## 4 Prediction Horizon Length and Convergence
### Convergence
To integrate Algorithm 4 into the DMPC scheme, in Algorithm 2, we include after line 11  
\[\mathbf{Call}\ Algorithm4.\]  
Note that, from the circular control law established by Algorithm 4, the robots are still obeying the collision avoidance constraints between the other robots as the DMPC scheme in Algorithm 2 is still executed and Assumption 1 is ensured for \(u_{p}^{c}\) with \(\cos\left(\arccos\left(\frac{z_{p}(n)-m_{x}}{r}\right)+u_{p}^{c}\right)=0\) and \(\sin\left(\arccos\left(\frac{z_{p}(n)-m_{x}}{r}\right)+u_{p}^{c}\right)=0\). Figure 7: Illustration of the circular control with centre \(\left(m_{x},m_{y}\right)\), and entry and exit points \(z_{p}^{s},z_{p}^{e}\) for one robot (**a**) and two robots with start and end points \(z_{1}^{s},z_{2}^{s}\) and \(z_{1}^{e},z_{2}^{e}\) leaving earlier due to the tangential condition (**b**). ```
1:for all\(p\in[1:P]\)do
2:if\(\exists q\in[1:P]\setminus\{p\}\) where Equation (28) holds for \(p,q\)then
3: Add\(\mathbf{Q}:=\mathbf{Q}\cup\{p,q\}\)
4:endif
5:endfor
6:if\(\mathbf{Q}\
eq\emptyset\) or \(w_{p}=true\)then
7:if\(w_{p}=false\)then
8: Set centre \(\left(m_{x},m_{y}\right)\) for \(z_{p}\) with Equation (29) and \(z_{p}^{s}\), \(z_{p}^{e}\)\(\forall p\in\mathbf{Q}\) with Equations (31) and (32)
9:endif
10: Set \(w_{p}:=true\)
11:for all\(p\in[1:P]\) and \(p\
otin\mathbf{Q}\)do
12: Keep\(u_{p}=\overline{u}_{p}\quad\forall k\in[n:n+N]\) according to Assumption 1
13:endfor
14:if\(z_{p}^{0}=z_{p}^{s}\) and \(p\in\mathbf{Q}\)then
15: Set \(u_{p}\) according to Equation (33)
16:endif
17:endif
18:if\(w_{p}=true\) and \(\left(\left(z_{p}^{0}-z_{p}^{s}\right)\times\left(z_{p}^{0}-\left(m_{x},m_{y} \right)^{\top}\right)=0\) or \(z_{p}^{0}=z_{p}^{e}\) for \(p\in[1:P]\right)\)then
19: Set \(u_{p}\) back to Equation (3)
20: Remove\(\mathbf{Q}:=\mathbf{Q}\setminus\{p\}\)
21: Set \(w_{p}:=false\)
22:endif
23:if\(\mathbf{Q}=\emptyset\)then
24:for all\(p\in[1:P]\)do
25: Reset\(u_{p}\) according to Equation (3)
26: Set \(w_{p}:=false\)
27:endfor
28:endif
```  
**Algorithm 4** Evaluation for switching control law of robot \(p\) to establish a collision avoidance circle  
**Theorem 4** (Collision avoidance via a fixed circular curve). : _Suppose a given number of \(P\) robots, each defined by Equation (1), with restrictions on state and control in Equations (2) and (3) and with initial feasible conditions and targets fulfilling Assumption 4. If the distance between two robots is such that Equation (28) is fulfilled and Assumption 5 is satisfied for all robots \(p\in[1:P]\), then the calculation of a circle and execution of the DMPC scheme in Algorithm 2 using the switched control by Algorithm 4 will allow the robots to let them converge to their targets, i.e., to steer the system practically stable._  
Proof. : From Assumption 4, the feasibility of the initial conditions (\(n=0\)) and targets for all \(P\) robots is guaranteed, which is weaker than Assumption 3 with additional allowance of points near to the bounds of the 2D plane. Then, with \(n>0\) each robot \(p\in[1:P]\) minimises the distance to the individual target \(z_{p}^{*}\). Then, if the condition in Equation (28) is fulfilled for at least two robots, the centre \(\left(m_{x},m_{y}\right)^{\top}\) of the circle for the conflicting robots is calculated with Equation (29) for the given radius \(r\) and the circular control law is applied for all conflicting robots according to Equation (33). All other robots are set to an immediate hold via \(\overline{u}_{p}\). As the condition in Equation (28) holds for any arbitrary chosen robots, the circle is created, when at least two predicted states are closer according to the given condition, and therefore, any other robot is outside this circle fulfilling Assumption 5 and additionally stops moving. With this assumption that no target is located inside the circle \(\left(z_{p}^{*}-\left(m_{x},m_{y}\right)^{\top}\right)\), each robot \(p\in\mathbf{Q}\) is able to calculate a feasible route via Algorithm 4 subject to the intersections \(z_{p}^{s}\) and \(z_{p}^{e}\).