Magnetic Continuum Robots: Principles & Applications

Updated 9 October 2025

Magnetic continuum robots are flexible systems that utilize external magnetic fields to achieve wireless, high-dexterity motion in challenging environments.
They integrate permanent magnets, soft magnetic composites, and ball-chain designs to enable controlled bending, twisting, and translation using advanced modeling and control algorithms.
Applications span minimally invasive surgeries, targeted drug delivery, and micro-assembly, with research advancing optimization, material innovation, and real-time feedback.

Magnetic continuum robots (MCRs) are a class of hyper-redundant, deformable robots actuated via externally applied magnetic fields. These systems integrate distributed or discrete magnetic elements within soft or articulated structures, enabling wirelessly powered, high-dexterity motion for navigation, manipulation, and therapeutic tasks in challenging or constrained environments. Recent advances in actuation principles, modeling, control algorithms, and integrated functionalities have positioned MCRs as promising platforms for minimally invasive interventions, targeted drug delivery, shape-sensing, and micro-assembly.

1. Fundamental Principles and Actuation Architectures

Magnetic continuum robots exploit magnetic torque and/or force generation at distributed locations or designated segments within a deformable structure, typically realized as:

Permanent-magnet-embedded sections (axial, radial, or arbitrary orientation), enabling modulation of bending, twisting, and translation by varying the external field's direction and strength (Zhang et al., 2 Oct 2025).
Distributed soft magnetic composites (magnetically induced metamorphic materials, or MIMMs), which respond to field gradients via volume-averaged magnetic stress and resultant continuum deformation (Shi et al., 3 Aug 2025).
Ball chain or chain-of-spheres designs, in which discrete spherical magnets connected by a flexible sleeve or sleeve-encased lumen achieve locally articulated motion and tight radii of curvature (down to the order of channel radii), with actuation induced by alignment of magnetic dipoles with the applied field (Pittiglio et al., 2023, Pittiglio et al., 30 Jan 2024, Pittiglio et al., 21 Oct 2024).

Actuation principles leverage:

Magnetic torque production ( $\mathbf{m} = \mu \times \mathbf{B}$ ), with $\mu$ the magnetic moment and $\mathbf{B}$ the applied field, for controlled bending or torsion.
Magnetic force ( $\mathbf{F} = (\mu \cdot \nabla) \mathbf{B}$ ), relevant when field gradients are significant, to achieve translation or enhance propulsion (Landers et al., 2023, Jeon et al., 26 Feb 2025).
Mode separation, as achieved via radial-embedded magnet arrays enabling either bending or twist (torsion) depending on the spatial orientation of $\mathbf{B}$ , thereby allowing dual-mode deformation with a single actuation source (Zhang et al., 2 Oct 2025).

2. Modeling Frameworks and Kinematics

MCR kinematics and statics are formulated using either continuum or discretized representations:

Model Type	Key Features	Typical Use Cases
Energy minimization (continuous)	Minimizes sum of magnetic + elastic (and gravitational) energies to solve for equilibrium shape	Ball chain robots, soft magnetic rods
Discretized pseudo-rigid-body	Robot as array of finite rigid segments with torsional/bending joints, each possibly hosting separate magnets	Multi-magnet embedded structures
Hard-magnetic elastic rod theory (DDG)	Continuum rod with hard magnetization, discretized into nodes/edges, capturing large deformations and contact	Real-time navigation, collision scenarios
Elastica-based energy functional	Euler–Bernoulli or Cosserat rod models with distributed or lumped magnetic actuation	Soft continuum sections and hybrid architectures

Key governing equations derive from energetic principles. For example, the equilibrium shape $\{\mathbf{p}_i^*, \mathbf{R}_i^*\}$ of a ball chain robot minimizes $U = U_b + U_e + U_s + U_g$ over all sphere positions and orientations (Pittiglio et al., 2023). In multi-magnet soft continuum robots, joint equilibrium is given by $\boldsymbol{\theta} = \Lambda^{-1} M(\boldsymbol{\theta}) \mathbf{b}$ , where $\Lambda$ is the joint stiffness matrix and $M(\boldsymbol{\theta})$ contains magnet dipole contributions (Wu et al., 15 Jul 2025). Maximum controllable degrees of freedom is shown to be $2N_m$ for $N_m$ embedded magnets.

Hybrid approaches combine kinematically distinct proximal and distal sections, e.g., tendon-driven tubes appended by magnetically actuated ball-chains, yielding tip positions modeled as

$\mathbf{p}_{\text{tip}} = \mathbf{p}_0 + \mathbf{R}_0 \mathbf{w} + nd_c \hat{\omega},$

enabling decoupled gross and fine orientation (Pittiglio et al., 30 Jan 2024).

Precise and safe navigation of MCRs in complex environments relies on advanced control and estimation algorithms:

Closed-loop quasi-static control (QSC), integrating a differential kinematic model (typically derived from energy or elastica theory) and a linear extended state observer (LESO) to estimate unmodeled dynamics and disturbances. Critical is the computation of the Jacobian $J_\psi = \frac{d\theta_L}{d\psi}$ relating actuator input to tip deflection, and the real-time update of input to minimize tracking error without overshoot (Wu et al., 6 Aug 2024).
Inverse design and optimal magnetic field computation, as implemented via a discrete elastic rod framework embedded in discrete differential geometry. A reduced-order contact model penalizes lumen wall contact, and a Newton-type search with finite-difference Jacobian updates the field to minimize tip constraint forces in real-time, even under high deformation and contact scenarios (Tong et al., 11 Mar 2025).
Swarm control via symmetry breaking, in millirobot collectives, assigns one "leader" to be directly actuated, while physical parameter heterogeneity (lengths or pivot separations) enables control of followers' relative displacement using only a global field. Trajectory planning precomputes sweep angles and step counts to ensure desired formation (Razzaghi et al., 2021).
Independent dual-robot control with external permanent magnets, using optimization of EPM poses (parameterized in spherical coordinates) to decouple wrenches on distinct robots, with crosstalk minimized at large separations (e.g., $3.9\%$ at 15cm for 5mm NdFeB robots). Mean positional errors of $8.7$mm and angular errors of $16.7^\circ$ are achieved at 60mm separation (Davy et al., 2023).
Shape estimation for feedback, employing magnetic ball chains whose time-varying spatial field is detected by external Hall sensor arrays, enabling >93% accurate tip position estimation with computation times suitable for real-time calibration and control (Pittiglio et al., 21 Oct 2024).

4. Structural Optimization and Performance Enhancement

Optimal placement and configuration of embedded magnets directly influence MCR dexterity, manipulability, and workspace characteristics:

Controllable DoF scaling: Maximum controllable degrees of freedom rises linearly with the number of magnets; at least three magnets are needed for full 6-DoF control (Wu et al., 15 Jul 2025).
Differential geometric optimization: Manipulability and dexterity measures are reparameterized over the actuation space, with performance density expressed as $z(\mathcal{H}(\theta); L_k)$ . Structural parameters (e.g., magnet positions/orientations) are optimized to shape the configuration space's Riemannian metric, achieving locally high dexterity and globally favorable workspace properties. In small-angle limit, closed-form solutions for joint angles as a function of magnet placement can be derived.
Graded-stiffness design: Sectional variation in Young’s modulus suppresses undesired sharp bending at clamped boundaries and achieves smoother, more circular curvature—beneficial for safe lumen navigation. Experimental studies confirm the graded approach leads to enhanced curvature control and avoidance of buckling (Shi et al., 3 Aug 2025).
Multi-parameter expansion models: Machine learning-based predictors (e.g., multi-branch fusion networks) trained on FEM datasets rapidly estimate bending behavior from field strength, modulus, morphogeometry, enabling near-instantaneous design iteration and library construction (Shi et al., 3 Aug 2025).

5. Sensing, Functional Integration, and Multi-Functional Behaviors

MCRs are increasingly designed for task-specific functional diversity and integrated sensing:

Shape sensing via embedded chains: Magnetic ball chains inserted into the lumen create unique field signatures; inverting magnetic field data via kinematic and dipole field models provides real-time curvature and tip position feedback (tip errors as low as 2.2mm or $2.4\%$ of length) without the cost or sterilization constraints of optical or electromagnetic methods (Pittiglio et al., 21 Oct 2024).
Reprogrammable magnetization profiles: Selective (de)magnetization of embedded modules allows switching between multiple surgical tasks (gripping, cutting, drug release, sample storage, remote heating) and full 6-DoF motion under uniform and weak fields ( $<65$ mT, $<1.5$ T/m) (Ng et al., 19 Sep 2025).
Superstructure and convoy strategies: Hierarchical microrobot assemblies boost propulsion (by increasing effective magnetic volume) and allow distributed force generation for large or heavy instrument carriage. On-demand structural transitions—e.g., gelatin/ion nanoparticle composites dissolving via magnetic hyperthermia—enable staged delivery or passage through narrower conduits (Landers et al., 2023, Jeon et al., 26 Feb 2025).
Dual-mode actuation for intervention: Radial IPM placement in catheters is shown to facilitate decoupled bending (for navigation) and torsion (for drug release). A dual-layer blockage mechanism triggers payload release when >160° twist is applied, as verified in phantom vasculature navigation experiments (Zhang et al., 2 Oct 2025).

6. Applications, Clinical Translation, and Limitations

MCR technologies have immediate relevance for:

Endoluminal and endovascular interventions: Cable-free, steerable catheters or tips that can reach deep targets in tortuous anatomy, execute local delivery (e.g., drug elution), perform electrocautery, or sample tissue (Pittiglio et al., 2023, Jeon et al., 26 Feb 2025, Zhang et al., 2 Oct 2025).
Minimally invasive surgery: Cooperative convoys of millirobots demonstrate enhanced force output, precision locomotion on slippery tissue, and the ability to transport electrically powered instruments or long catheters in dynamic biological environments (Jeon et al., 26 Feb 2025).
Micro-assembly and manipulation: Swarm-controlled ensembles can assemble, reconfigure, and manipulate micro-objects or structures at otherwise inaccessible scales (Razzaghi et al., 2021).
Shape feedback and navigation: Low-cost shape sensing methods facilitate navigation in dynamic and cluttered environments by delivering real-time curvature estimates for closed-loop control (Pittiglio et al., 21 Oct 2024).
Active drug delivery: Dual-mode actuation for precise, on-demand therapeutic agent release at target sites without onboard actuation or cabling (Zhang et al., 2 Oct 2025).

Limitations highlighted in experimental work include sensitivity of shape estimation to sensor noise, performance degradation in high-contact or rapidly changing environments, maximum tip errors ( $\sim$ 7% of robot length), and potential for unmodeled friction to affect fine formation or positioning tasks.

7. Future Directions

Ongoing and prospective research avenues for magnetic continuum robots include:

Material innovation: Advanced magneto-mechanically tunable polymers and bio-compatible composites for increased field response and functional integration, including biodegradable or self-healing materials (Shi et al., 3 Aug 2025).
Precise multi-agent control: Further reduction of crosstalk, scaling to more agents, or developing global field modulation techniques for decoupled actuation in rich clinical settings (Davy et al., 2023).
Hybrid architectures: Seamless integration of distinct actuation and sensing modalities (e.g., tendon-magnetic, ball chain–shape sensor synergies) to boost dexterity and robustness (Pittiglio et al., 30 Jan 2024, Pittiglio et al., 21 Oct 2024).
Real-time model-based feedback: More accurate, dynamically adaptive model-based controllers and estimation algorithms for in vivo deployment under imaging constraints (Tong et al., 11 Mar 2025, Wu et al., 6 Aug 2024).
Staged or combinatorial therapies: Multi-step agents capable of serial or parallel delivery, sequential tool deployment, or cooperative maneuvers within the same procedure (Landers et al., 2023, Ng et al., 19 Sep 2025).

The field is converging towards more dexterous, robust, and functional wirelessly actuated soft robots that can address the demands of next-generation minimally invasive surgery, shape-aware navigation, targeted treatment, and micro- or meso-scale assembly in complex settings.