# Constraint-Aware Physics-Informed Neural Networks for Static Shape Estimation of Co-Manipulative Continuum Robots

arXiv: https://arxiv.org/abs/2608.26273 · HTML: https://mlsanity.com/paper/2608.26273/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 32 references that are in the map.
- First posted on arXiv 26 Aug 2026.

## Builds on (most cited first, 15 of 15)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Soft Robots Modeling: a Structured Overview](https://mlsanity.com/paper/2112.03645/) (2021)
- [Physics-informed Neural Networks to Model and Control Robots: a Theoretical and Experimental Investigation](https://mlsanity.com/paper/2305.05375/) (2023)
- [Data-driven Methods Applied to Soft Robot Modeling and Control: A Review](https://mlsanity.com/paper/2305.12137/) (2023)
- [Discrete Cosserat Approach for Multi-Section Soft Robots Dynamics](https://mlsanity.com/paper/1702.03660/) (2017)
- [A MATLAB Toolbox for Hybrid Rigid Soft Robots Based on the Geometric Variable Strain Approach](https://mlsanity.com/paper/2107.05494/) (2021)
- [PINN-Ray: A Physics-Informed Neural Network to Model Soft Robotic Fin Ray Fingers](https://mlsanity.com/paper/2407.08222/) (2024)
- [Generalizable and Fast Surrogates: Model Predictive Control of Articulated Soft Robots using Physics-Informed Neural Networks](https://mlsanity.com/paper/2502.01916/) (2025)
- [Adaptive Model-Predictive Control of a Soft Continuum Robot Using a Physics-Informed Neural Network Based on Cosserat Rod Theory](https://mlsanity.com/paper/2508.12681/) (2025)
- [Backstepping Control of Tendon-Driven Continuum Robots in Large Deflections Using the Cosserat Rod Model](https://mlsanity.com/paper/2412.12035/) (2024)
- [Hybrid Visual Servoing of Tendon-driven Continuum Robots](https://mlsanity.com/paper/2502.14092/) (2025)
- [Collaborative Continuum Robots: A Survey](https://mlsanity.com/paper/2601.10721/) (2025)
- [Real-Time Projected Adaptive Control for Closed-Chain Co-Manipulative Continuum Robots](https://mlsanity.com/paper/2604.04286/) (2026)
- [Few-Shot Physics-Informed Neural Network for Shape Reconstruction of Concentric-Tube Robots](https://mlsanity.com/paper/2605.12790/) (2026)
- [Active Stiffness Control of a Supportive Continuum Robot](https://mlsanity.com/paper/2608.03677/) (2026)

## Abstract (arXiv)

Static shape estimation of co-manipulative continuum robots (CCRs) is challenging because the continuum arms and manipulated flexible object form a closed chain that must satisfy both static equilibrium and geometric loop-closure constraints. This paper presents a constraint-aware physics-informed neural network (PINN) for static shape estimation of a tendon-driven CCR modeled using the geometric variable strain formulation. The proposed method incorporates a projected static equilibrium residual and a configuration-level geometric residual to enforce the governing mechanics and closed-chain geometry. In simulation, the PINN is compared with a purely data-driven artificial neural network (ANN) under limited and noisy training data. With 140 samples and 50% label noise, the PINN reduces the relative configuration error, equilibrium residual, and closed-chain residual by 67.88%, 67.35%, and 88.06%, respectively. Using the full dataset, the PINN achieves 0.1597% relative configuration error with an inference time of 0.1773 ms, compared with 17.97 s for an iterative nonlinear solver. Experimental fine-tuning reduces the marker RMSE from 2.657 mm to 0.497 mm and increases R2 from -0.788 to 0.937. These results demonstrate accurate, physically consistent, and computationally efficient static shape estimation of closed-chain CCRs.