Underwater Robots - Gianluca Antonelli.pdf

38 2. Modelling of Underwater Robots
Knowing ν 1 , ν 2 , ˙ν 1 , ˙ν 2 ,(vehicle linear and angular velocities and acceleration
in body fixed frame), ˙q , ¨q ,(joint velocities and acceleration) it is
possible to calculate, for every link, the following variables:
ω i i ,angular velocity ofthe frame i ,
˙ω i i ,angular acceleration ofthe frame i ,
v i i ,linear velocity ofthe origin ofthe frame i ,
v i ic ,linear velocity ofthe center of mass of link i ,
a i i ,linear acceleration ofthe origin offrame i ,
by resorting to the following relationships:
ω i i = R i � i − 1
i − 1 ω i − 1 +˙q �
i z i − 1
˙ω
(2.59)
i i = R i � i − 1 i − 1
i − 1 ˙ω i − 1 + ω i − 1 × ˙q �
i z i − 1 +¨q i z i − 1
(2.60)
v i i = R i i − 1
i − 1 v i − 1 + ω i i × r i i − 1 ,i
(2.61)
v i ic = R i i − 1
i − 1 v i − 1 + ω i i × r i i − 1 ,c
(2.62)
a i i = R i i − 1
i − 1 a i − 1 + ˙ω i i × r i i − 1 ,i + ω i i × ( ω i i × r i i − 1 ,i) (2.63)
where z i is the versor offrame i , r i i − 1 ,i
of frame i − 1toward the origin offrame i expressed in frame i .
Since the task of UVMS missions is usually force/position control of the
end effector frame, it is necessary to consider the position of the end effector
in the inertial frame, η ee1 ∈ IR 3 ;this is afunction ofthe system configuration,
i.e., η ee1 ( η 1 , R B I , q ). The vector ˙η ee1 ∈ IR 3 is the corresponding time
derivative.
Let us further define η ee2 ∈ IR 3 as the orientation of the end effector in
the inertial frame expressed by Euler angles: also η ee2 is afunction ofthe
system configuration, i.e., η ee2 ( R B I , q ). Again, the vector ˙η ee2 ∈ IR 3 is the
is the constant vector from the origin
corresponding time derivative.
The relation between the end-effector posture η ee =[η T ee1 η T ee2 ] T and
the system configuration can be expressed by the following nonlinear equation:
η ee = k ( η , q ) . (2.64)
The vectors ˙η ee1 and ˙η ee2 are related to the body-fixed velocities ν ee via
relations analogous to(2.1) and (2.2), i.e.,
ν ee1 = R n I ˙η ee1
ν ee2 = J k,o ( η ee2 ) ˙η ee2
(2.65)
(2.66)
where R n I is the rotation matrix from the inertial frame to the end-effector
frame (i.e., frame n )and J k,o is the matrix defined as in (2.3) with the use