Underwater Robots - Gianluca Antonelli.pdf

168 7. Dynamic Control of UVMSs
Table 7.3 shows the computational load of each control law, in terms of
required floating point operations, in the case of a6-DOF vehicle equipped
with a3-DOF manipulator. Where required, inversion of the inertia matrix
has been obtained via the Cholesky factorization since M ( q )issymmetric
and positive definite; of course, the inverse of the constant matrix � M is computed
once off-line. As shown by the results in Table 7.3, the computational
load is reduced by about80% when thecontrol law(7.43),(7.44) is considered.
Table 7.3. Computational burden for different output feedback controllers
mult/div add/sub
Control law (7.38),(7.40) 1831 1220
Control law (7.41),(7.42) 1216 849
Control law (7.43),(7.44) 354 147
7.8.1 Stability Analysis
In order to derive the closed-loop dynamic equations, it is useful to define
the variables
where
σ d = ζ r − ζ = � ζ d + Λ d e de
(7.45)
σ e = ζ o − ζ = � ζ e + Λ e e e , (7.46)
�ζ d = ζ d − ζ (7.47)
�ζ e = ζ e − ζ . (7.48)
Combining (2.71) with the control law (7.38), (7.39), and using the equality
a r = ˙ ζ r + S PO( �ν 2 ,d) ζ d + Λ d S P ( �ν 2 ,d) e de ,
the tracking error dynamics can be derived
M ( q ) ˙σ d + C ( q , ζ ) σ d + K v σ d − K p e d = K v σ e − C ( q , σ e ) ζ r +
− M ( q ) S PO( �ν 2 ,d) ζ d − M ( q ) Λ d S P ( �ν 2 ,d) e de +
D ( q , ζ ) ζ − 1
2 D ( q , ζ r )(ζ r + ζ o ) . (7.49)
The observer equation (7.40), together with (7.49), yields the estimation
error dynamics
M ( q ) ˙σ e +
�
L v A ( � �
Q e ) − K v σ e − L p e e = − K v σ d − C ( q , ζ ) σ e +