Unicycle-like Robots with Eye-in-Hand Monocular Cameras: From PBVS towards IBVS - Visual Servoing via Advanced Numerical Methods - page 332

in the Lypunov derivative one gets V =

2 cos 2 2

0. By using LaSalle's
invariant set theorem, the controlled dynamics obtained plugging (18.7) in (18.6)
turns to be asymptotically stable.

− ρ β − λβ ≤

18.3.3 Visual Servoing with FOV Constraint

The basic idea to be applied in this section is rather simple, and is based on the fact
that the Lyapunov-based control described in the previous section is not uniquely
defined. Rather, a whole family of controllers can be defined by simply redefining
the control Lyapunov function candidate. It can be expected that for such different
candidates, the resulting stabilizing control laws and ensuing trajectories are differ-
ent, and that switching among these control laws should be enabled when the FOV
constraint for the i -th feature

w z i I x i

α x ∈ [ − Δ , Δ]

− ρ

sin φ

γ(ρ , φ , β)=φ − π − β − arctan = arctan (18.8)

w x i − ρ

cos φ

is about to be violated (see [26] for further details). Notice that the limited FOV is
described by a symmetric cone centered in the optical axis Z c with semi-aperture .

The switching controller is expressed in a set of different polar coordinates, which
are conveniently denoted by introducing the two vectors

Δ

˜

β =[ β , β − π , β − π , β +

] and ˜ π , β ]. Correspondingly, a set of five distinct

candidate Lyapunov functions can be written as V i (

+ π φ =[ φ − π , φ − 2 π , φ , φ − 2 π , φ

i + ˜

)= 2 (

2 + ˜ i ), with

ρ , α , β ρ φ β

i = 1 ,..., 5. The control law choice, i.e.

β i + sin

β cos β

˜

φ i + ˜ ( ˜ u = cos β , and ω = λ β i ) ,

˜

β i

is such that all the Lyapunov candidates have negative semi-definite time derivatives
and (by LaSalle's invariant set principle) asymptotically stable. These five different
control laws (parameterized by

) define in turn five different controlled dynamics
(analogous to (18.6)) that are globally asymptotically stable in the state manifold
R

λ

S 2 . Although none of these control laws alone can guarantee that the FOV
constraint is satisfied throughout the parking maneuver, it is shown that a suitable
switching logic among the control laws achieves this goal. The switching law is
triggered when, during the stabilization with one of the five control laws, a fea-
ture approaches the border of the FOV by a threshold

+

×

Δ j < Δ , i.e. when | γ |≥ Δ j .

It should be noticed that a dead zone is introduced in the controller for ρ ≤ ρ D ,

within which the forward velocity control u is set to zero and the so-called Zeno
phenomenon is avoided.

18.3.4 Visual Servoing in the Large

The classic approach for visual servoing, however, so far have focussed on local
stabilization, in the sense that the initial and desired conditions of the system are