# Quaternion Signal Analysis for Detection of Broken Rotor Fault Degrees in Induction Motors
## 2 Methodology
### QSA Method  
Quaternions model the three-dimensional movements of a point in space. In a complementary way, the QSA method allows modeling the signals that generate the movements, facilitating a multidimensional analysis adapted to spatial and temporal problems. This approach detects spatial signatures using quaternion arithmetic and temporal signatures by estimating statistical displacement models. Combining both elements enables the method to achieve greater precision and robustness in fault detection than the classic techniques. This section outlines the methodology for defining quaternion signals and calculating their rotations. Subsequently, our approach computes modulus and statistical measures based on these rotations. The quaternion \(q\) is constructed as shown in (1), using the coefficients \(q_{0}\), \(q_{1}\), \(q_{2}\), and \(q_{3}\)[26]. \[q=q_{0}+q_{1}i+q_{2}j+q_{3}k \tag{1}\]  
The quaternion signal \(q(t)\) shown in section A of Figure 1 is constructed from the vibration signals \(V_{h}(t)\), \(V_{d}(t)\), \(V_{a}(t)\), and \(V_{s}(t)\), as formulated in (2). Signals \(V_{h}(t)\), \(V_{d}(t)\), and \(V_{a}(t)\) were chosen to assess vibrations along the \(x\)-, \(y\)-, and \(z\)-axes of the induction motor, while \(V_{s}(t)\) provides insight into rear motor vibrations. By selecting signals from different motor sections, our methodology enables a more comprehensive analysis, as Quaternion Signal Analysis (QSA) integrates the distinct characteristics of each signal. The vibration signals are represented discretely with a total of 29,999 samples. \[q(t)=V_{h}(t)+V_{d}(t)i+V_{a}(t)j+V_{s}(t)k \tag{2}\]  
Then, the shifted quaternion signal \(q_{d}(t)\) is obtained by the displacement of \(\Delta t\) samples in \(q(t)\). Equation (3) presents the equality [22]. \[q_{d}(t)=q(t+\Delta t) \tag{3}\]  
The coefficients \(V_{hd}\), \(V_{dd}\), \(V_{ad}\), and \(V_{sd}\) form the quaternion \(q_{d}(t)\) shown in Figure 1 with 29,998 samples, as defined in (4). These coefficients are the vibration coefficients \(V_{h}\), \(V_{d}\), \(V_{a}\), and \(V_{s}\) in \(t+\Delta t\). \[q_{d}(t)=V_{hd}(t)+V_{dd}(t)i+V_{ad}(t)j+V_{sd}(t)k \tag{4}\]  
Figure 1: Proposed workflow diagram. (**A**) QSA method. (**B**) Fitting and evaluation procedures. A quaternion model can represent the transition from a current quaternion sample, \(q\), to a delayed quaternion sample, \(q_{d}\). This means that we can identify the quaternion model that enables \(q\) to achieve the values of \(q_{d}\) through quaternion rotation operations, as illustrated in Figure 2.  
Rotation quaternions represent three-dimensional rotations and they are related to the axis and angle. Based on [PERSON]'s rotation theorem, any rotation can be defined using unit vectors \(\hat{x},\hat{y}\), and \(\hat{z}\) and angle \(\theta\). The axis-angle components can be converted into components of a rotation quaternion \(q_{r}\) by Equations (5)-(8). \[q_{r0}=\cos\left(\frac{\theta}{2}\right) \tag{5}\]  
\[q_{r1}=\hat{x}\sin\left(\frac{\theta}{2}\right) \tag{6}\]  
\[q_{r2}=\hat{y}\sin\left(\frac{\theta}{2}\right) \tag{7}\]  
\[q_{r3}=\hat{z}\sin\left(\frac{\theta}{2}\right) \tag{8}\]  
The above-mentioned process generates a rotated quaternion for each sample and its corresponding delayed value. This operation is applied to all samples of the signals to characterize the rotational behavior of the signal over time. Taking into account the above described, \(q\) and \(q_{d}\) are algebraically related by (9) [21], where \(q_{r}^{-1}\) is the inverse of rotation quaternion. \[q(t)=V_{h}(t)+V_{d}(t)i+V_{a}(t)j+V_{s}(t)k \tag{2}\]0  
The operator \(\otimes\) refers to the Hamilton product which is described as in (10). \[q(t)=V_{h}(t)+V_{d}(t)i+V_{a}(t)j+V_{s}(t)k \tag{2}\]1 \[q(t)=V_{h}(t)+V_{d}(t)i+V_{a}(t)j+V_{s}(t)k \tag{2}\]2 \[q(t)=V_{h}(t)+V_{d}(t)i+V_{a}(t)j+V_{s}(t)k \tag{2}\]3 \[q(t)=V_{h}(t)+V_{d}(t)i+V_{a}(t)j+V_{s}(t)k \tag{2}\]4  
By the properties of quaternions, the relation of \(q_{r}\), \(q_{d}\), and \(q\) can be written as (11) [23].