## 2 Materials and Methods
### Federated Learning for Articulated Robot Control
#### 2.3.1 FL Module
 \tag{7}\]

where \(b_{x}\) and \(b_{y}\) are the coordinates of the centroid of a ground-truth bounding box; \(b_{x}\) and \(b_{y}\) are the coordinates of the centroid of an estimated bounding box; \(b_{w}\) and \(b_{h}\) are the width and height of a ground-truth bounding box; \(b_{\oplus}\) and \(b_{h}\) are the width and height of an estimated bounding box; \(C_{i}\) is the ground-truth confidence score at the \(i\)th grid; \(\hat{C}\) is the estimated confidence score at the \(i\)th grid; \(\lambda_{coord}\) and \(\lambda_{noobj}\) are multipliers to increase or decrease the weight of the loss to resolve the class imbalance problem; \(\mathbb{1}_{ij}^{obj}\) is a binary indicator for object recognition (0: not recognized; 1: recognized); \(p_{i}(c)\) is the ground-truth conditional class probability for class \(c\) in \(i\)th grid; \(\hat{p}_{i}(c)\) is the predicted conditional class \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Type** & \(C\) & \(B\) & \(E\) & **Rounds 1** & \(T_{f}\) **(s)** & \(R_{f}\) **(\%)** & \(N_{f}\) **(Mb)** \\ \hline FedSGD & 1.0 & \(\infty\) & 1 & NA & 82,980 & 45.8 & 14,125.62 \\ \hline \multirow{6}{*}{0.25} & \multirow{6}{*}{0.25} & \multirow{6}{*}{5} & 5 & 823 & 199,140 & 32.5 & 14,625.37 \\ \cline{3-6}  & & & 10 & NA & 493,140 & 33.7 & 14,255.37 \\ \cline{3-6}  & & & 5 & NA & 265,560 & 35.7 & 14,154.28 \\ \cline{3-6}  & & & 10 & NA & 503,940 & 38.6 & 15,053.75 \\ \cline{3-6}  & & & 5 & NA & 323,940 & 31.7 & 29,046.56 \\ \cline{3-6}  & & & 10 & 106 & 581,220 & 33.1 & 3000.76 \\ \cline{3-6}  & & & 5 & 700 & 228,780 & 36.2 & 19,828.44 \\ \cline{3-6} FedAVG & & & 10 & 76 & 570,480 & 37.5 & 2303.5 \\ \cline{3-6}  & & & 5 & 415 & 368,452 & 32.6 & 17,622.48 \\ \cline{3-6}  & & & 10 & 30 & 235,325 & 32.8 & 1363.89 \\ \cline{3-6}  & & & 5 & 393 & 364,380 & 38.3 & 17,864.67 \\ \cline{3-6}  & & & 10 & 26 & 423,423 & 38.2 & 1259.88 \\ \hline \multirow{6}{*}{1.0} & \multirow{6}{*}{5} & \multirow{6}{*}{10} & 5 & 93 & 46,324 & 32.3 & 6008.68 \\ \cline{3-6}  & & & 10 & 190 & 203,285 & 32.8 & 10,983.64 \\ \cline{3-6}  & & & 5 & 244 & 86,325 & 37.3 & 15,764.74 \\ \cline{1-1} \cline{2-6}  & & & 10 & 133 & 232,152 & 37.2 & 8593.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Federated learning results (Rounds, \(T_{f}\) (s), \(R_{f}\) (%), and \(N_{f}\) (Mb)) in different hyperparameter sets (\(C\), \(B\), and \(E\)). probability for class \(c\) in \(i\)th grid; \(S^{2}\) is the set of grids in the task place; and \(B\) is the set of anchor box. Equation (7) is the train loss function given by [41], and it consists of three parts: (1) the first and the second terms represent the localization loss between the detected bounding box and the ground truth; (2) the third and fourth terms represent the confidence loss between the estimated selection probability of the class (or an articulated robot) and the observed probability from the actual selection of that class; (3) the fifth term represents the classification loss between the detected class and the ground truth. There are four cases in each graph, with different \(C\) values. As mentioned in Table 4, there is only one case (\(B=0.5\) and \(E=0.5\)) that can complete model training with 823 rounds when \(C=0.25\); however, most of the other cases complete their modeling training within 1000 rounds. Due to the required object detection accuracy of 80% (i.e., the stopping condition of modeling training), the train loss in different cases reaches similar values of 11. However, each case has a different conversion pattern of training loss. In particular, when \(E=5\), cases with \(B=5\) require a lesser number of rounds to reach the stopping condition of the modeling training than other cases with \(B=10\), as shown in Figure 11a,c,d. On the other hand, when \(E=10\), all cases in Figure 11b-d only require less than 200 rounds. Under the given scenario, the best performance of 26 rounds is achieved when \(C=0.75\), \(B=10\), and \(E=10\) (see Figure 11c).