Unicycle-like Robots with Eye-in-Hand Monocular Cameras: From PBVS towards IBVS - Visual Servoing via Advanced Numerical Methods - page 329

For IBVS the image error between two different postures is computed directly
on the image measurements. Hence, an IBVS controller relates to Definition 18.2,
which implies the satisfaction of Definition 18.1 if pathological image feature pos-
tures (singularities) are avoided.

18.2.3 Optimal Trajectories

Path planning solutions to vision-based parking have been proposed in literature
([23]). Among all the possible choices for robot trajectories, a particular mention
deserves those one that are optimal, e.g. , minimum length. Moreover, optimal path
planning turns to be particularly challenging when the nonholonomicity of the plat-
form combines with the FOV constraint. A solution to this problem has been pro-
vided very recently by [2], where shortest paths are shown to be comprised of three
maneuvers: a rotation on the spot, a straight line and a logarithmic spiral. The re-
sults there proposed are additionally refined in [30], where a global partition of the
motion plane in terms of optimal trajectories has been derived.
As the optimal trajectories are retrieved, a visual servoing scheme can be used
to control the robot along the optimal trajectories. For example, [20] proposes a
PBVS homography-based controller. Alternatively, IBVS approaches can be used,
as reported in what follows.

18.3 PBVSintheLarge

As described previously, the PBVS comprises two major components that are de-
scribed briefly in what follows: a localization algorithm and the controller design.
The quantities referred in this section are reported in Figure 18.1(a), where the fixed
frame W and the camera frame C are reported.

18.3.1 Robot Localization

ξ 1 , ξ 2 , ξ 3 ] T be the set of Cartesian coordinates of the robot (see Figure
18.1(a)). The current position of the feature c P i in the camera frame is related to w P i
in the fixed frame by a rigid-body motion c H w , which can be computed assuming
the knowledge of the height of the features

w

Let ξ =[

c y i ,

∀ i . Indeed, with respect to Figure

18.1(b), we have

c x i
c z i

= b

w z i w x i 10

, (18.5)

w x i w z i 01

−

ξ 3 ] T . Equation 18.5
can be regarded as providing two nonlinear scalar equations in the 3 unknowns
(

with b =[ − cos ξ 3 , sin ξ 3 , ξ 2 cos ξ 3 − ξ 1 sin ξ 3 ,− ξ 1 cos ξ 3 − ξ 2 sin

ξ 1 , ξ 2 , ξ 3 ), for each feature observed in the current and desired images. Assuming
a number n ( t ) of

the unicycle can be evaluated by solving for b in a least-squares sense (see [26] for
further details).

≥ 4 of features, the actual unknown position and orientation ξ