In this section, we copared the perforance of ASM with EBDA to the classic MAA and IMM algoriths by siulation. MBDA based of ASM is also siulated. 6. Design of Siulation With the origin O at the radar center of the three axis (range, bearing, elevation), we assue the target oves in a plane with fixed z = he state vector is [ x x y y ] in Cartesian CS, whereas [ r r r ], [ b b b ] and [ e e e ] in spherical CS. Siulation tie is 0~00s. he odel-set of IMM has one CV odel and two C (Constant-urn) odels. here is always one odel in the odel-set which is atched to the syste otion ode copletely. In this point, the result of IMM algorith in this siulation is preferable to the adaptive IMM algorith. he transition probability atrix is: p ij = (4) and the odel initialized probability is: µ 0 = [ ] (5) he design of MAA and ASM siulation is a little ore coplex. o copare the perforance exactly, we use the easureents which were created in Cartesian CS containing process noise and easureent noise, after Cartesian-to-Spherical transforation, and obtain the easureents in Spherical CS. When finished the siulation, we transfor the estiating result in Spherical CS to Cartesian CS, and then copare to the true trajectory. In this siulation = 0.05, = 5 for EBDA, = 0 for MBDA. Pay attention that the filtering noise atrix in Spherical CS is relative to the noise in Cartesian CS but not equal to. he perforance of MAA and ASM siulation is very conservative and will be better in practice engineering in this point. We give the index of root ean square ( RMS ), noralized position error ( NPE ) and coputational load ( CL ). he CL of the algorith is the ean value which is evaluated in ters of calculating tie with respect to Monte Carlo runs. he forulas of calculating RMS and NPE are: M RMS( x, k) = ( [ ~ x ~ x ]) k k M (6) N k = j= RMS( x) = RMS( x,k) (7) N ISSN: Issue 8, Volue 8, August 009

8 WSEAS RANSACIONS on SYSEMS N / NPE = ( [ xk + yk + z k]/ N) / k= N ( [( xk zx, k) + ( yk zy, k) k= / ( zk zz, k) ] / N) M NPE NPE M j = + (8) = (9) where j =,,, M is the Monte Carlo runs, M = 00 ; k =,,, N is the siulation steps, N = 000 ; ~ xk ( j ), ~ y k, and ~ z k are respectively the estiation position error of k tie and the j th siulation in Cartesian CS; x k ( j ), y k, and z k are respectively the true position; z x, k, z y, k, and z z, k are respectively the easureents of three directions. Four different trajectories (Fig. ) of the target are tracked in the siulation. y.5 x rajectory rajectory rajectory 3 rajectory x Fig. he target trajectories () rajectory : x0 = [ ], constantly turns with ω( t ) = 5 / s. rajectory : x0 = [ ], 0~40s constantly turns with ω( t ) = 5 / s (turn left ), 55~75s constantly turns with ω( t ) = 7 / s (turn right), other tie oves in straight line with constant velocity (unifor otion). rajectory 3: x0 = [ ], 5~45s constantly turns with ω( t ) = 9 / s (turn right ), 60~75s constantly turns with ω( t ) = 7 / s (turn left), other tie oves in straight line with constant velocity. rajectory 4: x0 = [ ], 0~5s constantly turns with ω( t ) = 9 / s (turn right ), 40~55s and 60~85s constantly turns with ω( t ) = 7 / s (turn left), other tie oves in straight line with constant velocity. his is a trajectory like the figure Siulation Results Siulation results are given by able to able 5. able and able 3 gave the RMS of x and y directions respectively with ASM, IMM and MAA. able 4 gave the NEP and able 5 gave the CL. able Root Mean Square of x Coordinate () RMS ( x ) EBDA IMM MAA rajectory rajectory rajectory rajectory able 3 Root Mean Square of y Coordinate () RMS ( y ) EBDA IMM MAA rajectory rajectory rajectory rajectory able 4 Noralized Position Error () NPE EBDA MBDA IMM MAA rajectory rajectory rajectory rajectory able 5 Coputational Load (s) CL EBDA MBDA IMM MAA rajectory rajectory rajectory rajectory he filtering paraeters of ASM and MAA are identical and invariant in this siulation. he aneuverability (in Cartesian CS) is stronger and stronger fro trajectory to 4. Fro able, able 3 and able 4, the tracking perforance of EBDA and MBDA based of ASM is iproved obviously to MAA, although fro able 5 we can see that the coputational load of EBDA and MBDA is larger than that of MAA. Here we have 000 steps of one Monte Carlo siulation, the tie of one cycle of ASM is less than 3s (which is relative to the ISSN: Issue 8, Volue 8, August 009

9 WSEAS RANSACIONS on SYSEMS coputer perforance), and to 50s sapling interval the resource is well enough. For the tracking perforance of EBDA, to trajectory, IMM is a little better than EBDA, to trajectory and 3 the perforances of the two algoriths are near to each other, whereas to trajectory 4 EBDA is better than IMM. EBDA is also better than MBDA. Fro able, able 3 and able 4 we can conclude that when the aneuverability is weak or the otion ode varies little, IMM is a little better, but when the aneuverability is strong or the otion ode varies uch, ASM is better. Furtherore to say relatively, the tracking perforance of ASM is ore stationary, and that of IMM varies ore obviously. In the ASM algorith the adaptability of α is not considered. We can see fro (8) that if the acceleration a is approxiately constant then the coefficient α has less effect. In fact, to trajectory, the aneuver tie of the relative angular otion should be longer, that is α could be saller. We take α = 0. 0 for another siulation. he results show that the perforance of ASM ( NPE = 0.64 by EBDA) is nearer to IMM but not so apparent. 5 Conclusion his paper studied and proposed a new iproved adaptive single odel and two algoriths for aneuvering target tracking based on the relative angular otion of aerocrafts and radar, which is also suitable for range tracking. o say relatively, the tracking perforance of ASM is ore stationary, and that of IMM varies ore obviously according to the aneuverability and the design of odel-set. he united velocity-acceleration estiationbased decision algorith and direct easureentbased decision algorith based of ASM see to be valid for M. he physical concept of these new odel algoriths is clear, and they are siple and valuable for practical engineering, especially for radar tracking. Here we ephasized particularly on steadily tracking and exactly estiating of the state in spherical CS, which about the estiation of the state in Cartesian CS is not the goal of this paper, we will study in another one. References: [] X. R. Li and V. P. Jilkov, A survey of aneuvering target tracking-part I: dynaic odels, IEEE ransactions on Aerospace and Electronic Systes, Vol.39, No.4, 003, pp [] X. R. Li and V. P. Jilkov, A survey of aneuvering target tracking-part II: ballistic target odels, Proc. 00 SPIE Conf. on Signal and Data Processing of Sall argets, San Diego, CA, USA, 00, pp [3] X. R. Li and V. P. Jilkov, A survey of aneuvering target tracking-part III: easureent odels, Proc. 00 SPIE Conf. on Signal and Data Processing of Sall argets, San Diego, CA, USA, 00, pp [4] X. R. Li and V. P. Jilkov, A survey of aneuvering target tracking-part IV: decisionbased ethods, Proc. 00 SPIE Conf. on Signal and Data Processing of Sall argets, Orlando, FL, USA, 00, pp [5] X. R. Li and V. P. Jilkov, A survey of aneuvering target tracking-part V: ultipleodel ethods, IEEE ransactions on Aerospace and Electronic Systes, Vol.4, No.4, 005, pp [6] C. Z Han, H. Y. Zhu and Z. S. Duan, Multisource inforation fusion, singhua University Press, 006. [7] W. D. Blair, G. A. Watson and. R. Rice, racking aneuvering targets with an interacting ultiple odel filter containing exponentially correlated acceleration odels, Proc. wenty-hird Southeastern Syposiu on Systes heory, Colubia, SC, USA, 99, pp [8] Y. He, Z. J. Guo and J. P. Jiang, Model set design of the adaptive interacting ultiple odel tracking algorith, Electronics optics & Control, Vol.9, No., 00, pp [9] M. Lei and C. Z. Han, Expectationaxiization (EM) algorith based on IMM filtering with adaptive noise covariance, Acta Autoatica Sinica, Vol.3, No., 006, pp [0] B. J. Lee, J. B. Park and Y. H. Joo, IMM algorith using intelligent input estiation for aneuvering target tracking, IEICE ransactions on Fundaentals of Electronics, Counications and Coputer Sciences, Vol.88, No.5, 005, pp [] Kouasis, A., and Angelis, C.., Ipleenting adaptive systes using a odified Kalan filter, WSEAS ransactions on Systes, Vol.4, No., Dec. 005, pp [] Y. H. Zhang, W. Zhang, B. Zan, J. Wang and H. Chang, A ethod of reducing angular error in tracking noise jaer with ono-pulse radar, WSEAS ransactions on Counications, Vol.6, No.9, Sep. 007, pp ISSN: Issue 8, Volue 8, August 009

10 WSEAS RANSACIONS on SYSEMS [3] J. Jiang, Sliding-ode variable structure control for the Position racking Servo Syste, WSEAS ransactions on Systes, Vol.6, No., Feb. 007, pp [4] Bar-Shalo Y and Biriwal K. Variable diension filter for aneuvering target tracking, IEEE rans. on AES, Sep. 98, Vol.8, No.5, pp [5] J. C. Fang, A odified Kalan filter for tracking aneuvering targets, Acta Electronica Sinica, Vol., No.6, 983, pp [6] H. Zhou and K. S. P. Kuar, A current statistical odel and adaptive algorith for estiating aneuvering targets, AIAA Journal of Guidance, Vol.7, No.5, 984, pp [7] Z. Q. Zhu, A new adaptive Kalan filtering algorith for aneuvering target tracking, Acta Aeronautica et Astronautica Sinica, Vol.3, No.4, 99, pp [8] X. X. Liu, F. Ding, Z.. Hu and L. Zhou, An adaptive filtering algorith of noise variancesbased on odified current statistical odel, Chinese Journal of Electronics, Vol.5, No., 006, pp [9] B. Ekstrand. racking filters and odels for seeker applications, IEEE ransactions on Aerospace and Electronic Systes, Vol.37, No.3, Jan., 00, pp ISSN: Issue 8, Volue 8, August 009