A predictive adaptive control system for multi-joint robotic manipulators is proposed, leveraging Long Short-Term Memory (LSTM) neural networks for trajectory tracking error prediction. The method is designed to mitigate dynamic uncertainties and external perturbations without explicit system identification, thus enabling robust and model-agnostic control.

The key finding of this study is that an encoder–decoder LSTM network, trained on joint angle increments (in delta-space), is capable of making accurate multi-step forecasts, ensuring smooth trajectory continuation. Delta-space representation is emphasized as a crucial factor for maintaining physical consistency, effectively eliminating discontinuities commonly encountered in absolute-value predictions.

In order to establish a theoretical basis, this work incorporates concepts derived from functional analysis, particularly the generalized Hölder-type conditions, which are formalized through the notion of a local modulus of continuity. These concepts serve as a promising framework for examining the approximation properties and stability of the predictor.

These mathematical instruments provide a structured framework for future rigorous analysis, encompassing the derivation of error bounds and guarantees for preserving smoothness. The discussion emphasizes how these analytical approaches can inform the selection of architecture and the regularization of training and ultimately contribute to enhancing the reliability of data-driven predictive controllers in real-time robotic applications and deployment scenarios.