# Formation Control of Nonlinear Multi-Agent Systems with Nested Input Saturation
## 5 Experimental Results on a Swarm of Aerial Robots
\begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(k_{x,i^{\prime}}^{1}\), \(i=1,\ldots,5\) & 0.02 & \(\rho_{i,i}^{2}(0)\), \(i=1,\ldots,5\), \(l=x,y\) & 3 \\ \(k_{y,i^{\prime}}^{1}\), \(i=1,\ldots,5\) & 0.01 & \(\rho_{i,i}^{2}(0)\), \(i=1,\ldots,5\), \(l=z\) & 2 \\ \(k_{z,i^{\prime}}^{1}\), \(i=1,\ldots,5\) & 0.01 & \(\rho_{i,i}^{1}(0)\), \(i=1,5\), \(l=x,y\) & 3 \\ \(k_{z,i^{\prime}}^{2}\), \(i=1,\ldots,5\), \(l=x,y,z\) & 2 & \(\rho_{z,i}^{1}(0)\), \(i=1,5\) & 2 \\ \(\lambda_{1,i^{\prime}}^{1}\), \(i=1,\ldots,5\), \(l=x,y,z\) & 0.3 & \(\rho_{i,i}^{1}(0)\), \(i=2,3,4\), \(l=x,y\) & 2 \\ \(\lambda_{2,i^{\prime}}^{2}\), \(i=1,\ldots,5\), \(l=x,y,z\) & 4 & \(\rho_{z,i}^{1}(0)\), \(i=2,3\) & 0.3 \\ \(m_{l,i}\), \(i=1,\ldots,5\), \(l=x,y\) & 0.1 & \(\rho_{z,4}^{1}(0)\) & 0.6 \\ \(m_{z,i}\), \(i=1,\ldots,5\) & 0.15 & \(\rho_{i,i}^{2,\infty}\), \(i=1,\ldots,5\), \(l=x,y,z\) & 0.1 \\ \(r_{l,i}\), \(i=1,\ldots,5\), \(l=x,y\) & 5 & \(\rho_{i,i}^{1,\infty}\), \(i=1,\ldots,5\), \(l=x,y\) & 0.1 \\ \(r_{z,i}\), \(i=1,\ldots,5\) & 1 & \(\rho_{z,i}^{1,\infty}\), \(i=1,\ldots,5\) & 0.15 \\ \hline \end{tabular}
\end{table} 
Table 3: Control Protocol Parameters. Phase A of the experiment is shown in Figure 4, where the white star denotes the position of the leader. The agents start from a straight line formation on the ground and aim to achieve the desired configuration, represented by the white pentagon, and then hover around their desired positions. The achieved position of agents 2 and 4 concerning the z-axis is observed to be smaller than the leader's setpoint. This deviation arises due to the MAS topology, as agents 2 and 4 lack direct access to the leader's state, requiring them to move relatively to their neighbors only. This limitation can be compensated by selecting larger values of the control gains \(k_{z,i}^{1},\ i=2,4\) at the expense of more intense saturation effects. Consequently, coordinating motion in a 3D MAS framework involves coordinating three distinct motions, posing considerable challenges in achieving precise results. These challenges stem not only from motion couplings but also from actuation constraints and delays induced by feedback limitations. In Phase B, the coordinated swarm is commanded to track the leader's trajectory as depicted in Figure 5. Only agents 1 and 5 have access to the MAS's reference trajectory, while the rest of the following agents should align themselves with the trajectory of these two agents. The demonstration of the experiment is available at ([[https://vimeo.com/8898](https://vimeo.com/8898)]([https://vimeo.com/8898](https://vimeo.com/8898)) 77313?share=copy, accessed on 3 December 2023). The neighborhood tracking errors w.r.t. to axes \(x,y,z\) for each agent throughout the entire experimental duration are depicted in the left subfigures of Figures 6-8, respectively. Note, that these figures show cumulative neighborhood errors, which are strongly coupled with the relative inter-agent motion. The performance boundaries expand as the corresponding control input, which is illustrated in Figure 9 for each agent, reaches saturation, ensuring that the closed-loop signals remain within bounds. When the saturation is inactive the performance boundaries retrieve their prescribed form with exponential rate \(\exp\left(-\lambda_{i,i}^{1}t\right)\), \(i=1,\ldots,N,\ l\in\{x,y,z\}\). Additionally, the right subfigures of Figures 6-8 depict the control tracking error incorporating the limitations on control signal rates, as the system's actuators are unable to instantaneously modify the control effort arbitrarily. In summary, the decentralized adaptive performance scheme presented in this work effectively manages the formation control of a realistic MAS under nested saturation nonlinearities. However, it is crucial to recognize certain limitations of this approach.