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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 11 of 18

" #
e 0

1
= 1 = (43)
1
0 4
e
where

0
1 = − (44)
−
2 1

e
1 = 1 (45)

2
(46)
e
e e
e
4 = ( + ) − Γ − Ψ ( + )Ψ
e
e

According to conditions (31) and (32), matrix is a positive definite matrix. One has that

= − ⊗ + 2 ⊗
¤
(47)
2 ¯
≤ − min ( ) k k +2 k kk kk k
in which

¯
=
" 2 #
− − (48)
e
e
=
− −
e
e
From condition (33), one has that

¤ 2 ¯
≤ − min ( ) k k +2 k kk kk k
2 2
≤ − min ( ) k k + min ( ) k k
(49)
2
= −(1 − ) min ( ) k k
< 0

Define an event-triggered function as follows:

min ( )
( ) =k k − k k (50)
¯
2 k k

At an event time ∈ { 1 , . . . , }, one can obtain:

( ) ( ) − ( ) 0
( ) = = = (51)
( ) ( ) − ( ) 0