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Page 6 of 16 Wang et al. Intell Robot 2023;3(3):479-94 I http://dx.doi.org/10.20517/ir.2023.26

¤ ( ) = ( ) ( ) + 1 ( −1) ( ) + · · · + −1 ¤( ) + ( )
(8)
¤
= ( ) + ( )

where = −1 · · · −1 , = 0 0 · · · , and ( ) = ( ) ¤ ( ) · · · ( −1) ( ) will converge
to the origin when = 0 .
For a nonlinear system with bounded uncertainties (6), the designed sliding mode controller is proposed to
ensure the system is asymptotically stable. It includes two parts: the former is used to realize input/output
linearization, and the latter is used for robust compensation. The control law is designed as (9),

h i

−1
( ) = ( ( )) − ( ( )) + ( ) ( ) + ( ) + sgn( ( )) (9)

where sgn(·) is the sign function. A simple proof of convergence of the controller is as follows. Firstly, taking
the splitting operation of the control law (9),

j k
−1

−1
( ) = ( ( )) − ( ( )) + ( ) ( ) + ( ) + ( ( )) sgn( ( )) (10)
Substitute the control law (11) into the nonlinear system (7); it will be denoted by

( ) ( ) = ( ( )) + ( ( )) ( ) + ( ( ))

= ( ( )) − ( ( )) + ( ) ( ) + ( ) + sgn( ( )) + ( ( )) (11)

= ( ) ( ) + ( ) + sgn( ( )) + ( ( ))

thus, it can get sgn( ( )) + ( ( )) = − ( ) − ( ) = −¤( ).

¤
The Lyapunov method is utilized to verify the stability of the control system. We select the candidate Lyapunov
function as follows:

1
( ( )) = ( ) ( ) (12)
2

By taking the derivative of ( ( )) with respect to , it follows that

¤
( ( )) = ( ) ¤( )

= − ( )[ sgn( ( )) + ( ( ))]

= − k ( )k − ( ) ( ( )) (13)
≤ − k ( )k + k ( )kk ( ( ))k
= −k ( )k( − ( ( )))
According to the assumption that ( ( )) is bounded, where k ( ( ))k ≤ , so that ( ( )) ≤ 0. Therefore,
¤
the SMC law (9) can make the robot system (6) asymptotically stable.