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Page 96 Li et al. Intell Robot 2022;2(1):89–104 I http://dx.doi.org/10.20517/ir.2022.02

, , , should meet the conditions of Lemma 1:
According to the authors of Ref [23] ® ® ®
∫

 ® k ® ( )kd )
k ( ® )( )k ≤ (k (0)k +
 0
  ∫
® (25)
k +1 ( ® )( )k ≤ G (k (0)k + k ® ( )kd )
 ∫ 0

 ®
k ( ® )( )k ≤ (k (0)k + k ® ( )kd )
 0
Let us define the operator , : [0, ] → [0, ] below:
® ®
{
® −1
( ® )( ) = [ + (1 − ) ( ) ( )] [ − ( ) ( )] ® ( )
(26)
® ® ®
( ® )( ) = ( + )( ® )( )
Equation (23) can be rewritten as:
®
®
® +1 ( ) + +1 ( ® +1 ( ))( ) = ( ® ( ))( ) (27)
Since +1 ( ® )( ) can meet Lemma 1, the following operators can be defined:
®
{
® ¯
®
®
+1 ( )( ) = +1 ( ® )( )
(28)
® ¯
®
®
®
+1 ( )( ) = +1 ( )( )
where ® ( ) + +1 ( ® )( ) = ( ), ∀ ( ) ∈ [0, ]. Comparing with Equation (27), the following relationship
®
®
®
can be obtained:
{
® ® ¯ ®
+1 ( ® )( ) = − +1 ( ( ® ))( )
(29)
® ¯ ® ®
+1 ( ( ® ))( ) = +1 ( ® +1 )( )
From Lemma 1, one obtains
∫
® ¯ ® ® ® (30)
k +1 ( ( ® ))( )k ≤ ¯ (k +1 (0)k + k ( ® )( )kd
0
From Equations (24), (26), and (30), the following equation can be derived
∫
®
®
®
k +1 ( ® )( )k ≤ (k +1 (0)k + k (0)k + k ® ( )kd (31)
0
· max( , 1).
where = ¯
Let us define the operator : [0, ] → [0, ] as follows:
®
®
®
®
( ® )( ) = ( + +1 )( ® )( ) (32)
Equation (27) can be expressed accordingly as:
® ® ® ® ® (33)
® +1 ( ) = +1 ( ® ( ))( ) + ( ® ( ))( ) = ( +1 + + )( ® )( )
Taking the norm on both sides of Equation (32) and substituting the inequalities in Equations (25) and (31)
into Equation (32) leads to

®
®
®
k ( ® )( )k ≤ k ( ® )( )k + k +1 ( ® )( )k