Underwater Robots - Gianluca Antonelli.pdf

34 2. Modelling of Underwater Robots
As an example, ODIN has the following TCM:
⎡
∗ ∗ ∗ ∗ 0 0 0
⎤
0
⎢ ∗
⎢ 0
B v = ⎢ 0
⎣
0
∗
0
0
0
∗
0
0
0
∗
0
0
0
0
∗
∗
∗
0
∗
∗
∗
0
∗
∗
∗
0 ⎥
∗ ⎥
∗⎥
⎦
∗
∗ ∗ ∗ ∗ 0 0 0 0
(2.49)
where ∗ means anon-zero constant factor depending on the thruster allocation.
Different TCM can be observed as in, e.g., the vehicle Phantom S3
manufactured by Deep Ocean Engineering that has 4thrusters:
⎡
∗ ∗ 0
⎤
0
⎢ 0
⎢ 0
B v = ⎢ 0
⎣
∗
0
0
0
∗
∗
∗
∗
0
∗ ⎥
∗ ⎥
∗⎥
⎦
0
(2.50)
∗ ∗ 0 0
in which itcan be recognized that not all the directions are independently
actuated.
On the other hand, ifthe vehicle is controlled by thrusters, each ofwhich
is locally fed back, the effects of the nonlinearities discussed above is very
limited and alinear input-output relation between desired force/moment and
thruster’s torque is experienced. This isthe case, e.g., of ODIN [35, 83, 215,
216] where the experimental results show that the linear approximation is
reliable.
2.7 Underwater Vehicles’ Dynamics in Matrix Form
By taking into account the inertial generalized forces, the hydrodynamic effects,
the gravity and buoyancy contribution and the thrusters’ presence, it is
possible to write the equations of motion ofanunderwater vehicle in matrix
form:
M v ˙ν + C v ( ν ) ν + D RB( ν ) ν + g RB( R I B )=B v u v , (2.51)
where M v = M RB + M A and C v = C RB + C A include also the added mass
terms. Taking into account the current, apossible, approximated, model is
given by:
M v ˙ν + C v ( ν ) ν + D RB( ν ) ν + g RB( R I B )=τ v − τ v,C . (2.52)
The following properties hold:
• the inertia matrix is symmetric and positive definite, i.e., M v = M T v > O ;
• the damping matrix is positive definite, i.e., D RB( ν ) > O ;