SCIENCE CHINA Information Sciences, Volume 64 , Issue 4 : 140306(2021) https://doi.org/10.1007/s11432-020-3013-1

A large-scale clustering and 3D trajectory optimization approach for UAV swarms

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  • ReceivedApr 1, 2020
  • AcceptedJul 27, 2020
  • PublishedMar 5, 2021



This work was supported in part by National Natural Science Foundation of China (Grant No. 61871211), Natural Science Foundation of Jiangsu Province Youth Project (Grant No. BK20180329), Innovation and Entrepreneurship of Jiangsu Province High-level Talent Program, Summit of the Six Top Talents Program of Jiangsu Province.



Proof of Lemma 4.2

The proof of Lemma 1 is quite similar to that of Proposition 1 in [29]. Let $f(z)=\log_2(1+\frac{c}{z})$ with $c=\frac{P_i\gamma_0}{\alpha_i[n]}>0$. It is not difficult to verify that $f(z)$ is a convex function with $z\geq~0$. Then, $f(z)$ can be globally lower-bounded by its first-order Tayler expansion at any point $z_0$ 1). That is, we have \begin{equation}f(z)\geq f(z_0)+f'(z_0)(z-z_0), \forall z, \tag{31}\end{equation} where $$f'(z_0)=\frac{-c\log_2e} {z_0(z_0+c)}.$$ Consequently, letting $z=\max(\|{\boldsymbol~q}[n]-{\boldsymbol~s}_i\|^2,d_{\min}^2)$ and $z_0=\max(\|{\boldsymbol~q}^l[n]-{\boldsymbol~s}_i\|^2,d_{\min}^2)$, it completes the proof.

Boyd S, Vandenberghe L. Convex Optimization. Cambridge: Cambridge University Press, 2004.

Proof of Lemma 4.3

Recalling the expression of $\hat{R}_i[n]$ in 23, only the term $-\phi^l_i[n]z_i[n]$ involves ${\boldsymbol~q}[n]$. We can easily obtain the convexity of $z_i[n]=\max(\|{\boldsymbol~q}[n]-{\boldsymbol~s}_i\|^2,d_{\min}^2)$ as the maximum of convex function $\|{\boldsymbol~q}[n]-{\boldsymbol~s}_i\|^2$ and a constant $d_{\min}^2$. Combined with $-\phi^l_i[n]<0$, it immediately yields the concavity of $-\phi^l_i[n]z_i[n]$ as well as $\hat{R}_i[n]$.


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  • Figure 1

    (Color online) Hierarchical framework for large-scale UAV clustering and 3D trajectory design in UAV swarms.

  • Figure 4

    (Color online) Transmission delay with different numbers of CHs in area 6.

  • Figure 5

    (Color online) Ferry UAV trajectories with throughput requirement $C~=~300$ Mbits. (a) Optimal 2D trajectory with fixed altitude; (b) optimal 3D trajectory.

  • Figure 6

    (Color online) Ferry UAV trajectories with throughput requirement $C~=~1500$ Mbits. (a) Optimal 2D trajectory with fixed altitude; (b) optimal 3D trajectory.

  • Figure 7

    (Color online) Completion time with different throughputs.


    Algorithm 1 Modified k-means algorithm for each area

    Input: UAV swarms $\mathcal{D}=\{{\boldsymbol~x}_1,~{\boldsymbol~x}_2,\ldots,~{\boldsymbol~x}_M\}$, clusters number $S$ from 5.

    Output: locations of Super-CHs.

    Randomly select $S$ UAVs from $\mathcal{D}$ as the initial mean vectors $\{{\boldsymbol\mu}_1,~{\boldsymbol\mu}_2,~\ldots,{\boldsymbol\mu}_S\}$.

    Initialize $\mathcal{C}_i=\emptyset,~i=1,\ldots,S$.

    for $j=1,2,\ldots,M$

    Calculate the distance between each UAV ${\boldsymbol~x}_j$ and each mean vector ${\boldsymbol\mu}_i~(1\leq~i\leq~S)$: $d_{ji}=\|{\boldsymbol~x}_j-{\boldsymbol\mu}_i\|$;

    Determine the cluster label of ${\boldsymbol~x}_j$ based on the nearest mean vector: $\lambda_j=\arg\min\nolimits_{i\in\{1,2,\ldots,S\}}d_{ji}$;

    Divide ${\boldsymbol~x}_j$ into the corresponding cluster: $\mathcal{C}_{\lambda_j}=\mathcal{C}_{\lambda_j}\cup~\{{\boldsymbol~x}_j\}$.

    end for

    for $i=1,2,\ldots,S$

    Calculate the new mean vector: ${\boldsymbol\mu}&apos;_i=\frac{1}{|\mathcal{C}_i|}\sum_{{\boldsymbol~x}_i\in~\mathcal{C}_i}{\boldsymbol~x}$;

    if ${\boldsymbol\mu}&apos;_i\neq{\boldsymbol\mu}_i$ then

    Update the current mean vector${\boldsymbol\mu}_i$ to ${\boldsymbol\mu}&apos;_i$;


    Keep the current mean vector ${\boldsymbol\mu}_i$ unchanged;

    end if

    end for

    Stop the Loop (Step 5–17) until all mean vectors are not updated.

    Select UAVs closest to the mean vector as the CHs.

    Choose the CH closest to all other CHs as the Super-CH.

  • Table 1  

    Table 1Main notations

    Notation MeaningNotation Meaning
    $K$ Total number of Super-CH UAVs (i.e., the number of areas) $\alpha_i(t)$ Fraction of total bandwidth allocated for Super-CH $i$
    $M_k$ Total number of UAVs in area $k$ $B$ Total available bandwidth
    $m$ Packet size $P_i$ Transmit power of Super-CH $i$
    $\mu$ Transmission rate of each UAV $R_i(t)$ Instantaneous normalized achievable rate
    $T_t$ Total transmission delay of each UAV$\gamma_0$ Reference signal-to-noise ratio (SNR) at the reference distance of $d_0~=~1$ m
    $T_m$ CM delay $C_i$ Throughput requirement for Super-CH $i$
    $T_h$ CH delay $N$ Time slot number
    ${\boldsymbol~x}_i$, ${\boldsymbol~s}_i$ Locations of UAVs and the Super-CH, respectively $\delta$ Length of time step
    $\mathcal{C}_i$ Cluster $i$ composed of CMs and one CH$T_{\rm~tr}$ Ferry UAV' traveling time
    $\mathcal{U}$ Set of Super-CH UAVs $\hat{\pi}$ Visiting order under TSP
    ${\boldsymbol~q}(t)$ Ferry UAV's trajectory$T_{\rm~tsp}$ Minimum traveling time under TSP
    $V_{\max}$ Maximum Ferry UAV speed$\bar{T}_i$ Time for Ferry UAV to satisfy the throughput requirement of Super-CH $i$
    $d_{\min}$ Minimum safe distance between the Ferry UAVand Super-CH UAVs$\tilde{T}_i$ Residence time of Ferry UAV at Super-CH $i$
    $d_i(t)$ Distance between Ferry UAV and Super-CH $i$$r$ Radius of the sphere centered at Super-CH
    $h_i(t)$ Channel power gains${\boldsymbol~g}_i$ Waypoint inside the sphere for Super-CH $i$

    Algorithm 2 BCD based algorithm for (P1.3)

    Require:A given $T$, initial trajectory of the Ferry UAV $\mathcal{Q}^0$,

    prescribed thresholds $\epsilon_1>0$, $\epsilon_2>0$, $l=0$.

    Output:$\mathcal{Q}^l$, $\mathcal{A}^l$, $\eta^l$.

    while $\frac{\eta^{l+1}-\eta^l}{\eta^l}\geq~\epsilon_2$ do

    For given $\mathcal{Q}^l$,obtain $\mathcal{A}^{l+1}$by solving problem (P1.4);

    Initialize the inner iterative index $r=0$and the inner initial trajectory$\mathcal{Q}^{l,0}=\mathcal{Q}^{l}$;

    while $\frac{\eta^{r+1}-\eta^r}{\eta^r}\geq~\epsilon_1$ do

    For given $\mathcal{A}^{l+1}$ and $\mathcal{Q}^{l,r}$, obtain $\mathcal{Q}^{l,r+1}$ and $\eta^{r+1}$ by solving problem (P1.6);


    end while



    end while

  • Table 2  

    Table 2Main simulation parameters

    Parameter ValueParameter Value
    Total bandwidth: $B$10 MHzMinimum safe distance between the Ferry UAV and Super-CH UAVs: $d_{\min}$ $50$ m
    Noise power spectrum density: $N_0$ $-169$ dBm/HzTime step length: $\delta$ 4 s
    Channel power gain at the reference distance of $d_0=1$ m: $\lambda_0$ $-50$ dBThresholds in Algorithm 2: $\epsilon_1$,$\epsilon_2$ $10^{-2}$
    Transmit power of each Super-CH: $P_i$ 10 dBThreshold in Algorithm initr: $\epsilon$ $10^{-3}$
    Maximum speed of the Ferry UAV: $V_{\max}$ $50$ m/s

    Algorithm 3 Initial trajectory design for given $T$

    Require:A given $T$, locations of Super-CHs $\{{\boldsymbol~s}_i\}$.

    Output:$r$, the initial trajectory.

    Solve TSP to obtain minimum traveling time$T_{\rm~tsp}$ and optimal visiting order $\hat{\pi}$based on $\{{\boldsymbol~s}_i\}$;

    if $T\geq~T_{\rm~tsp}$ then

    Design initial trajectory based on Case 1;


    Let $r_l=0$, $r_u$ be sufficiently large andtolerance $\epsilon>0$;

    while $|T_{\rm~tr}~-T|\leq~\epsilon$ do


    Solve problem (P2.1) with visiting order $\hat{\pi}$ to derive traveling time $T_{\rm~tr}$ and waypoints $\{{\boldsymbol~g}_i\}$;

    if $T_{\rm~tr}~>T$ then

    Let $r_l=r$;


    Let $r_u=r$;

    end if

    end while

    Construct the initial trajectory based on $\{{\boldsymbol~g}_i\}$;

    end if