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Expanding the Workspace of Electromagnetic Navigation Systems Using Dynamic Feedback for Single- and Multi-agent Control

Jasan Zughaibi, Denis von Arx, Maurus Derungs, Florian Heemeyer, Luca A. Antonelli, Quentin Boehler, Michael Muehlebach, Bradley J. Nelson

arXiv:2511.18486v2cs.ROcs.SYeess.SY

TL;DR

E MNS workspace is constrained by rapid field decay and thermally demanding currents, while many field configurations can produce the same torque. The paper compares field-alignment with torque/force control and reports the same stabilization using 0.1–0.2 A rather than 8–14 A.

  • Problem

    Magnetic fields decay rapidly with distance, and many field configurations can produce the same torque while requiring substantially different coil currents, which complicates workspace and power constraints.

    Source: §I · pp. 1–2

  • Method

    The paper compares field-alignment with torque/force control and analyzes how required torques and forces are allocated to coil currents.

    Source: §VI · p. 15

  • Results

    0.1–0.2 A versus 8–14 A: torque/force-based control with optimal energy allocation achieves the same inverted-pendulum stabilization task.

    Source: C. Discussion · p. 11

  • Takeaways & Limitations

    Feedback expands workspace in single- and multi-agent settings, while the orthogonal field solution does not generally minimize energy.

    Source: §VI · p. 15

    Dynamic current allocation depends on instantaneous dipole orientation, making state estimation a core challenge.

    Source: §VI · p. 16

Abstract

from arXiv · show

Electromagnetic navigation systems (eMNS) enable a number of magnetically guided surgical procedures. A challenge in magnetically manipulating surgical tools is that the effective workspace of an eMNS is often severely constrained by power and thermal limits. We show that system-level control design significantly expands this workspace by reducing the currents needed to achieve a desired motion. We identified five key system approaches that enable this expansion: (i) motion-centric torque/force objectives, (ii) energy-optimal current allocation, (iii) real-time pose estimation, (iv) dynamic feedback, and (v) high-bandwidth eMNS components. As a result, we stabilize a 3D inverted pendulum on an eight-coil OctoMag eMNS with significantly lower currents (0.1-0.2 A vs. 8-14 A), by replacing a field-centric field-alignment strategy with a motion-centric torque/force-based approach. We generalize to multi-agent control by simultaneously stabilizing two inverted pendulums within a shared workspace, exploiting magnetic-field nonlinearity and coil redundancy for independent actuation. A structured analysis compares the electromagnetic workspaces of both paradigms and examines current-allocation strategies that map motion objectives to coil currents. Cross-platform evaluation of the clinically oriented Navion eMNS further demonstrates substantial workspace expansion by maintaining stable balancing at distances up to 50 cm from the coils. The results demonstrate that feedback is a practical path to scalable, efficient, and clinically relevant magnetic manipulation.

NOMENCLATURE

System-level control design is presented as a way to reduce current demands and expand eMNS workspace. The paper compares field-alignment and torque/force control for single- and multi-agent pendulum stabilization.

NOMENCLATURE

  • The nomenclature defines symbols for currents, fields, magnetic moments, forces, torques, state, actuation matrices, allocation matrices, and common mathematical operators.It also specifies notation for setpoints and body-fixed coordinates.Source: NOMENCLATURE · p. 1

I. INTRODUCTION

  • Power and thermal limits constrain eMNS workspace; the proposed architecture combines motion-centric torque/force objectives, energy-optimal current allocation, real-time pose estimation, feedback, and high-bandwidth hardware.The field-alignment and torque/force paradigms are compared as alternatives for single-agent stabilization.Source: §I · pp. 1–2
  • A result reported in can be achieved using the torque/force-based approach with substantially lower coil currents: 0.1–0.2 A instead of 8–14 A.The work compares the two control paradigms and analyzes the current disparity through electromagnetic workspaces and current-allocation strategies.Source: §I · p. 2
  • Two identical inverted pendulums are simultaneously stabilized in a shared OctoMag workspace by exploiting magnetic-field nonlinearity for independent actuation.The benchmark involves coupling between coils and magnets and between the two magnets.Source: §I · p. 2, Figure 1 · p. 2
  • 50 cm from the coils, the Navion eMNS stabilizes an inverted pendulum, demonstrating workspace expansion on a clinically oriented platform.The setup uses a pendulum pole tip balanced on a support plate while the arm is moved through space.Source: §I · p. 2, Figure 2 · p. 2

II. RELATED WORK

  • Prior multi-agent approaches use object differences or magnetic-field inhomogeneity; this work's identical pendulums do not rely on heterogeneity or time-scale separation.Prior examples include identical magnetic robots controlled by PID and soft microrobots controlled with a nine-coil system.Source: §II · p. 3

III. EXPERIMENTAL PLATFORMS, DYNAMICS, AND CONTROL

The study uses inverted-pendulum platforms and two eMNSs to evaluate control paradigms. It describes their hardware, bandwidth, and the position-dependent mapping from coil currents to magnetic fields.

A. Inverted Pendulum Platforms

  • Three platform variants support single-agent 3D, multi-agent planar, and multi-agent 3D experiments; the 3D multi-agent setup records all four angles on each unit in real time.The Navion large-workspace experiment uses a joint-less setup in which the pendulum is balanced on a plate.Source: A. Inverted Pendulum Platforms · p. 4

B. Electromagnetic Navigation Systems

  • OctoMag has eight independently driven coils with 26.4 Hz closed-loop current bandwidth; Navion has three coils and 24.5 Hz bandwidth.Current setpoints are updated at 200 Hz for OctoMag and 125 Hz for Navion.Source: B. Electromagnetic Navigation Systems · pp. 4–5

C. Relation between Magnetic Fields and Currents

  • The actuation matrices map coil currents to magnetic field and gradient, which are assumed to depend linearly on current.A(p) is updated in real time from measured dipole position; the field and gradient matrices are derived from a calibrated multipole-expansion model.Source: C. Relation between Magnetic Fields… · p. 5

D. Common Dynamics

The paper models the pendulum dynamics across control paradigms and uses decoupled LQRI state feedback for stabilization. The feedback implementation addresses steady-state error and coupling effects.

D. Common Dynamics

  • The 3D pendulum is approximated by two decoupled 2D pendulums for controller design, an approximation also used for the multi-agent configurations.The pendulum equation remains unchanged across paradigms, while actuator dynamics vary with the control input.Source: D. Common Dynamics · p. 5
  • The actuator's generalized-coordinate equation depends on whether the controller uses field-alignment or torque/force inputs.The equations of motion are derived with Lagrangian formalism.Source: D. Common Dynamics · p. 5

E. Feedback Control

  • LQRI state feedback is applied in a fully decoupled manner, with field angles or control torques as the controller outputs.Feedback gains are computed from discrete-time Riccati equations for the corresponding linearized dynamics.Source: E. Feedback Control · pp. 5–6
  • Integral controllers remove steady-state offsets, and the decoupled controller rejects coupling effects as external disturbances.Integrators are typically switched off after stabilization offsets are removed; a practical guideline is at least 20 Hz sampling and delays below 40 ms.Source: E. Feedback Control · p. 6

IV. SINGLE-AGENT CONTROL

This section frames field-alignment as a magnetic proportional controller: the dipole tends to follow the commanded field direction. Its open-loop stability can help without sensing, but depends on conditions and does not suit every dynamic system.

1) Dynamics and Stabilization without Digital Feedback:

  • Field-alignment treats the commanded field direction as a proportional-like input that generates torque toward dipole alignment.The field vector’s orientation is the operator input, and the dynamics are derived from the dipole’s magnetic potential energy.Source: 1) Dynamics and Stabilization without… · p. 6
  • When −ηg + |˜m||b| > 0, the system is open-loop stable and can align without explicit feedback, but gradients can destabilize the desired orientation.In Navion, an inherent gradient favors lateral coil configuration, making upright orientation difficult.Source: 1) Dynamics and Stabilization without… · p. 7
  • Field-alignment suits quasi-static settings but may excite oscillatory modes in dynamic systems such as actuators or catheters.The analysis assumes a homogeneous, gradient-free field; incorporating gradients directly adds complexity.Source: 1) Dynamics and Stabilization without… · p. 7

2) Field-Alignment with Digital Feedback:

The section contrasts field-alignment and torque/force control with feedback. Field-alignment’s current allocation targets a magnetic task, while torque/force control directly targets mechanical motion and requires real-time state information.

2) Field-Alignment with Digital Feedback:

  • 8–14 A are required for closed-loop field-alignment stabilization of the 3D pendulum, making it an inefficient comparative baseline once state feedback is available.The authors characterize field-alignment’s practical utility as sensorless, open-loop use; the pendulum itself remains dynamically unstable and requires active feedback.Source: 2) Field-Alignment with Digital Feedback · p. 7

3) Allocation Strategy:

  • Field-alignment maps field-angle commands to currents through a least-squares pseudoinverse, but this does not minimize electrical energy for a motion task.Its allocation specifies a desired magnetic field and zero gradient.Source: 3) Allocation Strategy · p. 7

B. Torque/Force-based Control

  • Torque/force control computes desired mechanical torque or force directly, without assuming dipole alignment or a homogeneous field.The field and any induced gradient emerge from the allocation process rather than being imposed by simplifying assumptions.Source: B. Torque/Force-based Control · p. 7

1) Dynamics and the Importance of Feedback Control:

  • Torque-based control requires digital feedback because free-dipole torque dynamics are unstable and torque or force commands are not directly observable motion quantities.The controller lets an operator issue high-level orientation targets while it modulates torque or force and rejects disturbances.Source: 1) Dynamics and the Importance of… · p. 7

2) Allocation Strategy:

The energy-optimal allocation maps motion objectives directly to coil currents by applying a pseudoinverse to the composed actuation map. Alternative staged allocations optimize different quantities and can require more current.

2) Allocation Strategy:

  • For a given torque task, this least-squares optimal current allocation directly targets mechanical torque generation and enables energy-efficient actuation, requiring coil currents of 0.1-0.2 A, up to two orders of magnitude lower than field-alignment.Source: 2) Allocation Strategy · p. 9

3) Alternative Torque-to-Current Allocations:

  • The one-step allocation minimizes current norm by applying the pseudoinverse to the entire composed map; reversing the mappings in stages does not yield the energy-optimal solution.The two-step method first minimizes field magnitude, which is a different objective from minimizing current magnitude.Source: 3) Alternative Torque-to-Current… · p. 9
  • The one-step allocation can require less current despite producing a larger field norm, by using parallel-field components in the torque mapping’s nullspace.Only the one-step method guarantees minimum current norm for a given torque task; even a suboptimal allocation still uses less current than field-alignment.Source: 3) Alternative Torque-to-Current… · p. 9

4) Expansion of Effective Workspace:

This section compares the paradigms’ feasible workspaces using current-domain analysis and a Feasibility Margin. It reports extended workspace with torque/force control, including stable Navion balancing at substantial coil distance.

4) Expansion of Effective Workspace:

  • 50 cm is the demonstrated Navion balancing distance from the coils under a 25 A current constraint; the authors note larger distances could be feasible with the same strategy.The current drivers support 45 A, but the experiment conservatively limited current to 25 A for safety.Source: 4) Expansion of Effective Workspace · p. 9
  • The Feasibility Margin measures current headroom at each position, while inverse-mapping both paradigms into the current domain makes their workspaces directly comparable.This extends a binary feasible/infeasible view with a measure of how robustly a position satisfies current constraints.Source: 4) Expansion of Effective Workspace · p. 10
  • The energy-efficient approach enables a significantly larger feasible region than field-alignment in the workspace comparison.The comparison matches baseline control authority by equating field-alignment magnetic stiffness with the torque controller’s active stiffness.Source: 4) Expansion of Effective Workspace · p. 10

5) Influence of Sampling Rate and Delays:

Energy-optimal torque/force control stabilizes the pendulum with much lower currents than field-alignment control. The sampling-rate and delay tests show that current allocation is unchanged within stability bounds, while delays and low sampling rates eventually destabilize the actuator.

5) Influence of Sampling Rate and Delays:

  • Within stability bounds, energy-optimal current allocation remains unaffected by reduced sampling rates and injected delays; passively stable clinical catheters are expected to tolerate lower frequencies and higher delays.The catheter implication is an expectation, not a direct test result.Source: 5) Influence of Sampling Rate and Delays · pp. 10–11

C. Discussion

  • Figure 8 compares field-alignment and energy-efficient workspaces using Feasibility Margin, where larger margins indicate more current headroom and a wider usable range.Low-margin regions near coil centers are identified as calibration-model artifacts.Source: Figure 8 · p. 11
  • 8–14 A for field-alignment control versus 0.1–0.2 A for energy-optimal torque/force control achieves the same stabilization task.The comparison is for stabilizing the inverted pendulum.Source: C. Discussion · p. 11
  • Field-alignment’s small angle between field and dipole requires a substantial field for significant torque, whereas energy-efficient allocation produces approximately orthogonal fields and needs less field magnitude.The orthogonal configuration makes the torque equation’s sinusoidal factor approximately 1.Source: C. Discussion · p. 11
  • The order-of-magnitude efficiency improvement reflects energy wasted in field-alignment’s torque-null parallel field component, which energy-efficient allocation minimizes.The parallel component is retained only when it lowers overall electrical energy consumption.Source: C. Discussion · pp. 11–12

5) Influence of Sampling Rate and Delays:

  • ≈40 ms of injected delay destabilizes the actuator, and marginal stability occurs at ≈20 Hz; currents remain unaffected across a wide range of delays and reduced sampling frequencies.The delay and sampling-rate tests used the actuator without the pendulum attached.Source: 5) Influence of Sampling Rate and Delays · p. 10, Figure 9 · p. 12

C. Discussion

  • The efficiency gain can support a smaller eMNS, potentially avoiding active cooling, or magnetic agents with lower magnetic volume.These are practical implications for the same motion task.Source: C. Discussion · p. 12
  • Navion’s lateral coils generate strong gradients that pull the object toward the coils, and its configuration cannot produce approximately uniform fields as OctoMag can.Without digital feedback, field-alignment does not guarantee zero steady-state error, even at 25 mT.Source: C. Discussion · p. 12

V. MULTI-AGENT CONTROL

The paper demonstrates independent control of two pendulums in a shared workspace and compares multi-agent workspaces under field-alignment and torque/force-based control. Torque/force control achieves a larger feasible workspace, while proximity, modeling assumptions, and hardware current limits constrain operation.

V. MULTI-AGENT CONTROL

  • OctoMag’s eight coils and magnetic-field nonlinearity support independent actuation of two agents despite magnetic interaction and shared-coil cross-talk.Field strength decays cubically with distance, so each coil predominantly affects the nearer agent.Source: §V · p. 12

A. Field-Alignment Control

  • Field alignment stabilizes the 2×2D setup at 65 mT at both positions, although one coil exceeds the 16 A saturation limit.The system remains stable through actuator redundancy and feedback; field alignment is limited to 2×2D for practical thermal reasons.Source: §V · p. 12, Figure 10 · p. 13, A. Field-Alignment Control · p. 13

B. Torque/Force-Based Control

C. Expansion of Effective Workspace

  • Torque/force-based control consistently yields a larger feasible two-agent workspace than field alignment, with feasibility defined by coil-current limits.The analysis fixes p2 while varying p1; the Feasibility Margin indicates sensitivity to current-constraint violations.Source: C. Expansion of Effective Workspace · p. 14, Figure 13 · p. 15
  • Workspace estimates may overstate close-proximity feasibility because they assume point dipoles and omit inter-magnet interactions.The point-dipole approximation neglects field variation across magnet volumes; the analysis also becomes less accurate near the coils.Source: C. Expansion of Effective Workspace · p. 14, Figure 13 · p. 15
  • Closer agent spacing rapidly raises commanded currents; with a conservative 4 A hardware limit, quasi-static analysis estimates a theoretical minimum separation of approximately 40 mm.The estimate reflects the threshold at which a single current constraint becomes active; relaxing the safety limit or using constraint-aware allocation could permit closer operation.Source: C. Expansion of Effective Workspace · p. 15

VI. CONCLUSION

The conclusion argues that system-level control choices can reduce actuation energy and expand electromagnetic workspaces. It highlights motion-centric objectives, energy-optimal allocation, feedback, pose estimation, and high-bandwidth hardware, while noting that more advanced control and non-visual state estimation remain future challenges.

VI. CONCLUSION

  • Torque/force-based control reduces actuation energy by up to two orders of magnitude without sacrificing performance and expands workspace in single- and multi-agent settings.The conclusion also reports independent stabilization of two 3D inverted pendulums and stabilization at large stand-off distances from a clinical-grade eMNS.Source: §VI · p. 15
  • The system architecture combines motion-centric torque/force objectives, energy-optimal current allocation, real-time pose estimation, feedback, and high-bandwidth hardware.The conclusion specifies real-time estimation above 20 Hz and feedback for rejecting disturbances and actuation-matrix errors.Source: §VI · pp. 15–16
  • Torque/force objectives direct current toward torque-generating field components, so current constraints become active only at greater stand-off distances.The conclusion states that stronger magnetic fields do not inherently enhance performance and that the strategy aligns magnetic energy with the mechanical task.Source: §VI · p. 16
  • Simple linear models with high-frequency feedback can suffice, while more elaborate architectures should preserve real-time constraints and show measurable benefits.The conclusion frames this as a principle of parsimony and says the architectural decisions are equally essential.Source: §VI · p. 16
  • Microscale application remains contingent on real-time non-visual state estimation because clinical operation in deep tissue lacks line-of-sight.The conclusion identifies this estimation bottleneck as a fundamental problem and a future direction.Source: §VI · p. 16

APPENDIX A INCORPORATING GRADIENTS IN FIELD-ALIGNMENT

The appendix models field gradients in field-alignment dynamics and summarizes the torque-generation structure of the magnetic dipole. It notes that the gradient approximation can fail for large gradients, while only field components perpendicular to the dipole generate torque.

APPENDIX A INCORPORATING GRADIENTS IN FIELD-ALIGNMENT

  • A position-dependent field model uses a first-order Taylor expansion to include gradients in field-alignment dynamics, but the approximation can fail when gradients are large.The resulting potential energy depends on field, position or orientation, and gradients; the authors note that embedding gradients requires additional modeling assumptions.Source: Appendix A · p. 16

APPENDIX B SINGULAR VALUE DECOMPOSITION

  • The dipole torque map has a one-dimensional nullspace along the dipole and a two-dimensional image plane perpendicular to it.Thus parallel field components cannot contribute to torque generation.Source: Appendix B · p. 17

APPENDIX C FEASIBILITY OF TORQUE-TO-CURRENT MAPPING

The appendices establish when the simplified torque-to-current mapping has an exact solution and compare one-step and two-step current allocations. The one-step method can require less current even when its field magnitude is larger, because it can use a torque-null parallel-field component.

APPENDIX C FEASIBILITY OF TORQUE-TO-CURRENT MAPPING

  • An exact current solution exists for the simplified torque mapping when the desired torque lies in the dipole’s torque range and the field-actuation matrix has full row rank.The guarantee is specific to the simplified mapping; adding gradient effects or a Jacobian transformation makes feasibility analysis more complex.Source: Appendix C · p. 17

APPENDIX D ANALYSIS OF TORQUE ALLOCATION METHODS

  • The one-step allocation requires no more current than the two-step allocation, although its field norm is at least as large.The one-step method gives the minimum-norm current solution, while the two-step method first minimizes field magnitude; the one-step method can use a parallel-field component that does not generate torque.Source: Appendix D · pp. 17–18
  • The two-step allocation strictly avoids parallel field components, which can make it attractive when minimizing field magnitude is the primary concern.In practice, the parallel component in the one-step solution is typically small, so both methods yield fields nearly orthogonal to the dipole.Source: Appendix D · p. 18

APPENDIX E INTER-AGENT MAGNETIC DISTURBANCE ESTIMATION

The appendix estimates inter-agent magnetic disturbances and describes practical considerations for stable simultaneous control of two pendulums. It also identifies likely sources of steady-state deflection.

APPENDIX E INTER-AGENT MAGNETIC DISTURBANCE ESTIMATION

  • Inter-agent dipole interactions produce torque and force, and the spherical joint converts translational force into mechanical torque included in the disturbance estimate.The combined disturbance torque is transformed into the body-fixed frame.Source: Appendix E · p. 18

APPENDIX F PRACTICAL CONSIDERATIONS FOR 2 × 3D PENDULUM

  • Direct torque-to-current allocation can simplify the single-agent case, but multi-agent field gradients make force-aware Jacobian mapping critical for stable simultaneous control.The gradients induce destabilizing forces at the actuators.Source: Appendix F · p. 18
  • Integral action removes large steady-state errors; stabilizing pendulums sequentially lets each integral term converge before stored values warm-start later simultaneous trials.Once calibrated, the integral values can be reused for simultaneous release and stabilization.Source: Appendix F · p. 18
  • After gravity-frame misalignment is analyzed and compensated, remaining deflections are likely tied to multipole-model inaccuracies, permanent-magnet interactions, and point-dipole omissions of finite-volume forces and torques.Spatial field variations across the finite magnet volume generate unmodeled forces and torques.Source: Appendix F · p. 18
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