###### Abstract  
Robotic manipulators play a key role in modern industrial manufacturing processes. Monitoring their operational health is of paramount importance. In this paper, a novel anomaly detection framework named U-TFF is introduced for energy consumption auditing of robotic manipulators. It comprises a cascade of Time-Frequency Fusion (TFF) blocks to extract both time and frequency domain features from time series data. The block applies the Fast Fourier Transform to convert the input to the frequency domain, followed by two separate dense layers to process the resulting real and imaginary components, respectively. The frequency and time features are then combined to reconstruct the input. A U-shaped architecture is implemented to link corresponding TFF blocks of the encoder and decoder at the same level through skip connections. The semi-supervised model is trained using data exclusively from normal operations. Significant errors were generated during testing for anomalies with data distributions deviating from the training samples. Consequently, a threshold based on the magnitude of reconstruction errors was implemented to identify anomalies. Experimental validation was conducted using a custom dataset, including physical attacks as abnormal cases. The proposed framework achieved an accuracy and recall of approximately 0.93 and 0.83, respectively. A comparison with other benchmark models further verified its superior performance. robotic manipulators; side-channel mechanism; energy consumption auditing; anomaly detection; fast Fourier transform +
Footnote †: journal: _Applied_  
To address this issue, the adoption of a side-channel energy consumption auditing approach has been proposed [5]. This method employs a physical mechanism to measure energy consumption through a separate module positioned alongside the robotic manipulator system. Such an approach allows for isolated monitoring, making it more robust to data manipulation and attacks [6]. The next step is to process the data for use in learning to identify anomalies when attacks occur, which is usually referred to as the anomaly detection task. In contrast to traditional approaches such as decision trees [7], deep learning-based methods can be employed in anomaly detection tasks by training neural networks to distinguish anomalies from normal samples automatically. This approach has found broad applications in AI-aided manufacturing [8]. Neural networks using these methods can vary from small models (e.g., [9; 10; 11]) to large models (e.g., [12; 13; 14]). Moreover, based on the model training methods, anomaly detection approaches can be generally classified into two main categories, namely, supervised and semi-supervised learning methods. In supervised learning, each instance in the dataset is labeled as either normal or anomalous for model training [15]. Because supervised methods require labeled data, they provide clear and precise information about both normal and anomalous instances. As a result, supervised methods have strong ability to tackle known anomalies that have been identified and labeled in the training dataset. They can learn the specific characteristics of these anomalies and accurately detect them during the testing phase. However, supervised learning relies heavily on labeled data for training, which may be expensive or difficult to obtain in many real-world scenarios. The resulting models may also struggle to detect anomalies that differ significantly from those observed during training. In other words, the generalization of supervised approaches is not guaranteed. On the other hand, semi-supervised learning typically uses normal instances only for model training to learn the underlying patterns of normal behavior. A well-trained model can identify instances that deviate significantly from the learned patterns of normal behavior, flagging these instances as anomalies. This process is also known as outlier detection.