Constraint vector

F = objective function, g = algebraic inequality constraint vector, c = algebraic equality constraint vector,... 

Collecting the orthogonal complement to the C ) in the (Vpam x Vg-eej-dimensional matrix L (note that the constraints vectors may be linearly dependent), we can express the requirements in Eq. (26) as a linear transformation to a new set of variables, x ... 

A constraint vector arising from the interaction between species... 

Because of their origin they may be called constraint vectors. 

We expect that Eqs. (2.43) and (2.46) are equivalent in the sense that they impose equivalent conditions on the vibrational 7-vectors. This indicates that we can form the rotational s-vectors of an atom a as linear combinations of the constraint vectors belonging to this atom,... 

Using this in Eq. (2.47) we then find the constraint vectors,... 

Recollecting Eq. (2.40) it is natural to check if axial constraints could be so formulated that differentiation results in equations of the form used in Eq. (2.46) with constraint vectors given by... 

With constraint vectors given by Eq. (3.33) the answer is noS4l We cannot totally eliminate Coriolis coupling, but, on the other hand, this is not too serious, since... 

However, with constraint vectors that are independent of the internal coordinates an integration can be performed. Furthermore, using... 

When linearized coordinates are considered, an additional reason arises for requiring structurally independent constraint vectors. This was also noticed by Eckart39 but, in the present context, it is most easily understood by considering that Eq. (2.46) requires constant constraint vector components, cg>ag-, since the f-vector components, tkag = tk ag, are constants. Hence, linearized internal coordinates are incompatible with constraint vectors like those of the PAS (Sect. 2.2.3.2.1). 

The two sets of rotational s-vectors are easily expressed using the corresponding constraint vectors of the Eckart conditions as described in Sect. 2.2.3.2.2,... 

Our previous experiences with the rotational s-vectors suggest a relation between the set of s-vectors applying to p and the constraint vectors of a p-convention. More-... 

As rotational constraints we retain the three Eckart conditions and the corresponding constraint vectors [Eq. (2.68) or (3.35)], here depending on p. These constraints have been discussed by Hougen17) as well. 

At the tip contact, the motion and constraint vector spaces may be defined using the general joint model discussed in Section 2.3. For convenience, we will assume that the two dual bases used to partitiai the spatial acceleration and force vectors at the tip. 

Here the constraint vector g(x) includes operational constraints and/or system models. The solution of problem [7.1] will yield an optimal solution x and a weighted cost for stakeholder s that we denote as wjf = wjf (x ). This weighted cost is ideal, or utopian, in the sense that it assumes that stakeholder 5 does not have to compromise with the rest of the stakeholders. When compromise is needed, as is often the case, we define the dissatisfaction of stakeholder 5 at an arbitrary compromise decision x as dj(x) = wj(f(x)—f ). Note that from optimality of X and associated weighted cost wjf, we have that ds x) > 0 for aU x and for all s S. 

The conditions are assembled in the constraint vector W and the loop closure conditions are written as... 

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