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Guo et al. Intell Robot 2023;3(4):596-613 I http://dx.doi.org/10.20517/ir.2023.32 Page 13 of 18
¤ = (1 + )(k k + k k )
(0, 0 ) = 0
From (33), the solution of the equation above also satisfies that
( , 0) = min ( )
2 k ¯ k
so that
= 1 ln 1+ ( ,0)
k k−k k 1+ k k ( ,0)
k k
The proof is, thus, completed.
Theorem 3. Consider system (27) and event-triggered consensus protocol (28), for any positive definite matrix
, if there exists a positive definite matrix satisfying = −( + ), suitable parameter and trig-
ger functions ( ) can always be designed so that the system achieves consensus under the event-triggered
conditions based on the above proof process.
Theorem 4. For the multi-UAV system, appropriate distance should be maintained between individuals. Con-
sensus protocol (2) can be transformed into
Í
¤
¤
( ) = − ∈ [( − ) − ( − )] + [( ¤ − ) − ( ¤ − )]
Í
= − [ ( − − ) + ( ¤ − ¤ )]
∈
in which is the constant offset. We define = − and ¤ = ¤ so that
0
0
Í
¤
¤
( ) = − ∈ [( − ) − ( − )] + [( ¤ − ) − ( ¤ − )]
Í 0 0 0 0
= − [ ( − ) + ( ¤ − ¤ )]
∈
Therefore, offset will not affect the consensus of the system.
4. SIMULATION
According to the scenario described in Remark 2, we consider a UAV group consisting of five individuals
whose dynamics are described by (1). The information interaction topologies of their velocity and position
are described in Figure 1A and B, respectively. The position information of other UAVs cannot be sensed by
individual 1 due to the damage of its position sensor.
One can obtain a Laplacian matrix of interaction topologies of velocity and position, respectively, as follows:
2 −1 0 0 −1 2 −1 0 −1 0
0 1 −1 0 0 0 1 −1 0 0
= 0 0 1 −1 0 = 0 0 2 −1 −1
0 0 0 1 −1 0 0 0 1 −1
−1 0 0 0 1 0 0 0 0 0
Consider = (1, 1, 1, 0.5, 0.25). According to Theorem 1 and Theorem 3, the system parameters can be
selected as follows: = 3, = 12, and = 0.99. The sampling period is set to 0.01. The initial state of the
system is set as follows:
