Sensor-based Trajectory Deformation: Application to Reactive Navigation of Nonholonomic Robots - Visual Servoing via Advanced Numerical Methods - page 309

the algorithm. In Section 17.3.1, we compute , τ j ) by restricting input perturba-

tion to a finite-dimensional subspace of functions. This restriction enables us in Sec-
tion 17.3.2 to take into account boundary conditions that force the initial and final
configuration of the deformation interval to remain unchanged. In Section 17.3.3, we
explain how to compute the direction of deformation that minimizes the variation
of the trajectory scalar value under constant L 2 -norm. The first order approximation
17.5 induces deviations of the nonholonomic constraints. Section 17.3.4 addresses
this issue and proposes a correction of this deviation.

η ( s

17.3.1 Finite-dimensional Subspace of Input Perturbations

As explained in Section 17.2, the control variables of a trajectory deformation pro-
cess are the input perturbation v and the initial condition , τ j ) belongs to

the infinite-dimensional space of smooth vector-valued functions defined over [0

η 0 . s → v ( s

S ].
To simplify the control of the trajectory deformation, we choose to restrict v to a
finite-dimensional subspace of functions. This restriction will make the boundary
conditions introduced later in Section 17.3.2 easier to deal with. Let p be a posi-
tive integer. We define e 1 ,...,

,

e p , a set of smooth linearly independent vector-valued
functions of dimension k , defined over [0 , S ]:

k

e i : [0 , S ] → R .

Various choices are possible for the e i 's ( e.g. truncated Fourier series, polynomials,
etc ) [16, 4, 3]. For each of these functions, we define E i ( s , τ j ) as the solution of

system (17.3) with initial condition η 0 = 0 and with e i ( s ) as input:

E i ( s

, τ

j )= A ( s

, τ

j ) E i ( s

, τ

j )+ B ( s

, τ

j ) e i ( s )

(17.7)

E i (0 , τ

j )=0

(17.8)

where matrices A and B are defined in Section 17.2.2. Let us notice that unlike e i ,
E i depends on τ j since system (17.3) depends on the current trajectory.

If we restrict v ( s , τ j ) in the set of functions spanned by the e i 's, that is for any

vector λ =( λ 1 ,..., λ p ):

p
i =1 λ i e i ( s )

v ( s , τ j )= (17.9)

as (17.3) is linear, the direction of deformation η corresponding to v is the same

linear combination of solutions E i

p
i =1 λ i E i ( s , τ j ) .

η ( s , τ j )= (17.10)

Using this restriction, the input perturbation v is uniquely defined by vector λ .