Human-machine Cooperative Manipulation with Vision-based Motion Constraints - Visual Servoing via Advanced Numerical Methods - page 54

4.2.2 Virtual Fixtures as Geometric Constraints

While it is clearly possible to continue to extend the notion of virtual fixture purely
in terms of compliances, we instead prefer to take a more geometric approach, as
suggested in [1, 2]. We will develop this geometry by specifically identifying the
preferred and non-preferred directions of motion at a given time point t . To this

end, let us assume that we are given a 6 × n time-varying matrix D = D ( t ) , 0 <

n Intuitively, D represents the instantaneous preferred directions of motion.

For example, if n is 1, the preferred direction is along a curve in SE (3);if n is 2 the
preferred directions span a surface; and so forth.
From D

< 6 .

we define two projection operators, the span and the kernel of the col-
umn space, as

,

[ D ]= D ( D D ) − 1 D Span( D ) ≡ (4.3)

Ker( D ) ≡ D = I − [ D ] . (4.4)

This formulation assumes that D has full column rank. It will occasionally be useful
to deal with cases where the rank of D is lower than the number of columns (in
particular, the case when D = 0). For this reason, we will assume [ · ] has been

implemented using the pseudo-inverse [29, pp. 142-144] and write

[ D ]= D ( D D ) + D Span( D ) ≡ (4.5)

Ker( D ) ≡ D = I − [ D ] . (4.6)

The following properties hold for these operators [29]:
1. symmetry: [ D ]=[ D ] ;
2. idempotence: [ D ]=[ D ][ D ] ;
3. scale invariance: [ D ]=[ kD ] ;
4. orthogonality: [ D ]=0;

5. completeness: rank(α

D

D + β[ D ]) = n where D is n × m and α , β = 0;

6. equivalence of projection: [ D f ] f = D f .

The above statements remain true if we exchange D and [ D ] . Finally, it is useful

to note the following equivalences:

• [[ D ]] = [ D ] ;

• D =[ D ] ;and

• [ D ]= [ D ] = D .

Returning to our original problem, consider now decomposing the input force
vector, f , into two components

f D ≡ [ D ] f and f τ ≡ f − f D = D f . (4.7)

It follows directly from property 4 that f D · f τ = 0 and from property 5 that f D + f τ = f .

Combining (4.7) and (4.2), we can now write