## 4 The Layout Designing of the Collaborative Workplace Supported by a Modeling Paradigm
### Proposed Approach
Constrained optimization problem solvingIn the previous subsection the goal function to be minimized was defined by Equation (2). That function is the sum of several terms related to specific features of the workplace. Currently, the present paper takes into account only the aspect related to the occupied floor space \(S(w)\). The occupied floor space minimization criterion guarantees the best solution when it is necessary to reduce the dimension of the workplace and it also goes to meet the need to reduce the distances covered by the human operator in order to enhance the ergonomic conditions during the execution of his tasks. Ergonomic aspects will be considered in more detail in the future developments of this approach. Therefore, considering the workplace floor space utilization [35] as:  
\[FS=(x_{max}-x_{min})+(y_{max}-y_{min}), \tag{3}\]  
where \(x\) and \(y\) are the maximum and the minimum coordinates reached by any resource in the plane, the goal function is defined as the sum of the Euclidean distances between the pairs of task centers:  
\[f=\sum_{k=1}^{b}\sqrt{(x_{k,i}-x_{k,j})^{2}+(y_{k,i}-y_{k,j})^{2}}. \tag{4}\]  
The \(x_{i}\), \(y_{i}\), and \(x_{j}\), \(y_{i}\) in Equation (4) are the Cartesian coordinates of the \(i\)-th and \(j\)-th task centers, respectively; \(b\) represents the number of simple combinations given by:\[\binom{n}{2}=\frac{n!}{2!(n-2)!}, \tag{5}\]  
where \(n\) is the number of tasks centers to be placed in the available space. Minimizing the Euclidean distances between the task centers means that the resources are placed closer together, and consequently, the occupied floor space is also reduced. Obviously, the problem is constrained to place the resources within the available space and to avoid overlap of them. Furthermore, the solution of the optimization problem is bound to compliance with the following provisions established by the reference standards:  
* The minimum distance between moving objects, the robot system and building areas, structures, users, and other machines should be at least 500 mm [24; 57]. * The minimum required separation distance between humans and robots established by the ISO/TS15066 [25] with regard to speed and separation monitoring is: \[S_{p}(t)=S_{r}+S_{r}+S_{s}+C+Z_{d}+Zr.\] (6) The standard clarifies well how each term is determined. * Safety distances are required to guarantee escape routes [58]. * The positioning and the orientation of the control device (e.g., the Human Machine Interface) should be such that the view on the robot is always unobstructed (as declared in the Annex A of ISO 10218-2 [24] and ISO 13850 [59]). * Maximum load carrying distance should be defined depending on the carried cumulative mass [60]. Moreover, logistic space, which contains the logistic resources, can be bound to assume one of the three configuration described in Section 4.1. In short, the mathematical modeling for the optimization problem is set up as the search for the minimum value of a nonlinear function of \(n\times 2\) variables subject to nonlinear constraints. The constraints are expressed by means of equalities and inequalities. The \(n\times 2\) variables correspond to the Cartesian coordinates \((x,y)\) of the plan projection of the n task centers associated with the relative passive resources. The problem is solved by means of a Matlab nonlinear programming solver. Rough layout definitionThis phase outcome is the optimized and standard-compliant principal solution to the HRC layout designing problem. Once the optimization problem is solved, and the passive resources are placed, the working areas are determined by defining the matrix \(F_{ap}\) that associates the active resources to the employed passive resources. The proposed approach identifies as the main workspaces the Human Space and the Robot Space, and using the definitions proposed in the paradigm, it is easy to mark them.