## 3 Methods
### Extraction of Human Pose Euler Angles  
#### 3.2.1 Initial Attitude Angle Calculation  
We use Euler angles to describe the orientation of several body parts [19; 20; 21], including the torso, left hand, right hand, left leg, and right leg, in order to calculate the initial torso rolling angle, pitch angle, and yaw angle. The acceleration vectors of various body parts must be converted into a single coordinate system in order to calculate the Euler angles with accuracy, assuring consistency in angle calculation and dimensionality reduction. To get the torso's Euler angles, the following procedures are conducted, assuming the common coordinate system is represented by \(xyz\):  
First of all, the roll angle of the torso, indicated as \(\varnothing_{t}^{torso}\), is calculated as follows:  
\[\mathcal{Q}_{t}^{torso}=tan^{-1}\frac{a_{y,t}^{torso}}{a_{z,t}^{torso}} \tag{1}\]  
Figure 1: Identification process of hybrid human activity recognition method using MLP neural network and Euler angle extraction based on IMU sensors. where, in the common coordinate system, \(a_{y,t}^{torso}\) denotes the acceleration along the \(Y\)-axis of the acceleration vector and \(a_{z,t}^{torso}\) signifies the acceleration along the \(Z\)-axis. Furthermore, the pitch angle of the torso \(\theta_{t}^{torso}\) is computed using the formula:  
\[\theta_{t}^{torso}=tan^{-1}\frac{-a_{x,t}^{torso}}{\sqrt{\left(a_{y,t}^{torso} \right)^{2}+\left(a_{z,t}^{torso}\right)^{2}}} \tag{2}\]  
In this equation, the term, \(a_{x,t}^{torso}\) denotes the acceleration of the torso's acceleration vector in the common coordinate system along the \(X\)-axis. The yaw angle of the torso, shown by the symbol \(\varphi_{t}^{torso}\), is then calculated. The initial yaw angle is set to zero during the computation since accelerometers can only detect an object's acceleration in relation to a fixed coordinate system and not its absolute direction:  
\[\varphi_{t}^{torso}=0 \tag{3}\]  
Once the Euler angles of the torso in the common coordinate system have been computed, the Euler angles for the other body parts can be calculated using specific methods tailored to each part. For instance, the acceleration vector of the right hand must be rotated so that it is perpendicular to the xz plane of the torso in order to calculate the Euler angles of the right hand. In order to accurately calculate the Euler angles in the torso coordinate system, this rotation seeks to remove the right hand's projection in the xz plane of the torso. Figure 2 is right hand coordinate transformation, gyroscope, and accelerometer schematic for the human body model, in which we describe the 3D human model. The process involves first calculating the rotation angle of the right hand vector, denoted as \(\alpha_{t}^{rightarm}\), relative to the coordinate system of the torso:  
\[\alpha_{t}^{rightarm}=tan^{-1}\frac{-a_{z,t}^{torso}}{a_{y,t}^{torso}} \tag{4}\]  
where, \(a_{z,t}^{torso}\) represents the acceleration along the \(Z\)-axis of the acceleration vector of the right hand position in the common coordinate system, and \(a_{y,t}^{torso}\) denotes the acceleration along the \(Y\)-axis. Figure 2: Right hand coordinate transformation, gyroscope, and accelerometer schematic for the human body model. Next, the acceleration vector of the right hand after rotation, represented as \(\alpha_{t}^{rightarm}\), is obtained by rotating the original acceleration vector \(\overset{\rightarrow}{\alpha_{t}}^{rightarm}\) about the \(x\)-axis of the torso coordinate system, using the rotation matrix \(R_{x}(\alpha_{t}^{rightarm})\)[22; 23]:  
\[\overset{\rightarrow}{\alpha_{t}}^{rightarm}=R_{x}(\alpha_{t}^{rightarm}) \overset{\rightarrow}{\alpha_{t}}^{rightarm} \tag{5}\]  
\[R_{x}(\alpha_{t}^{rightarm})= \begin{bmatrix}1&0&0\\ 0&cos(\alpha_{t}^{rightarm})&-sin(\alpha_{t}^{rightarm})\\ 0&sin(\alpha_{t}^{rightarm})&cos(\alpha_{t}^{rightarm})\end{bmatrix}