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

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.Source: §I · p. 1
• 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.Source: §I · p. 1

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.Source: §I · pp. 1–2
• 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.Source: §I · p. 2

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.Source: §I · p. 2

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.Source: A. GVS Kinematics · p. 2

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.Source: B. Static Equilibrium · p. 2
• 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.Source: B. Static Equilibrium · p. 3

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.Source: B. Static Equilibrium · p. 3

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.Source: B. Static Equilibrium · p. 3

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.Source: B. Static Equilibrium · p. 3

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.Source: C. Closed-Chain Geometric Constraints… · p. 3
• 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.Source: C. Closed-Chain Geometric Constraints… · p. 3

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.Source: C. Closed-Chain Geometric Constraints… · p. 3
• 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.Source: C. Closed-Chain Geometric Constraints… · p. 3

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.Source: C. Closed-Chain Geometric Constraints… · p. 3
• 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.Source: C. Closed-Chain Geometric Constraints… · p. 3

Constrained equilibrium

• Closed-chain connections generate internal reaction forces and moments represented by the Lagrange multiplier vector λ.These reactions maintain the geometric constraints.Source: D. Constraint-Consistent Static… · p. 3

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.Source: D. Constraint-Consistent Static… · p. 3
• 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.Source: D. Constraint-Consistent Static… · p. 3

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.Source: D. Constraint-Consistent Static… · p. 4
• 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.Source: D. Constraint-Consistent Static… · p. 4

• The PINN learns the mapping from tendon tension vector u to equilibrium GVS coordinates q, replacing per-input iterative nonlinear solving.Source: §III · p. 4
• Tendon tensions are normalized to [−1, 1], while generalized coordinates are standardized using training-set means and standard deviations.Source: §III · p. 4
• The network maps normalized tendon inputs to standardized GVS coordinates, which are transformed back to physical coordinates for strain and backbone reconstruction.Source: §III · p. 4
• 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.Source: §III · p. 4
• Physics residuals are evaluated through the differentiable GVS model, and automatic differentiation backpropagates gradients through the network and forward mechanics.Source: §III · p. 4
• The training objective combines supervised data, projected static equilibrium, and closed-chain geometric constraint losses with separately scaled weights.Source: §III · p. 4

• The simulation has two stages: comparison with an ANN under varied data and label noise, followed by benchmarking against an iterative nonlinear static solver.Source: §IV · p. 4

• Each CR is modeled as a single soft link, while the flexible object uses bending and spatial-motion coordinates in the complete CCR model.Source: A. Dataset Generation and Training Setup · p. 5
• Four tendon inputs vary independently from −5 to 5 N in 1 N increments, representing differential actuation of antagonistic tendon pairs.Source: A. Dataset Generation and Training Setup · p. 5
• Training sets contain 28, 70, or 140 configurations, with identical samples for ANN and PINN and noise-free held-out evaluation data.Source: A. Dataset Generation and Training Setup · p. 5
• Training-label robustness is tested with 0%, 25%, and 50% Gaussian noise applied only to GVS-coordinate labels.Source: A. Dataset Generation and Training Setup · p. 5
• 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.Source: A. Dataset Generation and Training Setup · p. 5
• The PINN augments the ANN's supervised loss with projected equilibrium and closed-chain geometric constraint losses.Source: A. Dataset Generation and Training Setup · p. 5
• At initialization, gradient balancing yields λphys ≈2899.8 and λcon ≈2392.3.Source: A. Dataset Generation and Training Setup · p. 5

PINN and ANN Comparison

• 67.88% relative configuration-error reduction is achieved with 140 samples and 50% label noise compared with the ANN.Source: B. PINN and ANN Comparison · p. 5
• 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.Source: B. PINN and ANN Comparison · p. 5
• 67.35% and 88.06% reductions are obtained for projected equilibrium and closed-chain residuals, respectively, with 140 samples and 50% noise.Source: B. PINN and ANN Comparison · pp. 5–6

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.Source: C. PINN and Nonlinear Static Solver… · p. 6
• 0.1773 ms PINN inference is substantially faster than the nonlinear solver's 17.97 s per prediction.Source: C. PINN and Nonlinear Static Solver… · p. 6

Experimental Setup

• The experimental platform uses two tendon-driven continuum arms rigidly connected to a flexible Nitinol rod, forming the closed-chain structure.Source: A. Experimental Setup · p. 6

• Static tendon actuation was varied within [−5, 5] N, and each configuration was allowed to settle before measurement.Source: B. Static Experimental Data Collection · pp. 6–7
• Experimental samples used tendon displacements and marker displacements relative to the zero-input equilibrium configuration.Source: B. Static Experimental Data Collection · p. 7
• 817 valid static configurations were recorded using tendon displacements and the three-dimensional positions of eight Vicon markers.Source: B. Static Experimental Data Collection · p. 7
• The experimental PINN uses tendon displacements as input and predicts marker positions plus auxiliary generalized coordinates.Source: C. Simulation Pretraining for… · p. 7
• The generalized coordinates connect predictions to GVS forward kinematics, tendon-length consistency, projected equilibrium, and closed-chain geometric constraints.Source: C. Simulation Pretraining for… · p. 7

• The 817 experimental configurations were partitioned into 571 training, 122 validation, and 124 held-out test samples without splitting repeated actuation inputs.Source: D. Experimental Fine Tuning · p. 7
• A frozen simulation-pretrained PINN provided auxiliary generalized-coordinate predictions because experimental labels for q were unavailable.Source: D. Experimental Fine Tuning · p. 7
• Fine tuning retained simulation-calibrated loss weights while reducing the physics contribution with αphys = 0.05.Source: D. Experimental Fine Tuning · p. 8
• Training alternated experimental marker and coordinate-consistency losses with physics-term evaluation on two samples in every third mini batch.Source: D. Experimental Fine Tuning · p. 8
• The selected model was obtained at epoch 149 using validation marker RMSE for model selection.Source: D. Experimental Fine Tuning · p. 8

• The held-out evaluation used 124 experimental configurations excluded from fine tuning and model selection.Source: E. Experimental Evaluation and Results · p. 8
• Marker RMSE decreased from 2.657 mm to 0.497 mm, an 81.3% reduction, after experimental fine tuning.Source: E. Experimental Evaluation and Results · p. 8
• R2 increased from −0.788 to 0.937, while predictions within 1 mm rose from 37.30% to 94.96%.Source: E. Experimental Evaluation and Results · p. 8
• 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.Source: E. Experimental Evaluation and Results · p. 8, Figure 4 · p. 8
• After fine tuning, predicted marker displacements closely followed measured values over the experimental range and retained the physics-informed formulation.Source: E. Experimental Evaluation and Results · p. 8, Figure 5 · p. 8
• The conclusion identifies future extensions to dynamic modeling, time-varying tendon inputs, online adaptation, and changing robot properties.Source: §VI · p. 9