Sensor-based Trajectory Deformation: Application to Reactive Navigation of Nonholonomic Robots - Visual Servoing via Advanced Numerical Methods - page 314

17.3.4.2 Correction of Nonholonomic Deviation

In order to make u i +1 ( s , τ ) ,···, u n ( s , τ ) tend toward 0 as τ increases, we apply the

following linear control:

∀ i ∈{ k + 1 ,···, n },∀ s ∈ [0 , S ] , v i ( s , τ )= − α u i ( s , τ )

where α is a positive constraint. We denote by η 1 the corresponding direction of

deformation for τ = τ j :

j )= A ( s η 1 ( s j )+ B ⊥ ( s j ) v ⊥ ( s

, τ , τ , τ , τ , τ

j ) η 1 ( s j ) (17.24)

, τ .

η 1 (0 j )=0 (17.25)

17.3.4.3 Deformation due to Obstacles

Following the procedure described in sections 17.3.1 and 17.3.3, we restrict input
functions ( v 1 ,···, v k ) to the finite dimensional subspace of functions spanned by

( e 1 ,···, λ 1 ,···, λ p ) according to (17.19). We denote by

η 2 the direction of deformation obtained with these coefficients:

e p ) and we compute λ =(

p
i =1 λ i E i ( s , τ j )

η 2 ( s , τ j )=

where the E i are now solution of system:

E i ( s , τ j )= A ( s

, τ j ) E i ( s , τ j )+ B ( s , τ j ) e ( s ) (17.26)

, τ .

E i (0 j )=0 (17.27)

17.3.4.4 Boundary Conditions

We wish the sum of

2 satisfies boundary conditions (17.13) and (17.14).
Again, (17.13) is trivially satisfied. (17.14) is an affine constraint over vector

η

1 and

η

λ :

η 2 ( S , τ j )= L λ = − η 1 ( S , τ j ) (17.28)

where L is the matrix defined in Section 17.3.2. Following the same idea as in Sec-
tion 17.3.2, we project vector λ over the affine subspace satisfying (17.28):

¯

P ( LP ) + P ( LP ) + L ) λ = − η 1 ( S , τ j )+( I p − λ .

We then get a direction of deformation satisfying the boundary conditions and mak-
ing the component of the velocity along additional vector fields converge toward 0:

p
i =1 λ i E i ( s , τ j )+η 1 ( s , τ j ) .

, τ

η ( s j )=