Unicycle-like Robots with Eye-in-Hand Monocular Cameras: From PBVS towards IBVS - Visual Servoing via Advanced Numerical Methods - page 338

the literature ( e.g. [4, 5]) that large image errors (hence, large robot movements)
decrease accuracy and robustness of the visual servo controller. In the proposed
architecture, the granularity of the topological map is then related to the visual ser-
voing accuracy.

18.4 Optimal Trajectories

This section presents a solution to the path planning problem of a unicycle-like
vehicle subject to limited FOV constraint. In particular, the paths derived are the
minimum length (optimal) paths between initial and desired positions.
Assume that the tracked feature position is coincident with the origin of the ref-
erence frame O w and that the total horizontal aperture of the camera's optical cone
is given by δ = 2 Δ . The desired position P of the robot on the plane of motion w.r.t.

W is assumed to lie on the X w axis, with polar coordinates ( ρ , 0). Hence, the de- P

sired state of the robot is ( ) with respect to the model (18.6). The objective

of the optimal trajectory problem with FOV constraint is the partition of the motion
plane into regions, depending on the desired position P . From all initial positions Q
inside a region, the structure of the shortest paths turns to be invariant.
Denote with “

ρ , 0 , π P

” a zero-length rotation on the spot, with SL a straight line and
with T a logarithmic spiral. The three different kinds of maneuvers that compose the
optimal paths, according to [2, 30], are then symbolically described. While the math-
ematic description of

∗

and SL is straightforward, logarithmic spirals deserve some
additional explanations. In general, the equation of the logarithmic spiral that passes
through a point Q (whose polar coordinates are (

∗

ρ Q , ψ Q ) w.r.t. W ) is expressed as

T Q :

,where g = cosα
sin

Q e (ψ Q − ψ) g

, ψ

ρ and α is the spiral's characteristic angle.

α

The spiral rotates clockwise if α < 0 (denoted with T 1 Q ) and counterclockwise if

α > 0 (denoted with T 2 Q ). Notice that if α = 0, the logarithmic spiral is a straight

line passing through the origin O W , and, if 2, it is a circumference.

Using standard tools from optimal control theory, it can be shown that optimal
words that minimizes the total length of the path

α = ± π /

= T L

0 | ν |

dt ,

are words of extremal arcs that can be covered forward or backward w.r.t. the direc-
tion of motion. Summarizing, extremal paths will be characterized by sequences of
symbols

.
Let us denote the two logarithmic spirals that pass through the desired position P
as T 1 P and T 2 P , with characteristic angles

{∗, SL , T 1 , T 2 }

, respectively. The
two spirals divide the plane into four disjoint regions. Using the geometric properties
of the problem, these regions are further subdivided considering the set of points Q
for which the optimal path is given by a straight-line from Q to P without violating
the FOV constraints ([2]). With this intuitive subdivision, eight regions are derived,
depicted in Figure 18.8(a). Defining a smooth transition between segments with the
symbol “-”, the optimal path from each region is defined as follows:

α = − Δ and α = Δ