## 6 Verification
### Sensory Pre-Processing  
As for our multimodal observation vector, it is comprised of 6 DoF force-torque signals, 6 DoF Cartesian velocity signals (from [PERSON]'s right-end effector), and 56 DoF tactile signals from both the left and right tactile sensors. Empirical feature extraction is performed on this observation vector to improve classification performance. We now describe how features for each modality at time \(t\) were engineered.  
Wrench modality: Assume the force and torque sequence is a time-series vector and that each element represents the magnitude of each dimension \((f_{x},f_{y},f_{z},t_{x},t_{y},t_{z})\). We wish to extract structural information from the wrench instead of simply using raw values. In this way, we can find signal patterns that may occur across the different DoFs. To this end, we computed the norm of both the force \(n_{f}\) and the torque \(n_{t}\) as features respectively:  
\[n_{f}=\sqrt{f_{x}^{2}+f_{y}^{2}+f_{z}^{2}},\quad n_{t}=\sqrt{t_{x}^{2}+t_{y}^{ 2}+t_{z}^{2}}. \tag{33}\]  
Velocity modality: Similarly, we take the Cartesian linear \((l_{x},l_{y},l_{z})\) and angular \((a_{x},a_{y},a_{z})\) velocities and compute their norm \(n_{l}\) and \(n_{a}\) respectively:  
\[n_{l}=\sqrt{l_{x}^{2}+l_{y}^{2}+l_{z}^{2}},\quad n_{a}=\sqrt{a_{x}^{2}+a_{y}^{ 2}+a_{z}^{2}}. \tag{34}\]  
Tactile modality: Due to the computational cost of processing the tactile sensor's high dimensionality, we empirically tested several features for each tactile sensor, which include the maximum taxel value, the largest five taxel values, the mean of all taxel values, and the standard deviation for all taxel values. It was the standard deviation, which proved to be the most useful feature for anomaly detection and classification. The standard deviation for each tactile sensor \(s_{l}\) and \(s_{r}\) are defined as:  
\[s_{l}=\sqrt{\frac{1}{28}\sum_{i=1}^{28}(l_{i}-\mu_{l})},\quad s_{r}=\sqrt{ \frac{1}{28}\sum_{i=1}^{28}(l_{i}-\mu_{r})}. \tag{35}\]  
where \(\mu_{l}=\frac{1}{28}\sum_{i=1}^{28}l_{i}\) and \(\mu_{r}=\frac{1}{28}\sum_{i=1}^{28}r_{i}\) are the mean of each tactile sensor, respectively. Equations (33)-(35) facilitate the identification of diverging signals during anomalous situations. We empirically concatenate all the valuable features for representing the robot executions both in nominal and anomalous cases. If raw signals are used directly, they result in high false-positive detection rates (FPR) in our experimentation. For instance, when a robot carries objects of varying weights, the F/T signals for such operations vary significantly, making it difficult to use raw values for detection. Hence, a standardization method is used to scale the extracted features through a normalization constant. We end with a feature vector of length 18 as shown below:  
\[y_{t}=[\frac{f_{x}}{10},\frac{f_{y}}{10},\frac{f_{z}}{10},\frac{t_{x}}{10}, \frac{t_{y}}{10},\frac{t_{z}}{10},\frac{n_{f}}{10},\frac{n_{t}}{0.4},l_{x},l_ {y},l_{z},a_{x},a_{y},a_{z},\frac{n_{l}}{0.4},n_{a},\frac{s_{l}}{250},\frac{s_ {r}}{250}]. \tag{36}\]  
Figure 6 illustrates an example of the multimodal signals in a whole nominal experiment with six skills.  
Note that all training and testing data are pre-processed in the same manner. We will also use the same pre-processing approach when compared to other baseline approaches.