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Page 2 of 14 Ma et al. Complex Eng Syst 2023;3:10 I http://dx.doi.org/10.20517/ces.2023.14

[2]
tracking control is a significant field in motion control, which has been studied extensively in recent years .
In practical engineering applications, a WMR is a highly coupled system with nonholonomic constraints and
external disturbances. Hence, it is significant to design an anti-interference trajectory tracking control scheme
with superior performance. At present, the design of the tracking controller of a WMR is mainly based on two
[3]
types: one is to consider only the kinematic model , while the other is to design on the basis of kinematic
[4]
and dynamic models . The kinematic model-based control only considers the linear velocity and angular
velocity as the control inputs. Compared with the kinematic model, the introduction of dynamic models can
[5]
solve the external disturbance problem and the crucial nonholonomic constraint problem .
[6]
In , the system with nonhonolomic constraints was transformed into an extended chain system by coordi-
nate transformation. On this basis, some scholars have designed the trajectory tracking control schemes by
[7]
transforming the kinematic model of a WMR into a chain structure . In practice, there is a problem called
“excellent velocity tracking” [8] when designing a trajectory tracking controller only based on a kinematic sys-
tem. Thus, it is more reasonable to take the force or torque as inputs of the control system instead of the
speed. Meanwhile, external disturbances can be further taken into account. Nevertheless, the design process
of the controller that simultaneously incorporates both the kinematic and dynamic models is complicated.
The work of Zhai and Song [9] transformed the dynamic error system into second-order and third-order sub-
systems. And an intermediate variable related to the position error is introduced to tackle the problem of
constructing a control method for a third-order system using the terminal sliding mode control. However, the
aforementioned control schemes can only achieve finite time stability. It is noteworthy that the upper limit of
the convergence time is unknown and dependent on the initial states of the control system. To overcome this
problem, fixed-time stable control methods are proposed [10] . In reference [11] , a new integral sliding mode-
based control (ISMC) scheme was developed and applied on the dynamic model of the WMR to enable the
WMR to track the desired trajectory in a fixed time. However, there exists the singularity problem, making the
WMR unable to track the arbitrary trajectories and limiting its practical application when the desired angular
velocity is zero.

In the practical motion environment, there are external disturbances and uncertainties that can deteriorate the
performance of the control system. To cope with the problem, an observer-based control scheme is an efficient
method with disturbance-rejection performance [12] . The traditional observers can only achieve asymptotic
stability of the observation errors, whereas the finite time disturbance observers were designed to improve
the performance of the observer [13] . On this basis, the fault-tolerant attitude control problem of spacecraft
under external disturbances was solved by the introduction of a continuous finite-time observer [14] , which
also restrains the chattering phenomenon. Zhang et al. put forward a novel continuous practical fixed-time
disturbance observer and applied it on a WMR, which can not only avoid the chattering problem but also
improvetheabilitytoattenuatedisturbance [15] . DifferentfromtheworkofZhang, theGaussianerrorfunction,
which is sometimes called probability integral [16] , can also be used to develop a control scheme that improves
the chattering problem [17] .

Motivated by the above discussions, an integral sliding mode-based fixed-time trajectory tracking control
scheme is proposed by combining the kinematic model with the dynamic model of a WMR in this paper. (1)
A continuous fixed-time disturbance observer using the Gaussian error function is proposed, which avoids
the chattering problem and estimates the external disturbance of a WMR accurately. (2) An auxiliary variable
incorporating variable exponential coefficients is introduced to simplify the design process of the controller
for the third-order subsystem and avoid the singularity problem simultaneously. (3) The reliability and effec-
tiveness of the designed control scheme are verified by a comparative experiment conducted on a wheeled
mobile experimental platform.

2. PRELIMINARIES AND PROBLEM STATEMENT