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As a result, an equations system (Fig. 11) is obtained, with the decomposition of
screws in 4 force equations and in 4 moment equations expressed in the arbitrarily
chosen point A:
The equations system is solved and the static equilibrium of the lift gate system is
expressed.
3.3
Computational Support for the Exploration of the Solution Space Based
on Constraint Programming and Interval Analysis
Special emphasis is also placed on interval-based computational methods [10] allow-
ing one to explore exhaustively the search space resulting from a declarative state-
ment of constraints [11]. Given the previous high level vector model linked to a given
topology, formal calculus and causal ordering based on bipartite graphs theory [12-
14] can be used to avoid part of the tedious work consisting in giving the mathemati-
cal expressions of some constraints as required to run dedicated solvers. The use of
interval computations within a constraint programming paradigm [15] also provides a
computational support to quantify uncertainties and to detect inconsistencies. From a
methodological point of view, the refinement inherent to the design process is under-
lined.
A Constraint Satisfaction Problem (CSP) is usually defined by ( X, D, C ) where X =
{x 1 , x 2 , …, x n } is a set of variables, D = {d 1 , d 2 , …, d n } is a set of domains such that
di , and C = {C1, …, Cm} is a set of constraints depending on the
variables in X . Each constraint includes information related to constraining the values
for one or more variables. When continuous variables are considered, the use of inter-
val analysis techniques naturally arises in order to represent the domains. Those
methods make it possible to explicitly take uncertainties (in the sense of deterministic
imprecision rather than probabilistic variability) into account in the preliminary de-
sign process. The use of an interval CSP solver (here, RealPaver) [16] allows an ex-
haustive search within the search space D which is partitioned into three sets, D = D 0
i
{ 1 ,…, n} , xi
D ? , the latter two being described by a box paving: D 0 is a sub-domains of D
where the constraints are never satisfied; D 1 is a sub-domains of D where the con-
straints are always satisfied; D ? is a sub-domains of D where the satisfaction of the
constraints has not been decided yet according to some stopping criterion (precision,
for instance).
From an engineering design point of view, the variables in X can be a set of design
parameters, the domains in D can be used to define the range of the search space of
interest, and the constraints in C can be concurrently stated by several engineers in
any order. Such a declarative modelling is a significant advantage of the CSP para-
digm throughout the life cycle of a Computer Aided Engineering (CAE) application
[17].
From a methodological point of view, the refinement inherent to the design process
can be supported as follows: the poor initial knowledge results in a small number of
constraints with few variables belonging to rather large intervals; then, the sequence
of assumptions, trials and evaluations constituting the heart of an iteration within the
design refinement loop allows the engineers to acquire knowledge, to organize it, and
to gradually converge toward what will become the detailed solution [18].
D 1
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