Underwater Robots - Gianluca Antonelli.pdf

7.9 Virtual Decomposition Based Control 185
The input force and moment acting onvehicle are instead given by the
vector h 0 t,0 .Notice that this vector was introduced inSection 2.6 with the
symbol τ v ,inthis Section, however, it was preferred to modify the notation
consistently with the serial chain formulation adopted here.
Let p d,o( t ), Q d,0 ( t ), q d ( t ), ν 0 d,0 ( t ), ˙q d ( t ), ˙ν 0 d,0 ( t ), ¨q d ( t )represent the desi-
red trajectory. Let define
ν 0 r,0 = ν 0 �
λ p,0 I 3
d,0 +
O 3
�
O 3
e 0 ,
λ o,0 I 3
(7.67)
˙q r,i =˙q d,i + λ i ˜q i i =1,...,n (7.68)
T
i +1 i
ν r,i +1 = U
where the (6 × 1) vector
� � T
R 0 ˜p 0
e 0 =
˜ε 0 0
i +1ν i i +1
r,i +˙q r,i +1z
i i =0,...,n− 1 , (7.69)
denotes position and orientation errors for the vehicle, λ p,0 ,λo,0 ,λi are positive
design gains.
It is useful considering the following variables:
s i i = ν i r,i − ν i i
s q,i =˙q r,i − ˙q i
s q =[sq,1 ... s q,n ] T
i =0,...,n
i =1,...,n
The discussed control law isbased onthe computation of the required
generalizedforce foreachrigid body in the system.Then, the input torquesfor
the manipulator and the input generalized force for the vehicle are computed
from the required forces according to(7.64) and (7.66).
In the following it is assumed that only anominalestimate ˆ θ i of thevector
of dynamic parameters is available for the i th rigid body. Hence, asuitable
update law for the estimates has to be adopted soastoensure asymptotic
tracking of the desired trajectory.
Forthe generic rigid body (including the vehicle) the required force has
the following structure
h i r,i = h i c,i − U i i +1
i +1h c,i+1
that, including the designed required force, implies
h i c,i = Y � R i , ν i i , ν i r,i , ˙ν i �
ˆθ
r,i i + K v,i s i i + U i i +1
i +1h c,i+1
(7.70)
with K v,i > O .The parameters estimate ˆ θ i is dynamically updated via
˙ˆθ
− 1
i = K θ,i Y T � R i , ν i i , ν i r,i , ˙ν i � i
r,i s i
(7.71)
with K θ,i > O .
The control torque at the i th manipulator’s joint isgiven by
τ q,i = z i T i
i − 1 h c,i . (7.72)