An Intelligent Prediction System for Mobile Source Localization Using Time Delay Measurements
aa r X i v : . [ c s . N I] A ug An Intelligent Prediction System for Mobile SourceLocalization Using Time Delay Measurements
Hengnian Qi, Xiaoping Wu, and Naixue Xiong
Abstract —In this paper, we introduce an intelligent predictionsystem for mobile source localization in industrial Internet ofthings. The position and velocity of mobile source are jointly pre-dicted by using Time Delay (TD) measurements in the intelligentsystem. To predict the position and velocity, the Relaxed Semi-Definite Programming (RSDP) algorithm is firstly designed bydropping the rank-one constraint. However, dropping the rank-one constraint leads to produce a suboptimal solution. To improvethe performance, we further put forward a Penalty FunctionSemi-Definite Programming (PF-SDP) method to obtain the rank-one solution of the optimization problem by introducing thepenalty terms. Then an Adaptive Penalty Function Semi-DefiniteProgramming (APF-SDP) algorithm is also proposed to avoid theexcessive penalty by adaptively choosing the penalty coefficient.We conduct experiments in both a simulation environment anda real system to demonstrate the effectiveness of the proposedmethod. The results have demonstrated that the proposed intel-ligent APF-SDP algorithm outperforms the PF-SDP in terms ofthe position and velocity estimation whether the noise level islarge or not.
Index Terms —mobile localization, semidefinite programming,time delay, rank-one solution, intelligent prediction, industrialInternet of things
I. I
NTRODUCTION
Nowadays, a popular trend in many application systems isthe using of mobile sources, such as Autonomous Vehicle (AV)and Unmanned Aerial Vehicle (UAV). The using of mobilesources shows their great advantages for the flexible mobilityability. Apparently, these mobile sources of AV and UAVcan complete some complicated tasks, such as location-basedservices, radar or sonar navigation, target tracking, wirelessnetwork coverage and sensing enhancement, data collection,etc [1]–[6]. It is very crucial to obtain the position parameteralong with the velocity of mobile source when the mobilesource assists to complete these tasks. Due to the mobilityof mobile source, the position obtaining of mobile source ismore complicated compared with immobile source [7]–[9]. Acommon way to obtain the position is to utilize the sensorswith known positions. Besides the sensors, the ranging infor-mation between source and sensor is also measured using sometechnologies, such as Time Of Arrival (TOA) [10], [11], Time
This work was supported by Zhejiang Key R&D Plan 2017C03047 andZhejiang Province Key Laboratory of Smart Management & Application ofModern Agricultural Resources under Grant 2020E10017. (Correspondingauthor: Xiaoping Wu)
Hengnian Qi and Xiaoping Wu are with the School of Informa-tion Engineering, Huzhou University, Huzhou 313000, China. (Email:[email protected]; [email protected]).Naixue Xiong is with the Department of Mathematics and ComputerScience, Northeastern State University, Tahlequah, OK, 74464, USA. (Email:[email protected]).
Difference Of Arrival (TDOA) [12], [13], Received SignalStrength (RSS) [14], [15], Angle Of Arrival (AOA) [16], andrecently popular Time Delay (TD) [17]–[19]. Among thesetechnologies, TD is considered to be a simple and effectivemethod to to predict the position and velocity of the mobilesource. [20].Most of recent researches on the mobile source are focusedon the obtaining of motion parameters by using the motionsensor or Doppler shift measurements [21]–[23]. In [24], theinertial navigation method is proposed to predict the positionof the mobile source using the motion sensors. When the mo-tion data is subjected to error, the filtering method is proposedto improve the performance in [25], [26]. A linear algebraicmethod is put forward to predict the motion parameters usingthe measurement of differential delay and Doppler shift [27].The direct acquisition of motion data depends on the hardwaredevices, and the measurements may exist errors. Therefore,recent researches are concentrated on how to utilize variousranging methods to directly predict the motion parameters ofmobile sources [15].In this paper we propose an intelligent practice predictionmethod of motion parameters including the position andvelocity of mobile source by only using the time delay (TD)measurements. The proposed method does not require anymotion sensors or Doppler shift measurements. To predict theposition and velocity of the mobile source, the optimizationmodel is firstly built by representing the measurement equationinto a linear form. Then a Relaxed Semi-Definite Programming(RSDP) form is designed by relaxing the non-convex modelinto a convex problem. However, the solution to the RSDPproblem is suboptimal due to the relaxation of rank-oneconstraint. To obtain the rank-one solution, we also proposethe Penalty Function (PF) method by introducing the penaltyterms. Then the APF-SDP algorithm is designed to obtain theoptimal performance of the prediction problem by adaptivelychoosing the penalty coefficient.We also deploy a simulation environment and a real ex-periment system to fully demonstrate the effectiveness of theproposed algorithms. The real system is composed of a mobilecar equipped with Ultrasonic module (UM) and nine sensors.Besides the UM, motion sensors are also equipped to themobile car and used to measure the true position and velocityof the mobile source. The experimental results show that APF-SDP significantly outperforms the PF-SDP in terms of theposition and velocity accuracy. 20% of the position error islarger than 0.05 m for RSDP and 0.04 m for PF-SDP. However,it is reduced to near 0.01 m for the APF-SDP. By adaptivelychoosing the penalty coefficient, the performance of APF-SDP is still sufficiently close to the expected Cram´er-Rao LowerBound (CRLB) of the prediction problem even if the numberof sensors is varied from 5 to 9. It confirms the advantage ofAPF-SDP by achieving the rank-one constraint.Our proposed TD-based localization model does not requireany motion sensors or Doppler shift measurements, so itdiffers from the existing mobile source localization problemin [24], [28]. Moreover, to obtain the well performance of theestimator, we propose the rank-one solution method using thepenalty function. Our rank-one solution of AFP-SDP methodis globally convergent by adaptively choosing the penaltycoefficient and essentially different from the existing rank-onesolution methods proposed in [29]–[31]. The contributions ofthis work are summarized as follows,1) To predict the position along with the velocity of the mo-bile source, dropping the rank-one constraint producesthe RSDP problem. To design a tighter convex model,we propose the penalty function method to obtain a rank-one solution by introducing the penalty terms.2) To avoid the excessive penalty, we also put forward theAPF-SDP algorithm by adaptively choosing the penaltycoefficient. We have theoretically proven that the APF-SDP can provide optimal performance of the predictionproblem by achieving the rank-one constraint.3) We have also developed an intelligent practice predictionsystem to demonstrate our proposed algorithms usingTD measurements. The results from both the simulatedand real experiments show that the APF-SDP algorithmprovides better performance than the PF-SDP whetherthe noise level is large or not.The rest of this paper is structured as follows. Related worksare introduced in Section II. Section III presents the systemmodel and definitions. Section IV in detail describes ourproposed intelligent prediction system. In Section V, thecomputational complexity of these proposed algorithms isderived. The numerical simulations and real experiments areanalyzed in Section VI. The conclusions and future work arepresented in Section VII. This paper contains a number ofsymbols. Following the convention, the matrix is representedby bold case letter. If the matrix is denoted by ( ∗ ) , ( ∗ ) − and ( ∗ ) T indicate the matrix inverse and transpose operator,respectively. k∗k denotes ℓ norm. A i,j is the ( i, j ) th elementof matrix A . m represents all-zero vector with the length m , and I p and m × n are m × m identity and m × n zeromatrices. Tr( A ) and rank( A ) stand for the trace and rank of A ,respectively. A (cid:23) indicates that A is positive semidefinite.II. R ELATED W ORKS
Based on the ranging information, various algorithms areproposed to predict the position and the velocity of mobilesource. The popular algorithms include Maximum LikelihoodEstimator (MLE) [32], [33], alternating direction method ofmultiplier (ADMM) method [34], [35], linear estimator [36],[37], and convex method [38], [39]. The numerical solutionof MLE requires a very good initial solution to guarantee itsglobal convergence. The ADMM method provides the optimalestimate of source position by converting the unconstrained nonlinear problem into an equivalent constrained form. Dueto the nonlinear nature of the optimization model, the MLEor ADMM method will be trapped in local optimum. Toavoid the problem, the linear estimator directly representsthe unknown variables as an algebraic analytic form by con-verting the nonlinear model into a linear problem. However,the constrained relationship among the variables is difficultto be exploited in the linear estimator, so the performanceis not well enough. Convex method does not depend onthe initialization for its global convergence and graduallybecomes a popular method for the source position predictionproblem. The convex methods can be achieved by Semi-Definite Programming (SDP) [40]–[42] and Second OrderCone Programming (SOCP) [43] which can be efficientlysolved by using existing algorithms such as interior pointmethods [44].A common method to obtain an SDP problem is to relaxthe non-convex optimization model into a convex form bydropping the rank-one constraint. The rank-one relaxationmay lead to produce a suboptimal solution that is not theoptimal solution of the original optimization problem. Toobtain the rank-one solution of the convex SDP problem,many mathematical methods are proposed to deal with thetroublesome problem [45]–[48]. To solve the low-rank SDPproblems, the factorization method is introduced by obtaininga reformulation of the original SDP problem in [49]. Amodified interior point method is proposed to solve low-rankSDP problem in [50]. Although these mathematical tools dealwith the low-rank SDP problems, the solutions are also notguaranteed to be global convergence for their nonlinear ornon-convex nature.To obtain the rank-one solution with global convergence,recently the two-stage method is proposed by refining theinitial solution of the relaxed SDP problem. In [29], a rank-reduction method is designed by solving an incremental matrixof the solution when an initial rank-maximum solution hasbeen obtained with the relaxed SDP problem. The relaxed SDPproblem provides a suboptimal solution, so a feasible methodto solve the rank-one solution is to continuously tighten therelaxed problem. In [30], a set of SOCP constraints is addedto tighten the convex model and find a rank-one solution ofthe original SDP problem. Above rank-one solutions stronglydepend on the initial solutions of the relaxed SDP problem.Therefore, if the initial solutions are not accurate enough, therank-one solution may fail. In [31], the Penalty Function (PF)method is firstly proposed to obtain the rank-one solution ofthe SDP problem by introducing the penalty term. The rank-one solution of PF method provides a global optimum solutionby choosing an appropriate penalty coefficient. However, toolarge penalty coefficient will lead to the occurrence of ex-cessive penalty, which badly affects the performance of thesolution.In this paper, we mainly propose an intelligent practicesystem, in which the motion parameters including the positionand the velocity of the mobile source are predicted by onlyusing the TD measurements. The proposed method does notrequire any motion sensors or Doppler shift measurements,and is thus applicable to track the mobile source, such as the i s i t v i su u i s ii su Fig. 1. A diagram of TD measurements between sensor and mobile source.
AV or UAV.III. S
YSTEM M ODEL AND D EFINITIONS
Assuming that M sensors with known positions are placedin a p -dimensional ( p = 2 or p = 3 ) scenario. Theknown positions of the sensors are denoted by s i ∈ R p , i = 1 , , . . . , M . In the same scenario, a mobile source startsfrom an initial position u ∈ R p at a constant velocity v ∈ R p .Fig. 1 shows the diagram of TD measurement between mobilesource and sensor. A mobile source transmits a signal at theinitial position. Then the signal is received by the sensors andimmediately reflected to the mobile source, thus forming atime delay that is formulated as: t oi = 1 c ( k u − s i k + k u i − s i k ) , (1)where t oi is the true time delay, c is the propagation speedof the signal, u i is the instantaneous position of the mobilesource when the signal from the i th sensor is received by themobile source. Apparently, we have u i = u + v t oi . (2)Substituting (2) into (1) and multiplying c on the both sidesproduce ct oi = k u − s i k + k u + v t oi − s i k . (3)Moving the term k u − s i k to the left side of (3) and squaringon the both sides give u − s i ) T v + 2 cd i + ( v T v − c ) t oi = 0 , (4)where d i = k u − s i k . Therefore, the true time delay is writtenas: t oi = 2( u − s i ) T v + 2 cd i c − v T v . (5)The measured time delay t i is always subject to error andgiven by t i = t oi + n i , (6)where i = 1 , , . . . , M , n i is the noise that constructs a vectorform n = [ n , n , . . . , n M ] T . The noise vector n is alwayszero mean Gaussian distribution with covariance matrix Q n .The goal of our system model is to predict the initialposition and velocity of the mobile source using the noisymeasurement t i . In the following, the relaxed SDP (RSDP)method is firstly proposed to predict the position and velocityof the mobile source by dropping the rank-one constraint. Thedrop of the rank-one constraint is considered as a relaxationmethod that leads to produce an SDP solution with the rankhigher than 1. So the performance of RSDP is suboptimal. To improve the performance, the penalty function method isproposed to obtain the rank-one solution by introducing thepenalty terms.IV. I NTELLIGENT P REDICTION S YSTEM
In this section, we in detail introduce the intelligent predic-tion system for predicting the position and velocity of mobilesource using TD measurement. The convex SDP problemshows the advantages for its global convergence and canbe solved efficiently using interior-point algorithms. In ourproposed system, we mainly concentrate on the convex SDPsolution to the mobile source localization problem. To obtaina convex SDP form, a general approach is to relax the non-convex optimization problem into a convex form, then theposition and the velocity of mobile source are predicted byextracting from the SDP solution.
A. RSDP Algorithm
Substituting (6) into (4) yields the following expression: − s Ti v + 2 u T v + t i v T v + 2 cd i − c t i = ε i , (7)where ε i = ( v T v − c ) n i , i = 1 , , . . . , M . To estab-lish the optimization model, a new unknown vector x isdefined by x = [ u T , v T , u T v , v T v , d T , T ∈ R M +2 p +3 , d = [ d , d , . . . , d M ] T . Then (7) is also represented by a linearmatrix form: Gx = ε , (8)where ε = [ ε , ε , . . . , ε M ] T . According to the expression of(7), ε ∈ R M and G ∈ R M × ( M +2 p +3) are also defined by ε = Bn , (9a) B = ( v T v − c ) I M , (9b) G = [ g T , g T , . . . , g TM ] T , (9c) g i = [ Tp , − s Ti , , t i , Ti − , c, TM − i , − t i c ] T , (9d)Therefore, the weighted least square (WLS) solution to theprediction problem is formulated as min x ( Gx ) T Q − ( Gx )s . t . k x p +1:2 p k = x p +2 , (10a) k x p − s i k = x p +2+ i , i = 1 , . . . , M, (10b) x T p x p +1:2 p = x p +1 , (10c) x M +2 p +3 = 1 , (10d)where Q ∈ R M × M is the covariance matrix with respect to ε , and Q is further obtained by Q = BQ n B . (11)The equality condition (10a) is also rewritten as k D x k = x p +2 , (12)where D = [ p , I p , p × ( M +3) ] . The equality condition (10b)is also given by k D i x k = x p +2+ i , (13) where D i = [ I p , p × ( M + p +2) , s i ] , i = 1 , , . . . , M . Whenthe equality conditions (10a) and (10b) are equivalent to theconditions (12) and (13), (10) is rewritten as min x ( Gx ) T Q − ( Gx )s . t . k D x k = x p +2 , k D i x k = u p +2+ i , i = 1 , . . . , M, x T p x p +1:2 p = x p +1 , x M +2 p +3 = 1 . (14)To further express (14) as a convex form, a new unknownmatrix is defined by X = xx T . It is obviously shown that rank(X) = 1 . Thus, (14) is given by min X Tr( CX )s . t . Tr( D T D X ) = X p +2 , p +2 , (15a) Tr( D Ti D i X ) = X p +2+ i, p +2+ i , i = 1 , . . . , M, (15b) X pi =1 X i,p + i = X p +1 , p +1 , (15c) X M +2 p +3 ,M +2 p +3 = 1 , X (cid:23) , (15d) rank( X ) = 1 , (15e)where C = G T Q − G . Due to the rank-one constraint of(15e), the expression of (15) is non-convex. However, droppingthe rank-one condition of X yields a convex SDP form, min X Tr( CX )s . t . A i X = b i , i = 1 , . . . , M + 3 , (16)where b i = [ × ( M +2) , T , the sparse matrices A i can beeasily obtained by the expression of (15), i = 1 , . . . , M + 3 .It is obviously shown that the problem depicted by (16) is atypical SDP problem that can be effectively solved using aconvex optimization package such as SEDUMI and SDPT3.Unfortunately, dropping the rank-one constraint relaxes theproblem depicted by (15) and leads to produce a suboptimalSDP solution with a rank higher than 1, which implies that thesolution of (16) may not be the optimal solution of the originalproblem depicted by (15). To obtain the optimal performance,we propose the penalty function method to meet the rank-oneconstraint. B. PF-SDP Algorithm
The convex problem depicted by (16) provides a relaxedSDP solution to the mobile source localization problem. How-ever, the relaxed SDP solution of (16) is suboptimal due tothe drop of rank-one constraint. To obtain the optimal perfor-mance, we attempt to find a rank-one solution by introducingthe penalty terms. To design the PF-SDP method, we firstlyprove the conclusion of Lemma 1.
Lemma 1.
For a given solution X ∈ R N × N to the problemdepicted by (16), if X i,i = X i,N ( N = M + 2 p + 3 , i =1 , , . . . , N − , then rank( X ) = 1 .Proof. If X is a positive semidefinite solution of (16), itsprincipal × submatrix is also a positive semidefinite matrix. It is noted that X N,N = 1 . By selecting the ( i, i ) and ( N, N ) entry of X as the principal diagonal elements, a new positivesemidefinite submatrix is constructed and given by (cid:20) X i,i X i,N X N,i (cid:21) (cid:23) , (17)where i = 1 , , . . . , N − . Since X i,N = X N,i , we have X i,i ≥ X i,N . When the equality condition is satisfied, it isalso modified as p X i,i = X i,N . (18)Every principal × submatrix of positive semidefinite matrix X is also positive semidefinite. Similarly, selecting the ( i, i ) , ( j, j ) , and ( N, N ) entry of X as the principal diagonalelements and using the equality (18), we also construct a × submatrix that is given by X i,i X i,j p X i,i X j,i X j,j p X j,j p X i,i p X j,j (cid:23) , (19)where i, j = 1 , , . . . , N − , i < j . (18) is also equivalent tothe following expression: (cid:20) X i,j − p X i,i X j,j X i,j − p X i,i X j,j (cid:21) (cid:23) , (20)where i, j = 1 , , . . . , N − , i < j . The equation (20) holds ifand only if X i,j = p X i,i X j,j for all i, j = 1 , , . . . , N − , i < j . Therefore, the positive semidefinite matrix X is alsorewritten as X = X , p X , X , . . . p X , p X , X , X , . . . p X , ... ... ... ... p X , p X , . . . . (21) It is obviously shown that rank( X ) = 1 .To achieve the equality (18), a feasible method is to restrictthe magnitude of X i,i by introducing the penalty terms.Therefore, a new PF-based SDP form is given by min X Tr( CX ) + η X N − i =1 X i,i s . t . A i X = b i , i = 1 , . . . , M + 3 , (22)where η is a strictly positive constant and called as penaltycoefficient. When η is gradually increased, X i,i will suffi-ciently close to X i,N . Eventually, large enough η will leadthat rank( X ) = 1 .When η is chosen to be large enough, the SDP problemdepicted by (22) provides a rank-one solution of the positivesemidefinite X . However, a concerned issue is that whetherthe rank-one solution is the original solution of (10) or not.It is directly shown that the problem depicted by (10) hasonly M + 2 p + 3 variables. Since there are M + 3 constraintsamong the variables in defined vector x , so the problem hasonly p independent variables. It is also observed that thedimension of X in the problem depicted by (22) is also M +2 p + 3 when rank( X ) = 1 . Due to the similar constraints,the problem depicted by (22) has also only p independentvariables. Therefore, the rank-one solution of the SDP problem −6 −5 −4 −3 −2 −1 0 1 2 30500 T r( C X * ) −6 −5 −4 −3 −2 −1 0 1 2 300.050.1 L o ca li za ti on e rr o r log( η )Tr(CX*)Localization error Fig. 2. Localization error and optimal objective
Tr( CX ∗ ) with a test. depicted by (22) is also the original solution of (10) sincethe convex SDP form of (22) is derived from the non-convexproblem depicted by (10).An appropriate penalty coefficient can ensure the effectiveworking of penalty terms. Although large enough η willachieve the rank of the solved X to be one, too large η maylead to a risk of excessive penalty, which will badly affect thepredicted result. Hence, the choosing of η is crucial to obtainthe well performance of the prediction problem. To choosean appropriate penalty coefficient, we propose an adaptivepenalty function-based SDP (AFP-SDP) algorithm to obtaina rank-one solution for the SDP optimization problem in thefollowing. C. APF-SDP Algorithm
Too large η increases the proportion of penalty terms tothe total cost function of (22), so it will weaken the originalobjective Tr( CX ) . To avoid the chosen η to be too large, η starts from a small value and increases gradually, until therank-one condition is satisfied. To clearly observe the effectof the penalty coefficient in the problem depicted by (22), weconducted a random test and the results are plotted in Fig.1,where X ∗ and Tr( CX ∗ ) represent the optimal solution andobjective, respectively. In the test, log η is gradually increasedfrom -6 to 3 (i.e. η is gradually increased from − to ).As can be seen that the penalty terms almost do not workwhen log η is increased from -6 to -4. The localization erroris gradually reduced for the working of penalty terms when log η is increased from -4 to -2. The optimal performance isachieved when log η is set to the range of (-2, 0). However,if log η is continuously to be increased, the performance willbecome worse. The localization error will sharply increaseddue to the occurrence of excessive penalty when log η is largerthan 1. The occurrence of excessive penalty badly affectsthe localization result, so it is crucial to detect whether thepenalty is excessive or not. It can be seen from Fig.2 thatthe optimal objective Tr( CX ∗ ) is simultaneously increasedwith the localization error when the excessive penalty occurs.Therefore, the optimal objective Tr( CX ∗ ) can be used todetect the occurrence of excessive penalty. Lemma 2. If X ∗ is an optimal rank-one solution of SDPproblem depicted by (16), then Tr( CX ∗ ) ∼ P Mi =1 λ i χ (1) .Proof. If X ∗ is an optimal rank-one solution of SDP problemdepicted by (16), a new vector x ∗ is defined by X ∗ = x ∗ x ∗ T ,where x ∗ ∈ R M +2 p +3 . Since the problem is derived from (10), x ∗ is also an optimal solution of problem depicted by (10).Then it yields Tr( CX ∗ ) = x ∗ T Cx ∗ . (23)The prediction error of x ∗ is denoted by △ x ∗ , then x ∗ + △ x ∗ = x o , where x o represents the true value of the defined x .The constrained optimization problem depicted by (10) hasonly p independent variables that constructs a vector z =[ u T , v T ] T . If the prediction error of z is denoted as △ z , wecan obtain ∂ ( Gx ) ∂ x ∂ x ∂ z △ z = ε , (24)where ∂ ( Gx ) ∂ x = G , ∂ x ∂ z is denoted as H and given by H = (cid:20) I p p × p v p ∂ d ∂ u p p × p I p u v p × M p (cid:21) T , (25)where ∂ d ∂ u = [ u − s k u k , u − s k u k , . . . , u − s M k u k ] . Therefore, (24) isrewritten as P △ z = ε , (26)where P = GH . Since ε = Bn , the WLS solution to (26)is △ z = ( P T W P ) − P T W Bn , (27)where W = Q − . Since △ x ∗ = H △ z , the prediction error △ x ∗ is obtained by △ x ∗ = En , (28)where E = H ( P T W P ) − P T W B . Since x ∗ = x o − △ x ∗ ,the optimal objective is approximately given by x ∗ T Cx ∗ ≈ x ∗ T Cx o . (29)Substituting x ∗ = x o − △ x ∗ into the right side of (29) yields x ∗ T Cx ∗ = x oT Cx o − x oT C △ x ∗ , (30)where C is interfered by the noise. Let C = C o + △ C , C o and △ C denote the true value and the error term caused bynoise, respectively. Similarly, we also define G = G o + △ G in which G o and △ G represent the true value and the errorterm, △ G = [ △ g T , △ g T , . . . , △ g TM ] T (31a) △ g i = [ T p +1 , n i , TM , − c n i ] T . (31b)According to the definition of C = G T W G , we can obtainthe true value C o and the error term △ C , C o = G oT W G o (32a) △ C = G oT W △ G + △ G T W G o + △ G T W △ G . (32b)Since G o x o = M , the first term of the right side in (30) isalso further written as x oT Cx o ≈ ( △ Gx o ) T W ( △ Gx o ) , (33) where △ Gx o is also given by △ Gx o = F n (34a) F = diag( T x o ) (34b) T = [ t T , t T , . . . , t TM ] T (34c) t i = [ T p +1 , , TM , − c ] T . (34d)Therefore, (33) is also rewritten as x oT Cx o = n T F T W F n . (35)Similarly, the second term of the right side in (30) can alsorewritten as x oT C △ x ∗ = x oT ( △ G T W G o + △ G T W △ G ) △ x ∗ . (36)Neglecting the high order term of (36) also yields x oT C △ x ∗ ≈ ( △ Gx o ) T W G o △ x ∗ . (37)Using the expression of (28) and (34a), we can rewrite (37)as x oT C △ x ∗ = n T F T W G o En . (38)According to the expressions (35) and (38), (30) is alsorewritten as x ∗ T Cx ∗ ≈ n T F T W ( F − GE ) n , (39)where G o is approximately equal to G . Let L = F T W ( F − GE ) . Therefore, x ∗ T Cx ∗ conforms to the chi-square distri-bution P Mi =1 λ i χ (1) , which is also denoted by x ∗ T Cx ∗ ∼ M X i =1 λ i χ (1) , (40)where λ i is the eigenvalue of the matrix LQ n , i =1 , , . . . , M .When η is chosen to be too small, the rank of X ∗ may belarger than 1. The increasing of η can ensure that the rankof X ∗ is gradually close to be one. To evaluate the rank of X ∗ , we firstly propose an eigenvalue method according to thefollowing threshold value, τ = e λ N − e λ N , (41)where e λ N − and e λ N are the N − th and the N th eigenvalueof X ∗ , N = M + 2 p + 3 . It is obviously shown that the rankof X ∗ tends to be one when τ is sufficiently close to zero.To ensure the well performance of the PF-SDP, it is crucialto choose an optimal penalty coefficient which depends onvarious factors including the positions of sensors and mobilesource, the number of sensors, and the noise level. Therefore,the optimal η is not invariable under the different situations. InAlgorithm 1, we propose an Adaptive Penalty Function-basedSDP (APF-SDP) algorithm by adaptively choosing an optimalpenalty coefficient.In Algorithm 1, two conditions is required to be judged.One is the condition τ < δ ( δ is a constant that is sufficientlyclose to zero), which is considered as rank-one condition andused to judge whether the rank of X ∗ is one or not. The Algorithm 1:
APF-SDP Algorithm
Input: A , b , C , η , α > , γ > , δ > , ǫ > Output: X ∗ while do Using (17) to calculate the optimal X ; if τ < δ then if Tr( CX ) < ǫ then X ∗ = X ; break; else δ = αδ ; goto 2; end else η = γη ; end end return X ∗ other is the condition CX < ǫ ( ǫ is also a threshold valuedetermined by the chi-square distribution of CX ∗ ), which isused to detect whether the penalty is excessive or not. η startsfrom a small value and increases gradually with η = γη ( γ > ), where γ is called as step length. When τ < δ is satisfied,it is considered to meet the rank-one condition. Then usingthe condition CX < ǫ , we further detect whether the penaltyis excessive or not. When meeting the condition CX < ǫ , weaccept the solution that is considered as a final solution X ∗ .If not, we relax the rank-one condition with δ = αδ ( α > )and resolve the problem depicted by (22) until two conditionshold. V. C OMPUTATIONAL C OMPLEXITY
The worst-case complexity of solving an SDP problem ineach iteration is O ( m n ) [51], [52], where m is the numberof the equality constraints, n is the dimension of the SDP cone.The number of iteration is bounded by O ( √ n ln(1 /ς )) , where ς is the solution precision, √ n is the number of iterationscaused by barrier parameter. For the proposed PF-SDP, wehave m = M + 3 and n = M + 2 p + 3 . Therefore, thecomputational complexity of the PF-SDP in (22) is given byPF-SDP Complexity ≃ O (cid:0) ( M + 3) N . ln(1 /ς )) (cid:1) , (42)where N = ( M + 2 p + 3) . To choose an appropriate penaltycoefficient, the APF-SDP requires κ PF-SDP solutions, where κ depends on η , the initial penalty coefficient, η ∗ , the optimalpenalty coefficient, and the step length γ . Therefore, it needsto be satisfied that η γ κ ≥ η ∗ . We can further obtain κ ≥ ln η ∗ − ln η ln γ . (43)The complexity of APF-SDP in Algorithm 1 is κ times of thatof the PF-SDP and given byPF-SDP Complexity ≃ O (cid:0) κ ( M + 3) N . ln(1 /ς )) (cid:1) . (44) TABLE IP
OSITIONS OF TEN SENSORS IN SIMULATIONS ( KM ) x y -0.385 0.199 -0.163 -0.464 0.765 -0.167 -0.473 0.409 0.736 -0.456 VI. P
ERFORMANCE A NALYSIS
The RSDP, PF-SDP and APF-SDP algorithms are proposedto predict the initial position and velocity of the mobilesource using the time delay measurements. To evaluate theperformance of these proposed algorithms, the simulationswere firstly conducted in a 2-D scenario. We have also donethe simulations for the 3-D case and the observations weresimilar. The positions of ten sensors are randomly generatedaccording to the uniform distribution U ( − , and listed inTab. I. In the same region, a mobile source starts from theinitial position (0.2, -0.4) km with a constant velocity (-1, 1)m/s. The propagation speed c is randomly drawn from therange (0 . , . km/s. TD measurements are generated basedon the model equation (1), and the noise in (6) is modeled aszero mean Gaussian distribution with covariance Q n = σ I M .The performance is evaluated using Mean Square Error (MSE)defined by ( MSE( u ) = K P Kk =1 k u k − u o k MSE( v ) = K P Kk =1 k v k − v o k , (45)where u o and v o represent the true initial position and velocity, u k and v k are the predicted results in the k th Monte Carlo(MC) run, K is the number of MC runs ( K is set to 1000 inour simulations). The proposed algorithms are also comparedwith the Cram´er-Rao Lower Bound (CRLB) of the predictionproblem. A. Penalty Coefficient
The increasing of η ensures that the rank of the SDP solutiongradually tends to be one. However, the performance of therank-one solution may be not optimal due to the occurrence ofexcessive penalty. Therefore, it is required to ensure that therank-one solution has no the excessive penalty. To evaluate theimproved performance of PF-SDP compared with the RSDP,the Logarithmic MSE difference (LOGMSED) is defined by LOGMSED = 10log(MSE) PF − R , (46)where PF and R represent the MSEin log-scale of PF-SDP and RSDP, respectively.The first eight sensors listed in Tab. I are used to determinethe initial position and velocity of the mobile source. Thenoise level σ is set to -40, -20, 0, respectively. Fig. 3(a)illustrates the LOGMSED( u ) performance when the penaltycoefficient η is increased from − to (i.e. log η isincreased from -6 to 3). The penalty terms do not basicallywork when η is too small. As can be seen that LOGMSED( u )is near zero when log η is set to -6. When η gradually increases,LOGMSED( u ) becomes a negative value, which illustratesthat the MSE of PF-SDP is smaller than that of the RSDPdue to the working of penalty terms. When log η is set to (0, −6 −5 −4 −3 −2 −1 0 1 2 3−10−5051015 log η l og ( M SE ( u )) σ =−4010log σ =−2010log σ =0 (a) logarithmic MSE difference of position. −6 −5 −4 −3 −2 −1 0 1 2 3−100102030 log η l og ( M SE ( v )) σ =−4010log σ =−2010loglog σ =0 (b) logarithmic MSE difference of velocity.Fig. 3. Performance comparison as the η varies.TABLE IIN UMBER OF RANK - ONE SOLUTIONS WHEN η VARIES (1000 MC
RUNS FOREACH VALUE OF η , σ = 0 )log η -6 -5 -4 -3 -2 -1 0 1 2Number 0 3 4 12 28 306 825 1000 1000
2) for σ = − , (-1, 1) for σ = − , and (-2,0) for σ = 0 , LOGMSED( u ) reaches the least value,indicating the optimal performance of PF-SDP. Therefore, theoptimal range of η is variable due to the different noise level.When log η is larger than the optimal range, LOGMSED( u )will be sharply increased, which confirms the occurrence ofexcessive penalty. For instance, when log η is set to (2,3),LOGMSED( u ) becomes a positive value for σ = 0 ,illustrating that the performance of PF-SDP is worse than thatof the RSDP.When the parameter setup is the same with that of Fig. 3(a),Fig. 3(b) shows the LOGMSED( v ) with the increasing of η .It is also observed that the LOGMSED( v ) becomes a negativevalue when log η is gradually increased from -6. Then theLOGMSED( v ) reaches a least value that manifests the optimalperformance of PF-SDP. For instance, when σ is set to-40, the optimal log η is set to the range of (0, 2), where thePF-SDP provides better performance and has almost 13 dBbias compared with the RSDP. When log η is larger than theoptimal range, the LOGMSED( v ) will be sharply increaseddue to the occurrence of excessive penalty. TABLE IIIN
UMBER OF RANK - ONE SOLUTIONS WHEN σ VARIES (1000 MC
RUNS FOR EACH VALUE OF σ )10log σ -60 -50 -40 -30 -20 -10 0 10 20Number ( log η = − ) 0 1 9 37 69 169 293 525 739Number ( log η = 0 ) 2 34 112 354 531 619 834 999 1000Number ( log η = 1 ) 205 542 675 769 872 999 1000 1000 1000 −60 −50 −40 −30 −20 −10 0 10 20−80−60−40−20020 10log( σ ) l og ( M SE ( u )) RSDPPF−SDP,log η =−6PF−SDP,log η =−4 PF−SDP,log η =−2APF−SDPCRLB (a) Performance of predicted position. −60 −50 −40 −30 −20 −10 0 10 20−100−50050 10log( σ ) l og ( M SE ( v )) RSDPPF−SDP,log η =−6PF−SDP,log η =−4 PF−SDP,log η =−2APF−SDPCRLB (b) Performance of predicted velocity.Fig. 4. Performance comparison as σ varies. The rank of PF-SDP solution will gradually tend to beone as η increases. If meeting τ < δ , it is considered as arank-one solution. When 10log σ is set to 0 dB, the numberof rank-one solutions is shown in Tab. II, where δ is set to − . As can be seen that the number of rank-one solutionsis increased with log η , since larger η provides tighter SDPcone constraints. When log η is larger than 1, all 1000 MCruns of PF-SDP solutions meet the rank-one condition. When10log σ is increased from -60 dB to 20 dB, the number ofrank-one solutions is also illustrated in Tab. III, where log η is set to -1, 0, and 1, respectively. The number of rank-onesolutions is also dramatically increased with 10log σ . When10log σ is set to -60 dB, the number of rank-one solutions is2 for log η = 0 , 205 for log η = 1 . However, all 1000 MC runsmeet the rank-one condition for log η = 0 and log η = 1 when10log σ is increased to 20 dB. B. Performance with Simulations
The parameters in APF-SDP are listed as follows, α = 5 , γ = 10 , and δ = 10 − . The first eight sensors listed in Tab. Iare used to locate the mobile source which starts from theinitial position (0.2, -0.4) km with a constant velocity (-1, 1)m/s. When the noise level 10log σ is increased from -60 dBto 20 dB, Fig. 4(a) and Fig. 4(b) illustrates the logarithmicMSEs of predicted initial position and velocity, respectively. l og ( M SE ( u )) RSDPPF−SDP,log η =−6PF−SDP,log η =−4 PF−SDP,log η =−2APF−SDPCRLB (a) Performance of predicted position. l og ( M SE ( v )) RSDPPF−SDP,log η =−6PF−SDP,log η =−4 PF−SDP,log η =−2APF−SDPCRLB (b) Performance of predicted velocity.Fig. 5. Performance comparison as c varies. l og ( M SE ( u )) RSDPPF−SDP,log η =−6PF−SDP,log η =−4 PF−SDP,log η =−2APF−SDPCRLB (a) Performance of predicted position. l og ( M SE ( v )) RSDPPF−SDP,log η =−6PF−SDP,log η =−4 PF−SDP,log η =−2APF−SDPCRLB (b) Performance of predicted velocity.Fig. 6. Performance comparison as number of sensors varies. It is obviously shown that the logarithmic MSE performancebecomes worse as 10log σ increases. When log η is set to -6 and -4, the logarithmic MSEs of PF-SDP are almost samewith that of the RSDP, indicating the no working of penaltyterms. However, the performance of PF-SDP becomes betterwhen log η is set to -2. Especially, the logarithmic MSE of PF-SDP is closer to the CRLB at large noise level, since the rankof PF-SDP solution is easier to be one. Compared with theRSDP, the AFP-SDP provides better performance in reachingthe CRLB accuracy. Hence, it confirms the advantage of theAPF-SDP in the position and velocity prediction.The noise level 10log σ is set to -40 dB and the otherparameters are the same with those in Fig. 4. Fig. 5(a)and Fig. 5(b) also shows the logarithmic MSEs of predictedinitial position and velocity when the propagation speed c is increased from 0.1 km/s to 1 km/s. It is shown that theperformance of predicted position or velocity becomes worseas the propagation speed increases. The performance of PF-SDP is always better that of the RSDP when log η is set to -2,indicating the working of penalty terms. The APF-SDP alwaysprovides the almost CRLB performance in the prediction ofposition and velocity when c is varied from 0.1 km/s to1km/s. It confirms the advantages of AFP-SDP which canadaptively choose the penalty coefficient and provides optimalperformance at the entire range of c .Finally, we also examine the impact of sensors on theperformance of these proposed algorithms when the sensorsare selected from Tab. I in the order of list. The other parametersetup is also the same with that in Fig.4. Fig. 6(a) and Fig. 6(b)show the logarithmic MSEs of predicted initial position andvelocity as the number of sensors varies. As can be seen thatthe logarithmic MSEs of PF-SDP (log η = -2) are the samewith those of RSDP at when there are only five or six sensors.However, the PF-SDP (log η = -2) provides better performancecompared with the RSDP when the number of sensors is largerthan seven. The proposed APF-SDP always reach the CRLBperformance. It is also shown that adding more sensors canreduce the logarithmic MSE. However, how much reductionin logarithmic MSE by using additional sensors depends onthe positions of the new sensors included. C. Real Experiments
To verify the performance of our proposed algorithms,we also conducted the real experiments using a mobile carequipped with Ultrasonic module (UM). Besides the UM,motion sensors are also equipped to the mobile car and usedto measure the parameters including the velocity and movingdirection. Nine sensors are placed at the positions listed inTab. IV. The UM transmits the ultrasonic signals to the sensorsat the initial position, then the signals are received by thesensors and reflected to the UM of mobile car. Extensive testsshow that the measurement noise in the range-difference issufficiently close to be zero-mean Gaussian distribution withan predicted noise variance . × − ms . The initial positionof the mobile car is set to (1, 2) m, and the velocity of themobile car is in the range of (1, 3) m/s.We collect 200 sampling data that is used to predict theinitial position and and velocity of the mobile car. Fig. 7(b) TABLE IVP
OSITIONS OF NINE SENSORS IN REAL EXPERIMENT ( M ) x y illustrates the cumulative distribution function (CDF) of posi-tion error with 200 runs with these different algorithms. Ascan be seen that 20% of the position error is larger than 0.05m for RSDP. However, it is reduced to near 0.01 m for theAPF-SDP, indicating the advantage of our APF-SDP in theperformance of position prediction. Fig. 7(c) shows the CDFof velocity error using 200 sampling data. 20% of the velocityerror is larger than 0.08 m/s for RSDP, but it is reduced to0.06 m/s for our proposed APF-SDP. Therefore, the APF-SDPperforms better than the RSDP in the prediction of velocity,confirming the advantage of the APF-SDP by achieving rank-one constraint.Selecting the sensors from Tab. IV in the order of list, wealso verify the performance of these proposed algorithm whenthe number of sensors is increased from 5 to 9. To clearlyillustrate the effect of these different algorithms, we use theroot mean square error (RMSE) to evaluate the performancewith 200 sampling data. The position RMSE performanceof these algorithms is plotted in Fig. 8(a), where “CRLBexpected” represents the best achievable performance expectedusing the predicted noise variance. The RSDP performs notwell even if the number of sensors increases. However, theproposed APF-SDP almost reaches the “CRLB expected”performance, which is consistent with the simulation results.Fig. 8(b) illustrates the velocity RMSE as the number ofsensors increases. The performance of PF-SDP is better thanthat of the RSDP when log η is set to -2. However, the PF-SDP can not provide the “CRLB expected” performance sinceit can not adaptively choose the penalty coefficient. D. Industrial Applications
There are many position and velocity prediction problemsin industrial Internet of things, such as the position obtainingof mobile AV and the tracking of UAV, which are illustratedin Fig. 9(a) and Fig. 9(b), respectively. Most of the cur-rent research on these problems is focused on the positionobtaining of mobile source using some ranging informationor the velocity prediction using motion sensors or Droppershift measurements. The motion sensor is an extra hardwaredevice which will increase the system cost. Since the velocityobtaining of Dropper shift measurement is subject to the errorof (2, 4) m/s, it is considered to be unacceptable in someapplication systems. Our proposed method does not requireany motion sensors or Dropper shift measurements and realizethe prediction of position along with the velocity of mobilesource.Our experimental results show that the mean position RMSEof APF-SDP is smaller than 0.01 m when the UM equippedto the mobile source is used to collect the TD information.Apparently, the signal can be acoustic or electromagnetic,and the signal can travels in the underwater or underground (a) The scene in real experiments. CD F ( × % ) RSDPPF−SDP, log η =−6PF−SDP, log η =−4PF−SDP, log η =−2APF−SDP (b) CDF of position error CD F ( × % ) RSDPPF−SDP, log η =−6PF−SDP, log η =−4PF−SDP, log η =−2APF−SDP (c) CDF of velocity error.Fig. 7. Performance comparison in real experiments. po s i t i on R M SE ( m ) RSDPPF−SDP,log η =−6PF−SDP,log η =−4 PF−SDP,log η =−2APF−SDPCRLB (a) RMSE performance of predicted position. v e l o c i t y R M SE ( m / s ) RSDPPF−SDP,log η =−6PF−SDP,log η =−4 PF−SDP,log η =−2APF−SDPExpected CRLB (b) RMSE performance of predicted velocity.Fig. 8. Performance comparison using experimental data as number of sensorsvaries. scenarios. Due to the different propagation speed of thesignal, our proposed algorithms provides distinct accuracyperformance in the prediction of position and velocity, whichis also demonstrated in Fig. 5. Therefore, a feasible method toimprove the accuracy performance is to reduce the noise levelat large propagation speed. Moreover, the precise timing isvery important to ensure the performance especially when thepropagation speed of electromagnetic signal reaches × m/s. (a) 2-D scenario of mobile AV.(b) 3-D scenario of mobile UAV.Fig. 9. Different scenarios of our proposed system in industrial applications. VII. C
ONCLUSIONS AND F UTURE W ORK
We introduce an intelligent practice prediction system formobile source using the TD measurements. To predict theposition and velocity of mobile source, the RSDP algorithm isfirstly proposed to obtain a convex SDP problem by droppingthe rank-one constraint. The performance of RSDP is verypoor due to the drop of rank-one constraint. Then the PF-SDP algorithm is put forward to obtain better performanceby introducing the penalty terms. In the PF-SDP, the penaltycoefficient is crucial to ensure the well performance of thePF-SDP and its optimal value is variable. Therefore, the AFP-SDP algorithm is proposed by adaptively choosing the penaltycoefficient. Compared with the RSDP and PF-SDP, the AFP-SDP shows its performance advantages in the prediction ofposition and velocity. We have also done the simulations andthe real experiments for the 2-D case. For the future work,our proposed methods are not applied in the 3-D scenario.Therefore, we will focus the future work on the positionprediction of UAV and extend the system to the 3-D case. R EFERENCES[1] Y. Wang, A. V. Vasilakos, J. Ma, and N. Xiong, “On Studying theImpact of Uncertainty on Behavior Diffusion in Social Networks,”
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