Underwater Robots - Gianluca Antonelli.pdf

3.2 Earth-Fixed-Frame-Based, Model-Based Controller 47
Table 3.1. Labels of the discussed controllers
label Authors Sect. frame
A Fjellstad &Fossen 3.2 earth
B Yuh et al. 3.3 earth
C Fjellstad &Fossen 3.4 vehicle
D Fossen &Balchen 3.5 earth/vehicle
E Antonelli et al. 3.6 earth/vehicle
F Sun &Cheah 3.7 earth/vehicle
3.2 Earth-Fixed-Frame-Based, Model-Based Controller
In 1994, O. Fjellstad and T.Fossen [122] propose an earth-fixed-frame-based,
model-based controller that makes use of the 4-parameter unit quaternion
(Euler parameter) to reach asingularity-free representation ofthe attitude.
The controller is obtained by extending the results obtained in [267] for robot
manipulators.
By defining
˜p = p d − p (3.1)
where p =[η T 1 Q T ] T ∈ IR 7 is the quaternion-based position/attitude vector
of the vehicle and p d =[η T 1 ,d Q T d ] T ∈ IR 7 is its desired value. The
following (7 × 1) vector can be further defined:
s = K D ˙˜p + K P ˜p + K I
� t
that implies the vector ˙p r ∈ IR 7 defined as
˙p r = K D ˙p d + K P ˜p + K I
0
˜p ( τ ) dτ = ˙p r − K D ˙p (3.2)
� t
0
˜p ( τ ) dτ (3.3)
where K D , K P and K I are (7 × 7) positive definite matrices of gains.
The following control law isproposed
τ � v = M � v ¨p r + C � v ˙p r + D � RB ˙p r + g � RB
+ Λs , (3.4)
where Λ is a(7 × 7) positive definite matrix of gains. Notice that the above
control law refers toaquaternion-based dynamic model in earth-fixed coordinates
whichcan be obtained from(2.53) by using the matrix J k,oq ∈ IR 4 × 3 instead
of the matrix J k,o ∈ IR 3 × 3 in the construction of the Jacobian J e ( R I B ).
Also notice that, with respect to the model detailed in Section 2.7 the dimension
of J e ( R I B )are different.