Sensor-based Trajectory Deformation: Application to Reactive Navigation of Nonholonomic Robots - Visual Servoing via Advanced Numerical Methods - page 306

C ∞ ([0

k )

v ∈ , S ] , R

C ∞ ([0

n )

q ∈ , S ] , R

Current
trajectory

n

η ∈ R

0

Controller senso r s

Fig. 17.1 A trajectory deformation process can be modelled as a dynamic control system
of time τ. At each time, the state is a feasible trajectory q , the input is a pair (η 0 , v )that
uniquely defines the time derivative of the state. The trajectory deformation algorithm we
describe in this chapter can be considered as a closed-loop controller that computes the input
of the dynamic control system with respect to the current trajectory and a task to achieve, for
instance avoiding obstacles, based on perceptual data

For each value of 0) is called the initial

trajectory . In order to keep notation light and intuitive, we use the same nota-
tion q to denote configurations, trajectories and trajectory deformations. We are
interested in deformations q ( s

τ , s → q ( s , τ ) is a trajectory. s → q ( s ,

) composed of only admissible trajectories. Such
deformations satisfy the following constraint: there exists a k -dimensional vec-
tor valued smooth mapping u =( u 1 ,...,

, τ

u k ) defined over [0 , S ] × [0 , ∞ ) such that

∀ , τ ∈ , × , ∞

( s ) [0 S ] [0 )

k
i =1 u i ( s , τ) X i ( q ( s , τ)) .

∂ q

s ( s

, τ )= (17.2)

∂

→ , τ → , τ

For each value of ).The

above equation simply expresses constraint (17.1) for each trajectory of the defor-
mation. As well as a trajectory is uniquely defined by the initial configuration and
the input function, a trajectory deformation is uniquely defined by the initial config-
uration q (0

τ , s u ( s ) is the input function of trajectory s q ( s

).
By differentiating (17.2), we get a relation between the input variation ∂ u
∂τ

, τ ) of each trajectory and by input functions u i ( s , τ

and the infinitesimal trajectory deformation when the deformation parameter τ

increases:

∂

)

2 q k

i =1

∂ u i

∂τ

) ∂

X i
∂

)) ∂

q
∂τ

( s , τ )= ( s , τ ) X i ( q ( s , τ )) + u i ( s , τ

q ( q ( s

, τ ( s , τ

∂ s ∂τ

We call respectively input perturbations and direction of deformation the following
vector valued functions: