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Li et al. Intell Robot 2022;2(1):89–104 I http://dx.doi.org/10.20517/ir.2022.02 Page 95

−1
®
®
®
®
where ( , ) 8×1 = [ ® ¤ ( , )] T , ( ® , ® ¤ ) 8×4 = [0, ( ® )] T , and = [0, ] T .
8×1 8×4 8×4
For the nonlinear system of Equation (17), based on the ILC law in Equation (8), if the system can meet the
following condition,
−1
([ + ( ) ( )] [ − ( ) ( )]) < 1, ∈ [0, ] (18)
the trajectory tracking error of the dynamic system converges to a certain small range with the increasing
iterations.

Let the system state, output, and input errors be set as:


 ® ® ®
 ( ) = ( ) − ( )


®
®
®
( ) = ( ) − ( ) (19)



 ® ( ) = ® ( ) − ® ( )

Definingthevariable 1 ( ( ), ) = ( ( ), )− ( ( )− ( ), ), thefollowinginequalitiescanbeobtained
®
®
®
® ®
® ®
by Lipschitz condition:
{
® ®
k 1 ( ( ), )k ≤ 1
(20)
®
®
®
®
®
®
k 1 ( 1 ( ), ) − 1 ( 2 ( ), )k ≤ 2 k 1 ( ) − 2 ( )k
Combining Equations (17) and (19) results in
 ® ¤
 ® ®
 ( ) = 1 ( ( ), ) + ( ) ® ( )


® ® (21)
( ) = ( ) ( )


 ® ¤ ¤ ® ® ¤
 ( ) = ( ) ( ) + ( ) ( )

with
® ® ® ¤
® +1 ( ) = ® ( ) − ( ) − +1 ( ) − ( ) ( )
® ¤ ® ® ¤ (22)
− ( ) +1 ( ) + +1 ( ) + ( ) +1 ( )
Substituting Equation (21) into Equation (22) yields
®
®
® +1 ( ) = ® ( ) − ( ) ( ) − (1 − ) ( ) +1 ( )
®
®
¤
®
− ( ( ) ( ) + ( ) 1 ( , ( )) + ( ) ( ) ® ( ))
¤
®
®
®
− (1 − ) ( ( ) +1 ( ) + ( ) 1 ( , +1 ( )) + ( ) ( ) ® +1 ( )) (23)
Let us define the operator , , : [0, ] → [0, ] as follows:
®
®
®

 ® ® ¤ ® ® ®
 ( ® )( ) = ( ) ( ) + ( ) ( ) + ( ) 1 ( ( ), )

 
® −1 ® (24)
+1 ( ® )( ) = [ + (1 − ) ( ) ( )] (1 − ) +1 ( ® )( )


 ® −1 ®
 ( ® )( ) = −[ + (1 − ) ( ) ( )] ( ® )( )
