# Trajectory Optimization of Multi-UAVs for Marine Target Tracking during Approaching Stage

Authors: Hao-Ran Shi; Fa-Xing Lu; Ling Wu; Jia-Wei Xia
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper focuses on a coordinated tracking planning method of multiple unmanned aerial vehicles (UAVs), which was deployed in the best positions to better fulfill the marine target localization tasks when approaching the target. The optimal planning of multi-UAVs was implemented using an online centralized nonlinear model predictive control (NMPC) based on the target state’s uncertainty criteria. The penalty function is used to solve UAV platform dynamic performance in the model predictive control method to consider the more realistic situation. The coordinated planning problems of multi-UAVs are numerically simulated and compared with the Lyapunov vector field guidance (LVFG) method under classical mission scenarios. Simulation results demonstrate that the algorithm can maintain the optimal observation configuration of multi-UAVs to improve the marine target positioning accuracy, verifying the feasibility and superiority of this method. Furthermore, the simulation results can provide a useful reference for the flight control law design of multi-UAVs with optimal observation configuration.


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## Body

## 1. Introduction

Recently, organized conflict occurred frequently in the maritime arena, threatening the safe passing of ships from the exclusive economic zones (EEZ). Therefore, powerful security surveillance is expected for the remote marine target in the EEZ [1]. UAVs are widely used in intelligence surveillance, early warning, communication, attack, and other aspects because of their various sensors, concealed actions, controllable cost, and needlessness for personnel, which has the potential to provide an accurate indication of the remote marine target [2, 3]. The sensor must obtain more target information to gain a precise state of the remote maritime target. As a result, this paper will study the location and tracking of the remote marine target by multi-UAVs based on angle and range measurements.Many works have been carried out on target location and tracking based on UAVs. To effectively use the observation information, Pack et al. [4] used the Kalman filter algorithm to estimate its state. Wang et al. [5] realized target fusion estimation using a distributed unscented information filter. The detection data was combined using an adaptive IMM-information filtering technique by Zhang et al. [6]. Another important field of multi-UAVs’ coordinated positioning is to enhance the target state estimation by studying multi-UAVs’ position configuration. The Fisher information matrix (FIM) and its inverse matrix, Cramer−Rao lower bound, (CRLB) are the popularly used criteria for evaluating target position accuracy. The optimal observation configuration can be classified into two types, namely, A-optimal and D-optimal. A-optimal seeks to minimize the trace of the FIM’s inverse, whereas D-optimal seeks to maximize the FIM’s determinant. Bishop et al. [7] investigated the best sensor-target geometries for range-only, time-of-arrival, and bearing-only localization using D-optimal. Zhong et al. [8] analyzed the optimal configuration of sensor-target geometries in three-dimensional space using D-optimal. Xu and Dogancay [9] explored the sensor placement problem for AOA observation in three-dimensional space utilizing A-optimal. However, the criteria based on FIM must calculate the expression of FIM, and the calculation becomes more complex as the dimension and number of UAVs increase. As a result, the information entropy is used in this paper. According to Grocholsky et al. [10], Hoffmann and Tomlin [11], and Darmon [12], entropy was defined to measure the amount of uncertainty in-state results and was used as the cost function to control the observation configuration of UAVs.Another problem of coordinated location and tracking based on multi-UAVs is controlling UAVs and maintaining the observation configuration to obtain the optimal target estimation. Many research studies have been carried out to solve the trajectory optimization problem of UAVs [13, 14]. Frew et al. [15] used the LVFG method to control multi-UAVs to keep standoff tracking of stationary and moving targets. Lim et al. [16] presented a strategy for standoff target tracking by a team of unmanned aircraft using vector field, which is only to maintain the optimal observation during the hovering phase. Using the vector field approach, Jung et al. [17] proposed path following, arrival position, angle, and time control. Sinclair et al. [18] proposed a suboptimal feedback guidance law that keeps the sight angle of the sensor and the target 90 degrees while ensuring that the sensor is close to the target. These are all based on the coordinated tracking of two UAVs. Zhao et al. [19] discussed the cooperative control of multi-UAVs in the case of only angle measurement. Hu et al. [20] and Wang et al. [21] realized the coordinated optimal target tracking of multi-UAVs using the receding horizon control method, which did not consider the observation of multi-UAVs based on angle and range measurements.Until now, most of the existing studies only focused on finding the necessary conditions for UAVs’ placement optimality, but very few have considered the concrete solution process of coordinated control multi-UAVs based on angle and range measurements. Moreover, most existing works only consider the optimal placement when UAVs circle around the target; very few studies in the literature have evaluated the optimal placement of multiple UAVs approaching the target. However, it is difficult for UAVs to make hovering motions for the remote marine target in practical applications. UAVs are expected to keep optimal observation while approaching the target. Naturally, two closely related problems can arise in the optimal UAV placement for remote marine target localization. The first is to determine the cost function of the observation based on angle and range measurements when the UAV approaches the target. The second is solving the optimal control command so that UAVs can be deployed in the best positions to better fulfill the target localization tasks.Motivated by the aforementioned aspects, this paper focuses on multi-UAVs’ coordinated optimal observation problems based on angle and range measurements for the remote marine target. Firstly, the cost function based on information theory is derived, and then the NMPC method is used to solve the optimal control problem of multi-UAVs with constraint variables. Finally, the simulation results of the coordinated tracking of multi-UAVs in four cases are displayed. The results show that the algorithm proposed in this paper can increase the measurement information and reduce the uncertainty of target state estimation to achieve accurate target localization.


## 2. Problem Definition

### 2.1. UAV Dynamic Model

Currently, the mathematical model of UAVs with autopilot is generally used in the coordinated control of UAVs, which has the advantage of engineering practicability. It is assumed that each UAV has a low-level flight controller, which controls the UAV’s yaw rate and velocity through control commands to control the UAV’s flight status. We consider a two-dimensional UAV kinematic model as [22](1)x˙y˙ψ˙v˙ω˙=fx,u=vcosψvsinψω−1τvv+1τvuv−1τωv+1τωuω,where x=x,y,ψ,v,ωT represent the inertial position, heading, speed, and yaw rate of the UAV, respectively. τv and τω are time-delay constants related to the UAV and its flight status. u=uv,uωT is the control input of the UAV, which represents the control command of the speed and the yaw rate, respectively, with the following dynamic limits of the UAV:(2)uv−v0≤Δvmax,uω≤ωmax,where v0 is a cruise speed of the UAV and Δvmax and ωmax represent the maximum change range of UAV cruise speed and turning rate, respectively. To design the nonlinear model predictive controller, the continuous UAV model in (1) can be discretized by Euler integration into(3)xk+1=fdxk,uk=xk+Tsfxk,uk,where xk=xk,yk,ψk,vk,ωkT, uk=uvk,uωkT, and Ts is a sampling time.


### 2.2. Target Model

The remote marine target is mainly aimed at the relatively slow-moving target such as the warship. Taking the uniform linear motion model as an example, the target’s motion model in the two-dimensional plane is established.xk=xk,x˙k,yk,y˙kT represent that the state of the target includes position and speed, and then the system state can be expressed in the following form [23]:(4)xk=Fkxk−1+Gkwk,where(5)Fk−1=1Ts000100001Ts0001,Gk−1=0.5Ts20Ts000.5Ts20Ts,wk is gaussian process noise with zero mean and the covariance matrix of process noise is Qk.


### 2.3. Sensor Model

The measurementz of the sensor can be modeled according to the position of the UAV x,yT and the target position xt,ytT as follows:(6)zk=hxk,xt,k+vk=xt,k−xk2+yt,k−yk2tan−1yt,k−ykxt,k−xk+vk,where vk is gaussian measurement noise with zero mean, and the measurement noise covariance matrix is Rk, Rk=σr200σϕ2.


### 2.4. Extended Information Filtering (EIF)

Compared with the extended Kalman filter (EKF) [24], EIF can easily realize the information matrix fusion. The simple addition of nature makes information filtering very suitable for multisensor, decentralized, and distributed estimation. Each sensor node only needs to generate an information matrix, and the global information estimation can be obtained by simple algebraic addition at the fusion center. The state estimate x∧ and covariance P are used in EKF, which are replaced by the information state y∧ and information matrix Y in the information form. The information state and information matrix are defined as y∧=P−1x∧, Y=P−1. EIF includes prediction process and the estimation process as follows [20]:Prediction process:(7)Yk|k−1=FkY−1k−1|k−1FTk+Qk−1,(8)y∧k|k−1=Yk|k−1Fkx∧k−1|k−1.Definition:(9)Iik=HiTkRi−1kHik,(10)iik=HiTkRi−1kzik−hixk,xt,k+Hikx∧k|k−1,(11)x∧k|k−1=Y−1k|k−1y∧k|k−1.Estimation process:(12)Yk|k=Yk|k−1+∑i=1nIik,(13)y∧k|k=y∧k|k−1+∑i=1niik,(14)x∧k|k=Y−1k|ky∧k|k,y∧k|k−1 and Yk|k−1 are prediction information state and prediction information matrix, respectively, Yk−1|k−1 is the information state matrix of k −  1 moment, Hik is the Jacobian matrix of the UAV i, and zik is measurement information of UAV i.


## 2.1. UAV Dynamic Model

Currently, the mathematical model of UAVs with autopilot is generally used in the coordinated control of UAVs, which has the advantage of engineering practicability. It is assumed that each UAV has a low-level flight controller, which controls the UAV’s yaw rate and velocity through control commands to control the UAV’s flight status. We consider a two-dimensional UAV kinematic model as [22](1)x˙y˙ψ˙v˙ω˙=fx,u=vcosψvsinψω−1τvv+1τvuv−1τωv+1τωuω,where x=x,y,ψ,v,ωT represent the inertial position, heading, speed, and yaw rate of the UAV, respectively. τv and τω are time-delay constants related to the UAV and its flight status. u=uv,uωT is the control input of the UAV, which represents the control command of the speed and the yaw rate, respectively, with the following dynamic limits of the UAV:(2)uv−v0≤Δvmax,uω≤ωmax,where v0 is a cruise speed of the UAV and Δvmax and ωmax represent the maximum change range of UAV cruise speed and turning rate, respectively. To design the nonlinear model predictive controller, the continuous UAV model in (1) can be discretized by Euler integration into(3)xk+1=fdxk,uk=xk+Tsfxk,uk,where xk=xk,yk,ψk,vk,ωkT, uk=uvk,uωkT, and Ts is a sampling time.


## 2.2. Target Model

The remote marine target is mainly aimed at the relatively slow-moving target such as the warship. Taking the uniform linear motion model as an example, the target’s motion model in the two-dimensional plane is established.xk=xk,x˙k,yk,y˙kT represent that the state of the target includes position and speed, and then the system state can be expressed in the following form [23]:(4)xk=Fkxk−1+Gkwk,where(5)Fk−1=1Ts000100001Ts0001,Gk−1=0.5Ts20Ts000.5Ts20Ts,wk is gaussian process noise with zero mean and the covariance matrix of process noise is Qk.


## 2.3. Sensor Model

The measurementz of the sensor can be modeled according to the position of the UAV x,yT and the target position xt,ytT as follows:(6)zk=hxk,xt,k+vk=xt,k−xk2+yt,k−yk2tan−1yt,k−ykxt,k−xk+vk,where vk is gaussian measurement noise with zero mean, and the measurement noise covariance matrix is Rk, Rk=σr200σϕ2.


## 2.4. Extended Information Filtering (EIF)

Compared with the extended Kalman filter (EKF) [24], EIF can easily realize the information matrix fusion. The simple addition of nature makes information filtering very suitable for multisensor, decentralized, and distributed estimation. Each sensor node only needs to generate an information matrix, and the global information estimation can be obtained by simple algebraic addition at the fusion center. The state estimate x∧ and covariance P are used in EKF, which are replaced by the information state y∧ and information matrix Y in the information form. The information state and information matrix are defined as y∧=P−1x∧, Y=P−1. EIF includes prediction process and the estimation process as follows [20]:Prediction process:(7)Yk|k−1=FkY−1k−1|k−1FTk+Qk−1,(8)y∧k|k−1=Yk|k−1Fkx∧k−1|k−1.Definition:(9)Iik=HiTkRi−1kHik,(10)iik=HiTkRi−1kzik−hixk,xt,k+Hikx∧k|k−1,(11)x∧k|k−1=Y−1k|k−1y∧k|k−1.Estimation process:(12)Yk|k=Yk|k−1+∑i=1nIik,(13)y∧k|k=y∧k|k−1+∑i=1niik,(14)x∧k|k=Y−1k|ky∧k|k,y∧k|k−1 and Yk|k−1 are prediction information state and prediction information matrix, respectively, Yk−1|k−1 is the information state matrix of k −  1 moment, Hik is the Jacobian matrix of the UAV i, and zik is measurement information of UAV i.


## 3. Nonlinear Model Predictive Control

### 3.1. Choice of Cost Function

Shannon defined entropy in information theory to measure the amount of uncertainty or randomness in-state results. Entropy is the most general measure of uncertainty in sensing and estimation tasks. The larger the entropy, the less accurate the state estimation. Conversely, the smaller the entropy, the more accurate the target state estimation will be. An additional benefit is choosing information entropy as a cost function in controlling multiple sensors. The entropy of the estimated distribution with multiple sensor observations can be decomposed according to the observed mutual information, which provides a beneficial structure for multiple sensor cooperation, as shown in EIF. Next, we will analyze the relationship between the information entropy and the covariance matrix.The probability density function of target stateX is px|z1,z2,zi; after EIF and obeying normal distribution Nμ,P, we can get the probability density function of n-dimensional random variable X as(15)px|z1,z2,zi=12πndetPexp−12x−μTP−1x−μ,where μ and P are Gaussian of mean and covariance after EIF, respectively. We define Hpx|z1,z2,zi to be the entropy of the estimate distribution, according to the definition of entropy:(16)Hpx|z1,z2,zi=−∫−∞∞px|z1,z2…,zilogpx|z1,z2,zidx|z1,z2…,zi=−∫−∞∞px|z1,z2…,zilog2π−n/2P−1/2exp−12x−μTP−1x−μdx|z1,…z2,zi=−∫−∞∞px|z1,z2…,zilog2π−n/2P−1/2−12x−μTP−1x−μlogedx|z1,z2…,zi=log2π−n/2P−1/2+loge2∫−∞∞x−μTP−1x−μdx|z1,z2…,zi.Letx=P1/2y+μ, we can get y∼N0,1 and(17)Hpx|z1,z2…,zi=log2π−n/2P−1/2+loge2∫−∞∞py×yTydy=log2π−n/2P−1/2+loge2∑i=1nEyi2=log2πn/2P1/2+nloge2=12log2πenP.Equation (17) shows that the target state’s entropy is proportional to the determinant of the covariance matrix. To obtain an accurate target state, the determinant of the covariance matrix needs to be as small as possible, corresponding to the determinant of the information matrix being as large as possible, and the performance function could be as follows:(18)Ji,k=−Yi,k.Therefore, the entropy minimization control problem is transformed into the control problem of minimizing the negative value of the information matrix’s determinant.


### 3.2. Model Predictive Control Method

The model predictive control (MPC) method is used to solve the problem of optimal control [25]. To relax requirements on the individual UAV’s processor capabilities, the centralized mode is adopted in this paper. At each discrete moment, predicting the system’s future state evolution through the system model and the predicted future state of the system is used to establish an optimization problem, which is solved online to obtain an optimal control sequence. By repeating the previous process at the next moment, this process will continue repeatedly roll as time goes on. The solution idea is shown in Figure 1.Figure 1
Diagram of model predictive control output.Figure2 shows the control structure of multi-UAVs’ coordinated tracking target based on information theory. By determining the optimal control ui,k∗ of UAV i at time k, we control UAV i at the optimal observation position. Through the observation of UAV i, the state entropy of the target at the current moment is minimized, thereby obtaining an accurate target state.Figure 2
Information-theoretic and NMPC tracking control structure.Combined with the model predictive control method, the cost function of UAVi for target observation at time k is(19)Ji,k=∑m=kk+NpLxt,m,xi,m=∑m=kk+Np−∑i=1nYi,m,where NP is the predicted number of steps, n is the number of UAVs, xi,m is the state variable of the UAV i at time m, xt,m is the target variable of the UAV i at time m, and Lxt,m,xi,m is the scalar function of the target and the UAV state at time m. Therefore, the multi-UAVs’ coordinated target tracking problem is transformed into a problem of optimal control that minimizes the objective function under constraints based on a nonlinear model. The mathematical model is as follows:(20)minui,k∗Ji,k,s.t.xi,k+1−fdxi,k,ui,k=0,Svui,k=ui,vk−v0−vmaxvmax≤0,Sωui,k=ui,ωk−ωmaxωmax≤0,where ui,k is the control input variable of the UAV i at time k.


### 3.3. Optimal Control Solution

To solve the problem of (20), the Lagrange multiplier vector λ is first introduced, and λ=λ1,λ2,λ3,λ4,λ5T based on the state of the UAV is five-dimensional. The optimal control problem of multi-UAVs’ system coordinated target tracking can be attributed to the problem of minimizing the following performance function by introducing a penalty factor μv and μw.(21)Ji,k=∑m=kk+NpLxt,m,xi,m+∑m=kk+Np−1λm+1Tfxi,m,ui,m−xi,m+1+μvlv,mSvui,m+μωlω,mSωui,m,k=1,2,…,N,l∗,m=0,S∗,m≤0,1,S∗,m>0,l∗,m shows that the penalty function is zero when the control variable meets the constraint; otherwise, the penalty function is sufficiently large.Let us define the Hamiltonian function as(22)Hi,m=Lxt,m,xi,m+λm+1Tfxi,m,ui,m+μvlv,mSvui,m+μωlω,mSωui,m,m=k,k+1,…,k+Np−1.We can get(23)Ji,k=Lxt,k+Np,xi,k+Np+∑m=kk+Np−1Hi,m−λm+1Txi,m+1=Lxt,k+Np,xi,k+Np−λk+NpTxi,k+Np+λkTxi,k+∑m=kk+Np−1Hi,m−λmTxi,m.We perform first variation on (23) to get(24)δJi,k=∂Lxt,k+Np,xi,k+Np∂xi,k+Np−λk+NpTTδxi,k+Np+λkTδxi,k+∑m=kk+Np−1∂Hi,m∂xi,m−λmTTδxi,m+∂Hi,m∂ui,mTδui,m.The necessary conditions for achieving optimal control are(25)λk+NpT=∂Lxt,k+Np,xi,k+Np∂xi,k+Np,(26)λm=∂Hi,m∂xi,m,where(27)∂Hi,m∂xi,m=∂Lxt,m,xi,m∂xi,m+∂fTxi,m,ui,m∂xi,mλm+1.LetY=∑i=1nYi,m, we can get Lxt,m,xi,m=−Y. It can be seen from the relationship between derivative and differential that(28)dLxt,m,xi,m=trace∂Lxt,m,xi,m∂YTdY.From the formuladY=YtraceY−1dY, we can know that(29)dLxt,m,xi,m=trace∂Lxt,m,xi,m∂YTdY=−tracedY=−traceYtraceY−1dY=−traceYtrace2Ri,m−1Hxt,mY−1∂Hxt,m∂xi,mdxi,m=−traceYtrace2Ri,m−1Hxt,mY−1TT∂Hxt,m∂xi,mdxi,m.According to formulatraceATB=vecATvecB, (29) becomes(30)dLxt,m,xi,m=−Ytracevec2Ri,m−1Hxt,mY−1TTvec∂Hxt,m∂xi,mdxi,m,where trace⋅ represents the trace of the matrix and vec⋅ represents the column vectorization of the matrix.According to the relationship between derivative and differential, we can get the following:(31)dLxt,m,xi,m=trace∂Lxt,m,xi,m∂xi,mTdxi,m.Combining (30) and (31), we can see that(32)∂Lxt,m,xi,m∂xi,m=−2YvecRi,m−1Hxt,mY−1T∂vecHxt,m∂xi,mTT=2Lxt,m,xi,mvecRi,m−1Hxt,m∑i=1nYi,m−1T∂vecHxt,m∂xi,mTT,where(33)Hxt,l=∂hixk,xt,k∂xt,kT=Δxi,kri,k0Δyi,kri,k0−Δyi,kri,k20Δxi,kri,k20,∂vecHxt,l∂xi,lT=1ri,k4−Δyi,k2ri,kΔxi,kΔyi,kri,k000−2Δxi,kΔyi,kΔxi,k2−Δyi,k20000000000000Δxi,kΔyi,kri,k−Δxi,k2ri,k000Δxi,k2−Δyi,k22Δxi,kΔyi,k0000000000000,∂fxi,l,ui,l∂xi,l=10−vi,lsinφi,lTscosφi,lTs001vi,lcosφi,lTssinφi,lTs00010Ts0001−Tsτv000001−Tsτω.Equation (24) becomes(34)δJi,k=λkTδxi,k+∑m=kk+Np−1∂Hi,m∂ui,mTδui,m.Using the gradient descent method to iteratively optimize the cost function, we get(35)ui,mt+1=ui,mt−Δi,m∂Hi,m∂ui,m,m=k,k+1,…,k+Np−1,where t is the number of iteration, Δi,m is the iteration step, and ∂Hi,m/∂ui,m is expressed as(36)∂Hi,m∂ui,m=λm+1T∂fxi,m,ui,m∂ui,m+μvlv,m∂Svui,m∂ui,m+μωlω,m∂Sωui,m∂ui,m,(37)∂fxi,m,ui,m∂ui,m=000Tsτv00000TsτωT.The iteration is terminated in the existing optimization method [13] when the cost function change is less than a certain threshold. However, this method has a slower convergence speed when the UAV’s initial position is far from the target. Therefore, this paper adopts the step size as an iteration termination condition to improve the convergence speed. The step size decreases when the cost function does not decrease but increases in the iterative process. The iteration is terminated when the step size reaches a smaller threshold, and the cost function changes little, indicating that the optimized result is obtained.In summary, the flow chart of the nonlinear model predictive control algorithm based on information entropy (IENMPC) is shown in Table1.Table 1
Nonlinear model prediction control algorithm based on information entropy (IENMPC).
Current UAViCommunicationCoordinated UAVCalculating optimal control ofk timeCurrent state of UAVi⟶ZikCurrent state of coordinated UAV.Observation information of sensor acquireszikZk↼Observation information of sensor acquireszk.Initialize control sequenceU0, cost function Jmin, predicted number of steps NP, iteration step Δi,k, α, ε and tWhileΔi,m>εWhileΔi,m>εUAV state propagation [equation (3)].Information matrix, target estimation state, jacobian matrix, and observation noise matrix were acquired through EIF. [equations (7)–(14)].λm computation [equation (25) and (26)].Ji computation [equation (23)].∂Hi,m/∂ui,m computation [equation (36)].IfJi<Jmin, Jmin=Ji.elseΔi,m=αΔi,m.end.ui,mt+1=ui,mt−Δi,m∂Hi,m/∂ui,m..t=t+1t=t+1endEndCalculating the status of UAV atk + 1 time [equation (3)]Calculating optimal control input ofk + 1 time


## 3.1. Choice of Cost Function

Shannon defined entropy in information theory to measure the amount of uncertainty or randomness in-state results. Entropy is the most general measure of uncertainty in sensing and estimation tasks. The larger the entropy, the less accurate the state estimation. Conversely, the smaller the entropy, the more accurate the target state estimation will be. An additional benefit is choosing information entropy as a cost function in controlling multiple sensors. The entropy of the estimated distribution with multiple sensor observations can be decomposed according to the observed mutual information, which provides a beneficial structure for multiple sensor cooperation, as shown in EIF. Next, we will analyze the relationship between the information entropy and the covariance matrix.The probability density function of target stateX is px|z1,z2,zi; after EIF and obeying normal distribution Nμ,P, we can get the probability density function of n-dimensional random variable X as(15)px|z1,z2,zi=12πndetPexp−12x−μTP−1x−μ,where μ and P are Gaussian of mean and covariance after EIF, respectively. We define Hpx|z1,z2,zi to be the entropy of the estimate distribution, according to the definition of entropy:(16)Hpx|z1,z2,zi=−∫−∞∞px|z1,z2…,zilogpx|z1,z2,zidx|z1,z2…,zi=−∫−∞∞px|z1,z2…,zilog2π−n/2P−1/2exp−12x−μTP−1x−μdx|z1,…z2,zi=−∫−∞∞px|z1,z2…,zilog2π−n/2P−1/2−12x−μTP−1x−μlogedx|z1,z2…,zi=log2π−n/2P−1/2+loge2∫−∞∞x−μTP−1x−μdx|z1,z2…,zi.Letx=P1/2y+μ, we can get y∼N0,1 and(17)Hpx|z1,z2…,zi=log2π−n/2P−1/2+loge2∫−∞∞py×yTydy=log2π−n/2P−1/2+loge2∑i=1nEyi2=log2πn/2P1/2+nloge2=12log2πenP.Equation (17) shows that the target state’s entropy is proportional to the determinant of the covariance matrix. To obtain an accurate target state, the determinant of the covariance matrix needs to be as small as possible, corresponding to the determinant of the information matrix being as large as possible, and the performance function could be as follows:(18)Ji,k=−Yi,k.Therefore, the entropy minimization control problem is transformed into the control problem of minimizing the negative value of the information matrix’s determinant.


## 3.2. Model Predictive Control Method

The model predictive control (MPC) method is used to solve the problem of optimal control [25]. To relax requirements on the individual UAV’s processor capabilities, the centralized mode is adopted in this paper. At each discrete moment, predicting the system’s future state evolution through the system model and the predicted future state of the system is used to establish an optimization problem, which is solved online to obtain an optimal control sequence. By repeating the previous process at the next moment, this process will continue repeatedly roll as time goes on. The solution idea is shown in Figure 1.Figure 1
Diagram of model predictive control output.Figure2 shows the control structure of multi-UAVs’ coordinated tracking target based on information theory. By determining the optimal control ui,k∗ of UAV i at time k, we control UAV i at the optimal observation position. Through the observation of UAV i, the state entropy of the target at the current moment is minimized, thereby obtaining an accurate target state.Figure 2
Information-theoretic and NMPC tracking control structure.Combined with the model predictive control method, the cost function of UAVi for target observation at time k is(19)Ji,k=∑m=kk+NpLxt,m,xi,m=∑m=kk+Np−∑i=1nYi,m,where NP is the predicted number of steps, n is the number of UAVs, xi,m is the state variable of the UAV i at time m, xt,m is the target variable of the UAV i at time m, and Lxt,m,xi,m is the scalar function of the target and the UAV state at time m. Therefore, the multi-UAVs’ coordinated target tracking problem is transformed into a problem of optimal control that minimizes the objective function under constraints based on a nonlinear model. The mathematical model is as follows:(20)minui,k∗Ji,k,s.t.xi,k+1−fdxi,k,ui,k=0,Svui,k=ui,vk−v0−vmaxvmax≤0,Sωui,k=ui,ωk−ωmaxωmax≤0,where ui,k is the control input variable of the UAV i at time k.


## 3.3. Optimal Control Solution

To solve the problem of (20), the Lagrange multiplier vector λ is first introduced, and λ=λ1,λ2,λ3,λ4,λ5T based on the state of the UAV is five-dimensional. The optimal control problem of multi-UAVs’ system coordinated target tracking can be attributed to the problem of minimizing the following performance function by introducing a penalty factor μv and μw.(21)Ji,k=∑m=kk+NpLxt,m,xi,m+∑m=kk+Np−1λm+1Tfxi,m,ui,m−xi,m+1+μvlv,mSvui,m+μωlω,mSωui,m,k=1,2,…,N,l∗,m=0,S∗,m≤0,1,S∗,m>0,l∗,m shows that the penalty function is zero when the control variable meets the constraint; otherwise, the penalty function is sufficiently large.Let us define the Hamiltonian function as(22)Hi,m=Lxt,m,xi,m+λm+1Tfxi,m,ui,m+μvlv,mSvui,m+μωlω,mSωui,m,m=k,k+1,…,k+Np−1.We can get(23)Ji,k=Lxt,k+Np,xi,k+Np+∑m=kk+Np−1Hi,m−λm+1Txi,m+1=Lxt,k+Np,xi,k+Np−λk+NpTxi,k+Np+λkTxi,k+∑m=kk+Np−1Hi,m−λmTxi,m.We perform first variation on (23) to get(24)δJi,k=∂Lxt,k+Np,xi,k+Np∂xi,k+Np−λk+NpTTδxi,k+Np+λkTδxi,k+∑m=kk+Np−1∂Hi,m∂xi,m−λmTTδxi,m+∂Hi,m∂ui,mTδui,m.The necessary conditions for achieving optimal control are(25)λk+NpT=∂Lxt,k+Np,xi,k+Np∂xi,k+Np,(26)λm=∂Hi,m∂xi,m,where(27)∂Hi,m∂xi,m=∂Lxt,m,xi,m∂xi,m+∂fTxi,m,ui,m∂xi,mλm+1.LetY=∑i=1nYi,m, we can get Lxt,m,xi,m=−Y. It can be seen from the relationship between derivative and differential that(28)dLxt,m,xi,m=trace∂Lxt,m,xi,m∂YTdY.From the formuladY=YtraceY−1dY, we can know that(29)dLxt,m,xi,m=trace∂Lxt,m,xi,m∂YTdY=−tracedY=−traceYtraceY−1dY=−traceYtrace2Ri,m−1Hxt,mY−1∂Hxt,m∂xi,mdxi,m=−traceYtrace2Ri,m−1Hxt,mY−1TT∂Hxt,m∂xi,mdxi,m.According to formulatraceATB=vecATvecB, (29) becomes(30)dLxt,m,xi,m=−Ytracevec2Ri,m−1Hxt,mY−1TTvec∂Hxt,m∂xi,mdxi,m,where trace⋅ represents the trace of the matrix and vec⋅ represents the column vectorization of the matrix.According to the relationship between derivative and differential, we can get the following:(31)dLxt,m,xi,m=trace∂Lxt,m,xi,m∂xi,mTdxi,m.Combining (30) and (31), we can see that(32)∂Lxt,m,xi,m∂xi,m=−2YvecRi,m−1Hxt,mY−1T∂vecHxt,m∂xi,mTT=2Lxt,m,xi,mvecRi,m−1Hxt,m∑i=1nYi,m−1T∂vecHxt,m∂xi,mTT,where(33)Hxt,l=∂hixk,xt,k∂xt,kT=Δxi,kri,k0Δyi,kri,k0−Δyi,kri,k20Δxi,kri,k20,∂vecHxt,l∂xi,lT=1ri,k4−Δyi,k2ri,kΔxi,kΔyi,kri,k000−2Δxi,kΔyi,kΔxi,k2−Δyi,k20000000000000Δxi,kΔyi,kri,k−Δxi,k2ri,k000Δxi,k2−Δyi,k22Δxi,kΔyi,k0000000000000,∂fxi,l,ui,l∂xi,l=10−vi,lsinφi,lTscosφi,lTs001vi,lcosφi,lTssinφi,lTs00010Ts0001−Tsτv000001−Tsτω.Equation (24) becomes(34)δJi,k=λkTδxi,k+∑m=kk+Np−1∂Hi,m∂ui,mTδui,m.Using the gradient descent method to iteratively optimize the cost function, we get(35)ui,mt+1=ui,mt−Δi,m∂Hi,m∂ui,m,m=k,k+1,…,k+Np−1,where t is the number of iteration, Δi,m is the iteration step, and ∂Hi,m/∂ui,m is expressed as(36)∂Hi,m∂ui,m=λm+1T∂fxi,m,ui,m∂ui,m+μvlv,m∂Svui,m∂ui,m+μωlω,m∂Sωui,m∂ui,m,(37)∂fxi,m,ui,m∂ui,m=000Tsτv00000TsτωT.The iteration is terminated in the existing optimization method [13] when the cost function change is less than a certain threshold. However, this method has a slower convergence speed when the UAV’s initial position is far from the target. Therefore, this paper adopts the step size as an iteration termination condition to improve the convergence speed. The step size decreases when the cost function does not decrease but increases in the iterative process. The iteration is terminated when the step size reaches a smaller threshold, and the cost function changes little, indicating that the optimized result is obtained.In summary, the flow chart of the nonlinear model predictive control algorithm based on information entropy (IENMPC) is shown in Table1.Table 1
Nonlinear model prediction control algorithm based on information entropy (IENMPC).
Current UAViCommunicationCoordinated UAVCalculating optimal control ofk timeCurrent state of UAVi⟶ZikCurrent state of coordinated UAV.Observation information of sensor acquireszikZk↼Observation information of sensor acquireszk.Initialize control sequenceU0, cost function Jmin, predicted number of steps NP, iteration step Δi,k, α, ε and tWhileΔi,m>εWhileΔi,m>εUAV state propagation [equation (3)].Information matrix, target estimation state, jacobian matrix, and observation noise matrix were acquired through EIF. [equations (7)–(14)].λm computation [equation (25) and (26)].Ji computation [equation (23)].∂Hi,m/∂ui,m computation [equation (36)].IfJi<Jmin, Jmin=Ji.elseΔi,m=αΔi,m.end.ui,mt+1=ui,mt−Δi,m∂Hi,m/∂ui,m..t=t+1t=t+1endEndCalculating the status of UAV atk + 1 time [equation (3)]Calculating optimal control input ofk + 1 time


## 4. Simulation Results

The mission of the UAV used in this paper is to observe the remote sea surface target, and it is known that the estimation accuracy depends not only on the filter performance but also on the relative geometry between the unmanned aerial vehicle (UAV) and the target. Therefore, the scenario of two, three, four, and five UAVs cooperating to observe the target is considered in this paper, and the constraint condition that UAV needs to meet is that the velocity and yaw rate of UAV are within the constraint range.


### 4.1. Assumptions and Conditions

Experiments were performed using MATLAB R2020a on the computer with CPU processor of Intel Corei7 and dominant frequency of 1.6 GHz. It is assumed that the target is moving with a constant speed at 8.5 m/s, and the mean square deviation of the velocity noise is 0.01 in the horizontal and vertical directions. The initial position of the target is at (80000, 80000). The initial position, speed, and performance parameters of the UAV are shown in Table2, and the other parameter settings required are shown in Table 3. The simulation time lasts for 1500 s.Table 2
Initial state information and performance constraints of UAVs.
NumberInitial statev0 (m/s)Δvmax (m/s)ωmax (rad/s)Observation error (distance (m), angle (°))1(0, 0, 0, 100, 0)100100.35[20, 0.3]2(1000, 0, 0, 100, 0)100100.35[20, 0.3]3(2000, 0, 0, 100, 0)100100.35[20, 0.3]4(−1000, 0, 0, 100, 0)100100.35[20, 0.3]5(−2000, 0, 0, 100, 0)100100.35[20, 0.3]Table 3
Simulation parameters.
ParameterValueUnitTs0.5sNp9N/Aτv, τω0.5sμc, μw1e5N/Aε0.001N/AJmininfN/AΔ0.05N/Aα0.9N/AInitial covariance matrix isP0=diag100,50,100,50 in filter and the initial control sequence is U0=100,0;100,0;100,0;100,0;100,0;100,0;100,0;100,0;100,0T.


### 4.2. Analysis of Simulation Results

To analyze the feasibility and benefits of the proposed method (IENMPC) in this paper, the same scenarios are discussed with the method of Lyapunov vector field guidance (LVFG), which is expressed as(38)gx,y=gxgy=x˙dy˙d=vdrr2+R02−xrr2−R02−yr2⋅r⋅R0−yrr2−R02+xr2⋅r⋅R0.Obtaining the expected heading angle of the UAV through the vector field guidance, we get(39)φd=arctany˙dx˙d.A proportional controller is designed using the principle of feedback control as follows:(40)uω=φ˙d−kiφd−φ,φ˙d=4vdR0r2r2+R022,where r=xi−xT2+yi−yT2=xr2+yr2, xT,yT is the position of the target that can be estimated from the filter, and R0 is a desired standoff distance from the UAV to the target. vd is a desired UAV speed. In this simulation set, R0=20km and vd=110m/s.Figure3 shows the simulation results of UAV1 and UAV2 when the simulation time lasted for 1500 s. It can be seen from Figures 3(a) and 3(b) that the UAV first performs angular separation so that the angle between the two UAVs and the target approaches 90 degrees and then moves towards the target. Figures 3(c) and 3(d) show the change curve of speed and yaw rate during the UAV flight, respectively. It is easy to see that the yaw rate control command and the speed control command are feasible according to the range of yaw rate, which is within the maximum range ±0.35, and the range of speed is within the maximum variation range ±10 m/s. Another phenomenon is that the UAV flies to the target at the maximum speed of 110 m/s in Figure 3(c) which indicates that the UAV separates the angle and flies to the target in the shortest time to reduce the uncertainty of the target position estimation.Figure 3
Simulation results of two UAVs. (a) Trajectory of IEMPC method. (b) The line-of-sight angle. (c) The variation curve of speed. (d) The variation curve of yaw rate.
(a)(b)(c)(d)Figure4(a) shows the trajectories of UAVs planned by the LGVF method and the UAVs directly moving towards the target without angle separation. Figure 4(b) displays the target position’s root mean squared error (RMSE) based on the IENMPC method and the LVFG method after 200 Monte Carlo simulations. The average RMSE of the target position is 6.5 m through the IENMPC method. However, the average RMSE of the target is 16.6 m using the LGVF method, and the error converges to 7.2 m until the simulation is in the 1400 s, when the UAV is already close to the target. Figure 4(c) shows the information entropy variation curve of the two methods. It can be seen from Figure 4(c) that the information entropy is −8.76 in the IEMPC method, and the information entropy is −8.02 in the LGVF method when the simulation is 300 s. The information entropy of the IENMPC method converges at −9.29, while the information entropy of the LGVF method maintains around −8.97 in the end. The information entropy of the IEMPC method, which is more accurate in estimating the target state, is significantly smaller than that of the LGVF method. We can conclude that the IENMPC method is better than the LGVF method in the position estimation of the remote marine target.Figure 4
Comparison results of IEMPC and LVFG. (a) Trajectory of LVFG method. (b) RMSE of the target position. (c) Information entropy.
(a)(b)(c)Figure5 shows the simulation results of three UAVs when the simulation time lasted for 1500 s. Figure 5(a) displays the trajectory of coordinated tracking of three UAVs. Figure 4(b) shows the line-of-sight angle between three UAVs and the target. UAV2 and UAV3 fly in the same direction for a while at first and then UAV2 turns back and finally moves towards the target to make UAV2 form a 60° angle between UAV1 and the target in Figure 5(b). It can be further obtained that the optimal observation configuration for three UAVs is that the line-of-sight angle between UAV2, UAV 1, and the target is 60 degrees, and the line-of-sight angle between UAV1, UAV3, and the target is 120 degrees from Figure 5(b). From Figures 5(c) and 5(d), UAVs’ speed and yaw rate are within the constraint range, proving the proposed method’s effectiveness.Figure 5
Simulation results of three UAVs. (a) Trajectory of three UAVs. (b) The line-of-sight angle between three UAVs and target. (c) The variation curve of speed. (d) The variation curve of yaw rate.
(a)(b)(c)(d)The simulation results of four UAVs and five UAVs are shown in Figure6. The optimal geometric configuration of four UAVs is that the two UAVs fly to the target at 90 degrees. The optimal geometric configuration of the observation of five UAVs is the configuration of 2-2-1. Therefore, the optimal observation geometric configuration is that the UAV formation angle is 90 degrees based on the fact that the number of UAVs is even. The UAV formation angles are 60 degrees and 120 degrees when the number of UAVs is odd for the multi-UAVs’ coordinated target tracking problem, which has guiding significance for designing multi-UAVs’ flight control laws that maintain the optimal observation configuration.Figure 6
Simulation results of multi-UAVs. (a) Trajectory of four UAVs. (b) Trajectory of five UAVs.
(a)(b)In Figure7, the RMSE and information entropy of multi-UAVs’ target position estimation are displayed. In Figure 7(a), the RMSE of the target position is decreasing as the number of UAV increases, which indicates that increasing the UAV number can improve the accuracy of target position estimation. However, through Figure 5(b), the information entropy is reduced by 2.83% from two UAVs to three UAVs. It is reduced by 2.41% from three UAVs to four UAVs. The information entropy is only reduced by 1.63% from four UAVs to five UAVs. This result shows that the target position’s improvement is getting smaller and smaller as the number of UAVs increases. Considering the use effectiveness and tracking accuracy, observing three UAVs or four UAVs is an excellent choice to track the target.Figure 7
RMSE and information entropy of multi-UAVs. (a) RMSE of the target position. (b) Information entropy.
(a)(b)To analyze the computational efficiency (CPU cost) of the proposed algorithm, the average elapsed time (AET) of the algorithm, the AET of the optimized solution, and the maximum running time of the algorithm are evaluated. The AET of the algorithm refers to the time taken by the algorithm to run once, and the results are shown in Table4.Table 4
Consume time of the algorithm with different UAVs.
Number of UAVsThe AET of the optimized solution (ms)The AET of the algorithm (ms)The maximum running time (ms)Two UAVs32.58662.27187.673Three UAVs38.211111.793125.527Four UAVs43.476175.781210.093Five UAVs48.489241.526265.688Table4 shows that the AET of the algorithm, the AET of the optimized solution, and the maximum run time increase with the number of UAVs. However, the maximum run time of UAVs does not exceed 300 ms, which indicates the good real-time performance of the proposed IENMPC method.


## 4.1. Assumptions and Conditions

Experiments were performed using MATLAB R2020a on the computer with CPU processor of Intel Corei7 and dominant frequency of 1.6 GHz. It is assumed that the target is moving with a constant speed at 8.5 m/s, and the mean square deviation of the velocity noise is 0.01 in the horizontal and vertical directions. The initial position of the target is at (80000, 80000). The initial position, speed, and performance parameters of the UAV are shown in Table2, and the other parameter settings required are shown in Table 3. The simulation time lasts for 1500 s.Table 2
Initial state information and performance constraints of UAVs.
NumberInitial statev0 (m/s)Δvmax (m/s)ωmax (rad/s)Observation error (distance (m), angle (°))1(0, 0, 0, 100, 0)100100.35[20, 0.3]2(1000, 0, 0, 100, 0)100100.35[20, 0.3]3(2000, 0, 0, 100, 0)100100.35[20, 0.3]4(−1000, 0, 0, 100, 0)100100.35[20, 0.3]5(−2000, 0, 0, 100, 0)100100.35[20, 0.3]Table 3
Simulation parameters.
ParameterValueUnitTs0.5sNp9N/Aτv, τω0.5sμc, μw1e5N/Aε0.001N/AJmininfN/AΔ0.05N/Aα0.9N/AInitial covariance matrix isP0=diag100,50,100,50 in filter and the initial control sequence is U0=100,0;100,0;100,0;100,0;100,0;100,0;100,0;100,0;100,0T.


## 4.2. Analysis of Simulation Results

To analyze the feasibility and benefits of the proposed method (IENMPC) in this paper, the same scenarios are discussed with the method of Lyapunov vector field guidance (LVFG), which is expressed as(38)gx,y=gxgy=x˙dy˙d=vdrr2+R02−xrr2−R02−yr2⋅r⋅R0−yrr2−R02+xr2⋅r⋅R0.Obtaining the expected heading angle of the UAV through the vector field guidance, we get(39)φd=arctany˙dx˙d.A proportional controller is designed using the principle of feedback control as follows:(40)uω=φ˙d−kiφd−φ,φ˙d=4vdR0r2r2+R022,where r=xi−xT2+yi−yT2=xr2+yr2, xT,yT is the position of the target that can be estimated from the filter, and R0 is a desired standoff distance from the UAV to the target. vd is a desired UAV speed. In this simulation set, R0=20km and vd=110m/s.Figure3 shows the simulation results of UAV1 and UAV2 when the simulation time lasted for 1500 s. It can be seen from Figures 3(a) and 3(b) that the UAV first performs angular separation so that the angle between the two UAVs and the target approaches 90 degrees and then moves towards the target. Figures 3(c) and 3(d) show the change curve of speed and yaw rate during the UAV flight, respectively. It is easy to see that the yaw rate control command and the speed control command are feasible according to the range of yaw rate, which is within the maximum range ±0.35, and the range of speed is within the maximum variation range ±10 m/s. Another phenomenon is that the UAV flies to the target at the maximum speed of 110 m/s in Figure 3(c) which indicates that the UAV separates the angle and flies to the target in the shortest time to reduce the uncertainty of the target position estimation.Figure 3
Simulation results of two UAVs. (a) Trajectory of IEMPC method. (b) The line-of-sight angle. (c) The variation curve of speed. (d) The variation curve of yaw rate.
(a)(b)(c)(d)Figure4(a) shows the trajectories of UAVs planned by the LGVF method and the UAVs directly moving towards the target without angle separation. Figure 4(b) displays the target position’s root mean squared error (RMSE) based on the IENMPC method and the LVFG method after 200 Monte Carlo simulations. The average RMSE of the target position is 6.5 m through the IENMPC method. However, the average RMSE of the target is 16.6 m using the LGVF method, and the error converges to 7.2 m until the simulation is in the 1400 s, when the UAV is already close to the target. Figure 4(c) shows the information entropy variation curve of the two methods. It can be seen from Figure 4(c) that the information entropy is −8.76 in the IEMPC method, and the information entropy is −8.02 in the LGVF method when the simulation is 300 s. The information entropy of the IENMPC method converges at −9.29, while the information entropy of the LGVF method maintains around −8.97 in the end. The information entropy of the IEMPC method, which is more accurate in estimating the target state, is significantly smaller than that of the LGVF method. We can conclude that the IENMPC method is better than the LGVF method in the position estimation of the remote marine target.Figure 4
Comparison results of IEMPC and LVFG. (a) Trajectory of LVFG method. (b) RMSE of the target position. (c) Information entropy.
(a)(b)(c)Figure5 shows the simulation results of three UAVs when the simulation time lasted for 1500 s. Figure 5(a) displays the trajectory of coordinated tracking of three UAVs. Figure 4(b) shows the line-of-sight angle between three UAVs and the target. UAV2 and UAV3 fly in the same direction for a while at first and then UAV2 turns back and finally moves towards the target to make UAV2 form a 60° angle between UAV1 and the target in Figure 5(b). It can be further obtained that the optimal observation configuration for three UAVs is that the line-of-sight angle between UAV2, UAV 1, and the target is 60 degrees, and the line-of-sight angle between UAV1, UAV3, and the target is 120 degrees from Figure 5(b). From Figures 5(c) and 5(d), UAVs’ speed and yaw rate are within the constraint range, proving the proposed method’s effectiveness.Figure 5
Simulation results of three UAVs. (a) Trajectory of three UAVs. (b) The line-of-sight angle between three UAVs and target. (c) The variation curve of speed. (d) The variation curve of yaw rate.
(a)(b)(c)(d)The simulation results of four UAVs and five UAVs are shown in Figure6. The optimal geometric configuration of four UAVs is that the two UAVs fly to the target at 90 degrees. The optimal geometric configuration of the observation of five UAVs is the configuration of 2-2-1. Therefore, the optimal observation geometric configuration is that the UAV formation angle is 90 degrees based on the fact that the number of UAVs is even. The UAV formation angles are 60 degrees and 120 degrees when the number of UAVs is odd for the multi-UAVs’ coordinated target tracking problem, which has guiding significance for designing multi-UAVs’ flight control laws that maintain the optimal observation configuration.Figure 6
Simulation results of multi-UAVs. (a) Trajectory of four UAVs. (b) Trajectory of five UAVs.
(a)(b)In Figure7, the RMSE and information entropy of multi-UAVs’ target position estimation are displayed. In Figure 7(a), the RMSE of the target position is decreasing as the number of UAV increases, which indicates that increasing the UAV number can improve the accuracy of target position estimation. However, through Figure 5(b), the information entropy is reduced by 2.83% from two UAVs to three UAVs. It is reduced by 2.41% from three UAVs to four UAVs. The information entropy is only reduced by 1.63% from four UAVs to five UAVs. This result shows that the target position’s improvement is getting smaller and smaller as the number of UAVs increases. Considering the use effectiveness and tracking accuracy, observing three UAVs or four UAVs is an excellent choice to track the target.Figure 7
RMSE and information entropy of multi-UAVs. (a) RMSE of the target position. (b) Information entropy.
(a)(b)To analyze the computational efficiency (CPU cost) of the proposed algorithm, the average elapsed time (AET) of the algorithm, the AET of the optimized solution, and the maximum running time of the algorithm are evaluated. The AET of the algorithm refers to the time taken by the algorithm to run once, and the results are shown in Table4.Table 4
Consume time of the algorithm with different UAVs.
Number of UAVsThe AET of the optimized solution (ms)The AET of the algorithm (ms)The maximum running time (ms)Two UAVs32.58662.27187.673Three UAVs38.211111.793125.527Four UAVs43.476175.781210.093Five UAVs48.489241.526265.688Table4 shows that the AET of the algorithm, the AET of the optimized solution, and the maximum run time increase with the number of UAVs. However, the maximum run time of UAVs does not exceed 300 ms, which indicates the good real-time performance of the proposed IENMPC method.


## 5. Conclusion

This paper proposed multi-UAVs’ coordinated target tracking control method, which uses a nonlinear model predictive control method to minimize the target state’s information entropy to obtain an accurate target position. This method’s advantage is that it can keep the UAV at the optimal observation position and is easy to extend to the coordinated tracking of multiple UAVs. It can effectively improve the remote marine target tracking performance under the performance constraints of the UAV at the same time. Also, the step size was used as the iterative termination condition to solve the problem of slower convergence speed when the UAV’s initial position is far from the target compared with the existing optimization methods. The convergence speed of the optimization method was improved. Comparative simulations of two UAVs with the LVFG method showed the superiority of this method for target positioning. It was possible to obtain the geometric configuration that maintained the optimal observation through the simulation of three UAVs, four UAVs, and five UAVs, which can reference the design of multi-UAVs’ flight control law based on the optimal observation configuration.Future works are planned to extend this method to three-dimensional space because the UAV position is more complex and changeable in three-dimensional space, leading to UAV’s optimal observation configuration that significantly impacts target tracking. An additional crucial future work will be performed to consider the factor of time, that is, how to obtain the accurate target state within finite time, which is also the focus of future research.



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*Source: 5472105-2022-12-26.xml*