Unicycle-like Robots with Eye-in-Hand Monocular Cameras: From PBVS towards IBVS - Visual Servoing via Advanced Numerical Methods - page 331

i.e. , initial covariance matrix for the vehicle, has its values set to half a meter for ξ 1

3 . P 0 ∈ R

2 × 2 , i.e. , the block diagonal matrices for

and ξ 2 , while it is one radian for ξ

≈

the features, have their entries set to 20 centimeters in our experimental setting.

Unicycle kinematic model, with odometric data, is assumed for state prediction.
Two different noise sources are taken into account. The prediction errors are mod-
elled as additive and zero mean Gaussian noises with covariance matrix Q = P 0 .
Systematic errors are assumed to be removed by suitable calibration, hence nonzero
mean errors are not modelled. The odometry errors

γ r , γ l , for the right and left wheel
respectively, are assumed to be zero mean and Gaussian distributed. They are also
assumed to be equal for both wheels. This is a simple but easily verified assumption
in respect to generic unicycle like vehicles, and is computed taking into account
lack of accuracy in odometry (typically due to wheels slipping and skidding). The
covariance matrix of the prior estimate (model prediction) is then calculated by the
formula

P k = A k P k − 1 A k +

2
γ

B k B k + Q σ ,

2
γ

where γ l by assumption), A k and B k are the model

Jacobians and P k − 1 is the model covariance matrix at the previous step.
During experiments, the EKF-based localization thus derived shows to outper-
form the least mean squares approach, as it may be expected.

σ is the input variance ( γ = γ r =

18.3.2 Visual Servoing with Omnidirectional Sight

Consider that only one feature has to be tracked, coincident with the origin O w .
Consider for this problem a new set of coordinates, which is better suited to describe
the angle by which the feature is observed from the vehicle, described by

2

Φ : R ×

+

S 2 (see Figure 18.1(a)) and the new dynamics

S → R ×

⎡
⎣

⎤
⎦ ,

⎡
⎣

⎤
⎦ =

⎡
⎣

⎤
⎦ =

⎡
⎣

⎤
⎦

ξ 2 2
arctan( ξ 2

ξ 1 2 +

u
ω

ρ ˙ ˙

ρ
φ
β

− ρ cos β 0

φ ˙

and sin β 0 , (18.6)

ξ 1 )

β sin β − 1

+ arctan( ξ 2

π

ξ 1 )

− ξ 3

v
ρ

where we let u = . Observe that this change of coordinates is a diffeomorphism

everywhere except at the origin of the plane (exactly where the feature point is). A
continuous, time-invariant control law can in principle stabilize the system (18.6).
Indeed, the two control vector fields are now linearly dependent at the origin, thus
making Brockett's negative result [3] unapplicable.
Consider the candidate Lyapunov function V = 2 (

2 + 2 + 2 ), with

ρ φ λβ λ > 0a

free parameter to be used in the following controller design. Substituting

= φ

sin β cos β + λβ sin β cos β

u = cos β , and ω + β (18.7)

λβ