# Formation Control and Distributed Goal Assignment for Multi-Agent Non-Holonomic Systems
## 8 On Some Generalization  
The condition (32) can be rewritten in more general form as follows:  
\[\sigma=\left\{ \begin{array}{ll}1&\quad\text{if}\;\;||p_{mk}||^{n}+||p_{nl}|| ^{n}>||p_{nk}||^{n}+||p_{ml}||^{n}\\ 0&\quad\text{otherwise}\end{array} \right. \tag{40}\]  
Taking \(n=1\) leads to the shortest path criterion that seems to be natural in many cases because the shortest path induces lower motion cost. Unfortunately this observation may not be true in a cluttered case. This will be shown later in this section. Note that in the cases for \(n\
eq 2\) the Lyapunovanalysis is much more complex. In [23] the similar method that results in the shortest total distance to the goals is presented. Several specific scenarios for the simple case of two robots are analysed further. These scenarios should be treated as an approximation of a real case because typically paths of real robots are not straight lines in the case of the platforms that are not fully actuated (like the differentially driven mobile platform considered here). In Figure 7 two robots are the same distance away from their goals. Initially the goals were assigned as marked with dashed arrows. The goal exchange procedure led to the assignment marked with continuous arrows. The resulting paths were less collisional or even non-collisional (as they are parallel). The new assignment was optimal using both the shortest path criterion (\(n=1\)) and quadratic criterion (\(n=2\)). Figure 8 shows a case in which two robots and their goals lie on a straight line. Initially goal 2 is assigned to robot 1 and goal 1 is assigned to robot 2. This situation caused a saddle point because during the motion to the goal \(R_{1}\) stays on the path of \(R_{2}\). One can observe that the shortest path criterion (\(n=1\)) produces exactly the same result for both possible goal assignments, while quadratic one (\(n=2\)) produces the result marked by continuous arrows. By the assignment \(R_{1}-G_{1}\) and \(R_{2}-G_{2}\) the goals can be reached by the robots without bypass manoeuvre. This is one of the examples showing the significant advantage of the quadratic criterion over the shortest path criterion proposed in [23]. In Figure 9\(R_{1}\) is exactly at the goal \(G_{2}\) assigned to it. \(G_{1}\) was assigned to \(R_{2}\). The quadratic criterion resulted in the opposite assignment (continuous arrows). Notice that for the shortest path criterion the collision avoidance interaction between \(R_{1}\) and \(R_{2}\) was possible. For this type of situation the quadratic criterion resulted in goal exchange for all locations in the hatched circle. The opposite situation is presented in Figure 10. Fields of squares shown in the figures represent values of cost  
Figure 8: Two robot-goal assignment—case 2. Figure 7: Two robot-goal assignment—case 1.  
functions for two possible goal assignments (compare two left-hand side squares with right-hand square). The quadratic criterion resulted (in comparison to the shortest path) in a higher cost function for assigning far goals to the robots. It favoured a larger number of short assignments instead of the lower number of farther ones. This promoted the reduction of collision interaction situations. In Figure 11 certain positions of the robot \(R_{1}\) and goals \(G_{1}\) and \(G_{2}\) have been assumed. If robot \(R_{2}\) is located in the hatched area the quadratic criterion assigns it to \(G_{2}\), otherwise it is assigned to \(G_{1}\). Some initial configuration may have led to temporary collision interaction states (i.e., when \(R_{2}\) initially is located to the left of the \(R_{1}\)) but the saddle point occurrence was not possible. Dashed circles on the sides represent examples of boundary locations (for goal exchange) of the robot \(R_{2}\). Figure 10: Two robot-goal assignment—case 4. Figure 9: Two robot-goal assignment—case 3.