Unicycle-like Robots with Eye-in-Hand Monocular Cameras: From PBVS towards IBVS - Visual Servoing via Advanced Numerical Methods - page 341

θ − θ 0 = arctan I x d
α x

arctan I x i
α x

− , (18.10)

where , the robot orientation. Therefore, the angle

variation can be computed for pure rotations by means of (18.10).
Pure translation : by plugging into the image Jacobian (18.4), the condition

θ 0 is the initial value of θ

ω = 0

= ¯ = constant and solving for the I y c feature coordinate, yields to and ν ν

I y i
I x i

I y c = I x c ,

(18.11)

which describes a straight line passing through the initial position of the feature
and the principal point.
Logarithmic spiral : consider the condition

I x c = 0, i.e. , I x c = I x c = constant.

Using (18.4), the condition reflects in

I x 2 + x

− α x
α

α

ω = v . (18.12)

w y I x I y

y

I x c is constant, if follows that the angle

Since in (18.8) between the forward

direction of the vehicle and the feature direction is constant. Since this is the
condition verified by a logarithmic spiral path, the condition (18.12) identifies
such a path.

γ

18.4.1.2 From Image Paths to Servoing

Once the alphabet of feature sub-paths has been defined, the rules to choose and
construct the right sequence of maneuvers (the word) that correspond to the shortest
path must be defined directly on the image plane. To this end, path of increasing
complexity, i.e. , increasing number of symbols, are computed. If a path is feasible,
i.e. , the feature path remains constrained inside the image view, than it is chosen.
Otherwise, it is assumed that the robot state pertains to a different region in space
and the algorithm is reiterated. The path solutions derived in Section 18.4.1.1 are
computed numerically using the Levenberg-Marquardt algorithm ([30]).
The optimal feature path for Region I or Region I' is shown in Figure 18.9 and
is composed of: a piece of a conic passing through the initial position of the feature
(rotation); a piece of straight line passing through the principal point (translation);
and a piece of conic passing through the final position of the feature (rotation).
Given that

is known (or estimated), then the path can be computed solving the
following equation

Ω

= arctan I x 1
α x

arctan I x i
α x

+ arctan I x d
α x

arctan I x 2
α x

θ i + θ d = Ω − − ,

where θ d are the orientation of the vehicle in the initial and final positions,

the variables I x 1 or I x 2 correspond to the intermediate (unknown) positions of the

θ

i and