Page 79 - Read Online

Page 4 of 18 Guo et al. Intell Robot 2023;3(4):596-613 I http://dx.doi.org/10.20517/ir.2023.32

thatanindividualislesssusceptibletotheinfluenceofneighbors. Therefore,theconclusionofthispapercanbe
extended to the heterogeneous system. In addition, this paper focuses on the consistency proof of large-scale
network topology based on the graph theory. Using traditional drone models will make the proof process
obscure and cumbersome. The control input in this paper can be considered as the expected acceleration.
Therefore, the dynamic model of UAVs has been simplified during the proof process.

Definition 1. The heterogeneous multi-UAV system (1) is said to reach consensus for any initial conditions,
when and only when we have lim k − k= 0 and lim k ¤ − ¤ k= 0 for ∀ , ∈ .

→+∞ →+∞
To achieve urgent task objectives, an event-triggered consensus protocol will be proposed based on the follow-
ing second-order consensus protocol:

( ( − ) + ( ¤ − ¤ )) (2)
( ) = −Σ ∈
where and are stiffness gain and damping gain, respectively. is the coupling coefficient of position
information interaction, and is the coupling coefficient of velocity information interaction. If > 0 or
> 0, it means that the relevant information of UAV can be captured by UAV . How to achieve consensus
in system (1) based on the above protocol and event triggering mechanism is the problem that needs to be
addressed in this paper.

Remark 2. The communication and sensor faults assumed in this paper refer to the inability of individuals to
obtain information sent by neighbors through wireless data transmission or other means. Therefore, in order
to cope with situations where wireless data transmission cannot be utilized due to strong interference, the
method of individuals acquiring information through sensors, such as position and velocity, is widely adopted.
We further assume that position sensors of some individuals are damaged, and they are unable to obtain the
positioninformationofsurroundingindividuals(infact, theprocessingmethodsfordamagedpositionsensors
and speed sensors are generally similar, and this article only discusses the former), which is reflected in the
Laplacian matrix that contains all-zero rows.

2.2. Preliminaries
Lemma 1. Communication topology can be represented as a weighted directed (undirected) graph =
( , , ) of order with a vertex set = {1, 2, , } and edge set ⊂ × and a non-negative sym-
metric matrix = [ ] × . ( , ) ∈ ⇔ > 0 ⇔ the information of individual can be captured by
individual ⇔ is the neighbor member of individual . We assume = 0. The neighbor set of individual
is represented by = { |( , ) ∈ }. The Laplacian matrix of the weighted diagraph is defined as = [ ],
where = − and = Σ ≠ .

Lemma 2 [25] . If graph contains at least one directed spanning tree, its corresponding Laplacian matrix
satisfies the following properties:

(a) ( ) = − 1;
(b) 0 is an eigenvalue of matrix , and [ , , , ] is its corresponding eigenvector;

(c) Re( ) ≥ 0, ∀ ∈ {1, 2, , }; and there is only one eigenvalue of 0;
(d) Laplacian matrix related to the strongly connected graph is an irreducible matrix.

Laplacian matrix = [ ] × and = [ℎ ] × are defined as: