2.1 Forming principle

Considering a GNSS-denied scenario, with the inter-flight mutual ranging, poorly equipped UAVs can also realize localisation by cooperation. To address this challenge, a two-layer scheme is investigated based on the composition of the heterogeneous swarm. According to their capabilities, UAVs in the swarm are divided into two types: A-UAV can still realise the localisation in GNSS-denied environment; B-UAV, which can only utilise low-precision IMU for localisation. Correspondingly, the scheme consists of measurement and communication layers. The two-layer schematic diagram for cooperative localisation is shown in Fig. 1.

Figure 1.

The schematic diagram of cooperative localisation.

The set of UAV swarm in the whole network is defined as $\mathcal{N}=\{1,2,\dots,N\}$ , where $N=n_A+n_B+n_c$ , $k\in\mathcal{N}$ , $i\in{1,2,\dots,n_A}$ and $j\in{n_A+1,\dots,N}$ . For explanation the forming principle, some preliminaries about graph theory are given. Let $\mathcal{G}=(\mathcal{V},\mathcal{E},G)$ be a weighted graph of order P, with $\mathcal{V}=\{v_1,v_2,\dots,v_N\}$ , $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$ and $G=(G_{pq})_{P\times P}$ . A directed edge $\mathcal{E}_{ij}$ in network $\mathcal{G}$ is denoted by the ordered pair of nodes $(v_p,v_q)$ , where node $v_p$ can receive information from $v_q$ , but not another way around.

Definition 2 [Reference Yu, Wen, Chen and Cao32]. An undirected network $\mathcal{G}$ is complete connected if it has a connection between each pair of distinct nodes $v_p$ and $v_q$ in $\mathcal{G}$ .

Figure 2.

Ranging principle incorporated with composition. (a) ranging principle of structure 1 and (b) ranging principle based on structure 2.

According to Assumption 3, following three basic forming structures are discussed

• Structure 1: $n_A=1$ and $n_B=5$ .

• Structure 2: $n_A=2$ and $n_B=3$ .

• Structure 3: $n_A\geq3$ and $n_B=1$ .

According to the composition of a heterogeneous swarm, the forming principle is divided into ranging principle and communication networks. For each proposed composition, the ranging principle is outlined as the principles for the basic structure and its additional structure. The diagrammatic ranging principles for basic structures are shown in Fig. 2. Based on the measurements, the corresponding network is essential to data transmission. A network supporting communication between heterogeneous and homogeneous UAVs is given.

1. 1. Structure 1 : 1 A-UAV and 5 minimum required B-UAVs.

   Basic composition with one A-UAV $L_1$ and the minimum required 5 B-UAVs satisfies the following principle:

   • Ranging principle of measurement layer:

     The ranging principle of basic structure 1 is similar to the complete graph, as shown in Fig. 2(a). Measure the mutual ranging between the A-UAV and each B-UAV. Besides, the mutual ranging between any two B-UAVs needs to be measured.

     The ranging principle of the additional structure requires each additional B-UAV to hold three new ranging links. Simultaneously, the additional structure at least contains a ranging link connected with the UAV in the basic structure.

   • Cooperative network of the communication layer:

     Taking the only A-UAV as the computing centre, the communication network of the $L_1$ -based swarm is undirected and strongly connected.

2. 2. Structure 2 : 2 A-UAVs and 3 minimum required B-UAVs.

   Basic composition with two A-UAVs $L_1,L_2$ and the minimum required 3 B-UAVs satisfies the following principle:

   • Ranging principle of measurement layer:

     As shown in Fig. 2(b), the mutual ranging relationships of L1 and L2 with the other three B-UAVs satisfy undirected and complete connections, respectively.

     The ranging principle of additional structure requires each additional B-UAV to hold three ranging links, at least one of which should be kept with the UAV in the basic structure.

   • Cooperative network of communication layer:

     Choose one of A-UAVs as the computing centre. At the very least, the communication network containing the selected A-UAV and all the B-UAV should be strongly connected. Simultaneously, there exists a directed connection from the other A-UAV to the computing centre.

3. 3. Structure 3 : more than three A-UAVs and 1 B-UAV.

   Basic composition with A-UAVs more than three and 1 B- UAV satisfies the following principle:

   • Ranging principle of measurement layer:

     For any B-UAV in the heterogeneous swarm, its localisation problem can be solved if there exist three ranging links connected to different A-UAVs in the basic structure 3.

     The ranging principle of additional structure first requires that three new ranging links are added correspondingly for each additional B-UAV. For the first additional B-UAV $M_{n_B+1}$ , there exist at least two ranging links connected with different A-UAVs. For the second additional B-UAV $M_{n_B+2}$ , at least one ranging link connected to any A-UAV is needed. When $n_C\geq 3$ , $M_j(j=3,\dots,n_B + n_C)$ only needs to ensure three-ranging-link principle.

   • Cooperative Network of Communication layer:

     Choosing one of A-UAVs as the computing centre, at the very least, the communication network containing the selected A-UAV and all the B-UAVs should be strongly connected. Simultaneously, there exist directed connections from the other A-UAVs to the computing centre, which means that the computing A-UAV can obtain the complete information through transmission.

2.2 Cooperative localisation functions

Based on the above compositions of the heterogeneous swarm, the forming principle are given for ensuring the data transmission and adequate information for localisation. Accordingly, the cooperative relative localisation functions are established and the fitness function is selected.

To describe the ranging principle as undirected graph $G_M$ , the basic structure based ranging principle is given as follows

(1) \begin{equation}G_{Mb}=[a_{hk}]_{n_b \times n_b}=\left[\begin{array}{c@{\quad}c} \mathit{0}_{n_{A} \times n_{A}} & \mathit{1}_{n_{A} \times n_{B}} \\[4pt] \mathit{1}_{n_{B} \times n_{A}} & \mathit{1}_{n_{B} \times n_{B}}- \mathit{I}_{n_{B} \times n_{B}} \end{array}\right]\end{equation}

and the additional structure based ranging principle of $v_k$ is given as follows

(2) \begin{equation} G_{Ma}(1:(k-1),k)=\left[ a_{hk}\right]_{(k-1) \times 1},(k>n_b,k>h,h = 1,2,\dots,k-1)\end{equation}

where $n_b=n_A +n_B$ , and if the mutual ranging between $v_k$ and $v_j$ needs to be measured, then the adjacency weight $a_{hk}=a_{kh}=1(k\neq h)$ , otherwise, $a_{hk}=a_{kh}=0$ . Hence, $G_M$ is expressed as

(3) \begin{align} G_{M b}=\left[ \begin{array}{c@{\quad}c|c@{\quad}c@{\quad}c} 0_{n_{A} \times n_{A}} & 1_{n_{A} \times n_{B}} & G_{M a}\left(1\,:\, n_{A}, n_{b}+1\right) & \ldots & G_{M a}\left(1\,:\, n_{A}, \textrm{N}\right) \\[4pt] 1_{n_{B} \times n_{A}} & 1_{n_{B} \times n_{B}}-I_{n_{B} \times n_{B}} & G_{M a}\left[\left(n_{A}+1\right)\,:\,\left(n_{b}\right), n_{b}+1\right] & \ldots & G_{M a}\left[\left(n_{A}+1\right)\,:\,\left(n_{b}\right), \textrm{N}\right] \\[4pt] \hline G_{M a}^{\textrm{T}}\left(1\,:\, n_{A}, n_{b}+1\right) & G_{M a}^{\textrm{T}}\left[\left(n_{A}+1\right)\,:\,\left(n_{b}\right), n_{b}+1\right] & 0 & \ldots & a_{n_{b}+1, k} \\[4pt] \vdots & \vdots & \vdots & \ddots & \vdots \\[4pt] G_{M a}^{\textrm{T}}\left(1\,:\, n_{A}, \textrm{N}\right) & G_{M a}^{\textrm{T}}\left[\left(n_{A}+1\right)\,:\,\left(n_{b}\right), \textrm{N}\right] & a_{k, n_{b}+1} & \ldots & 0 \end{array}\right]\nonumber\\[5pt]\end{align}

Thus, let $X=[x_{(n_A+1)},y_{(n_A+1)},z_{(n_A+1)},\dots,x_k,y_k,z_k,\dots,x_N,y_N,z_N]^T \in N^{3(n_B+n_C)}$ ,the cooperative localisation function is described as follows

(4) \begin{equation} F(X) = [f_{hk}]_{3(n_B+n_C)\times 1}=0, k= n_A+1,\dots,N\ {\textrm{where}}\ k>h\ {\textrm{and}}\ a_{hk}=1\end{equation}

with

(5) \begin{equation} f_{hk}=\sqrt{\left(x_{k}-x_{h}\right)^{2}+\left(y_{k}-y_{h}\right)^{2}+\left(z_{k}-z_{h}\right)^{2}}-R_{h k}\end{equation}

Remark 2. Based on Assumption 3, ranging principle of the measurement layer is established, where the number of localisation functions is equal to the number of position coordinates to be solved. When the number of well-equipped A-UAVs is given, the basic structure can ensure that the number of functions is the same as the number of state variables. Furthermore, as the basically additional ranging principle requires, the additional structure with each UAV holding three new ranging links means that the number of swarm localisation functions satisfies the following equation:

(6) \begin{equation} n_A \times n_B +\sum_{p=1}^{n_B-1}p +3 \times n_C=3 \times (n_B+n_C) \end{equation}

Sufficient and non-repetitive measurements can avoid the existence problem of the overdetermined or underdetermined equations. Therefore, the equation set constructed by the localisation functions is well-determined and has a solution.

In the cooperative localisation engagement, with known upper bound of IMU drift error $e_r$ , the measurements transmitted to the centre A-UAV are divided into three types:

• position coordinates of A-UAVs: $[x_k,y_k,z_k], \ k=1,2,\dots,n_A$ ;

• mutual ranging based on the ranging principle: $R_{h k},\ k= n_A+1,\dots,N,\ k>h$ ;

• inaccurate position coordinates measured by IMU $[\hat{x}_k,\hat{y}_k,\hat{z}_k], \ k=n_A+1,\dots,N$ .

Therefore, the localisation problem can be transformed into how to find a solution to minimize the error effectively. we define the fitness function as follows

(7) \begin{equation} F_{\max}=\max(|f_{hk}|),k= n_A+1,\dots,N\ {\textrm{where}}\ k>h\ {\textrm{and}}\ a_{hk}=1\end{equation}

3.1 PSO algorithm

Practical swarm optimisation (PSO) algorithm is one of swarm intelligence algorithms. Inspired by the social behaviour of birds, the PSO algorithm was first proposed by James Kennedy and Russell Eherhart [Reference Kennedy and Eberhart18, Reference Eberhart and Kennedy19]. Each single particle represents a bird and the birds flocked around a ‘roost’. Similar to birds looking for attractive food, the particle searches for the global optimal space. To implement the concept, the position of each particle is updated based on the social behaviour of a population of individuals that adapts to its environment by returning to previously discovered promising regions. The process is stochastic in nature and makes use of the memory of each particle, as well as the knowledge gained by the swarm as a whole. Firstly, initialise a set of particles randomly distributed throughout the design space, in which the position of each particle is a feasible solution. Then, the fitness value of the k-th particle can be recorded in the process of iterating, and the optimal fitness value of previous iterations will be memorised, which is called the individual extreme value( $p_{best}$ ). Simultaneously, other particles exchange information with each other to find the global extreme value ( $g_{best}$ ), which represents an optimal value among all particles in the current iteration process. Finally, influenced by the memory of two extreme values, the dynamical update is made for the speed of the next iteration, which drives the particles to move closer to the best point.

The schematic diagram of particle motion is shown in Fig. 3.

Figure 3.

Particle motion diagram of the PSO algorithm.

The attributes of each particle include position, speed and fitness function. Suppose a group $\mathcal{P}$ composed of P particles in the $\mathcal{D}$ -dimensional search space, where the position of q-th particle is $p_q=(p_{q1},p_{q2},\dots,p_{qd},\dots,p_{q\mathcal{D}})^T$ , the speed is expressed as ${S}_q=(s_{q1},s_{q2},\dots,s_{qd},\dots,s_{q\mathcal{D}})^T$ , the fitness function is defined as $f_{p_q}$ and the number of iterations is $iter\in[1,2,\dots,\mathcal{I}]$ . In the iterative process of the PSO algorithm, the calculation formulas of the velocity and position are described as follows,

(8) \begin{equation} \left\{\begin{array}{l} s_{q d}( { iter }+1)=w s_{q d}( { iter })+c_{1} r_{1}\left(p_{ {best }, d}-p_{q d}({ iter })\right)+c_{2} r_{2}\left(g_{ {best }, d}-p_{q d}( { iter })\right) \\[4pt] p_{q d}( { iter }+1)=p_{q d}( { iter })+s_{q d}( { iter }) \end{array}\right. \end{equation}

where w represents the weight of inertia, $c_1$ and $c_2$ are non-negative constants, $r_1$ and $r_2$ are random numbers and obey the uniform distribution on [0,1]. Denote $s_{\max}$ as the bounds of particle search range and the speed $s_{qd}\in[{-}s_{\max},s_{\max}]$ .

The weight of inertia w in the PSO algorithm adjusts with the number of iteration as follows

(9) \begin{equation} w(iter)=w_{\max}-\frac{iter}{\mathcal{I}}(w_{\max}-w_{\min}) \end{equation}

where $w_{\max}$ and $w_{\min}$ are upper and lower bounds of the weight on [0,1].

The outline of a basic PSO algorithm is given in the Fig. 4.

Figure 4.

Diagram of the PSO algorithm.

3.2 Adaptive weight for the PSO algorithm

In view of the weak convergence speed of the PSO algorithm, an adaptive weight is introduced as a further improvement. An adjustable parameter $\beta$ is added into it to adjust the inertia weight factor w and the acceleration factors $c_1$ and $c_2$ in real-time. The adaptive principle is described as follows,

(10) \begin{equation} \left\{\begin{array}{l} \alpha( { iter })=\dfrac{\sum_{{q}=1}^{P} \sqrt{\sum_{{d}=1}^\mathcal{D}\left[p_{{qd}}( { iter })-g_{best, {d}}( { iter })\right]^{2}}}{N} \\[8pt] \beta( { iter })=\dfrac{\alpha( { iter })}{\max (\alpha)} \\[9pt] w( { iter })=\beta( { iter }) \\[4pt] c_{1}( { iter })=2 \times \beta({ iter }) \\[4pt] c_{2}( { iter })=2-c_{1}( { iter }) \end{array}\right.\end{equation}

3.3 Adaptive SA-PSO algorithm-based cooperative localisation

In order to further improve the performance of the adaptive PSO algorithm, a simulated annealing(SA) is introduced to it.

Although the convergence speed is improved by the adaptive-improved PSO algorithm, limited by the inner characteristic of the PSO algorithm, falling into a local extreme point is an inevitable problem in the calculation process. Particles may converge to the minimum value prematurely during the search process, and falling into the local extreme point easily will lead to an obvious reduction of global search ability.

The SA algorithm faces the drawbacks of relying too much on the process of cooling. On the other hand, if global convergence is achieved, harsher temperature-limiting conditions are required, which leads to a reduction of optimisation efficiency.

On the other hand, the SA algorithm is a heuristic algorithm that allows a solution worse than the current one in a certain probable situation. Thus, it is possible for the SA algorithm to break away from the local optimal solution to obtain the global optimal solution, which can effectively overcome the aforementioned disadvantage of the PSO algorithm.

Therefore, combined with the adaptive weight, the SA and the PSO, the proposed algorithm will not only improve the local optimisation of the cooperative localisation problem but also realize the requirement for high efficiency. The diagram of cooperative localisation based on adaptive SA-PSO algorithm is given in Fig. 5.

Figure 5.

The diagram of cooperative localisation based on adaptive SA-PSO algorithm.

The adaptive SA-PSO algorithm for cooperative localisation is proposed as follows:

1. 1. Data collection. Transmit the necessary data to the centre A-UAV by forming principle organised in Section 2.1. The data includes the positions of A-UAVs, the mutual ranging, inaccurate positions of B-UAVs measured by IMU and the upper bound of IMU drift error.

2. 2. Parameter initialisation. Initialise the following parameters: population size P, maximum number of iterations $\mathcal{I}$ , speed limitation $s_{\max}$ , initial weight w and initial acceleration factors $c_1$ and $c_2$ .

3. 3. Population initialisation. Initialise the states on the basis of inaccurate localisation measurements $[\hat{x},\hat{y},\hat{z}]$ and the upper bound of IMU drift error $e_r$ .

4. 4. Fitness calculation. Evaluate the fitness values of each particle and swarm. The current particle’s position and fitness value are stored in $\gamma_q$ . Store the individual optimum $p_{best}$ and global optimum $g_{best}$ .

5. 5. Temperature initialisation. Initialise the annealing temperature t(0) as

   (11) \begin{equation} t(0)=\frac{F_{\max}(p_{best})}{ln5} \end{equation}

6. 6. Temperature calculation.

   (12) \begin{equation} t_{F}\left(\gamma_{q}\right)=\dfrac{\textrm{e}^{-\left(F_{\max}\left(\gamma_{q}\right)-F_{\max}\left(p_{best}\right)\right) / t( { iter }-1)}}{\sum_{{q}=1}^{{P}} \textrm{e}^{-\left(F_{\max}\left(\gamma_{{q}}\right)-F_{\max}\left(p_{best}\right)\right) / t({ iter }-1)}} \end{equation}

7. 7. Cumulative probability calculation. Calculate the cumulative probability of each individual

   (13) \begin{equation} T_F(q)=\sum_{l=1}^{q}{t_F\left(\gamma_l\right)} \end{equation}

8. 8. Roulette judgement. Apply the roulette rule to determine the global optimal, substitute value $p^{*}_{best}$ from all the $p_{best}$ . Generate a random number $\epsilon\in[0,1]$ , if $\epsilon<T_F(1)$ , the first particle is chosen; else if $T_F(q-1)<r<T_F(q)$ , the q-th particle is chosen.

9. 9. States updating. Update the states of each particle according to Equations (8) and (10).

10. 10. Temperature reduction. Apply the attenuation coefficient method to realise the temperature-reducing process as

    (14) \begin{equation} t(iter+1)=0.4t(iter) \end{equation}

11. 11. Judgement. Stop the optimisation process and output the localisation results when the termination conditions are satisfied; otherwise, return to Step 6 and continue the optimisation search.

4.1 Simulation of basic forming structure

In the case of solving the localisation problem by forming principle of three basic structures, comparative experiments were conducted with the PSO algorithm to verify the feasibility and superiority of the proposed adaptive SA-PSO cooperative relative localisation algorithm.

4.1.1 Simulation of basic forming structure 1

In this GNSS-degraded engagement, 5 poorly equipped UAVs’ integrated navigation lost the GNSS information for a period of time. According to the performance of the equipped IMU, the upper bound of the drift errors in this interrupted time period is $e_r=60$ m.

Generate randomly the IMU measurements within a reasonable range, the initial conditions are listed in Table 1.

Table 1.

Initial conditions of Case 4.1.1

Hence, according to Equation (1) the symmetric matrix of mutual ranging can be measured as follows,

(15) \begin{equation} R_{ideal}=\left[ \begin{array}{l@{\quad}|@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l} 0&6,473.634&8,594.780&6,547.223&9,045.309&9,646.087\\[4pt] \hline \dots&0&5,081.110&5,624.563&9,178.430&7,455.768\\[4pt] \dots&&0&2,891.439&6,338.085&4,816.879\\[4pt] \dots&&&0&4,646.996&4,689.856\\[4pt] \dots&&&&0&3,407.461\\[4pt] \dots&&&&&0\\[4pt] \end{array} \right] \end{equation}

Choose the common parameters as $P=120$ , $\mathcal{I}=800$ , $V_{\max}=60$ and $P_e=1$ m. Since all randomly generated particles are located in a spherical space, so we took IMU measurements $[\hat{x},\hat{y},\hat{z}]$ as centre and $e_r$ as radius. The initial $p_{best}$ started from the IMU measurements. The constant parameters of PSO algorithm were designed as $w_{\max}=0.8$ , $w_{\min}=0.6$ and $c_1=c_2=1.49445$ .

The simulation results are shown in Fig. 6.

Figure 6.

Optimisation results of basic structure 1. (a) Fitness value, and (b) adaptive acceleration factors.

The cooperative localisation results of basic structure 1 are shown in Table 2.

Table 2.

Localisation results of Case 4.1.1

4.1.2 Simulation of basic forming structure 2

Similar to the Section 4.1.1, when $e_r=80$ m, A-UAVs $L_1$ and $L_2$ locate in (3,000m, 6,000m, 3,500m) and (4,000m, 4,500m, 3,000m), the initial conditions of B-UAVs were listed in Table 3.

Table 3.

Initial conditions of Case 4.1.2

According to Equation (1), the symmetric matrix of the measured mutual ranging was given as follows,

(16) \begin{equation} R_{ideal}= \left[ \begin{array}{l@{\quad}l@{\quad}|@{\quad}l@{\quad}l@{\quad}l} 0&0&2,348.583&5,025.350&3,790.795\\[4pt] 0&0&2,989.455&3,586.941&3,239.310\\[4pt] \hline \dots&&0&4,391.319&2,146.067\\[4pt] \dots&&&0&2,755.054\\[4pt] \dots&&&&0\\[4pt] \end{array} \right] \end{equation}

Some parameters were selected as $P=100$ and $\mathcal{I}=500$ . The simulation results are shown in the Fig. 7 and Table 4.

Table 4.

Localisation results of Case 4.1.2

Figure 7.

Optimisation results of basic structure 2. (a) Fitness value, and (b) adaptive acceleration factors.

4.1.3 Simulation of basic forming structure 3

In this case, the only B-UAV $M_4$ is located at (1,200m, 1,200m, 2,221m). The known IMU drift error $e_r$ is less than 70m and the measured location is (1,203.759m, 1,262.639m, 2,226.235m). The A-UAVs $L_1$ , $L_2$ and $L_3$ are located at (2,000m, 4,000m, 3,500m), (3,000m, 4,500m, 3,000m) and (4,000m, 3,400m, 4,000m), respectively. The number of particles and iterations were set as 30 and 100. $P_e$ was chosen as $0.001$ m.

According to Equation (1), the symmetric matrix of the measured mutual ranging was given as follows,

(17) \begin{equation} R_{ideal}= \left[ \begin{array}{l@{\quad}l@{\quad}l@{\quad}|@{\quad}l} 0&0&0&3,180.541\\[4pt] 0&0&0&3,838.859\\[4pt] 0&0&0&3,980.558\\[4pt] \hline 3,180.541&3,838.859&3,980.558&0\\[4pt] \end{array} \right] \end{equation}

The simulation results are shown in the Fig. 8. The calculation time of three algorithms are $0.0476$ , $0.0174$ and $0.0096$ s, respectively.

Figure 8.

Optimisation results of basic structure 3. (a) Fitness value, and (b) adaptive acceleration factors.

4.2 Simulation of additional forming structure

Based on the simulation of basic structure, the expandability of proposed ranging principle for additional structure is demonstrated in this section.

4.2.1 Simulation of the general additional ranging principle

For the additional ranging principle, the general basic condition requires three unrepeated ranging links.

Take Structure 2 for example, the applicability of additional principle is verified in this simulation.

The common parameters were given as $P=100$ , $\mathcal{I}=800$ and $V_{\max}=60$ .

The real position conditions of basic structure are same as that in Section 4.1.2. One additional B-UAV located in (2,000m, 2,500m, 4,828m) was added. The additional-structure-based ranging connection is $G_{Ma}=[0;0;1;1;1;0]$ .With $e_r=70$ m, generate randomly the IMU measurements at (2,017.054m, 2,474.198m, 4,851.443m) and the corresponding mutual ranging matrix is measured as follows,

(18) \begin{equation} R_{ideal}= \left[ \begin{array}{l@{\quad}l@{\quad}|@{\quad}l@{\quad}l@{\quad}l@{\quad}|@{\quad}l} 0&0&2,348.583&5,025.350&3,790.795&0\\[4pt] 0&0&2,989.455&3,586.941&3,239.310&0\\[4pt] \hline \dots&&0&4,391.319&2,146.067&3,837.506\\[4pt] \dots&&&0&2,755.054&3,258.812\\[4pt] \dots&&&&0&3,833.670\\[4pt] \hline \dots&&&&&0\\[4pt] \end{array}\right]\end{equation}

The simulation results are shown in the Fig. 9 and Table 5.

Table 5.

Localisation results of Case 4.2.1

Figure 9.

Optimisation results of additional structure based on structure 2. (a) Fitness value, and (b) adaptive acceleration factors.

4.2.2 Simulation of the additional ranging principle for Structure 3

According to the ranging principle of structure 3, the first two additional B-UAVs are required to satisfy the corresponding ranging connection. With the conditions in Section 4.1.3, the cases were given as follows,

• Case 1 : one additional UAV

  Parameter selection: $N=100$ , $\mathcal{I}=800$ , $V_{\max}=60$ , $P_e=0.5$ m, $e_r=70$ m.

  $M_5$ is located at (2,000m, 1,500m, 1,828m) and $G_{Ma}^T=[1 \quad 0 \quad 1 \quad 1 \quad 0]$ .

• Case 2 : two additional UAVs

  Parameter selection: $N=100$ , $\mathcal{I}=800$ , $V_{\max}=60$ , $P_e=0.5$ m, $e_r=70$ m.

  $M_6$ is located at (1,700m, 1,600m, 1,320m) and $G_{Ma}^T=\left[\begin{array}{l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l}1 &0& 1& 1& 0& 0\\0& 1& 0& 1& 1& 0\end{array}\right]$ .

• Case 3 : three additional UAVs

  Parameter selection: $N=120$ , $\mathcal{I}=800$ , $V_{\max}=80, $ , $P_e=1$ m, $e_r=60$ m.

  $M_7$ is located at (2,600m, 1,123m, 3,400m) and $G_{Ma}^T=\left[\begin{array}{l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l}0 &1& 1& 1& 0& 0&0\\[4pt]0& 1& 0& 1& 1& 0&0\\[4pt]0& 0& 0& 1&1&1&0 \end{array}\right]$ .

The simulation results are shown in the Tables 6–8.

Table 6.

Localisation results of additional Case 1

Table 7.

Localisation results of additional Case 2

Table 8.

Localisation results of additional Case 3