## 2 Materials and Methods
### Stability Labeling
#### 2.2.1 Wavelet Transform Maxima Modulus (WTMM)  
The stability of the manipulator can be measured from two aspects. One is the motion characteristics of the manipulator, i.e., the main components of vibration signals. The other is the influence of the mechanical arm structure itself. The vibration components caused by the main motion of the manipulator are characterized by large amplitude, low frequency, and low singularity. On the other hand, the vibration caused by structure deformation has high frequency and small amplitude as well as strong randomness. WTMM is suitable for separating the components with large singularity and is used to separate the two vibration components according to their different properties.  
The WTMM was proposed by [PERSON] [18]. In WTMM, the modulus maxima in each layer are filtered when the corresponding Lipschitz exponent \(\alpha\) is less than 0. The remaining modulus maxima are used to reconstruct the filtered signal by the cross-projection method [19]. The Lipschitz exponent is often used to measure the singularity of a function. The signal that is differentiable at a high order at a certain point or interval is smooth in this region and has a small singularity and large \(\alpha\). On the contrary, the smaller the \(\alpha\), the larger the singularity of the signal. When the signal satisfies Equation (1), the signal is admitted to have a consistent Lipschitz exponent in the interval, where \(k\) is an integer and \(W_{f}\) is the wavelet transform modulus maxima of signal \(x\) at layer \(j\).  
\[\left|W_{f}\left(2^{j},x\right)\right|\leq k2^{2\alpha} \tag{1}\]  
As shown in Equation (2), the Lipschitz exponent \(\alpha\) can be expressed as the logarithm of the ratio of the modulus maxima \(W_{f}\) between different scales \(j\) and \(j+1\).  
\[\alpha=\log_{2}\frac{\left|W_{f}\left(2^{j+1},x\right)\right|}{\left|W_{f} \left(2^{j},x\right)\right|} \tag{2}\]  
When \(\alpha>0\), the maxima increase as \(j\) increases. When \(\alpha<0\), the maxima decrease as \(j\) increases. The maxima do not vary with \(j\) when \(\alpha=0\).  
When it comes to vibration signal analysis of the manipulator, the vibration component brought by the motion itself, denoted as \(a_{main}\), has a lower frequency and is smoother. The component caused by structural deformation, denoted as \(a_{vib}\), has a higher frequency and stronger singularity. The WTMM is used to separate these two vibration components:  
1. Perform binary wavelet transform on the signal, select db3 wavelet, and calculate the modulus maximum points of the wavelet coefficient on each scale;
2. On the largest scale \(j\), set the threshold of the modulus maximum points, filter out the modulus maximum points below the threshold, and obtain new modulus maxima;
3. Starting from the scale \(j-2\), filter out the points corresponding to the position where the modulus maximum value is 0 at the higher scale. Repeat until \(j=2\), and only keep the modulus maxima that are not zero when \(j-2\);
4. Use the cross-projection method to reconstruct the signal \(a_{main}\) with the remaining modulus maximum points; the filtered part is \(a_{vib}\).