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PINN-Ray: A Physics-Informed Neural Network to Model Soft Robotic Fin Ray Fingers
Xing Wang, Joel Janek Dabrowski, Josh Pinskier, Lois Liow, Vinoth Viswanathan, Richard Scalzo, David Howard
TL;DR
Soft-robotics models face challenges from complex deformation and a sim-to-real gap. PINN-Ray combines elastic mechanics with machine learning and assimilates real-finger data; on a Fin Ray finger, the authors report improved displacement accuracy after assimilation.
Problem
Complex geometry, large deformation, and material nonlinearity challenge accurate soft-robotics surrogates, while model estimates can differ from reality.
Source: §I · p. 1Method
PINN-Ray incorporates minimum potential energy from elastic mechanics and real Fin Ray measurements into neural-network training.
Source: §I · pp. 1–2Results
PINN-Rayasm achieved a displacement MAE of 0.03 compared with 1.463 for displacement, with nine assimilated data points.
Source: §VI · p. 8Takeaways & Limitations
The authors report that assimilation improves convergence speed and stress and strain estimates relevant to evaluating soft-robot durability.
Source: §VI · p. 8
Abstract
from arXiv · showhide
Modelling complex deformation for soft robotics provides a guideline to understand their behaviour, leading to safe interaction with the environment. However, building a surrogate model with high accuracy and fast inference speed can be challenging for soft robotics due to the nonlinearity from complex geometry, large deformation, material nonlinearity etc. The reality gap from surrogate models also prevents their further deployment in the soft robotics domain. In this study, we proposed a physics-informed Neural Networks (PINNs) named PINN-Ray to model complex deformation for a Fin Ray soft robotic gripper, which embeds the minimum potential energy principle from elastic mechanics and additional high-fidelity experimental data into the loss function of neural network for training. This method is significant in terms of its generalisation to complex geometry and robust to data scarcity as compared to other data-driven neural networks. Furthermore, it has been extensively evaluated to model the deformation of the Fin Ray finger under external actuation. PINN-Ray demonstrates improved accuracy as compared with Finite element modelling (FEM) after applying the data assimilation scheme to treat the sim-to-real gap. Additionally, we introduced our automated framework to design, fabricate soft robotic fingers, and characterise their deformation by visual tracking, which provides a guideline for the fast prototype of soft robotics.
I. INTRODUCTION
Soft robotics’ flexibility creates indefinite degrees of freedom, making accurate, robust models difficult to construct; model estimates can also differ from reality. PINN-Ray combines elastic mechanics and machine learning, assimilating real-finger measurements to model deformation under external actuation.
I. INTRODUCTION
- Soft robotics’ indefinite degrees of freedom make accurate, robust models difficult, while nonlinear structure and material properties contribute to a sim-to-real gap.Data-driven models can require large training datasets and may be inaccurate where training data is scarce.Source: §I · p. 1
- PINN-Ray combines elastic mechanics theory with machine learning, using minimum potential energy in its loss and assimilating real Fin Ray measurements during training.The approach is presented as robust in data-scarce scenarios.Source: §I · pp. 1–2
- The authors compare PINN-Ray, its data-assimilation-free variation, and FEM for estimating a Fin Ray finger’s nonlinear displacement field.Source: §I · pp. 1–2
- The work also presents image processing for real-time intrinsic key-point tracking to characterize soft robotic deformation efficiently.Source: §I · p. 2
II. RELATED WORK
Prior approaches include geometry-based and continuum-mechanics models, data-driven models, and hybrid physics-guided learning. Their reported constraints include geometry-specific modelling challenges and data deficiencies; prior PINN work in soft robotics is described as limited.
II. RELATED WORK
- Geometric models capture shape variation but are constrained by robot shape, while continuum-mechanics approaches require precise geometry and are difficult to generalize across designs.Examples include constant-curvature modelling and continuum mechanics using Neo-Hookean elasticity and Cosserat rod theory.Source: §II · p. 2
- A data-driven soft-robot model used cable-tension training data and infrared-camera tip positions, but data-driven methods are challenged by data deficiencies.Source: §II · p. 2
- Physics-informed learning can embed physical laws through equation-generated training data, network architecture, or loss-function constraints.Source: §II · p. 3
- Prior soft-robotics examples include a physics-constrained surrogate for soft-fish swimming and physics-augmented sensing of pneumatic-actuator bending angle or force.The passage describes PINN endeavors in soft robotics as scant.Source: §II · p. 3
A. Elastic theory
PINN-Ray models Fin Ray deformation using linear elasticity and the minimum-potential-energy formulation, under stated material assumptions and static equilibrium.
A. Elastic theory
- The material is assumed continuous, uniform, and isotropic, and its deformation is characterized with a linear elasticity model.The two-dimensional constitutive equations use plane strain and Hooke’s law.Source: A. Elastic theory · p. 3
- Total potential energy is the sum of internal energy U and external-actuation energy W.U arises from stresses and strains in the body material.Source: A. Elastic theory · p. 3
- Internal energy is expressed using the stress and strain tensors, whose components are indexed by coordinate directions.The stated expression integrates their product over the design domain.Source: A. Elastic theory · p. 3
- Under static equilibrium, the total-energy formulation gives a PDE in displacement field (u, v), which is solved to obtain deformation under external actuation.The constitutive equations relate stress to strain using Young’s modulus E and Poisson’s ratio µ.Source: A. Elastic theory · p. 3
B. physics-informed Neural Network
PINN-Ray represents the displacement field with neural networks and optimizes their parameters to minimize system energy, using automatic differentiation to evaluate derivatives.
B. physics-informed Neural Network
- A neural network ψ(x, y; θ) estimates displacement field (u, v), and its parameters are optimized to minimize total system energy.The energy is represented by the loss function under static equilibrium conditions.Source: B. physics-informed Neural Network · p. 3
- The network takes coordinates (x, y) as inputs and uses automatic differentiation to compute displacement derivatives needed for stress, strain, and energy calculations.The resulting energy is used to optimize the network parameters.Source: B. physics-informed Neural Network · pp. 3–4
- Two separate neural networks predict u and v, and the minimum-potential-energy method is implemented in the loss function.Both networks use hyperbolic tangent activation, four layers, and 64 neurons per layer.Source: Figure 2 · p. 4, B. physics-informed Neural Network · p. 4
- The fully connected architecture used in this study is attributed to.The network input tensor X represents (x, y).Source: B. physics-informed Neural Network · p. 4
C. Loss function
PINN-Ray trains against total-energy PDE, boundary-condition, and data-assimilation losses, with separate weights for the latter two components.
C. Loss function
- The objective combines total-energy PDE loss, boundary-condition loss, and data-assimilation loss, weighted by λBC and λASM for the latter two terms.The weighting parameters control their respective loss components.Source: C. Loss function · p. 4
- Data-assimilation loss compares predicted and measured displacements at available locations, typically measured on a physical body.The passage describes this as a mean squared error loss.Source: C. Loss function · p. 4
- Boundary loss incorporates zero displacement at fixed boundaries and specified displacement at forced boundaries.The forced boundary condition represents an applied force that causes a known boundary displacement.Source: C. Loss function · pp. 4–5
- The PDE loss is the mean squared error of total energy, computed from network outputs and automatic differentiation.For the forced-boundary setup, W is set to zero and the integral for internal energy is approximated by Monte Carlo sampling over domain points.Source: C. Loss function · p. 5
A. Design space & manufacturing
The Fin Ray finger is geometrically designed for two-dimensional modelling and fabricated in a single 3D-printing step with integrated tracking markers.
A. Design space & manufacturing
- The finger geometry is constructed in CAD and governed by design parameters, with a triangle-element discretization used for sampling points.The bottom edge is fixed, and additional markers are assigned for deformation tracking.Source: A. Design space & manufacturing · p. 5
- The design is one-shot 3D printed with markers integrated during printing, using a digital material made by mixing VeroUltra black and Agilus30 clear.The selected material has Shore hardness 85A, Young’s modulus E = 11.4 MPa, and Poisson’s ratio µ = 0.45.Source: A. Design space & manufacturing · p. 5
B. Experiment setup
The experiment evaluates a Fin Ray finger under contact loading, using visual tracking to measure marker displacements and FEM-mesh nodes as sampling points.
B. Experiment setup
- A Jimstron rig moved a rigid object toward the finger to induce deformation, with contact positions adjustable along the finger.The forced boundary condition specified displacement at (35,0).Source: B. Experiment setup · pp. 5–6
- A calibrated RGB camera and image processing tracked printed markers, detecting their centers and converting pixel positions to the finger’s world frame.The processing included Gaussian blur and Hough Circle detection.Source: B. Experiment setup · pp. 5–6, Figure 4 · p. 6
- Marker positions were averaged across frames at static initial and final states, and their difference was used to calculate measured displacement.Nine markers were indexed left to right.Source: B. Experiment setup · p. 6
- For comparison with FEM, its mesh generated collocation points: 90,959 domain points, 1,000 evenly spaced base points, and one forced point at (35,0) producing (0 mm, -10 mm); assimilation used nine initial-position marker measurements.Source: B. Experiment setup · p. 6
V. RESULTS & DISCUSSION
PINN-Ray with data assimilation produced lower marker displacement errors than FEM and the standard PINN-Ray variant. The results also indicate lower PDE loss and highlight the material-design relevance of stress and strain estimates.
V. RESULTS & DISCUSSION
- Both PINN-Ray variants were trained for 60,000 epochs with a 10^-3 learning rate; boundary and assimilation loss weights were each 1000.The larger weights were chosen to enforce those conditions more strongly.Source: §V · p. 6
- PINN-Rayasm achieves very low absolute errors across all markers, significantly improving over FEM and PINN-Raystd; Table I reports MAE.Fig. 7 compares per-marker absolute errors for FEM and PINN-Rayasm.Source: §V · p. 6, Figure 7 · p. 7, Table I · p. 7
- Including data assimilation significantly lowers the PDE loss, which the authors associate with a lower energy level and a more accurate PDE solution.The loss curves are plotted over training epochs in Fig. 8.Source: §V · p. 6, Figure 8 · p. 7
- PINN-Raystd underestimates material stress and strain, which the authors warn can be detrimental when designers consider yield points, elasticity, and toughness.PINN-Rayasm’s estimates are considered likely more accurate because it finds a more optimal PDE solution.Source: §V · p. 7, Figure 9 · p. 7, Figure 10 · p. 8
VI. CONCLUSION
PINN-Ray combines elastic mechanics with a neural network and assimilates experimental measurements to model Fin Ray finger deformation. With data assimilation, it reports lower displacement MAE and faster convergence, alongside more accurate stress and strain estimates.
VI. CONCLUSION
- The evaluated PINN-Ray combines physical laws from elastic theory with a nonlinear neural-network approximator.The approach is tested on the classic Fin Ray robotic finger under external actuation.Source: §VI · p. 8
- Displacement MAE is 0.03 for PINN-Rayasm versus 1.463 for the comparison after assimilating experimental data.Only 9 data points were assimilated.Source: §VI · p. 8
- PINN-Raystd achieves competitive accuracy compared with high-fidelity FEM on the Fin Ray finger deformation task.The evaluation captures deformation under external actuation using FEM, PINN-Raystd, and PINN-Rayasm.Source: §VI · p. 8
- Experimental data assimilation improves modelling accuracy, convergence speed, and the accuracy of latent stress and strain estimates.The paper identifies stress and strain as critical indices for durability evaluation of soft robotics designs.Source: §VI · p. 8