## 4 Analysis of the Properties of LCPSO
### Removing the Simplifying Hypotheses  
#### 4.6.1 Non-Zero Mass \(c_{0}\)  
In a real robot, the weight \(c_{0}\) must be taken into account and is set according to its geometry and mass. This parameter will influence the speed norm of the agents when the formation is stable according to Definition 2. To illustrate our point, we keep the hypothesis that the attractor \(\mathbf{p}^{g}\) is always the same agent. Thanks to Theorem 1, we have \(\bar{\mathbf{v}}_{eq}=-c_{2}\big{(}\bar{\mathbf{p}}(t)-\mathbf{p}^{g}(t)\big{)}\) when \(c_{0}=0\). \(\bar{\mathbf{v}}_{eq}\) is invariant in time because the swarm formation is stable. When \(c_{0}\
eq 0\), we have:  
\[\bar{\mathbf{v}}(t+\Delta t)= c_{0}\cdot\bar{\mathbf{v}}(t)-c_{2}\big{(}\bar{\mathbf{p}}(t)- \mathbf{p}^{g}(t)\big{)}\] \[= c_{0}\cdot\Big{(}c_{0}\cdot\bar{\mathbf{v}}(t-\Delta t)-c_{2} \big{(}\bar{\mathbf{p}}(t-\Delta t)-\mathbf{p}^{g}(t-\Delta t)\big{)}\Big{)}+ \bar{\mathbf{v}}_{eq}\] \[= \bar{\mathbf{v}}_{eq}\cdot\sum_{k=0}^{\frac{t_{max}}{\Delta t}}c _{0}^{k}\underset{t_{max}+\infty}{=}\frac{\bar{\mathbf{v}}_{eq}}{1-c_{0}}. \tag{19}\]  
Hence, if \(c_{0}\) influences the speed norm of the swarm, it does not influence its direction, which depends only on the position of the attractor. If the mathematical analysis is not impacted by the pseudo-mass when \(|c_{0}|<1\)[39, 44], the pseudo-mass \(c_{0}\) greatly influences the convergence time of the swarm during its transitory phase, since it smoothes the trajectory of the agents by taking into account the previous velocity vector.  
We support what we have just said by Monte-Carlo simulations as a function of \(c_{0}\) in Figure 4, for \(N=7\) and \(N=15\) agents in dimension 2. For \(N=7\), the agents are distributed on the same circle around the center of gravity, while for \(N=15\), we have a distribution on 2 levels, as in Figure 3f. As our simulation time is \(t_{max}=100\) s, a convergence time  
Figure 3: Swarm evolution and formation following the Local Charged Particle Swarm Optimization (LCPSO) Equation (18). \(r_{eq}=7\) m, \(c_{0}=0\), \(c_{2}=0.5\). (**a**) \(\rho(N)\) in dimension 1; (**b**) Radius \(r_{max}\) in dimension 1 and 2; (**c**) Example of swarm trajectory; (**d**) \(\rho(N)\) in dimension 2. Boxplots with 1000 samples; (**e**) \(r_{max}\) in dimension 2. Boxplots with 1000 samples; (**f**) Example of emergent swarm formation.  
close to this number shows a high failure rate of the agents to converge, based on the fact that it takes between 10 s and 30 s on our samples for them to converge when \(c_{0}=0\). The maximal simulation time is arbitrary, but increase its value does not change the results, the swarm is still unable to converge.  
We have displayed only the most interesting results in Figure 4. We can see that the number of agents \(N\) as well as the weight \(c_{0}\) influence our convergence time Figure 4. For \(-0.1\leq c_{0}\leq 0.5\), we have 100% convergence for \(N=15\), but for \(N=7\), we can extend the range of \(c_{0}\) to \(-0.5\leq c_{0}\leq 0.7\). Hence, the convergence to a stable formation is more difficult when \(N\) increases. These conclusions are more restrictive than those of [44] on the analysis of the PSO algorithm, since when \(|c_{0}|\geq\)1, the PSO is unable to converge. Indeed, when \(c_{0}\) is too important, the previous velocity vector is so influent that it will prevent the swarm from stabilizing efficiently.