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

Rana Danesh, Pari Qarehdaghi, Farrokh Janabi-Sharifi

arXiv:2608.26273v1cs.ROcs.LG

TL;DR

Closed-chain CCRs require static shape estimates that satisfy both equilibrium and geometric loop closure, while purely data-driven models may violate these constraints under limited or noisy data. The paper develops a constraint-aware PINN with projected equilibrium and configuration-level geometric residuals, achieving improved robustness and physical consistency, fast simulation inference, and substantially better experimentally fine-tuned marker predictions.

  • Problem

    Closed-chain CCR shape estimation is difficult because coupled arms and a flexible object must satisfy static equilibrium and geometric loop-closure constraints, while data-driven models may violate mechanics under noisy or unseen inputs.

  • Method

    A constraint-aware PINN uses projected static equilibrium and configuration-level geometric residuals, with simulation pretraining followed by experimental fine-tuning.

  • Results

    After fine-tuning, marker RMSE decreases from 2.657 mm to 0.497 mm and R2 increases from -0.788 to 0.937 on held-out experimental data.

  • Takeaways & Limitations

    The learned mapping provides accurate, mechanically consistent, and computationally efficient static shape estimation for closed-chain CCRs within the evaluated simulation and experimental settings.

Abstract

from arXiv · show

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.

I. INTRODUCTION

Co-manipulative continuum robots form mechanically coupled closed chains whose static shape estimation must satisfy both robot mechanics and geometric constraints. The paper develops a constraint-aware PINN using GVS modeling to predict physically consistent equilibrium configurations from tendon inputs.

  • Motivation: Closed-chain CCRs couple multiple continuum arms through a manipulated object, imposing geometric constraints and internal reaction forces that complicate static modeling.The arms and object form a closed-chain mechanism, so individual arm configurations are not independent.
  • Motivation: Purely data-driven ANNs can evaluate actuation-to-configuration mappings quickly but may violate static equilibrium or loop-closure constraints under unseen or noisy inputs.Large representative experimental datasets are also difficult to collect.
  • Contributions: The proposed constraint-aware PINN incorporates static equilibrium and geometric loop-closure constraints into learning to predict physically consistent equilibrium configurations from tendon actuation inputs.The formulation targets closed-chain CCRs rather than individual continuum robots.
  • Contributions: The formulation enforces static equilibrium in a constraint-consistent subspace and loop closure through a configuration-level constraint residual.This distinguishes the method from approaches focused primarily on individual continuum or soft robots.
  • Evaluation: The study evaluates the PINN against a data-driven ANN under limited and noisy data and against an iterative nonlinear solver for accuracy, consistency, robustness, and efficiency.The framework also pretrains in simulation and fine-tunes using measured tendon displacements and marker positions.
  • GVS modeling: The GVS model represents continuous strain fields with finite spatial basis functions and reconstructs nonlinear poses using exponential-map spatial integration.Angular and linear strain components account for bending, torsion, shear, and axial deformation.

B. Static Equilibrium

The static equilibrium model combines elastic, tendon-actuation, and gravitational generalized forces for the GVS coordinates. Constitutive stiffness maps strain deviations into elastic forces, while tendon routing determines configuration-dependent actuation forces.

  • Elastic forces: At static equilibrium, internal elastic wrench is related to strain deviation from the reference configuration through a linear elastic constitutive law.The cross-sectional stiffness matrix specifies the elastic response.
  • Elastic forces: The global generalized elastic force is expressed as Qe = Kq after assembling stiffness contributions from the deformable bodies.Each body contributes a corresponding stiffness block to the global matrix.
  • Actuation: The generalized tendon actuation force depends on tendon inputs through the configuration-dependent matrix Bq(q), which incorporates routing geometry and its variation with configuration.Each column of Bq represents the generalized force from a unit input of the corresponding tendon.
  • Gravity: Gravity is mapped to generalized coordinates using the geometric Jacobian and a spatial-frame gravity twist.The resulting generalized gravitational force contributes to the static equilibrium equations.
  • Equilibrium: With no external wrench other than gravity, the unconstrained static equilibrium combines tendon actuation, gravitational loading, and elastic resistance.This equation precedes incorporation of the closed-chain constraints.

C. Closed-Chain Geometric Constraints and Jacobian

Closed-loop connections impose configuration-level geometric closure constraints between paired bodies. Their Jacobian captures compatible differential directions and supports the treatment of internal reaction forces in constrained equilibrium.

  • Loop geometry: Each closed-loop connection relates two bodies whose connection-frame transformations determine relative position and orientation differences.Fixed offsets between body frames and actual connection frames are included through constant transformations.
  • Loop geometry: The relative transformation between connection frames is used to define the geometric closure error.The error measures the difference between the two sides of each closed-loop connection.
  • Geometric constraints: Geometric closure requires the constrained components of each relative transformation to be zero.The matrix Bp,i selects the wrench directions constrained by connection i.
  • Geometric constraints: Individual closure errors are combined into a global geometric constraint vector for all closed-loop connections.The total number of scalar constraints is denoted by nc.
  • Constraint Jacobian: The global constraint Jacobian A(q) stacks connection-level Jacobian blocks, and its null space defines differential directions compatible with closed-chain constraints.The Jacobian also supports accounting for unknown closed-chain reaction forces.
  • Constraint Jacobian: The geometric constraint vector e(q) measures configuration-level closure error, whereas A(q) defines differential constraint directions for reaction-force treatment.These quantities therefore serve distinct roles in the closed-chain model.

D. Constraint-Consistent Static Equilibrium

Closed-chain reactions are represented with Lagrange multipliers but need not be solved explicitly. Projection onto the constraint Jacobian’s null space removes their contribution, after which the projected equilibrium residual is combined with geometric closure error.

  • Constrained equilibrium: Closed-chain connections generate internal reaction forces and moments represented by the Lagrange multiplier vector λ.These reactions maintain the geometric constraints.
  • Projection: The constrained equilibrium equation is projected onto the null space of the constraint Jacobian to eliminate unknown reaction forces.The projection assumes A(q) has full row rank.
  • Projection: The projector satisfies P(q)A^T(q) = 0, so constraint reactions do not contribute to the projected equilibrium.This removes the need to determine reaction forces explicitly.
  • Residual construction: The projected physics residual enforces static equilibrium only in directions consistent with the closed-chain constraints.It does not independently guarantee geometric loop closure.
  • Residual construction: The complete static residual combines the projected equilibrium residual with the configuration constraint vector e(q).For a prescribed tendon input u, the static configuration q* satisfies the resulting constrained conditions.

III. PHYSICS-INFORMED STATIC SHAPE ESTIMATION

The proposed PINN learns tendon-input-to-equilibrium-GVS mappings while enforcing projected static equilibrium and closed-chain geometric consistency through differentiable physics residuals.

  • The PINN learns the mapping from tendon tension vector u to equilibrium GVS coordinates q, replacing per-input iterative nonlinear solving.
  • Tendon tensions are normalized to [−1, 1], while generalized coordinates are standardized using training-set means and standard deviations.
  • The network maps normalized tendon inputs to standardized GVS coordinates, which are transformed back to physical coordinates for strain and backbone reconstruction.
  • The fully connected network uses four hidden layers of 128 neurons with hyperbolic tangent activations and a linear output layer for 16 standardized GVS coordinates.
  • Physics residuals are evaluated through the differentiable GVS model, and automatic differentiation backpropagates gradients through the network and forward mechanics.
  • The training objective combines supervised data, projected static equilibrium, and closed-chain geometric constraint losses with separately scaled weights.

IV. SIMULATION RESULTS

The simulation study evaluates the PINN against both a data-driven ANN and an iterative nonlinear static solver across accuracy, physical consistency, robustness, and computation time.

  • The simulation has two stages: comparison with an ANN under varied data and label noise, followed by benchmarking against an iterative nonlinear static solver.

A. Dataset Generation and Training Setup

The study generates constrained static GVS data over tendon actuation ranges and tests ANN and PINN performance with limited samples and corrupted training labels.

  • Each CR is modeled as a single soft link, while the flexible object uses bending and spatial-motion coordinates in the complete CCR model.
  • Four tendon inputs vary independently from −5 to 5 N in 1 N increments, representing differential actuation of antagonistic tendon pairs.
  • Training sets contain 28, 70, or 140 configurations, with identical samples for ANN and PINN and noise-free held-out evaluation data.
  • Training-label robustness is tested with 0%, 25%, and 50% Gaussian noise applied only to GVS-coordinate labels.
  • Both models share four 128-neuron hidden layers, tanh activations, a linear output, and training with Adam for 500 epochs at a 10−3 learning rate.
  • The PINN augments the ANN's supervised loss with projected equilibrium and closed-chain geometric constraint losses.
  • At initialization, gradient balancing yields λphys ≈2899.8 and λcon ≈2392.3.

B. PINN and ANN Comparison

The PINN is more robust than the ANN under noisy supervision and offers much faster inference than the nonlinear solver, while the solver remains more accurate in equilibrium prediction.

  • PINN and ANN Comparison: 67.88% relative configuration-error reduction is achieved with 140 samples and 50% label noise compared with the ANN.
  • PINN and ANN Comparison: At 50% noise, PINN relative error falls from 2.38% with 28 samples to 1.04% with 140 samples, while ANN errors remain substantially larger.
  • PINN and ANN Comparison: 67.35% and 88.06% reductions are obtained for projected equilibrium and closed-chain residuals, respectively, with 140 samples and 50% noise.
  • PINN and Nonlinear Solver Comparison: The nonlinear solver achieves 0.0053% relative ℓ2 error versus 0.1597% for the PINN and produces smaller physical residuals.
  • PINN and Nonlinear Solver Comparison: 0.1773 ms PINN inference is substantially faster than the nonlinear solver's 17.97 s per prediction.
  • Experimental Setup: The experimental platform uses two tendon-driven continuum arms rigidly connected to a flexible Nitinol rod, forming the closed-chain structure.

B. Static Experimental Data Collection

The experimental dataset uses static tendon actuation and Vicon marker measurements to represent CCR configurations. The PINN maps tendon displacements to marker positions and generalized coordinates while embedding GVS kinematics, tendon-length, equilibrium, and closed-chain constraints.

  • Static tendon actuation was varied within [−5, 5] N, and each configuration was allowed to settle before measurement.
  • Experimental samples used tendon displacements and marker displacements relative to the zero-input equilibrium configuration.
  • 817 valid static configurations were recorded using tendon displacements and the three-dimensional positions of eight Vicon markers.
  • The experimental PINN uses tendon displacements as input and predicts marker positions plus auxiliary generalized coordinates.
  • The generalized coordinates connect predictions to GVS forward kinematics, tendon-length consistency, projected equilibrium, and closed-chain geometric constraints.

D. Experimental Fine Tuning

Fine tuning adapts the simulation-pretrained PINN to measured marker data while retaining selected physical consistency terms. Because experimental generalized-coordinate labels are unavailable, a frozen pretrained model supplies auxiliary coordinate references.

  • The 817 experimental configurations were partitioned into 571 training, 122 validation, and 124 held-out test samples without splitting repeated actuation inputs.
  • A frozen simulation-pretrained PINN provided auxiliary generalized-coordinate predictions because experimental labels for q were unavailable.
  • Fine tuning retained simulation-calibrated loss weights while reducing the physics contribution with αphys = 0.05.
  • Training alternated experimental marker and coordinate-consistency losses with physics-term evaluation on two samples in every third mini batch.
  • The selected model was obtained at epoch 149 using validation marker RMSE for model selection.

E. Experimental Evaluation and Results

Evaluation on held-out experimental configurations shows that fine tuning substantially improves marker prediction accuracy and agreement across both continuum robots. The results support the PINN’s simulation-to-experiment transfer while retaining its physics-informed formulation.

  • The held-out evaluation used 124 experimental configurations excluded from fine tuning and model selection.
  • Marker RMSE decreased from 2.657 mm to 0.497 mm, an 81.3% reduction, after experimental fine tuning.
  • R2 increased from −0.788 to 0.937, while predictions within 1 mm rose from 37.30% to 94.96%.
  • Fine tuning reduced prediction error at all eight measured marker locations, with the largest post-tuning RMSE equal to 0.827 mm at CR1 M4.
  • After fine tuning, predicted marker displacements closely followed measured values over the experimental range and retained the physics-informed formulation.
  • The conclusion identifies future extensions to dynamic modeling, time-varying tendon inputs, online adaptation, and changing robot properties.
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