## 3 Materials and Methods
### Validation Methodology for a Main Instance of the Generic Planning Problem (IGPP)
#### 3.3.1 Scene Initial States for a Main Instance of the Generic Planning Problem (IGPP)  
As mentioned in Section 3.2.3, a scene for an IGIS is a specific distribution of the _things_ to the _thingposes_ and of the _agents_ to the _agentposes_. Therefore, more scenes are possible for an IGIS. To reduce the number of possible distributions and to guarantee that only distributions that make logical and physical sense are selected, our validation methodology splits the _things_, _agents_, and _poses_ sets into subsets and introduces mappings between these sets. The splitting procedures and the mappings are also dependent on the selected IGIS. To better understand the approaches introduced above, an example is given for a simplified IGIS. The following sets of _items_ and _thingposes_ are given:  
\[ \begin{split} I&=\{connector\_i,corner\_i|i\in\{1, \dots,4\}\}\\ T&=\{itembench\_i\_thingpose\_j|i\in\{1,2\},j\in\{1, \dots,4\}\}.\end{split}  \tag{2}\]  
These two sets are split as follows:  
\[ \begin{split} I_{1}&=\{connector\_i|i\in\{1, \dots,4\}\};I_{2}=\{corner\_i|i\in\{1,\dots,4\}\}\\ T_{1}&=\{itembench\_1\_thingpose\_j|j\in\{1, \dots,4\}\}\\ T_{2}&=\{itembench\_2\_thingpose\_j|j\in\{1, \dots,4\}\}.\end{split}  \tag{3}\]  
The elements of set \(I_{1}\) are mapped to the elements of set \(T_{1}\) and the elements of set \(I_{2}\) to the elements of set \(T_{2}\). These splitting and mapping procedures guarantee that the physical and logical constraints of the IGIS are satisfied. In the example, the connectors can be placed only on _itembench_1_, and the _corners_ can be placed only on _itembench_2_ (see Figure 7). In the next step, the mapping functions between all subsets must be defined. One possibility would be to determine all combinations of allowed assignments of _things_ and _agents_ to _poses_. However, the number of these assignments increases exponentially with the number of _things_, _agents_, or _poses_. For the custom grippers IGIS, the number of possible configurations is still extremely high. For this reason, the mapping procedures are based on a random method, in which each _thing_ and _agent_ is randomly allocated to only one _pose_. This mapping approach has thus only one degree of freedom: the number of randomly generated scenes \(n_{scene}\). This number corresponds to the number of scene initial states and it spawns a subset of all situational IGPPs for the selected main IGPP. Finally, each scene is translated into PDDL with _at_ grounded predicates, resulting in a set \(s_{0,scene}\), which is part of an initial state \(s_{0}\). 3.2 Process Steps Initial States and Process Steps Goals for a Main Instance of the Generic Planning Problem (IGPP)  
This subsection presents our methodology used to determine allowed process steps initial states and process steps goals for a given main IGPP. These states are the second type of degrees of freedom that spawn the set of situational IGPPs. At the beginning, the subsets \(s_{0,const}\) and \(s_{0,ps}\) from the PDDL problem formulation of the main IGPP are identified. From these, three types of information are extracted and used in the validation method: the configuration of all process steps, the start process steps of the processing chain \(st_{ps}\), and the process steps that must be finished by the end of the processing chain \(fi_{ps}\). The configuration of the process steps is interpreted from the grounding predicates from set \(s_{0,const}\) of the PDDL formulation of the main IGPP and further encoded in a graph structure that allows an easy representation of these process steps and their dependencies. **Definition 3**. : _A directed simple graph \(G=(V,E,\phi)\), where \(V\) is the set of vertices, \(E\) the set of edges, and \(\phi:E\rightarrow\{(x,y)|(x,y)\in V^{2},x\
eq y\}\) a function that maps each edge to an ordered pair of vertices._  
**Definition 4**. : _A dependency graph \(DG=(\tilde{V},\tilde{E},\tilde{\phi})\) for a main IGPP is a directed simple graph.