## Abstract

In order to achieve the cooperative standoff tracking of target in non-wide area by multiple unmanned aerial vehicles (UAVs) with installed optical cameras, this paper proposes the constrained interacting multiple model (CIMM) filter to estimate the target state, as well as the time optimal guidance vector field (TOGVF) to optimize the UAV trajectory. Firstly, the geographical constraint equation deduced from the non-wide area is introduced into the traditional interacting multiple model, aiming to improve the estimation accuracy of the motion state of moving target measured from the optical cameras. According to the target motion information, the TOGVF method is then adopted to generate the velocity in the vertical plane guiding each UAV to the optimal observation height, as well as the velocity in the horizontal plane with which each UAV will converge to the standoff distance along the tangent of the limit cycle. On this basis, the speed of each UAV is adjusted in order to balance the phase difference and achieve the cooperation among multi-UAVs. The experimental results show that our method is effective in non-wide area. The estimation accuracy of target motion via CIMM increases by more than 30% compared to the single-model based filter, and the UAVs transfer to the limit cycle along the shortest path by TOGVF.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The unmanned aerial vehicles (UAVs) installed with optical cameras are increasingly used in military and civilian fields, such as surveillance, search and rescue [1–5], to perform repetitive or dangerous tasks instead of humans. Usually the most basic requirement is to improve the response capacity to various cases, requiring little or even no human intervention [6–7]. Monitoring and tracking a static or moving ground target is one of the important applications of UAV, and it is important to increase the capacity of understanding the surrounding environment [8–10]. The detected targets (such as enemy vehicles, missing persons, and urban hazard sources) are tracked by UAVs to obtain the optimal monitoring effect. However, most relevant researches focus on the target in wide area such as sea surface, flatland and so on, and there are not many researches on tracking a target in narrow areas such as roads, rivers and other special ones (called non-wide area in this paper).

To perform the mission of target tracking, the target motion first needs to be estimated using the multi-UAV optical cameras. Take as an example that a ground vehicle travels on a road, whose topographic coordinates are usually accessible with certain accuracy. This roadmap information can be used to improve the tracking accuracy of the target significantly by limiting the state of the ground target of interest, in particular its position, velocity, and acceleration in the geometry of the non-wide area. This is called the constrained target tracking problem in non-wide area, and there are three main types of technologies using the information of non-wide area. The first is the post-processing correction technology, which first runs the tracking algorithm without the information of the non-wide area, but then corrects it. Tang et al. [11] propose a Bayesian filtering method based on hospitability map, which provides each point with the possibility of proportional to the ability of the target to move at that location. Along with this approach, Kassas et al. [12] introduce the concept of synthetic inclination diagrams, describing how the target is integrated and tends to move in different directions with a certain velocity component. The second is the preprocessing of the target state or sensor measurements. This method takes advantage of the non-wide area information by defining the target states in the coordinates of non-wide area, and performs the transformation between non-wide-area environment and ground coordinate system to consider sensor measurements in filter update steps [13–14]. Herero et al. [15] utilize the pretreatment of sensor measurements, given the environmental constraints of non-wide area. The third is to use the topographic information of non-wide area as the constraint equation. Zhang et al. [16] propose the pseudo measurement method to consider the non-wide area constraints as additional fictitious measurements based on the work of Tahk [17]. Traditional filters, such as the extended Kalman filter (EKF) or particle filter, can be used for state estimation [18–19], but may be less accurate if the target maneuver is complex. In addition, the information fusion between multiple UAVs can significantly improve the estimation accuracy compared to a single UAV [20]. In many cases, the complex movement can be reduced to typical segmented motions at each specific time, so the interacting multiple model (IMM) filter [21–22] is more efficient than the abovementioned single-model filter. However, the traditional IMM filter does not utilize the geographic information. Hence the constrained IMM (CIMM) algorithm is proposed in this paper, which not only fully makes use of the geographical environment information, but also can estimate the complex target state with strong mobility more accurately. It understands the environment well and makes it possible to design the efficient guidance for UAVs next. The CIMM filter achieves the multi-models based estimation defined by proper expanding models, and has more detailed modeling information than a single model. It has good adaptability by continuously adjusting the model probability and increasing or decreasing the number of models.

According to the target motion information from the network of multi-UAV sensors, the optimal trajectories of UAVs then need to be optimized. The widely-used tracking mode can be divided into two categories, i.e., the persistent tracking and the standoff tracking. The standoff tracking requires that the horizontal distance between the UAV and the target remains constant, thus ensuring the UAV sensor’s coverage of the target and reducing the positioning error or the risk of UAV exposure. In the target-referred frame, the UAV motion can be manifested as hovering over the target. The standoff tracking mode is a common practice with high application value. The Lyapunov guidance vector field (LGVF) is an effective method to realize the standoff tracking task [23–24]. This method is divided into two levels: firstly, the Lyapunov distance function is defined and the desired course is determined to guide the UAV to the limit cycle above the target; then the Lyapunov phase function is defined to adjust the speed of each UAV, so that the multi-UAVs will distribute on the limit cycle uniformly according to the optimal space configuration among multi-UAVs. Oh et al. [25] further realize the tracking of multi-targets group using a single UAV by means of variable standoff trajectory, and use the active perception and guidance method to obtain the measurement of the maximum number of multi-targets by multi-UAVs. Reference [26] introduces the information structure to further expand the Lyapunov vector field, through the design of variable airspeed controller to achieve multi-vehicle cooperative control in the desired angular spacing control. In order to further improve the convergence speed, Chen et al. [27] propose the tangential guidance vector field, which can make the UAV converge to the limit cycle along the shortest path, but the strategy is only effective for UAV outside of the limit cycle. On this basis, Ref. [28] adopts a hybrid method using the tangential guidance vector field outside of the limit cycle and the Lyapunov guidance vector field inside of the limit cycle. Oh et al. [29] design the standoff tracking guidance law based on differential geometry method, which improves the tracking accuracy by introducing the target velocity term. Reference [30] generates the UAV trajectory by route construction method, which satisfies the turning radius constraint. Compared with Dubins method, this method generates a route with fewer discontinuous points. Kim et al. [31] propose a distributed nonlinear model predictive control framework. The noise covariance matrix of the target motion is introduced, and the relative phase error is replaced by the inner product of the relative position vector of the UAV to improve the tracking accuracy. In addition, there are other relevant algorithms such as rudder behavior, controlled collective motion, backstepping theory, partially observable Markov decision process [32–34]. However, these algorithms generate a long path, and most of them are only effective in two-dimensional plane. The time optimal guidance vector field (TOGVF) proposed in this paper can plan the shortest tracking path for UAV in three-dimensional space, shorten the transition time of UAVs to the limit cycle, and improve the tracking efficiency.

On the basis of the above analysis, this paper will first introduce the constraint equation into the traditional IMM filter in order to further improve the estimation accuracy of the target motion state. This is to say, the target motion changing among various models (such as acceleration dynamics model, left or right turning model) at different time is estimated by the CIMM filter. In addition, the estimation results are fused among multi-UAVs, reflecting the state information of the target more truthfully. On the basis of the estimation results, the TOGVF is then proposed in order to obtain the shortest path. It is assumed that the UAV motion is decoupled in the vertical plane and horizontal plane. The movement in the horizontal plane can take the heading control idea along the shortest tangential path. In the vertical plane, the vertical speed control method based on the maximum vertical acceleration can be adopted to reach the optimal height. Besides, the horizontal speed of each UAV is corrected through Lyapunov phase function to guide each UAV distributing uniformly on the limit cycle, so as to realize the cooperative monitoring of a target.

The remaining paper is organized as follows. In the second section, the problem of standoff target tracking is modeled. The third section describes the CIMM filter. The fourth section introduces the TOGVF method. The simulation results are analyzed in the fifth section. The final section makes a summary of this article.

## 2. Modeling of standoff tracking of target

#### 2.1. Problem description

First, on the basis of fully considering and utilizing the information of non-wide area constraints, the motion information of the target (such as position, velocity) measured from optical cameras is estimated with high accuracy. Then the trajectory optimization is carried out to make the UAVs converge to the required limit cycle exactly above the target and distribute uniformly according to the phase.

#### 2.2. Modeling of UAV

The UAV is simplified to be a typical three-degree-of-freedom point mass model with stable flight control system, as this paper mainly focuses on the trajectory planning but not the precise control of UAVs. Assuming that the horizontal and vertical motions of UAV are decoupled, the kinematic model of UAV in the three-dimensional inertial frame can be described by the following equations:

#### 2.3. Modeling of target

In fact, this paper only needs to get the location and speed information of the target in order to track the target. For the convenience of research, we simplify the motion model of the target. In the three-dimensional coordinate system, we define the position of target as ${P_t} = ({x_t},{y_t},{z_t})$ and the velocity of the target as $({v_{tx}},{v_{ty}},{v_{tz}}) = ({\dot{x}_t},{\dot{y}_t},{\dot{z}_t})$. We assume that the target is always on the ground with ${z_t}=0$ and ${\dot{z}_t}=0$, and the speed of target is less than the maximum speed of the UAV. The motion models of UAV in the inertial coordinate system and the relative coordinate system are shown in Fig. 1, and the velocity of UAV relative to target is

*v*and ${v_t}$ the speed of UAV and target in the inertial coordinate system respectively. In Fig. 1, the horizontal distance between the UAV and the target is $r = \sqrt {{{(x - {x_t})}^2} + {{(y - {y_t})}^2}} $, and the vertical distance is $h = z - {z_t}$. The problem of standoff tracking is studied in this paper. According to Ref. [24], it is known that UAVs should converge to the limit cycle above the target and distribute uniformly according to the phase. The standoff distance

*R*is taken as the radius of the required limit cycle, and the optimal observation height

*H*is taken as the height of limit cycle.

## 3. Estimation of target motion

#### 3.1. Constrained extended Kalman filter (CEKF)

Since the traditional Kalman filter only deals with a linear system, it is necessary to use the extended Kalman filter (EKF) to estimate the nonlinear motion of the target [18]. In this paper, assuming that the target has been moving in a non-wide area and simplifying the observation model of the UAV optical camera, the state equation and observation equation of the EKF in a horizontal plane can be expressed by

*i*th curve segment. Depending on the curve shape of non-wide area, different mathematical equations can be expressed. For example, if the route is a straight line, the constraint equation above can be expressed as where $\theta $ is the line direction,

*b*the intercept between straight line and vertical axis. Other curved routes can be divided into several arcs with different curvatures, and the arc of each segment can be expressed as the following form of constraint:

*i*th segment, respectively. This paper uses the pseudo-measurement method, a constrained Kalman filter algorithm using the constraint equation as an additional fictitious model [17]. Unlike other algorithms such as the maximum probability method and the projection method [35], this method has the advantage of considering the degree of compliance limitation by monitoring the magnitude of additional pseudo-measurement noise. This pseudo-measurement model can be expressed by the constraint equation: where ${\boldsymbol z}_k^{{r_i}} = 0$, ${{\boldsymbol h}_{{r_i}}}({\boldsymbol x}_k^{{r_i}})={r_i}({{\boldsymbol x}_t})$, $v_k^{{r_i}}\sim N(0,{\boldsymbol R}_k^{{r_i}})$ is zero mean white Gaussian noise reflecting the uncertainty of the constraint equation, and ${\boldsymbol R}_k^{{r_i}} = {({\sigma _{road}})^2}$ . Then the pseudo-measurement model is added to the previous measurement model of Eq. (5): where ${\boldsymbol z}_k^a = {\left[ {\begin{array}{{cc}} {{{\boldsymbol z}_k}}&{{\boldsymbol z}_k^{{r_i}}} \end{array}} \right]^\textrm{T}}$, ${\boldsymbol h}_k^a({{\boldsymbol x}_k}) = {\left[ {\begin{array}{{cc}} {{{\boldsymbol h}_k}}&{{{\boldsymbol h}_{{r_i}}}({\boldsymbol x}_k^{{r_i}})} \end{array}} \right]^\textrm{T}}$, ${\boldsymbol v}_k^a={\left[ {\begin{array}{{cc}} {{{\boldsymbol v}_k}}&{{\boldsymbol v}_k^{{r_i}}} \end{array}} \right]^\textrm{T}}$, and the measurement noise covariance matrix is ${\boldsymbol R}_k^a = diag({{\boldsymbol R}_k},{\boldsymbol R}_k^{{r_i}})$. The estimation of the target can be accomplished by the EKF with constraint measurement equation, which is called CEKF in this paper. The time update and measurement update of CEKF mainly include five core formulas:

#### 3.2. Constrained interacting multiple model (CIMM)

The above CEKF method only estimates one type of nonlinear motion. In the non-wide area, however, the target may have different constrained motion models at different time. Firstly, non-wide-area constraint equations are introduced into the filters corresponding to each constrained model. Using the CIMM algorithm, various constrained models corresponding to CEKFs are selected to match the different motion states of the target. At the same time, each CEKF works in parallel. The final result of CIMM is the weighted sum of the estimated results from these different CEKFs. The weight is the probability that the constrained model correctly describes the motion of the target at the current moment. The initial values of each filter at time *t* are based on the weighted synthesis of each constrained model filter at time *k*−1.

- 1) Input interaction
The mixed interaction probability of the constrained model is:

$$\mu _{k - 1|k - 1}^{i,j} = \frac{1}{{\sum\limits_{t = 1}^n {{p_{tj}}\mu _{k - 1|k - 1}^t} }}{p_{ij}}\mu _{k - 1|k - 1}^i$$where ${p_{ij}}$ is the transition probability from constrained model*i*to constrained model*j*. The mixing state and covariance matrix of each constrained model are then calculated, which will be taken as the inputs of corresponding CEKF: - 2) Conditional filtering of constrained model
Given the observation ${{\boldsymbol z}_k}$ and mixture results from Eqs. (18)–(19), the state estimation of each constrained model is updated using the CEKF process in 3.1.

- 3) Probability updating of constrained model
The probability functions are given:

$$\Lambda _k^j = N\{{\tilde{{\textbf z}}_k^j;0,{\textbf S}_k^j} \}= \frac{1}{{\sqrt {|{2\pi {\textbf S}_k^j} |} }}\exp \left( { - \frac{1}{2}{{(\tilde{{\textbf z}}_k^j)}^{\rm T}}{{({\textbf S}_k^j)}^{ - 1}}\tilde{{\textbf z}}_k^j} \right)$$Then we update the probability of each constrained model at the current moment where $c = \sum\limits_{t = 1}^n {\left( {\Lambda _k^t\sum\limits_{i = 1}^n {{p_{it}}\mu_{k - 1|k - 1}^i} } \right)}$ is the normalizing factor, ${p_{ij}}$ the Markov transition probability. - 4) Estimation fusion
Combining the probability estimation and state estimation of each constrained model, the state as well as the covariance matrix at the current moment is estimated by weight sum:

#### 3.3. Data fusion among multi-UAVs

Since multiple UAVs can track a target cooperatively, each UAV sensor can measure the ground target independently and perform the CIMM algorithm. Assuming that UAVs can establish communication, the estimation results by each UAV are fused in a distributed manner to further improve the estimation accuracy. Given that each sensor is independent and synchronous, this paper adopts the state vector fusion algorithm [20,22,36], which ignores the cross-covariance between different agents:

*t*th UAV respectively, $\hat{{\textbf x}}_{k|k}^q$ and ${\textbf P}_{k|k}^q$ the sum of state estimation and the sum of covariance estimation of the other UAVs. $\hat{{\textbf x}}_k^t$ is taken as the estimated target state required for UAV.

The flow diagram of the CIMM filter for multi-UAVs is shown in Fig. 2. CIMM is recursive, and each cycle consists of five steps, i.e., input interaction, conditional filtering of constrained model, probability updating of constrained model, estimation fusion, and data fusion among multi-UAVs. It can be regarded as a collection of multiple models, so it rapidly increases, decreases and transforms models with strong interaction and self-adaptability. It will effectively adjust the probability of each model and transfer between multiple models by the Markov process, especially for the positioning and tracking of maneuvering targets.

## 4. Target tracking based on TOGVF

In order to obtain the optimal monitoring effect and reduce the exposure risk of UAV, this paper adopts the standoff tracking mode, which requires the horizontal distance between UAV and target to be *R*, and the vertical distance to be a constant *H*. The traditional LGVF solves the problem of standoff tracking in the horizontal plane [37]. The Lyapunov function is used to guide the UAV to the limit cycle with radius *R*. However, this process takes long time. In many cases, it is necessary for UAV to converge to the limit cycle as soon as possible. Assuming that the motions of UAV in horizontal and vertical planes are decoupled, this paper proposes the TOGVF method, which guides the UAV not only to the limit cycle along the shortest path in the horizontal plane, but also to the optimal observation height along the shortest path in the vertical plane.

#### 4.1. Tracking a static target by one UAV

This paper focuses on continuously adjusting the heading of the UAV to track the target in the horizontal plane. The desired flying velocity ${{\boldsymbol v}_d} = {({\dot{x}_d},{\dot{y}_d})^\textrm{T}}$ is determined so that the UAV can converge to a target-centered limit cycle. When the UAV starting point is outside the limit cycle, it is obvious that the tangent is the shortest path, so it can be used as the desired route directly, as shown in Fig. 3(a). In this case, there are two tangent points which could decide the clockwise or counterclockwise flight:

*R*the radius of the limit cycle, $r = \sqrt {{{(x - {x_t})}^2} + {{(y - {y_t})}^2}} $ the distance from the UAV to the cycle center. In these two tangent points, we select the one with a smaller deviation from the UAV velocity vector.

When the UAV is inside the limit cycle, the inner tangential circle with the minimum turning radius ${r_{\min }}$ is first determined. Then the outer tangential line from the starting point to the inner tangential circle, as well as a part of the inner tangential circle, is selected to compose the desired route, as shown in Fig. 3(b). There are usually two circles and we choose the one closer to UAV, whose center is

In order to fly in the tangent direction, the required horizontal velocity is as follows

In the vertical plane, it is required that the longitudinal velocity of UAV is exactly 0 when it reaches the optimal altitude *H* (suppose ${z_t} = 0$). The longitudinal velocity control method based on the maximum longitudinal acceleration can be adopted. Firstly, the critical altitude is determined. Before reaching the critical altitude, the UAV flies at the maximum longitudinal velocity. When reaching the critical altitude, the UAV starts to decelerate uniformly with the maximum longitudinal acceleration. The longitudinal velocity of UAV is exactly 0 at the optimal altitude, so that the transition distance of UAV in the vertical plane is the shortest. The critical height is obtained (taking downward motion as an example) by

#### 4.2. Tracking a dynamic target by one UAV

In order to track a dynamic target, it is necessary to modify the UAV velocity to offset the effect of absolute velocity of target. Assuming the absolute velocity of the moving target on the ground is ${({\dot{x}_t},{\dot{y}_t},0)^\textrm{T}}$, the UAV velocity relative to the target is

#### 4.3. Multi-UAVs tracking a target

In order to eliminate the blind spots when a single UAV detects target, we can dispatch the multiple UAVs to enhance the monitoring accuracy. By controlling the phase, multiple UAVs are evenly distributed on the limit cycle. Even if the target is trying to escape, the UAV in a favorable position can continue tracking it. Figure 4 shows the relationship between the phases of three UAVs in cooperative tracking. The UAV phase refers to the angle between the vector from the target to the UAV and the *x* axis. Take UAV1 with the absolute position ${({x_1},{y_1},{z_1})^\textrm{T}}$ as an example, and ${\theta _1}$ represents its phase, and the distance between UAV1 and target is

The derivative of Lyapunov phase function with respect to time is

*R*the set standoff distance, ${k_2}$ and ${k_3}$ the positive values defining the convergence rate to the set phase. Taking Eq. (38) into Eq. (37), we have

In the case of tracking a dynamic target, the velocity of each UAV can be further adjusted as 4.2, so the horizontal speed in the relative frames is $\alpha {v_0}$ but not ${v_0}$. Thus, the condition $d{V_q}/dt \le 0$ may not hold in some cases, and the phase differences between UAVs will not reach the set value stably, but oscillate around the set value. It is also noticed that the oscillation enlarges if the target has a larger maneuver.

Figure 5 shows the flow chart of the cooperative standoff tracking of a non-wide target in three-dimensional space.

## 5. Simulation results

The algorithms are simulated in MATLAB R2016a on a computer with Intel Core i5-7500 CPU with 3.40 GHz frequency and 8G RAM. The configurations and designing of simulation experiments are determined on the basis of Fig. 5. Table 1 gives the necessary parameters. The first work shows the high estimation accuracy of the CIMM filter and the state vector fusion, compared to other filters. Then the standoff tracking of a target by one UAV is displayed. Finally, the cooperative standoff tracking by multi-UAVs is simulated on this basis. In addition, we make a comparison between the TOGVF method and the well-known LGVF.

#### 5.1. Estimation results by CIMM filter

Assume that the initial state of the target is (200 m, 6 m/s, 0, 300 m, 8 m/s, 0). The target moves along a straight line at a speed of 10 m/s in 0–140s, and makes right-turn motion with a turning rate of 0.008 rad/s in 141–360s, and then makes left-turn motion with a turn rate of 0.01 rad/s in 361–700s. As shown in Fig. 6, the real trajectory includes three motion models, i.e., linear, left-turn, and right-turn motion. The probability of the initial model is assumed to be ${\boldsymbol \mu }_{0|0}^{} = [0.98,0.01,0.01]$, and the Markov transition probability matrix is

*x*-axis or

*y*-axis is shown in Fig. 7(a) and Fig. 7(b) respectively. It is seen that the estimated results by CIMM is accurate in both position and velocity.

Table 2 shows the estimation errors of target position and velocity under various filtering conditions. Obviously, the estimation errors by the CIMM filter are smaller than those by the conventional IMM filter. The geographical constraint is fully considered and utilized in the CIMM filter, so the estimation accuracy is improved. Table 2 shows that the estimated accuracy by CIMM increases at least 30% in comparison to the traditional IMM under the same condition. The estimation accuracy by a sensor network of multi-UAVs is higher than that by a single UAV. The information fusion among UAVs can fully integrate the information of other UAVs and improve the estimation accuracy. In addition, when the uncertainty of the non-wide-area constraint becomes larger, the estimation accuracy will increase.

#### 5.2. Target tracking simulation results

Assume that the position of static target is (1500,1800,0)m, and the start position of UAV is (1100,1000,400)m. In order to verify the superiority of TOGVF, it is compared with the existing LGVF. The planned paths can be seen in Fig. 8. Although the LGVF can make the UAV converge to the limit cycle, the flight path is relatively long. The TOGVF will guide the UAV to the limit cycle in accordance with the shortest path, so as to reach the goal of tracking and monitoring target as soon as possible.

The tracking paths of UAV in plane $x - y$ and plane $s - z$ are shown in Fig. 9, where *s* denotes the forward distance. Using the TOGVF method, the UAV eventually converges to the upper limit circle along the tangent direction in the horizontal plane, and the standoff distance remains 300 m. The convergence distance by TOGVF is much shorter than that by LGVF. In the vertical plane, the UAV first flies at a uniform speed, and then starts to decelerate uniformly once reaching the critical altitude (312.5 m), and finally converges to the optimal observation altitude. The transition distance to the optimal height is much shorter than that by LGVF. The experimental results show that the tracking efficiency greatly improves via TOGVF.

On the basis of estimation results from 5.1, the standoff tracking of a moving target is carried out. The initial position of the UAV is (−500, −500, 400) m. Figure 10 shows the tracking path of the UAV in the three-dimensional coordinate system by TOGVF. In the inertial coordinate system, the general movement trend of the UAV is the same as the target, but at the same time there is a hovering motion. The reason is that the UAV velocity consists of these two parts. One part is to correct the velocity of the UAV's hovering movement, and the other part is the absolute velocity of the target. In the relative coordinate system, the UAV makes a hovering motion because the target velocity is subtracted. Figure 11 shows that the UAV converges to the limit cycle along the tangent direction relative to the moving target in the horizontal plane. Figure 12 shows the tracking path in the vertical plane.

When multiple UAVs are cooperatively tracking a moving target, it is necessary not only to ensure that each UAV can converge to the limit cycle in the horizontal plane and to the optimal height in the vertical plane, but also to ensure that the phase difference between the UAVs should approach to the set value. Figure 13 shows the trajectories of three UAVs in the three-dimensional coordinates. If the target moves faster, the hovering motion of the UAV will become sparser. From Fig. 14, it can be seen that each UAV can converge to the limit cycle along the tangent direction relative to the moving target. Since the speed of the UAV is corrected based on the target speed, the distance can reach the set value of 300 m. The larger the UAV speed is, the shorter time it takes to converge to the limit cycle. If the standoff distance becomes larger, the turning rate of the UAV will decrease. It is obvious that each UAV can converge to the optimal observation height in the vertical plane according to the shortest path from Fig. 15. For UAVs at the same height, the critical height is only related to the maximum longitudinal velocity and the maximum longitudinal acceleration.

Figure 16 shows the phase difference between the UAVs during the tracking process. The two phase differences always oscillate near the set value. This is because the UAVs are affected by the correction coefficient of flight speed when tracking a dynamic target, but the phase difference can still keep within a certain range, and the cooperative task can be accomplished. The convergence rate to the set phase is defined by the coefficients ${k_2}$ and ${k_3}$. By comparing the two figures in Fig. 16, it can be seen that the bigger coefficient means the faster convergence rate to the set phase. Figure 17 shows the horizontal speed of each UAV. It can be seen that the horizontal speed of UAV1 or UAV3 is constantly adjusted under the effect of phase control, and the bigger coefficient means the faster adjustment or the greater fluctuation of the UAV speed.

## 6. Conclusion

This paper focuses on the high-precision estimation of target motion measured from UAV optical cameras in non-wide area, as well as the standoff guidance of multi-UAVs for tracking a target on this basis. Firstly, the target motion switching between different models is estimated by introducing the geographic constraint equation into the interacting multiple models, along with the state vector fusion from a sensor network of multi-UAVs. The simulation results show that the CIMM filter greatly improves the estimation accuracy of the target motion. When it comes to target tracking, the TOGVF method is proposed. The experimental results show that the UAV can converge to the limit cycle and the optimal observation height in the three-dimensional environment. Compared with the traditional Lyapunov guidance vector field, this method makes the UAV converge to the limit cycle directly along the shortest path and reduce the tracking time greatly.

## Funding

Natural Science Foundation of Shandong Province, China (ZR2018BF016); China Postdoctoral Science Foundation (2017M622278); Natural Science Foundation of Beijing Municipality (4194074).

## Disclosures

The authors declare that they do not have any conflicts of interest to this work.

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