## 2 Materials and Methods
### Federated Learning for Articulated Robot Control
#### 2.3.1 FL Module
To this end, the articulated robot anatomy condition is considered to be a closed form by the geometric analytic inverse kinematics method (see Figure 6). Figure 5: Example of sub-area \(s\) with \(3\times 3\) resolution. In Figure 6, \(\theta_{1}\), which is \(\theta_{2}\), and \(\theta_{3}\) are the three joint angles; the end-effector position is represented as \(P_{\alpha e}=(P_{X},P_{Y},P_{Z});l_{1},l_{2}\), and \(l_{3}\) are the lengths of the three links. Radius \(r\) of the circle that contains coordinates \((P_{X},P_{Y})\) implies \(\sqrt{P_{X}^{2}+P_{Y}^{2}}\). \(\theta_{1}\), which is the angle for joint 3, can be derived from Equation (3). The length of \(D\), which is the confronting side of angle \(\pi-\theta_{3}\) in the triangle that is created from \(l_{2}\) and \(l_{3}\), can be described as Equation (4). According to this triangle, \(\theta_{3}\) can be derived as Equation (5). Finally, the last angle, \(\theta_{2}\), is derived from the relationship between \(\theta_{3}\) and \(\varnothing\), the angle contained between the two sides \(w_{1}\) and \(D\), as described in Equation (6). The inverse kinematics formulation establishes the relationship between the angles and the link lengths: \(\theta_{1}\) depends on the projection of the end-effector onto the XY plane; distance \(D\) is calculated using the Pythagorean theorem in 3D space; angle \(\theta_{3}\) follows from the cosine rule applied to the triangle formed by \(l_{2}\), \(l_{3}\), and \(D\); and \(\theta_{2}\) is determined by considering both \(\theta_{3}\) and \(\varnothing\), which account for the vertical and radial positioning of the end-effector. \[\theta_{1}=tan^{-1}\frac{P_{X}}{P_{Y}} \tag{3}\]

\[D=\sqrt{r^{2}+\left(l_{1}-P_{z}\right)^{2}} \tag{4}\]

\[\theta_{3}=cos^{-1}\frac{l_{2}^{2}-l_{3}^{2}-D^{2}}{2l_{2}l_{3}} \tag{5}\]

\[\theta_{2}=tan^{-1}\frac{l_{1}-P_{Z}}{r}-tan^{-1}\frac{l_{3}sin\theta_{3}}{l_ {2}+l_{3}cos\theta_{3}} \tag{6}\]

After the kinematics solver processor computes each joint angle for the operating task, the PWM controller processor sets the duty cycle of each joint actuator to operate the picking task (i.e., a pick-and-place task). Note that PWM is an analog modulating method using the width, duration, and time of the pulse [6] to control the joint angle to a specific target angle. Due to its advantages (i.e., the convenience of implementation and control, high compatibility with a recent and past digital microprocessor, low power consumption,

Figure 2:0 Articulated robot in spherical coordinates for inverse kinematics analysis. and robustness under stable temperature variation, machine aging, and degradation), it is widely used to control the motors controlling the joint angles of the articulated robot [45]. ## 3 Results

This section discusses a quantitative analysis of the computational and a power efficiency aspects of the robot control process and the FL process. The analysis is conducted on low-end computing hardware through the framework proposed in this study for learning. It also compares the proposed object detection model with standard CNN models. Section 3.1 introduces the experimental conditions and quantitative indicators. These include a standard CNN model and the object detection model used in this study, the hardware resources of the robot manipulator, the working environment of the robot manipulator, and the hyperparameters of the FL process. ### Scenario

The object detection model trained by the FL module in Section 2.3.1 is compared to a standard CNN model in a comparative experiment to analyze whether it can be trained effectively on low-end hardware equipment. The standard CNN models to be compared are Faster R-CNN [46], SSD [42], and YOLO-LITE [47]. The experiments were conducted on a GPU-equipped hardware setup (Intel Xeon(r) Silver 4210 Processor (Intel, Santa Clara, CA, USA), NVIDIA RTX Quadro 4000 (NVIDIA, Santa Clara, CA, USA)) with the objective of comparing not only the proposed object detection model but also popular existing CNN models.