Enhancing Machine Learning-Based Feedforward Control of 2-DOF Flexure Manipulator - Benefits of Time-Delay Embedding

Aug 5, 2026

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Machine learning-based feedforward control is developed for a fully actuated 2-DOF flexure-joint manipulator to improve trajectory tracking using a data-driven inverse dynamics framework. The controller combines a Lagrangian Neural Network to model conservative forces with a Feedforward Neural Network to predict force variances and uses residual modeling with weighted recursive least-squares to discover DID errors. Time-Delay Embedding is evaluated by augmenting training snapshots into Hankel matrices, reducing mean residue errors and exponentially decreasing uncertainty while increasing computational cost.

LanguageEnglish

CategoryResearch & Report

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Pages2

Reading Time5 min

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Introduction

Integrating Time-Delay Embedding into Data-Driven Inverse Dynamics

Reducing the Feedforward Control Error

Acknowledgments

References

The 7th International Conference on Multibody System Dynamics June 9-13, 2024, Madison, WI, USA Enhancing Machine Learning-Based Feedforward Control of 2-DOF Flexure Manipulator: Benefits of Time-Delay Embedding Maciej Pikuliski \#, Paweł Malczyk\#, Ronald Aarts† \# Institute of Aeronautics and Applied Mechanics † Applied Mechanics and Data Analysis Warsaw University of Technology University of Twente Nowowiejska 24, 00-665, Warsaw, Poland P.O. Box 217, 7500 AE, Enschede, The Netherlands [maciej.pikulinski.dokt, pawel.malczyk]@[pw.edu.pl](pw.edu.pl) [r.g.k.m.aarts@utwente.nl](r.g.k.m.aarts@utwente.nl) 1 Introduction The rapid evolution of machine learning offers new opportunities to improve model-based control systems. Our research focuses on using these techniques to enhance a feedforward controller for a fully actuated 2 degrees of freedom (DOF) manipulator with flexure joints (Fig. 1) . Feedforward control is a powerful technique to improve the tracking performance of a manipulator. This study’s feedforward controller is built solely based on data-driven techniques. The foundation of the controller is a combination of the Lagrangian Neural Network (LNN) and the Feedforward Neural Network (FNN), primarily introduced in [1] . We utilize the LNN to simulate the system’s conservative forces, whereas the FNN is tasked with forecasting the variances between the total necessary forces and those generated by the LNN, particularly the non-conservative forces. It is essential to note the inherent limitation of both networks in precisely modeling the reproducible part of these forces. To address this limitation, we introduce residual modeling as a secondary layer in the controller, enabling us to integrate forecasted errors into the controller’s original output, as illustrated in Fig. 2. We call this approach the Data-Driven Inverse Dynamics (DID) error discovery as we use the weighted recursive least-squares method with regularization, which maps the system’s configurations to the residue of control signals [2] . Its efficiency comes from rank-1 updates and the ability to outperform the NN-based part in updating. One of the promising methods to improve these state-of-the-art results is Time-Delay Embedding (TDE), which has already seen applications in robotics modeling [3] . This study assesses whether TDE can enhance our DID error modeling and ultimately improve control performance. 2 Integrating Time-Delay Embedding into Data-Driven Inverse Dynamics The DID method develops a linearized error model D ∈R2 ×8. It maps from zj = hxTj rTj+1iT ∈R8 , which comprises the current state of the system xj ∈ R4 (position and velocity for each degree of freedom) and the desired next state rj+1 ∈ R4 to the residual error ej = [e1,j e2,j]T ∈ R2 of the control signals, which drive the actuators moving the system’s arms. This can be formulated as Here, the residual is definedeja= Ds thezjd,iffwerenee be jeteF j−en thjacfotul c=ntr,l i, g.n. a. ls Fj delivered to the full(1y-) actuated robotic system and the control forces j predicted by the combined action of the LNN and FNN. Figure 1: Photo of the 2 DOF manipulator with flexure joints (photo by Ger Folkersma) . The model D is derived from D = EZ 􀀀 ZZT + αI􀀁 −1, where E, Z constitute the training dataset and α is the reguralization parameter. This dataset is constructed to mimic ideal tracking. Specifically, it is created from measurements assuming the desired state ri+1 ← xj is precisely reached from the current state xi ← xj−1 (accordingly ei ← ej−1) . If we build the model with such snapshot pairs (ei , zi) from steps i = 1 , . . . , k, these matrices are formed as follows Ek = 􀀂e1 e2 ... ek−1 􀀃 , Z k = 􀀔xx12 x2 ... x3 ... xkx−k1􀀕 . (2) The TDE involves augmenting the matrices Ek , Zk with time-shifted copies of the measurements, transforming them into Hankel matrices. This process can be represented as follows 􀀲 e 1 e2 ... ek−NTDE−1􀀳 􀀲􀀶 x 1x 2 x 2x 3 .

FAQs

What controller architecture is used for feedforward control in the study?

The controller combines a Lagrangian Neural Network (LNN) for conservative forces with a Feedforward Neural Network (FNN) that forecasts variances, complemented by residual modeling for DID error discovery.

How does Data-Driven Inverse Dynamics (DID) define and learn the residual error?

DID maps system states and desired next states to residual control-signal errors using a weighted recursive least-squares method with regularization, allowing forecasted errors to be integrated into the controller output.

What effect does adding Time-Delay Embedding (TDE) have on control error and computation?

Increasing embeddings reduces mean residue error, notably for e1, and decreases standard deviation in an exponential-like manner; however, computational cost rises as the DID model updates become more expensive.

Aug 5, 2026

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