Underwater Robots - Gianluca Antonelli.pdf

7.4 Nonlinear Control for UVMSs with Composite Dynamics 147
the vehicle/manipulator dynamics are used as abasis for the control design.
This has been developed having in mind aVortex vehicle together with a
PA10, 7-DOF, manipulator for which Tables 7.1 and 7.2 reports some time
response of the subsystems.
Table 7.1. Vortex/PA10’s sensor time response
measured variable time response
surge and sway x , y 400 ms
depth z 1000 ms
yaw ψ 1000 ms
pitch and roll φ , θ 100 ms
joint position q 1ms
Table 7.2. Vortex/PA10’s actuators time response
type time response
vehicle thruster 160 ms
manipulator’s electrical CD motor 0 . 5ms
Let consider astate vector composed by the earth-fixed-frame-based coordinates
of the vehicle and the joint position. Inthe following the dependencies
will be dropped out toincrease readability. It can be demonstrated that its
(6 + n ) × (6 + n )inertia matrix has the form [255]:
� � �
M v + M qq M vq
M T vq
M q
where M � v ∈ IR 6 × 6 is the earth-fixed inertia matrix of the sole vehicle
(see eq. (2.53)), M q ∈ IR n × n is the inertia matrix of the sole manipulator,
M qq ∈ IR 6 × 6 is the contribution of the manipulator onthe inertia matrix
seen from the vehicle, M vq ∈ IR 6 × n is the coupling term between vehicle
and manipulator; all the matrices include the added mass. Its inverse can be
parameterized in:
M − 1 =
� M − 1
11
− M T 12
− M �
12
− 1
M 22
(7.4)
where the block diagonal matrices M 11 and M 22 are ofdimension 6 × 6and
n × n ,respectively, and M 12 ∈ IR 6 × n .