## 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)
Each vertex \(v\in\tilde{V}\) is generated by parsing the first two parameters of the grounded predicates process_step_NAME from \(s_{0,const}\) in form of itemX_stepNR. Each edge \(e\in\tilde{E}\) and the function \(\tilde{\phi}\) are defined by parsing the pairs of each two consecutive parameters of the grounded predicates process_step_precedence_typeX. The first pair corresponds to the parent vertex, while the following pairs correspond to the child vertices connected to that parent vertex._  
Each vertex \(v\in\tilde{V}\) can have zero, one, or more output edges \(e\in\tilde{E}\) and has at least one input edge. Vertices with more than one output edge enable the parallel execution of process steps after their execution, while vertices with more than one input edge transform the parallel execution to a sequential one. The configuration of the process steps as presented in Listing 11 is translated in the DG from Figure 8, while Appendix A shows the dependency graph of the custom gripper main IGPP. Figure 7: Sub-areas of two scenes of the custom gripper IGIS. ```
1(:init
2(process_step_connectitem1 step1 workbench2_thingpose1)
3(process_step_precedence_type1 item1 step1 item1 step2)
4(process_step_connectitem1 step2 workbench2_thingpose1)
5(process_step_precedence_type1 item1 step2 item2 step1)
6(process_step_precedence_type1 item1 step2 item3 step1)
7(process_step_connectitem2 step1 workbench2_thingpose1)
8(process_step_connectitem3 step2 workbench2_thingpose1)
9(process_step_todoitem1 step1)...)
10(:goal(and
11(process_step_doneitem1 step2)
12(process_step_doneitem2 step1)
13(process_step_doneitem3 step1)...))
```  
**Listing 11**Excert from the PDDL problem formulation for a simple IGPP. Furthermore, the start-process-steps set \(st_{ps}\) and the finish-process-step set \(fi_{ps}\) also contain elements in the form \(itemX\_stepNR\). The elements of set \(st_{ps}\) are obtained by parsing the first two parameters of the grounded predicates _process_step_todo_ from the initial-state sub-set \(s_{0,ps}\). The elements of set \(fi_{ps}\) are obtained by parsing the first two parameters of the grounded predicates _process_step_done_ from the goals set \(g\). For the example presented in Listing 11, the two sets have the following elements:  
\[ \begin{split} st_{ps}&=\{item1\_step1\}\\ f_{ips}&=\{item1\_step1,item1\_step2,item2\_step1, item3\_step1\}.\end{split}  \tag{4}\]  
Using the definition of the dependency graph and of the sets \(st_{ps}\) and \(fi_{ps}\), the validation methodology can be introduced. In a simplified case, a possible validation approach for the IGPP checks if a plan can be generated only for the main IGPP. The main IGPP has in its initial state corresponding grounded predicates for all the elements of the set \(st_{ps}\), and in its goals, grounded predicates for all elements of the set \(fi_{ps}\). This simplified approach is not complete enough for the validation of the main IGPP for two reasons. First, the configuration and dependencies of all process steps can be so complex that the selected planning problem cannot be solved within the set deadline. This is the case, for example, when the set \(fi_{ps}\) has many elements. The second reason is related to the scenarios considered in this work. Especially due to the involvement of human actors, it is possible that during the execution of the process steps, errors occur and new plans must be generated. These plans must be computed from new initial states that imply new start-process-steps sets \(\tilde{st}_{ps,i}\). The sets \(\tilde{st}_{ps,i}\) are different from the original set \(st_{ps}\), where nothing has yet happened. Each of the two issues presented above requires a special approach. The first challenge, which is related to the complexity of a main IGPP, can be tackled by selecting only a sub-set of all process steps that should be planned for in a planning iteration. Subsets \(\tilde{f}^{i}_{ps,i}\) with cardinality \(n_{ps}\) (where \(n_{ps}\leq|fi_{ps}|\)) can be determined for the original set \(fi_{ps}\).