Underwater Robots - Gianluca Antonelli.pdf

8.4 External Force Control 207
be fulfilled along with the primary task as long as they do not conflict. For
example, we can ask the system not to change the vehicle orientation, or not
to use the vehicle at all as long asthe manipulator is working in adexterous
posture.
Let us define x p as the primary task vector and x s as the secondary task
vector. The frame in which they are defined depends onthe variables weare
interested in: if the primary task is the position of the end-effector it should
be normally defined in the inertial frame.
The task priority inverse kinematics algorithm is based onthe following
update law [78]
ζ d = J # p [ ˙x p,d + Λ p ( x p,d − x p )] +
+(I − J # p J p ) J # s [ ˙x s,d + Λ s ( x s,d − x s )] , (8.7)
where J p and J s are the configuration-dependent primary and secondary
task Jacobians respectively, the symbol # denotes any kind of matrix inversion
(e.g. Moore-Penrose), and Λ p and Λ s are positive definite matrices. It
must be noted that x p , x s and the corresponding Jacobians J p , J s are functions
of η d and q d .The vector ζ d is defined in the body-fixed frame and a
suitable integration must be applied to obtain the desired positions: η d , q d
(see (2.58)).
Let us assume that the primary task is the end-effector position that
implies that J p = J pos. Tointroduce aforce control action (8.7) is modified
as follow: the vector ˙x c is given as areference value to the IK algorithm and
the reference system velocities are computed as:
where
ζ d = J # p [ ˙x p,d + ˙x c + Λ p ( x p,d − x p )] +
+(I − J # p J p ) J # s [ ˙x s,d + Λ s ( x s,d − x s )] , (8.8)
˙x c = k f,p ˜ f e − k f,v ˙ f e + k f,i
� t
0
˜f e ( σ ) dσ (8.9)
being ˜ f e = f e,d − f e the force error. The direction in which force control is
expected are included in the primary task vector, e.g., if the task requires to
exert aforce along z ,the primary task includes the z component of x .
8.4.2 Stability Analysis
Pre-multiplying (8.8) by J p ∈ IR 3 × (6+n ) we obtain:
J p ζ d = ˙x p,d + ˙x c + Λ p ( x p,d − x p ) . (8.10)
Let us consider aregulation problem, i.e., the reference force f e,d and
the desired primary task x p,d are constant. In addition, let us assume that
f e,d ∈R( K ), where K is the stiffness matrix defined in (2.77), and that, as