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Physics informed neural network for parameter identification and boundary force estimation of compliant and biomechanical systems

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Abstract

This paper presents the framework of a physics-informed neural network (PINN) with a boundary condition-embedded approximation function (BCAF) for solving common problems encountered in flexible mechatronics and soft robotics; both forward and inverse problems are considered. Unlike conventional PINNs that minimize a lumped loss function including the errors contributed by the initial or boundary conditions (ICs or BCs), the BCAF-PINN completely satisfies the ICs and/or BCs while minimizing a loss function for parameter identification and boundary force estimation, overcoming a common erroneous-convergence problem due to unbalanced gradients in training a PINN. The formulation and implementation of a BCAF-PINN are illustrated with three practical applications, including a nonlinear system where solutions are available for numerical verification and for comparing with conventional PINNs, and a biomechanical system where a BCAF-PINN uses multiple cycles of natural foot flexion to identify its dynamic parameters experimentally. While overcoming several problems associated with traditional studies based on perturbation models with certain level of muscle contraction, the damping ratio identified by the BCAF-PINN indicates that the ankle joint is an overdamped system during flexion, consistent with that observed in published experiments.

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Abbreviations

ANN:

Artificial neural network

BC:

Boundary condition

BCAF:

BC-embedded approximation function

BCLF:

BC loss function

BVP:

Boundary value problem

DOF:

Degree of freedom

DF:

Dorsiflexion

FM:

First metarsal heads

FSM:

Full state measurement

HF:

Head of fibula

IBK:

Inertia I, viscous B and elastic K model

IC:

Initial condition

IVP:

Initial value problem

LM:

Lateral malleolus

MM:

Medial malleolus

MSO:

Multiple single output

ODE:

Ordinary differential equation

PDE:

Partial differential equation

PF:

Plantarflexion

PINN:

Physics informed neural network

PSM:

Partial state measurement

RC:

Ridge of the calcaneus

SMO:

Single multiple output

SS:

State space

VM:

Fifth metarsal heads

A :

ICs/BCs satisfied function entry

E, G :

Elastic and shear modulus

F :

ICs/BCs no contribution function entry

I :

Area moment of inertia

J :

Average angular moment of inertia of foot

L :

Length

N :

Network output entry

M :

Number of measurements

A :

ICs/BCs satisfied function vector

F :

ICs/BCs no contribution function vector

N :

Network output vector

P C :

Position vector of beam end point

a :

Initial/boundary conditions

b, h :

Beam width, thickness

s, t :

Path-length,, time

x, y :

Independent, dependent variable

y a :

BCAF dependent variable entry

\(\hat{y}_{a}\) :

BCLF dependent variable entry

u :

Applied force

u 1, u 2 :

Applied force in y1, y2 directions

u M :

Ankle joint net torque

e 0 :

Initial error

e b :

Boundary error

e i :

Measurement error

p :

Network weights and biases

r p :

Residual

x, y :

Independent, dependent variable vectors

y a :

BCAF dependent variable vector

\({\hat{\mathbf{y}}}_{a}\) :

BCLF dependent variable vector

Γ:

Structural rigidity

\({{\mathcal{G}}}\) :

Differential operator

\({{\mathcal{L}}}_{\text{F}}\) :

BCAF forward loss function

\({{\mathcal{L}}}_{\text{I}}\) :

BCAF inverse loss function

\({\hat{\mathcal{L}}}_{\text{F}}\) :

BCLF forward loss function

\({\hat{\mathcal{L}}}_{\text{I}}\) :

BCLF inverse loss function

α, α :

Coefficient entry/vector

λ :

Magnitude tuning factor

ρ,ν :

Density and Poisson ratio

ψ,κ :

Angle of rotation and curvature

η :

Initial angle of rotation

θ :

Foot flexion angle

ζ,ω n :

Damping ratio and natural frequency

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Acknowledgements

This work was supported in part by the U.S. National Science Foundation under Grant CMMI-1662700.

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  1. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 30332-0405, USA

    Wenjing Li & Kok-Meng Lee

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  1. Wenjing Li
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  2. Kok-Meng Lee
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Correspondence to Kok-Meng Lee.

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Li, W., Lee, KM. Physics informed neural network for parameter identification and boundary force estimation of compliant and biomechanical systems. Int J Intell Robot Appl 5, 313–325 (2021). https://doi.org/10.1007/s41315-021-00196-x

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