PINNs-enabled inverse programming of magnetic soft continuum robots: Shape morphing and tip trajectory

Magnetic soft continuum robots (MSCRs) offer transformative potential for minimally invasive procedures due to their high flexibility and magnetic responsiveness. However, reliable and efficient programming of MSCRs for anatomical adaptability and precise tip manipulation remains a key challenge, particularly in navigating tortuous pathways and targeting hard-to-reach lesions. Addressing this, we propose a unified inverse programming framework based on Physics-Informed Neural Networks (PINNs) that simultaneously tackles two critical design objectives in MSCR applications: shape morphing and tip trajectory control. The shape morphing problem involves programming magnetization distributions during fabrication to achieve desired global geometries, while trajectory control is realized by designing time-varying magnetic fields to guide the robot tip along prescribed paths. Leveraging the hard-magnetic elastica model, we reformulate the inverse design challenge into solving a nonlinear ordinary differential equation (ODE). The proposed PINN-based framework seamlessly integrates physical priors into the learning process, enabling rapid convergence while requiring only sparse data. We validate our approach using complex geometries, including shapes resembling the letters “USTC”, and benchmark the results against finite difference (FDM) and finite element method (FEM) simulations. The strong agreement across methods confirms the reliability and accuracy of the PINN-based framework. Our method offers a versatile and computationally efficient tool for the inverse design and control of programmable MSCRs and opens new pathways for data-free, high-fidelity, multi-objective optimization in magnetically actuated soft robotics.