Underwater Robots - Gianluca Antonelli.pdf

184 7. Dynamic Control of UVMSs
Hereafter, asuperscript will denote the frame to which avector is referred
to, the superscript will be dropped forquantities referred to the inertial frame.
Notice that some differences may arise inthe symbology of the vehicle’s
variables due to the different approach followed inthis Section. Coherently
with the virtual decomposition approach, the vehicle is considered aslink
number 0.
The (6 × 1) vector of the total generalized force (i.e., force and moment)
acting on the i th body is given by
h i t,i = h i i − U i i +1
i +1h i +1
, i =0,...,n− 1
h n t,n = h n n , (7.64)
where h i i +1
i is the generalized force exerted by body i − 1onbody i , h i +1
is the
generalized force exerted bybody i +1on body i .The matrix U i 6 × 6
i +1 ∈ IR
is defined as
�
R i �
i +1 O 3 × 3
.
U i i +1 =
S ( r i i,i+1 ) R i i +1
R i i +1
where R i i +1 ∈ IR 3 × 3 is the rotation matrix from frame T i +1 to frame T i ,
S ( · )isthe matrix operator performing the cross product between two (3 × 1)
vectors, and r i i,i+1 is the vector pointing from the origin of T i to the origin
of T i +1.
The equations of motion ofeach rigid body can bewritten inbody-fixed
reference frame in the form [314, 127]:
M i ˙ν i i + C i ( ν i i ) ν i i + D i ( ν i i ) ν i i + g i ( R i )=h i t,i , (7.65)
where ν i i ∈ IR6 is the vector of generalized velocity (i.e., linear and angular
velocities defined in Section 2.8), R i is the rotation matrix expressing the
orientation of T i with respect to the inertial reference frame, M i ∈ IR 6 × 6 ,
C i ( ν i i ) ν i i ∈ IR6 , D i ( ν i i ) ν i i ∈ IR6 and g i ( R i ) ∈ IR 6 are the quantities introduced
in (2.51) referred to the generic rigid body. InChap. 2, the details on
the dynamics of arigid body moving in afluid are given.
According tothe property oflinearity inthe parameters (7.65) can be
rewritten as:
Y ( R i , ν i i , ˙ν i i ) θ i = h i t,i
where θ i is the vector of dynamic parameters of the i th rigid body. Notice
that, for the vehicle, i.e., for the body numbered as 0, the latter is exactly
(2.54); only for this Section, however, the notation of the vehicle forces
and regressor will be slightly different from the rest ofthe book.
The input torque τ q,i at the i th joint ofthe manipulator can be obtained
by projecting h i on the corresponding joint axis via
T
i
τ q,i = z i − 1 h i i , (7.66)
where z i i − 1 = R T i z i − 1 is the z -axis of the frame T i − 1 expressed in the frame
T i .