Nomenclature

UAV

Unmanned Aerial Vehicle

QUAV

Quadrotor Unmanned Aerial Vehicle

2.0 Graph theory

QUAV formation consensus control covers multiple aspects such as formation reconstruction, formation maintenance, obstacle avoidance and so on, all of which are designed based on graph theory. In graph theory, a graph is composed of vertexes and edges, where the vertexes represent the QUAVs and the edges represent the communication between the QUAVs.

Assuming the total number of QUAVs is n. The graph $G = \left( {V,E} \right)$ consists of a vertex set $V = (1,2, \cdots, n)$ and an edge set $E \subseteq V \times V$ . If vertex $i$ and vertex $j$ are connected, then there’s an edge $({v_i},{v_j}) \in E$ , and vertex $j$ is called a neighbour of vertex $i$ . Graphs are divided into directed graphs and undirected graphs, if all the edges in a graph are undirected, it is called an undirected graph; otherwise, it is called a directed graph.

The relationship between vertex $i$ and vertex $j$ is representedby ${a_{ij}}$ . If vertex $i$ and vertex $j$ are connected by an edge, then ${a_{ij}} = 1$ ; otherwise, ${a_{ij}} = 0$ . The adjacency matrix $A = [{a_{ij}}](i,j \in \left\{ {1,2, \cdots, n} \right\})$ is a method used in graph theory to represent a graph as a two-dimensional array, where the rows and columns represent the vertexes in the graph, and the elements of the array represent whether there are edges between the vertices.

The degree matrix $D = diag({d_1},{d_2}, \ldots, {d_n})$ is diagonal arrays, and the elements on the diagonal represent the degrees of a single vertex. In an undirected graph, the degree $d({v_i})$ of a vertex is the number of edges of vertex $i$ . In a directed graph, degree is divided into two concepts: in-degree and out-degree. The in-degree is the number of directed edges that end at the vertex; the out-degree is the number of directed edges starting from the vertex.

The Laplace matrix of a graph is related to the degree matrix and the adjacency matrix by $L = D - A$ .

3.0 Mathematical models

The UAV model used in this paper is a QUAV, assuming that the safety radius of the centre of the QUAV is ${R_S}$ and the communication radius is ${R_C}$ . The distance between the centre of the QUAV and the surface of the obstacle needs to be greater than the safety radius; otherwise, a collision may occur, resulting in a crash. The distance between any two QUAV centres needs to be less than the communication radius; otherwise it will cause the QUAV to lose contact. The safety radius and communication radius are illustrated in Fig. 1.

Figure 1. Illustration of QUAV safety radius profile.

The continuous-time multi-QUAVs linear model used in this paper is:

(1) \begin{align} {\dot x_i}(t) = {u_i}(t),{\rm{ }}i \in N \end{align}

where, ${x_i}(t) \in R$ and ${u_i}(t) \in R$ denote the state and control input of QUAV $i$ at moment $t$ , respectively.

Due to the fact that QUAV is a linear model, it is treated as a mass point in the artificial potential field method, which contradicts the safety radius that the QUAV needs to maintain. It is necessary to attach the safety radius of the QUAV to the obstacle, which the QUAV is treated as a mass point and no longer requires a safety radius. Assuming the actual radius of the spherical obstacle is ${R_O}$ , the total radius $R = {R_O} + {R_S}$ . The radius of the obstacle is shown in Fig. 2.

Figure 2. Illustration of obstacle safety radius profile.

4.0 Finite-time consensus with time delay

The consensus protocol with equal communication delay, input delay and undirected connected graph network topology is as follows:

(2) \begin{align} {u_i}(t) = \mathop \sum \limits_{{V_j} \in {N_i}} {a_{ij}}\!\left( {{x_j}(t - \tau ) - {x_i}(t - \tau )} \right) \end{align}

where, $\tau $ denotes the time delay; when $ \tau \in [0,{{\;\pi } / {2{\lambda _n}}}) $ the system can achieve consistent; ${\lambda _n}$ is the maximum eigenvalue of the Laplace matrix of the system.

Definition 1 [Reference Xiao, Wu and Peng30] If there exists an expected time ${T_0} \in [0, + \infty )$ , such that for any initial state, the multi-QUAVs satisfy $\mathop {\lim }\limits_{t \to {T_0}} \left\| {{x_i}(t) - {x_0}(t)} \right\| = 0$ and ${x_i}(t) = {x_0}(t),{\rm{ }}\forall t \geqslant {T_0},{\rm{ }}i \in I$ , then it can be said that finite-time consensus control has been achieved.

In Ref. (Reference Sun and Zhu31), Sun and Guan proposed the following finite-time consensus protocol:

(3) \begin{align} {u_i}(t) = \mathop \sum \limits_{j \in {N_i}} {a_{ij}}si{g^\alpha }\!\left( {{x_j}(t) - {x_i}(t)} \right) + {a_{i0}}si{g^\alpha }\!\left( {{x_{0\vphantom{j}}}(t) - {x_i}(t)} \right) \end{align}

where $0 \lt \alpha \lt 1$ .

Definition 2 $si{g^\alpha }({x_i}) = {\left| {{x_i}} \right|^\alpha }{\mathop{\rm sgn}} \left( {{x_i}} \right)$ , ${\mathop{\rm sgn}} ()$ is a symbolic function, defined as $sgn(r) = \left\{ {\begin{array}{l@{\quad}l}{1,} & {}{r \gt 0}\\[3pt] {0,} & {}{r = 0}\\[3pt] { - 1,} & {}{r \lt 0}\end{array}} \right.$ .

Considering the existence of time delay based on protocol (3), protocol (4) is obtained:

(4) \begin{align} {u_i}(t) = \mathop \sum \limits_{j \in {N_i}} {a_{ij}}sig{\left( {{x_j}(t - \tau ) - {x_i}(t - \tau )} \right)^\alpha } + {a_{i0}}sig{\left({x_{0\vphantom{j}}}(t - \tau ) - {x_i}(t - \tau )\right)^\alpha } \end{align}

In order to accelerate the convergence speed of protocol (4), this paper has made improvements and proposed protocol (5):

(5) \begin{align} {u_i}(t) =\, & \mathop \sum \limits_{j \in {N_i}} {a_{ij}}sig{\left( {{x_j}(t - \tau ) - {x_i}(t - \tau )} \right)^\alpha } + {a_{i0}}sig{\left( {{x_{0\vphantom{j}}}(t - \tau ) - {x_i}(t - \tau )} \right)^\alpha }\\[3pt] & {\rm{ }} + \beta \mathop \sum \limits_{j \in {N_i}} {a_{ij}}\left( {{x_j}(t - \tau ) - {x_i}(t - \tau )} \right) + \beta {a_{i0}}\left( {{x_{0\vphantom{j}}}(t - \tau ) - {x_i}(t - \tau )} \right) \nonumber \end{align}

where, $\beta \gt 0$ , the larger $\beta $ is, the faster system converges.

Next, it will be proven that protocol (5) can achieve finite-time consensus and has a faster convergence speed than protocol (4). Before that, the following lemmas need to be introduced.

Next, using the Lyapunov function method, it is shown that the control of multi-QUAVs through protocol (5) can achieve state consensus within a finite-time.

Proof: Define the position error

(6) \begin{align} {e_i}(t) = {x_i}(t - \tau ) - {x_0}(t - \tau ),\quad \forall i \in I \end{align}

This is obtained through Equations (1) and (6):

(7) \begin{align} {\dot{e}}_{i}(t) & = \sum \limits_{j \in N_{i}} {a_{ij}}sig{{\left( {{e_j}(t) - {e_i}(t)} \right)}^\alpha } - {a_{i0}}sig{{\left( {{e_i}(t)} \right)}^\alpha } + \beta \sum \limits_{j \in {N_i}} {a_{ij}}\left( {{e_j}(t) - {e_i}(t)} \right) - \beta {a_{i0}}{e_i}(t)\\[3pt] & { = \mathop \sum \limits_{j = 0}^n {a_{ij}}sig{{\left( {{e_j}(t) - {e_i}(t)} \right)}^\alpha } + \beta \mathop \sum \limits_{j = 0}^n {a_{ij}}\left( {{e_j}(t) - {e_i}(t)} \right)} \nonumber \end{align}

The Lyapunov function ${V_1}(t) = \mathop \sum \limits_{i = 0}^n e_i^2(t)$ chosen according to Lemma 1 is given by Equation (7):

(8) \begin{align} {{\dot V}_1}(t) & = 2\mathop \sum \limits_{i = 0}^n {e_i}(t){{\dot e}_i}(t)\nonumber\\[3pt] & { = 2\mathop \sum \limits_{i = 0}^n {e_i}(t)\left[ {\mathop \sum \limits_{j = 0}^n {a_{ij}}sig{{\left( {{e_j}(t) - {e_i}(t)} \right)}^\alpha } + \beta \mathop \sum \limits_{j = 0}^n {a_{ij}}\left( {{e_j}(t) - {e_i}(t)} \right)} \right]}\nonumber\\[3pt] & { = 2\mathop \sum \limits_{i = 0}^n {e_i}(t)\mathop \sum \limits_{j = 0}^n {a_{ij}}sig{{\left( {{e_j}(t) - {e_i}(t)} \right)}^\alpha } + 2\beta \mathop \sum \limits_{i = 0}^n {e_i}(t)\mathop \sum \limits_{j = 0}^n {a_{ij}}\left( {{e_j}(t) - {e_i}(t)} \right)} \nonumber\\[3pt] & { = - \mathop \sum \limits_{i = 0}^n \mathop \sum \limits_{j = 0}^n {a_{ij}}sig{{\left( {{e_j}(t) - {e_i}(t)} \right)}^\alpha }\left( {{e_j}(t) - {e_i}(t)} \right) - \beta \mathop \sum \limits_{i = 0}^n \mathop \sum \limits_{j = 0}^n {a_{ij}}{{\left( {{e_j}(t) - {e_i}(t)} \right)}^2}} \\[3pt] & { = - \mathop \sum \limits_{i = 0}^n \mathop \sum \limits_{j = 0}^n {a_{ij}}{{\left( {{e_j}(t) - {e_i}(t)} \right)}^{1 + \alpha }} - \beta \mathop \sum \limits_{i = 0}^n \mathop \sum \limits_{j = 0}^n {a_{ij}}{{\left( {{e_j}(t) - {e_i}(t)} \right)}^2}}\nonumber\\[3pt] & { \lt - \mathop \sum \limits_{i = 0}^n \mathop \sum \limits_{j = 0}^n {a_{ij}}{{\left( {{e_j}(t) - {e_i}(t)} \right)}^{1 + \alpha }}}\nonumber\\[3pt] & { = - \mathop \sum \limits_{i = 0}^n \mathop \sum \limits_{j = 0}^n {{\left[ {a_{ij}^{\frac{2}{{1 + \alpha }}}\left( {{e_j}(t) - {e_i}(t)} \right)} \right]}^{\frac{{1 + \alpha }}{2}}}} \nonumber \end{align}

From Lemma 2, it can be concluded that:

(9) \begin{align} {\dot V_1}(t) \le - {\left[ {\mathop \sum \limits_{i = 0}^n {{\left( {\mathop \sum \limits_{j = 0}^n a_{ij}^{\frac{2}{{1 + \alpha }}}\left( {{e_j}(t) - {e_i}(t)} \right)} \right)}^2}} \right]^{\frac{{1 + \alpha }}{2}}} \end{align}

Let $B = L(A) + Diag({a_{10}}, \ldots, {a_{n0}})$ , from Lemma 3, we can conclude that:

(10) \begin{align} \frac{{\mathop \sum \limits_{i = 0}^n {{\left[ {\mathop \sum \limits_{j = 0}^n a_{ij}^{\frac{2}{{1 + \alpha }}}\left( {{e_j}(t) - {e_i}(t)} \right)} \right]}^2}}}{{{V_1}(t)}} = \frac{{2e_i^T(t)B{e_i}(t)}}{{e_i^T(t){e_i}(t)}} \geqslant 2{\lambda _2}\left( {{L_B}} \right) \end{align}

Combining Equations (9) and (10) yields ${\dot V_1}(t) \le - {\left[ {2{\lambda _2}\left( {{L_B}} \right)} \right]^{\frac{{1 + \alpha }}{2}}}V{(t)^{\frac{{1 + \alpha }}{2}}}$ , and by Lemma 4 $T(x(0)) \le \frac{{2{V_1}{{(0)}^{\frac{{1 - \alpha }}{2}}}}}{{(1 + \alpha ){{\left[ {2{\lambda _2}\left( {{L_B}} \right)} \right]}^{\frac{{1 + \alpha }}{2}}}}}$ . And that’s the end of the proof.

In the following, by choosing the same Lyapunov function ${V_2}(t) = \mathop \sum \limits_{i = 0}^n e_i^2(t)$ , it is shown that the convergence of the multi-QUAV formation under protocol (5) is faster than that under protocol (4) in the same way as described earlier.

\begin{align} {{\dot V}_2}(t) & = 2\mathop \sum \limits_{i = 0}^n {e_i}(t){{\dot e}_i}(t)\nonumber\\[3pt] & { = 2\mathop \sum \limits_{i = 0}^n {e_i}(t)\left[ {\mathop \sum \limits_{j = 0}^n {a_{ij}}sig{{\left( {{e_j}(t) - {e_i}(t)} \right)}^\alpha }} \right]}\nonumber\end{align}

(11) \begin{align} & { = 2\mathop \sum \limits_{i = 0}^n {e_i}(t)\mathop \sum \limits_{j = 0}^n {a_{ij}}sig{{\left( {{e_j}(t) - {e_i}(t)} \right)}^\alpha }}\nonumber\\[3pt] & { = - \mathop \sum \limits_{i = 0}^n \mathop \sum \limits_{j = 0}^n {a_{ij}}sig{{\left( {{e_j}(t) - {e_i}(t)} \right)}^\alpha }\left( {{e_j}(t) - {e_i}(t)} \right)}\\[3pt] & { = - \mathop \sum \limits_{i = 0}^n \mathop \sum \limits_{j = 0}^n {a_{ij}}{{\left( {{e_j}(t) - {e_i}(t)} \right)}^{1 + \alpha }}} \nonumber \end{align}

From Equation (8), we can obtain ${\dot V_1}(t) = - \mathop \sum \limits_{i = 0}^n \mathop \sum \limits_{j = 0}^n {a_{ij}}{\left( {{e_j}(t) - {e_i}(t)} \right)^{1 + \alpha }} - \beta \mathop \sum \limits_{i = 0}^n \mathop \sum \limits_{j = 0}^n {a_{ij}}{\left( {{e_j}(t) - {e_i}(t)} \right)^2}$ , and from $\beta \gt 0$ easily know ${\dot V_2}(t) \lt {\dot V_1}(t)$ . Therefore, the convergence speed of protocol (5) is faster than that of protocol (4).

5.0 Artificial potential field method

5.1 Traditional artificial potential field method

The traditional artificial potential field method is created based on the principle of force in physics. It regards the controlled object as a particle and studies its motion in a virtual potential field. The QUAV moves towards the target position under the influence of potential field and avoids obstacles during this process. The whole potential field consists of two parts: gravitational potential field and repulsive potential field. By calculating the negative gradient of the gravitational potential field and repulsive potential field acting on the QUAV, gravity and repulsive force are obtained, where the force at the target point is gravity and the force at the obstacle is repulsive force. Under the combined action of gravity and repulsion, the QUAV can move toward the target point while avoiding obstacles, and a smooth and continuous motion trajectory can be generated during the whole movement process. The principle of artificial potential field method is shown in Fig. 3.

Figure 3. Illustration of artificial potential field method.

The traditional artificial potential field method has the following shortcomings [Reference Luan, Tan and You36]:

1. (1) Collision with obstacles. When the distance between the QUAV and the target point is too far, the attractive force on it will be much greater than the repulsive force, and it may collide with obstacles during flight.

2. (2) Target unreachability. When the target point is located near an obstacle and the QUAV approaches the target point, the repulsive force applied to the QUAV is too strong relative to the attractive force, causing the QUAV to hover or remain stationary in a certain area.

3. (3) Falling into a local minimum. When the QUAV moves to a non-target point location, it will fall into a local minimum due to force equilibrium, causing it to oscillate or come to rest repeatedly near the local minimum point.

5.2 Improved artificial potential field method

The attractive potential field function is improved to address the problem of excessive attractive force caused by the QUAV being too far away from the target. Set the attraction action threshold for segment to avoid the problem of excessive attraction force. The modified attractive potential field function is shown in Equation (12):

(12) \begin{align} {U_{att}}(q) = \left\{ \begin{array}{l@{\quad}l}{\frac{1}{2}{k_{att}}{d_{goal}}^2,} & {}{{d_{goal}} \le {d_0}}\\[3pt] {{d_0}{k_{att}}{d_{goal}} - \frac{1}{2}{k_{att}}d_0^2,} & {}{{d_{goal}} \gt {d_0}}\end{array} \right. \end{align}

where, ${k_{att}}$ is the attractive coefficient, ${d_0}$ is the impact distance threshold of the target point, $q$ is the spatial position of the QUAV, and ${d_{goal}} = {\left\| {q - {q_{goal}}} \right\|_2}$ is the Euclidean distance between the QUAV and the target point.

The attraction force is the negative gradient of the attractive potential field, as shown in Equation (13):

(13) \begin{align} {F_{att}}(q) = - \nabla {U_{att}}(q) = \left\{ \begin{array}{l@{\quad}l} {{k_{att}}\left( {{q_{goal}} - q} \right),} & {}{{d_{goal}} \le {d_0}}\\[3pt] {\frac{{{d_0}}}{{{d_{goal}}}}{k_{att}}\left( {{q_{goal}} - q} \right),} & {}{{d_{goal}} \gt {d_0}}\end{array} \right. \end{align}

For the target unreachability problem, an improved repulsive potential field function is introduced, which combines the distance between the target point and the QUAV to ensure that the potential field is globally minimised at the target point only. The improved repulsive potential field function is shown in Equation (14):

(14) \begin{align} {U_{rep}}(q) = \left\{ \begin{array}{l@{\quad}l}{\frac{1}{2}{k_{rep}}{{(\frac{1}{{{d_{obs}}}} - \frac{1}{{{d_{\max }}}})}^2}d_{goal}^2,} & {}{{\rm{ }}{d_{obs}} \le {d_{\max }}}\\[3pt] {0,} & {}{{\rm{ }}{d_{obs}} \gt {d_{\max }}}\end{array} \right. \end{align}

where, ${K_{rep}}$ is the repulsive coefficient, ${d_0}$ is the maximum impact distance of an obstacle, and ${d_{obs}} = {\left\| {q - {q_{obs}}} \right\|_2}$ is the Euclidean distance between the QUAV and the obstacle.

The repulsive force is the negative gradient of the repulsive potential field, as shown in Equation (15):

(15) \begin{align} {F_{rep}}(q){\rm{ }} = - \nabla {U_{rep}}(q) = \left\{ \begin{array}{l@{\quad}l} {{F_{rep1}}{{\bf{n}}_1} + {F_{rep2}}{{\bf{n}}_2},} & {}{{\rm{ }}{d_{obs}} \le {d_{\max }}}\\[3pt] {0,} & {}{{\rm{ }}{d_{obs}} \gt {d_{\max }}}\end{array} \right. \end{align}

where, ${{\bf{n}}_1}$ and ${{\bf{n}}_2}$ denote the unit direction vectors along the directions ${F_{rep1}}$ and ${F_{rep2}}$ . The expressions for ${F_{rep1}}$ and ${F_{rep2}}$ are shown in Equation (16):

(16) \begin{align} \left\{ \begin{array}{l}{{F_{rep1}} = {k_{rep}}\left( {\frac{1}{{{d_{obs}}}} - \frac{1}{{{d_{\max }}}}} \right)\frac{{d_{goal}^2}}{{d_{obs}^2}}}\\[3pt] {{F_{rep2}} = \frac{n}{2}{k_{rep}}{{\left( {\frac{1}{{{d_{obs}}}} - \frac{1}{{{d_{\max }}}}} \right)}^2}{d_{goal}}}\end{array} \right. \end{align}

where ${F_{rep1}}$ and ${F_{rep2}}$ are the components of the repulsive force in two different directions, respectively.

The direction of vector points from the obstacle to the QUAV, which is the repulsive force component; the direction of vector points from the QUAV to the target point and is the attractive force component. As shown in Fig. 4.

Figure 4. Illustration of repulsive force calculation method.

It is important to note that the QUAVs in the formation also need to avoid collisions with each other. QUAVs can consider each other as dynamic obstacles, obtain the positions of other QUAVs, and avoid collisions through artificial potential field methods.

The total potential field consists of attractive potential field and repulsive potential field, as shown in Equation (17):

(17) \begin{align} U(q) = {U_{att}}(q) + {U_{rep}}(q) \end{align}

The combined force is equal to the negative gradient of the total potential field, as shown in Equation (18):

(18) \begin{align} F(q) = - \nabla U(q) \end{align}

For the local minimum problem, introduce a random perturbation and set the minimum velocity as the condition for triggering the random perturbation. If the velocity components ${v_x},{v_y},{v_z}$ of the QUAV in all three directions of $x,y,z$ are less than the minimum velocity, the random perturbation is triggered, as shown in Equation (19):

(19) \begin{align} {v_{rand}} = \left[ {\begin{array}{*{20}{c}}{{v_x}}\\[3pt] {{v_y}}\\[3pt] {{v_z}}\end{array}} \right] = \left[ {\begin{array}{*{20}{l}}{\lambda (2{\rm{rand}} - 1)}\\[3pt] {\lambda (2{\rm{rand}} - 1)}\\[3pt] {\lambda (2{\rm{rand}} - 1)}\end{array}} \right],\,\,\,{v_x} \lt {v_{\min }}\ {} {v_y} \lt {v_{\min }}\ {} {v_z} \lt {v_{\min }} \end{align}

where, is a random function, is the minimum velocity, $\lambda$ is a coefficient with a small absolute value, and ${\rm{2rand}}$ is a random number in the range of 0 to 1.

5.3 Constraint conditions

In order to avoid collision between QUAVs and obstacles due to excessive speed, a maximum speed constraint is introduced and the constraints are ${v_x},{v_y},{v_z}$ .

Taking the velocity component of the X-axis as an example,and calculate the velocity change at each sampling time, as shown in Equation (20):

(20) \begin{align} \Delta {v_x}(t) = {v_x}(t) - {v_x}(t - 1) \end{align}

where, ${v_x}(t)$ is the speed of the QUAV at the current sampling time, ${v_x}(t - 1)$ is the speed of the QUAV at the last sampling time, and $\Delta {v_x}(t)$ is the amount of change in the velocity component.

Then, the velocity component of the QUAV at the next sampling time is calculated, as shown in Equation (21):

(21) \begin{align} \left\{ {\begin{array}{*{20}{l}}{{v_{0x}}(t + 1) = \left\{ {\begin{array}{*{20}{c}}{\min \left\{ {v(t) + \Delta v(t),v(t) + {a_x} \cdot dt} \right\},{\rm{ }}\Delta v(t) \gt 0}\\[3pt] {\max \left\{ {v(t) + \Delta v(t),v(t) - {a_x} \cdot dt} \right\},{\rm{ }}\Delta v(t) \lt 0}\end{array}} \right.}\\[3pt] {{v_x}(t + 1) = \left\{ {\begin{array}{*{20}{c}}{\min \left\{ {{v_{0x}}(t + 1), + {v_{x\max }}} \right\},{\rm{ }}{v_{0x}}(t + 1) \gt 0}\\[3pt] {\max \left\{ {{v_{0x}}(t + 1), - {v_{x\max }}} \right\},{\rm{ }}{v_{0x}}(t + 1) \lt 0}\end{array}} \right.}\end{array}} \right. \end{align}

where, ${v_{x\max }}$ is the maximum velocity component of the QUAV, ${a_x}$ is the acceleration component. The values of ${v_{x\max }}$ and ${a_x}$ need to meet the requirement that the QUAV can avoid collision by decelerating when encountering obstacles. ${v_{0x}}(t + 1)$ plays a transitional role in determining the ${v_x}(t + 1)$ of the velocity component of the QUAV at the next sampling time.

6.0 Mutli-quavs formation consensus obstacle avoidance algorithm

The formation control adopts the leader-following method in distributed control, and its basic idea is that: the leader receives commands and controls the entire formation to perform the expected task. The followers follow the leader based on their expected positions relative to the leader. The leader-following method is illustrated in Fig. 5:

Figure 5. Illustration of leader-following method.

Figure 6. Illustration of QUAV formation control structure.

The control protocol of the leader is shown in Equation (24):

(24) \begin{align} {u_0}(t) = \sum\limits_{i \in {N_i}} {{a_{i0}}} \Delta {x_{i0}}(t) + {F_0}(t) + {v_{rand}} \end{align}

where $\Delta {x_{i0}}(t)$ is the relative position of the leader and the follower $i$ at moment $t$ , ${F_0}(t)$ is the combined force on the leader at moment $t$ .

Table 1. Position coordinates of six QUAVs

Figure 7. Formation consensus of six QUAVs.

The control protocol of the follower is shown in Equation (25):

(25) \begin{align} {u_i}(t) = \sum\limits_{j \in {N_i}} {{a_{ij}}} \left( {{x_j}(t - \tau ) - {x_i}(t - \tau ) - \Delta {x_{ij}}(t)} \right) + {F_{rep(i)}}(t) \end{align}

where $\Delta {x_{ij}}(t)$ denotes the relative position of QUAV $i$ and QUAV $j$ at moment $t$ , and ${F_{rep(i)}}(t)$ the obstacle repulsive force on QUAV $i$ at moment $t$ .

Table 2. Position coordinates of four QUAVs

Figure 8. Formation consensus of four QUAVs.

Figure 9. Distance of multi-QUAVs from obstacles.

Figure 10. Comparison of QUAV formation consistency obstacle avoidance.

Figure 11. Position error of multi-QUAVs.

Figure 12. Position consensus of multi-QUAVs.

Figure 13. Positional errors of leader and followers.

Figure 14. Comparison of position errors of Z-axis.

Figure 15. Speeds of multi-QUAVs.

After receiving the command signal, the leader QUAV sends control command to the followers through the formation consensus obstacle avoidance algorithm. The followers enter the range of influence of obstacles and avoid them autonomously, thus realising a consistent state throughout the formation. The QUAV formation control structure is shown in Fig. 6.

In order to realise the QUAV formation to maintain formation and avoid obstacles autonomously, adaptive inter-QUAV communication weights are considered in this paper. Add a weight to the algorithm ${\omega _{ij}}(t)$ that varies with the position of QUAV $i$ and QUAV $j$ , and the system reaches state consensus when $\left\| {{x_i}(t) - {x_j}(t)} \right\| \to 0$ . The designed multi-QUAVs finite-time consensus obstacle avoidance protocol is shown in Equation (26):

(26) \begin{align} \begin{array}{l}{u_i}(t) = {\omega _{ij}}(t)[\mathop \sum \limits_{j \in {N_i}} {a_{ij}}sig{({x_j}(t - \tau ) - {x_i}(t - \tau ))^\alpha } + {a_{i0}}sig{({x_0}(t - \tau ) - {x_i}(t - \tau ))^\alpha }\\[3pt] {\rm{ }} + \gamma \mathop \sum \limits_{j \in {N_i}} {a_{ij}}({x_j}(t - \tau ) - {x_i}(t - \tau )) + \gamma {a_{i0}}({x_0}(t - \tau ) - {x_i}(t - \tau ))] + {F_i}(t)\end{array} \end{align}

where ${\omega _{ij}}(t) = 2 - {e^{ - {{({x_j}(t - \tau ) - {x_i}(t - \tau ))}^2}}}$ , ${F_i}(t)$ denotes the combined force of the artificial potential field on the QUAV $i$ at moment $t$ .

7.0 Numerical simulation

The effectiveness of the algorithm in an environment without obstacles is verified by MATLAB simulation. QUAVs have a safety radius of ${R_S} = 0.25{\rm{m}}$ and a communication radius of ${R_C} = 3.5{\rm{m}}$ , and the safety radius between two QUAVs is $R = 2 \times {R_S} = 0.5{\rm{m}}$ . The desired positions in Table 1 indicate the position of each QUAV in the desired formation. Based on the position of the leader as a reference, the followers move towards the desired position to form the desired formation.

Each QUAV quickly forms the desired formation under the control of the finite-time consensus protocol. The leader autonomously plans the flight trajectory, and the followers maintain the desired formation and follow the movement of the leader under the control of the consensus agreement. The simulation results of QUAV formation consensus are shown in Fig. 7.

Then, the effectiveness of the algorithm(26) is verified by MATLAB simulation, and both the network topology and the desired formation of the QUAV formation are shown in Fig. 4. The radius of the spherical obstacle ${R_O} = 0.25{\rm{m}}$ . The safe radius between the QUAV and the obstacle $R = {R_O} + {R_S} = 0.5{\rm{m}}$ . The attraction gain coefficient ${K_{att}} = 2$ , repulsion gain coefficient ${K_{rep}} = 0.0002$ , target influence distance threshold ${d_0} = 2{\rm{m}}$ , maximum obstacle influence distance ${d_{\max }} = 1{\rm{m}}$ , and the coordinates of the target point are $(20,20,10)$ . Several spherical obstacles are randomly generated on the flight trajectory of QUAV formation, with the spherical centre coordinates are ${\rm{(6 8 3)}}$ , ${\rm{(8 6 3)}}$ , ${\rm{(9 8 4)}}$ , ${\rm{(9 10 5)}}$ , ${\rm{(11 8 5)}}$ , ${\rm{(11 9 5)}}$ , ${\rm{(11 11 6)}}$ , ${\rm{(12 11 6)}}$ , ${\rm{(12 13 7)}}$ , ${\rm{(13 11 7)}}$ , ${\rm{(13 14 8)}}$ . Maximum velocity constraint is ${v_{x\max }} = {v_{y\max }} = {v_{z\max }} = 3{\rm{m/s}}$ , ${v_{\max }} = 3\sqrt 3 {\rm{m/s}}$ , acceleration ${a_x} = {a_y} = {a_z} = 1{\rm{m/}}{{\rm{s}}^{\rm{2}}}$ . The sampling time is $\Delta t = 1{\rm{ms}}$ , time delay $\tau = 10{\rm{ms}}$ , finite-time coefficient $\alpha = 0.5$ , finite-time coefficient $\gamma = 4$ . The initial and desired position coordinates of each UAV are shown in Table 2.

The algorithm requires that multi-QUAVs to quickly assemble to form a desired formation, and the QUAV formation should avoid obstacles in the airspace and reach the target position while maintaining the desired formation. The simulation results are shown in Fig. 8. The spherical shape in Fig. 8 represents an obstacle, and its radius is equal to the sum of the actual radius of the obstacle and the safety radius. Multi-QUAVs fly to the target position under the guidance of the leader, and the end of the mission is signaled when the leader reaches the target position. From the four angles of Fig. 8, it can be clearly seen that the multi-QUAVs can quickly assemble to form the desired formation. The QUAV formation can autonomously avoid obstacles while flying to the target position, and the formation is well maintained.

Figure 9 shows the distance between the QUAVs and the obstacles. Observe the minimum value of the Y-axis in the figure. If it is greater than the total radius of the obstacle, it indicates that there is no collision with the obstacle. The distance between the centres of the 4 QUAVs and the centres of the 11 obstacles is always greater than 0.5m, indicating that each QUAV can avoid obstacles.

Figure 10 shows a comparative simulation of QUAV formation consistency obstacle avoidance under the action of traditional artificial potential field method, with the same other conditions. The results indicate that multi-QUAVs cannot maintain the desired formation during obstacle avoidance, demonstrating the effectiveness of the algorithm proposed in this paper.

Figure 11 shows the adjacent distance between any two QUAVs, where ${D_{12}}$ denotes the distance between follower 1 and follower 2; ${D_{10}}$ denotes the distance between follower 1 and the leader, and so on. The distance between any two QUAVs is greater than the safe radius and less than the communication radius. This indicates that QUAVs do not collide with each other and are always able to establish communication. After about 1.6s, the distance between any two QUAVs remains unchanged, indicating that the formation is well maintained.

To reflect the QUAV formation consensus motion under this algorithm more clearly, the convergence of the formation’s positions on X-axis, Y-axis and Z-axis are simulated by MATLAB. The simulation results are shown in Fig. 12. The leader arrives at the target position at about 11.3s. From Fig. 12(c), it can be seen that the multi-QUAVs complete desired formation at about 1.6s, and the heights of each QUAV on the Z-axis are always synchronised during the subsequent flight.

Simulation of position error curves of each follower and leader, the spacing between the follower and the leader in the desired formation on the axis is $\Delta x = [{-}2, -{\rm{1,0,0]}}$ , $\Delta y = [0, -1, -2{\rm{,0]}}$ , $\Delta z = [0,{\rm{0,0,0]}}$ , respectively, and the QUAV position error $\Delta = [{-}2, -\sqrt 2, -2,0]$ . In Fig. 13, the position error of each follower and leader is the same as the predetermined position, which indicates that the multi-QUAVs can form and maintain the desired formation.

Figure 14 shows a simulation comparison of the Z-axis position error without trajectory segmentation strategy and improved finite-time consensus control, with the same other conditions. The Z-axis position error are selected for comparison, as the positions of each QUAV in the required formation are different on the X-axis and Y-axis, while and the required heights for the Z-axis are the same, but the initial heights are different. The multi-QUAVs can only form the desired formation at about 8s, and the leader reaches the target position at about 12.3s. It is verified that the multi-QUAV finite-time consensus protocol enables multi-QUAVs to quickly assemble to form a desired formation, and the trajectory segmentation strategy can improve task execution efficiency.

Figure 15(a)–(c) show the velocity components of the QUAVs with the maximum constraint of $3{\rm{m/s}}$ , and Fig. 15(d) shows the velocity of the QUAVs with the maximum constraint of $3\sqrt 3 {\rm{m/s}}$ . None of the QUAVs exceeded the maximum velocity constraint and converged uniformly in approximately 1.6s. Afterwards, each QUAV has the same speed, indicating that the formation is well maintained.