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Page 211 Zhu et al. Intell Robot 2022;2(3):200­222 I http://dx.doi.org/10.20517/ir.2022.13

The transformation matrix J(p) is
[ ]
J 1 O 3×3
J(p) = , (4)
O 3×3 J 2
where J 1 and J 2 are
cos cos cos sin sin − sin cos cos sin cos + sin sin 
 
J 1 = sin sin sin sin sin + cos cos sin sin cos + cos sin , (5)


 
 − sin cos sin cos cos 
 
1 tan sin cos tan 
 
J 2 = 0 cos − sin  . (6)

 
 0 sin /cos cos /cos 
 
Among the six DOFs of the underwater vehicle, surge, sway, heave, roll, pitch, and yaw, roll and pitch can be
neglectedsincethesetwoDOFsbarelyhaveaninfluenceontheunderwatervehicleduringpracticalnavigation.
Therefore, whenestablishingthetrajectorytrackingmodeltokeepacontrollableoperationoftheUUV,usually
only four DOFs, namely surge, sway, heave, and yaw, are involved (see the DOFs shown in Figure 6). Hence,
for the kinematic equation, the position vector can be simplified as

¤
   cos − sin 0 0   cos − sin 0 0   
       
   sin cos   sin cos   
 ¤  0 0 0 0   (7)
   ,
p =   = J(p)v =   v = 
 ¤   0 0 1 0  0 0 1 0  
       
 ¤   0 0 0 1   0 0 0 1   
       

where J is a transformation matrix derived from the physical structure of the UUV body, while [ ]
represents the velocities at the chosen four axes of a UUV, as presented in Figure 6.
For UUVs, a generally accepted dynamic model has be defined as
Mv + C(v)v + D(v)v + g(p) = , (8)
where M is the inertia matrix of the summation of rigid body and added mass; C(v) is the Coriolis and cen-
tripetal matrix of the summation of rigid body and added mass; D(v) is the quadratic and linear drag matrix;
g(p) is the matrix of gravity and buoyancy; and is the torque vector of the thruster inputs.

The torque vector of the thruster input is represented by
[ ]
= τ τ τ τ τ τ , (9)
where τ , τ , τ , τ , τ , and τ represent torques of the UUV in the surge, sway, heave, pitch, roll, and yaw
directions, as shown in Figure 6. In addition, as mentioned in the previous section, in some practical cases,
the torques in pitch and roll directions can be neglected.

Due to the nonlinearity of the UUV system, the typical linearization method, i.e., proportion–integration–
differentiation (PID) control, does not work very well and is less studied for UUV trajectory tracking [83] .
Hence, in this section, the major methods that are used for UUV trajectory tracking are discussed and catego-
rizedintoconventionalcontrol(consistingofbacksteppingcontrol,slidingmodecontrol,andmodelpredictive
control), intelligent control, and fault-tolerant control.

3.1. Conventional control
In this section, some conventional control methods such as backstepping control, sliding mode control, and
model predictive control are described. Studies regarding their applications in the trajectory tracking control
of the UUV are stated. The summary of the features for these conventional controls is given in Table 4.