## 3 DSML-Based Methodology Proposal
### SysML Profile for Tolerance Analysis (SYSML4 TA)
#### 3.1.3 TTRS-Based Modeling Profile Package (TTRS_MPP)  
TTRS_MPP includes stereotypes to support the analysis-oriented representation of the system based on TTRS concepts, that is, the mathematical definition of the position and orientation of the geometries according to their invariance classes. As proposed in the TTRS model, each invariance class is characterized by a particular MGRE, and a MGRE is  
Figure 5: Abstract syntax of artifact definition (“*” is a “zero to many” multiplicity). Figure 6: Abstract syntax of feature-based artifact specification (“*” is a “zero to many” multiplicity). composed of one, two, or three RGEs and RGE_constraints (ConstraintBlock specializations) between them when necessary (Figure 7). The TTRS-based representation of a feature (F_TTRS_repr) includes the \"local\" property, typed by MGRE_repr, that contains the mathematical definition of the feature with respect to the local coordinate system. This MGRE_repr owns two <<MGRE_usage>> Ports typed by MGRE, one for the nominal definition and other for the computed deviated definition. As mentioned in previous sections, MGREs must be also expressed with respect to a global coordinate system that is common for all the parts participating in the assembly. For this reason, the F_TTRS_repr also includes the \"global\" Port, typed by MGRE_repr, which is computed from \"local\" through a coordinate transformation. In this proposal, the transformation is governed by an homogeneous transformation matrix owned by the part. This matrix is defined as a <<TransformationMatrix>> property of each part, which is connected to the <<TM_Port>> Port of an <<F_TTRS_repr>> Block. Moreover, to characterize the 35 combinations of related MGREs considered in this work, the TTRS concept was also included as an entity with two Ports typed by MGREs and the mathematical constraints between their RGEs (RGE_constraints). Another important concept is the AssemblyMGRE Port, which is the TTRS-based representation of the previously described AssemblyFeature Port. <<AssemblyMGRE>> Ports are always typed by <<F_TTRS_repr>> Blocks, and they support the assembly relationships through connection between TTRS-based representations of different Artifacts. The set of concepts and relationships mentioned up to this point are represented in Figure 7.  
TTRS_MPP also includes the mathematical definition of geometric tolerances (GT_TTRS_definition), supporting the equations that define a tolerance zone (TZ_TTRS_definition), limiting the position and orientation deviations in target geometries (MGREs) with respect to datum geometries. It is important to point out that form deviations/tolerances were not considered in this work. As depicted in Figure 8, GT_TTRS_definition is composed of: (a) TZ_TTRS_definition; (b) a Port of type TTRS_repr_usage named \"target\"; (c) 1 to 3 Ports of type TTRS_repr_usage named \"datum\" (d) mathematical constraints (TTRS) to construct the TZ_TTRS_definition with respect to the \"datum\" Ports; (e) a ConstraintBlock to validate the correctness of the GT specification (SpecifValidation); (f) a ConstraintBlock to constructs a deviated target (TargetConstruction) with respect to the TZ_TTRS_definition; (g) a deviation vector (TargetDeviation), whose components specify the deviation value in each variant DoF. Regarding the commented ConstraintBlocks, it is important to clarify that these constructions include as constraints the equations defining the mathematical relationships between geometries, that is, MGREs and its RGEs. Specifically, SpecifValidation defines equations by combining some of the 13 simple cases of MGRE associations, while TargetConstruction defines equations to construct deviated geometries according to the values of the considered SDT. Figure 7: Abstract syntax of TTRS-based concepts (“*” is a “zero to many” multiplicity). As in the FMPP previously discussed, although some of the concepts have an associative nature, they have been modeled as Block specializations.