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Zhu et al. Intell Robot 2022;2(3):200­222 I http://dx.doi.org/10.20517/ir.2022.13 Page 212

Figure 6. Degrees of freedom and corresponding axes for a UUV.

3.1.1. Backstepping control
In the backstepping method, control functions for each subsystem are designed based on the Lyapunov tech-
niques and generated to form the complete control law [84] . However, the actuator saturation is induced by the
speed-jump problem, which usually occurs in the backstepping control methods for trajectory tracking [85] .
The excessive speed references affect the robustness of the UUV trajectory tracking by introducing excessive
fluctuations of velocities at initial states or other large error states during the kinematic controlling procedure.
Therefore, asharpspeedchangeisderivedfromthelargeerrorsaccumulatedfromthegenerationofthesubsys-
tems, where speed-jump issues are induced when the deviation occurs. As the UUV cannot provide infinite
driving inputs such as torques/forces due to its underwater workspace and limited electric power, actuator
saturation, has to be considered during the trajectory tracking process of the vehicle, with the torques/forces
constraints applied [86–88] .

3.1.2. Sliding mode control
As one of the most basic robust controlling strategies, sliding mode control (SMC) is widely used due to its
simpleandrobustmechanism; hence, SMCisoftenchosentoconstructthetrajectorytrackingcontrollerofthe
vehicle [89,90] . InSMC,aslidingsurfacemodeissupposedtofollowthedesiredtrackingandkeepthecontrolled
outputs remaining on the surface. Once the controlled trajectory is out of the perfect sliding surface mode,
SMC will push the trajectory slide back to the surface with addition or subtraction on the original controlling
equation [91,92] . Therefore, SMC restricts the fluctuation of controlled outputs in an acceptable range through
a simple operation, which is highly applicable in trajectory tracking problems [93] .

However, SMC suffers chattering issues, although it is robust to variable changes, which is a critical factor that
needs to be considered when designing the control strategy [94] . Xu refined SMC with a bio-inspired neural
network algorithm such that the chattering problem can be alleviated, but it is limited to the application of
land vehicles where fewer degrees of freedom are involved [95] .

3.1.3. Model predictive control
Model predictive control (MPC) is appropriate for the UUV system that navigates in the mode of slow velocity.
MPC is not demanding on the model accuracy and provides in-time feedback, and constraints can be added
to the control strategy to alleviate the jumps of the speed. Therefore, motivated by the requirements of in-time
reaction and restriction of velocities within physical constraints throughout the whole tracking process, MPC
controlstandsouttobeoneofthemostfeasiblesolutionsforconstructingthetrackingcontrolfortheUUV [96] .

The MPC resolves the online optimization problem at each timeslot and derives in-time predictions with min-