Cascade-TCN-BiLSTM: accurate prediction of long-term transmission error curves in multi-stage transmission system

Accurately forecasting long-term transmission error trends in multi-stage transmission systems is essential for ensuring high motion accuracy in mechanical systems. Effectively modeling the nonlinear propagation and inter-stage coupling of errors to enhance predictive capabilities remains a significant challenge. This research introduces a cascaded deep learning framework, termed Cascade-Temporal Convolutional Network-Bidirectional Long Short-Term Memory, designed to estimate long-term transmission error curves across planetary and harmonic stages. By building a three-stage cascade aligned with intrinsic errors of the planetary reducer, inter-stage assembly errors at the planetary–harmonic interface, and operational errors of the harmonic reducer, we establish a one-to-one mapping between network modules and the corresponding error sources, thereby ensuring physical interpretability. The model incorporates both static assembly features and short-term dynamic input signals. A stage-specific cascaded configuration is embedded into a comprehensive sequence-to-sequence structure, consisting of an encoder-decoder network. Each encoder and decoder component consists of stacked temporal convolutional networks and bidirectional long short-term memory layers, followed by a multi-head attention module designed. Experimental results indicate that the proposed model consistently achieves low mean squared error and mean absolute error, typically below 0.22 and 0.33, respectively. The coefficient of determination exceeds 0.97 in most cases, demonstrating that the model significantly outperforms both traditional machine learning methods and baseline deep learning architectures. Ablation studies further confirm the critical contributions of the unified architecture, temporal modeling, and attention mechanism to the model’s performance. In addition to multi-stage transmissions, the method applies to series elastic actuators, surgical and industrial robot joints, and rotating machinery.