Enhancing Kinematic Performances of Soft Continuum Robots for Magnetic Actuation

TL;DR

This work tackles the challenge of improving kinematic performance in magnetically actuated soft continuum robots by unifying equilibrium mechanics with kinematic metrics. It introduces a framework that uses Riemannian Jacobian spectra on the equilibrium manifold and an energy-based equilibrium computation to evaluate and optimize structural design under magnetic actuation. The authors derive analytical results for weak uniform fields, and develop a two-level gradient-based optimization for general dipole fields and multi-magnet configurations, with validation from simulations and physical experiments. Key findings show that constructive torque interactions and cancellation zones, dictated by magnet placement and orientation, organize the global kinematic behavior and provide scalable design principles across actuation regimes.

Abstract

Soft continuum robots achieve complex deformation through elastic equilibrium, making their reachable motions governed jointly by structural design and actuation-induced mechanics. This work develops a general formulation that integrates equilibrium computation with kinematic performances by evaluating Riemannian Jacobian spectra on the equilibrium manifold shaped by internal/external loading. The resulting framework yields a global performance functional that directly links structural parameters, actuation inputs, and the induced configuration space geometry. We apply this general framework to magnetic actuation. Analytical characterization is obtained under weak uniform fields, revealing optimal placement and orientation of the embedded magnet with invariant scale properties. To address nonlinear deformation and spatially varying fields, a two-level optimization algorithm is developed that alternates between energy based equilibrium search and gradient based structural updates. Simulations and physical experiments across uniform field, dipole field, and multi-magnet configurations demonstrate consistent structural tendencies: aligned moments favor distal or mid-distal solutions through constructive torque amplification, whereas opposing moments compress optimal designs toward proximal regions due to intrinsic cancellation zones.

Figures (5)

• Figure 1: Illustration of the basic concept of the PRB model on describing the SCR. The soft body is discretized into rigid rods connected by joints with local frames. The configuration is represented by the joint variables $\theta_\mathbf{t}^i$, $\theta_\mathbf{u}^i$, and $\theta_\mathbf{v}^i$. The system reaches an equilibrium state (②) when the torques of the actuators (①) are balanced by the elastic restoring moments, and the joint deformation is stored as elastic strain.
• Figure 2: a.) Normalized manipulability and dexterity of MeSCRs with two embedded magnets as functions of normalized magnet position $L_{k_0}/L\in[0, 1]$: (left) aligned magnetic moments, (right) opposing magnetic moments. b.) Distal end trajectories under various magnet placements $L_{k_0}/L$ and moments. Color gradient encodes normalized local manipulability.
• Figure 3: Experimental setup: A permanent magnet attached to a 6-DoF robot arm generates a spatially varying dipole field. The MeSCR is placed inside a transparent container, while a downward-facing camera records its deformation for subsequent shape extraction and Jacobian estimation. Insets show details of the MeSCR geometry (outer diameter 1.2mm, embedded magnet length 5mm) and an example of the extracted centerline.
• Figure 4: Performance evaluation under three dipole-field task paths. a.) Illustration of the three task paths. b.)–f.) Normalized kinematic performances with respect to the movable magnet position $L_{k_0}/L$ with same and opposite directions of orientation: b.) $\mathcal{Z}_{\mathrm{vol}}$, c.) $\mathcal{Z}_{\mathrm{dis}}$, d.) $\mathcal{Z}_{\kappa}$, e.) $\mathcal{Z}_{\mathrm{vol}}/\mathcal{Z}_{\mathrm{dis}}$, and f.) $\mathcal{Z}_{\mathrm{vol}}/\mathcal{Z}_{\kappa}$. Solid curves correspond to numerical results for the three paths, and dashed lines indicate the optimal placements. Experimental data are plotted as discrete markers.
• Figure 5: Optimal movable magnet position under varying values of the nondimensional parameter $\nu$. a.) Same direction magnetization. b.) Opposite direction magnetization.

Theorems & Definitions (17)

• Proposition 1
• Proposition 2
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3: Material-twist-free
• proof
• Corollary 1