Underwater Robots - Gianluca Antonelli.pdf

40 2. Modelling of Underwater Robots
The total forces and moments acting on the generic body of the serial
chain are given by:
f i i = R i i +1
i +1f i +1 + F i i − m i g i + ρ ∇ i g i + p i
µ i i = R i i +1
i +1µ i +1 + R i i +1
i +1r i − 1 ,i × R i i +1
i +1f i +1 + r i i − 1 ,c × F i i + T i i +
O i − 1
f i , µ i
+ r i i − 1 ,c × ( − m i g i + d i )+r i i − 1 ,b × ρ ∇ i g i
r i − 1 ,B
r i − 1 ,C
d i
− ρ ∇ i g
B i
r i − 1 ,i
Fig. 2.4. Force/moment acting on link i
C i
m i g
The torque acting on joint i is finally given by:
τ q,i = µ i T i
i z i − 1 + f disign(˙q i )+f vi ˙q i
r i,C
O i
f i +1 , µ i +1
(2.69)
with f di and f vi the motor dry and viscous friction coefficients.
Let us define τ q =[τ q,1 ... τ q,n ] T ∈ IR n and τ ∈ IR
the vector of joint torques
6+n
� �
τ v
τ =
(2.70)
τ q
the vector of force/moment acting on the vehicle as well as joint torques. It
is possible to write the equations of motions of an UVMS in amatrix form:
M ( q ) ˙ ζ + C ( q , ζ ) ζ + D ( q , ζ ) ζ + g ( q , R I B )=τ (2.71)
where M ∈ IR (6+n ) × (6+n ) is the inertia matrix including added mass
terms, C ( q , ζ ) ζ ∈ IR 6+n is the vector of Coriolis and centripetal terms,
D ( q , ζ ) ζ ∈ IR 6+n is the vector of dissipative effects, g ( q , R B I ) ∈ IR 6+n
is the vector of gravity and buoyancy effects. The relationship between the
generalized forces τ and the control input is given by: