Underwater Robots - Gianluca Antonelli.pdf

7.9 Virtual Decomposition Based Control 187
that is positive definite in view of positive definitiveness of V i ,i=0,...,n.
Its time derivative issimply given by the time derivatives of all the scalar
functions:
˙V =
n�
i =0
�
− s i T
i ( K v,i + D i ) s i i + s i �
T i
˜h
i t,i ,
where the last term is null. In fact, let us consider the two equations:
h i t,i = h i i − U i i +1
i +1h i +1 ,
T
i +1 i
ν i +1 = U
i +1ν i i +1
i +˙q i +1z i
it is possible to observe that the same relationships hold for ˜ h i
t,i and s i i :
˜h i
t,i = ˜ h i
i − U i i +1 ˜ i +1
h i +1 ,
T
i +1 i
s i +1 = U
i +1s i i +1
i + s q,i +1z i
.
where ˜ h i
i = h i c,i − h i i .Recalling that τ = J T w h e ,the last term of the time
derivative ofthe Lyapunov function candidate is then given by:
n�
n�
i =0
s i T i
˜h
i t,i = s 0 T 0
˜h
0 0 +
= s 0 T 0
˜h
0 0 +
= s 0 T 0
˜h
0
i =1
s q,i z i T i
˜h
i − 1 i
n�
s q,i ˜τ q,i
i =1
0 + s T q ˜τ q
� �
0 T
s 0 = J
s q
T w ˜ h e
=0 (7.75)
since, in absence of contact at the end effector, ˜ h e = 0 .Tounderstand the
first equality let rewrite the first term for two consecutive links:
s i T i
˜h
i
t,i + s
i +1T
i +1
˜h
i +1
t,i+1 =(U
i − 1 T i − 1
i s i − 1 + s q,i z i i − 1 ) T i
h ˜
i − s i T i i +1
i U ˜
i +1h i +1 +
( U i T i i +1
i +1 s i + s q,i +1z i ) T i +1
h ˜ i +1T
i +1
i +1 − s i +1 U i +2 ˜ i +2
h i +2
=(U
i − 1 T i − 1
i s i − 1 + s q,i z i i − 1 ) T i
h ˜
i +
( s q,i +1z
i +1
i ) T i +1
h ˜ i +1T
i +1
i +1 − s i +1 U i +2 ˜ i +2
h i +2,
that, considering that the first and the last term are null for the first and the
last rigid body respectively, gives the relation required in (7.75).
It is now possible to prove the system stability inaLyapunov-Like sense
using the Barbălat’s Lemma. Since
• V ( s 0 0 ,...,s n n , ˜ θ 0 ,..., ˜ θ n )islower bounded;