# On the Development of a Surrogate Modelling Toolbox for Virtual Assembly
## 2 State of the Art
### Methods for Relative Positioning  
In contrast to data fusion registrations such as the commonly known Best Fit registration, by VA not similarity in point sets, but corresponding contact faces are searched. This makes it necessary to implement constraints to the registration algorithm that represent physical effects of the corresponding real-world assembly. The underlying basic optimization problem stated in Equation (1) is a minimization problem of the sum of \(i\in[1;N]\) squared signed distances \(d_{sig,i}\) according to:  
\[f\left(t_{x},t_{y},t_{z},\phi,\theta,\psi\right)=\sum_{i=1}^{N}\ d_{sig,i}^{2}= min! \tag{1}\]  
with \(t_{x},t_{y},t_{z},\phi,\theta,\psi\) as the three translational and rotational optimization parameters of the affine transformation, respectively. As a convention, negative signed distances \(d_{sig,i}\) denote a virtual intersection of the point sets. The sign is determined from the normal vectors of the measured surfaces, where the normal by convention points away from the surface's material side. The most relevant constraint is the avoidance of intersection of point sets, which comprises the impossibility of parts to intersect. Commonly, the non-intersection is formulated as hard constraint allowing only signed distances \(d_{sig,i}\geq 0\) between the point set to register and the rigid point set [30; 38; 39]. [PERSON] linearizes this constraint in order to simplify the underlying optimization problem [40]. By formulation as a soft constraint, a certain intersection of parts can be tolerated, e.g., to simulate the flattening of surfaces [41]. Using the Lagrange multiplier method, for the Lagrangian function \(\mathcal{L}\) as per:  
\[\mathcal{L}\left(t_{x},t_{y},t_{z},\phi,\theta,\psi,\lambda\right)=\sum_{i=1}^ {N}\ d_{sig,i}^{2}+\lambda\cdot n_{neg}, \tag{2}\]  
the optimality condition is found by setting the partial derivatives equal zero according to:  
\[\
abla\mathcal{L}\left(t_{x},t_{y},t_{z},\phi,\theta,\psi,\lambda\right)=0. \tag{3}\]In Equation (2), \(\lambda\) is the Lagrange multiplier and \(\mathrm{n_{neg}}=\left|\mathrm{N_{neg}}\right|\) is the cardinality of a set of \(\mathrm{N_{neg}}=\left\{\mathrm{d_{sig,i}}\mid\mathrm{d_{sig,i}}<0\right\}\) negative signed distances. The work of [PERSON] and [PERSON] at the National Institute of Standards and Technology (NIST) includes such constraint registration methods for datum definitions considering \(\mathrm{L_{1}}\) norm (Manhattan) [42] and \(\mathrm{L_{2}}\) norm (Gaussian) distance metrics [29] and can be considered as preparatory research for future ISO and ASME standards as indicated e.g., in draft ISO/DIS 5459.2:2017 [43]. The basic optimization problem in our approach is extended by _pre-alignment SOs_ and _optimization constraints_ (during optimization) to consider the influence of the joining process on the geometrical quality of the assembly.  
Another approach for relative positioning is the difference surface approach from [PERSON] et al. [44]. The difference surface equals to the differences of corresponding form deviations of both contact faces. Form deviations are separated by modal decomposition as described in [45]. The contact points of both faces are computed by the intersection of the difference surface with a contact triangle that achieves a maximal surface area. As a constraint, the assembly force vector must intersect the contact triangle [44; 46]. A further approach stated by [PERSON] is the minimization of the convex hull of the difference volume (gap volume) between the contact faces.