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Page 10 of 18 Guo et al. Intell Robot 2023;3(4):596-613 I http://dx.doi.org/10.20517/ir.2023.32

∈ (0, 1) (36)

Proof: Choose the Lyapunov function as

= ( ⊗ ) (37)

in which

" #
e e Γ
e
( + )
= (38)
e Γ e
Γ
Define = , where

e

0
= (39)
−

The contract transformation does not change the positivity of the matrix, and

" #
2
e e Γ
e
( + ) − 0
= 2 (40)
e
0 Γ
e
Since condition (31) implies that Γ is a positive definite matrix, one has that > 0. Therefore, is a positive
e
definite matrix.
Differentiate to obtain that

= ( + ) ⊗ + 2 ⊗ (41)
¤

= − ⊗ + 2 ⊗
in which

= −( + )

1 2
=
(42)
2 4
" #
2 ( + ) ( − )
e
e
e
e
=
e
e
e e
Γ
( − ) ( + ) − 2 e

And = . Performing the contract transformation on matrix , one can obtain that