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Page 4 of 14 Ma et al. Complex Eng Syst 2023;3:10 I http://dx.doi.org/10.20517/ces.2023.14

Figure 1. Physical model of a WMR. WMR: wheeled mobile robot.

Assumption2: Suppose the 1, 2, andtheir derivatives exist with bounds, which is given by | 1 | ≤ 1 , | 2 | ≤
2 , where 1 and 2 are all positive constants.

Then, the tracking errors of the WMR are expressed as

 cos sin
  0   − 
     
  =  − sin cos 0    . (5)
     − 
   0 0 1   
     − 
Furthermore, the error dynamics system could be transformed in the form of

 ¤ = − + cos




 ¤ = sin −


 
¤ (6)
= −



 ¤ = 1 + 1




 ¤ = 2 + 2

To simplify the whole design process, the system (6) can be divided into two subsystems, which contain a
second-order subsystem:
(
¤
= −
(7)
¤ = 1 + 1
and a third-order subsystem:
 ¤ = − + cos


 
¤ = sin − (8)


 ¤ = 2 + 2

3. FIXED-TIME TRAJECTORY CONTROL
In this section, a fixed-time sliding mode control scheme is developed to realize the fast and high-accuracy
trajectory tracking control of a WMR under external disturbances. Firstly, a fixed-time disturbance observer