OctoMag eMNS: Advanced EM Navigation

• OctoMag eMNS is a modular electromagnetic navigation system with eight independently controlled coils arranged around a cubic workspace, enabling precise manipulation of surgical microrobots.
• It employs motion-centric control and energy-optimal current allocation to achieve up to a threefold expansion of the manipulation workspace while minimizing power consumption.
• Incorporating high-bandwidth feedback and machine learning-based nonlinear field modeling, the system delivers sub-millisecond control and high positioning accuracy.

OctoMag eMNS refers to a class of electromagnetic navigation systems based on the OctoMag device, which employs eight independently controlled electromagnets arranged around a cubic workspace. These systems enable precise magnetic manipulation of surgical microrobots and tools, with applications in remote and minimally invasive procedures. A central challenge is maximizing the manipulation workspace while respecting stringent power and thermal constraints. Advanced control and modeling techniques—including motion-centric objectives, energy-optimal current allocation, high-bandwidth feedback, and nonlinear field modeling—have driven significant performance gains in workspace, efficiency, and multi-agent capability.

1. OctoMag Electromagnetic Architecture and Field Modeling

The OctoMag system comprises eight coils centered at the vertices of a cube, each coil having a soft-magnetic core (length 15 cm, radius 1.5 cm) and wound with 500 turns of wire (coil radius 3 cm, length 10 cm). Coil axes are aligned from the cube center to each corner, establishing a precisely defined 10 cm cube workspace. The system is described by:

$$\mathbf{B}(p) = \sum_{i=1}^8 I_i\,\mathbf{b}_i(p) = F_B(p)\,\mathbf{I}, \qquad F_B(p)\in\mathbb{R}^{3\times8},\quad \mathbf{I}=[I_1,\dots,I_8]^T$$

where $\mathbf{B}(p)$ is the field at position $p$, each $I_i$ is the current in coil $i$, and $F_B(p)$ encodes the linearized superposition principle ("point-dipole" approximation). The spatial gradient is similarly given by

$$\nabla\mathbf{B}(p) = \sum_{i=1}^8 I_i\,\nabla\mathbf{b}_i(p) = F_G(p)\,\mathbf{I}, \qquad F_G(p)\in\mathbb{R}^{5\times8}$$

where $F_G(p)$ captures the five independent components of $\nabla\mathbf{B}$ at $p$ (Zughaibi et al., 23 Nov 2025, Yu et al., 2019). These system matrices are typically obtained by analytic multipole expansion followed by calibration [Petruska et al. 2017].

To address significant nonlinearity—especially at high coil currents and in the presence of core saturation—recent work replaces the linear approximation with data-driven models. Random forests and artificial neural networks trained on Hall-sensor data reduce field-magnitude RMSE by 40–80% compared to the linear multipole model, achieving under 10 mT error even at 30–35 A, with sub-millisecond inference suitable for real-time control (Yu et al., 2019).

2. Motion-Centric Control Objectives: Torque and Force Mapping

Traditional eMNS controllers align the magnetic moment $\mathbf{m}$ of a tool with the local field $\mathbf{B}(p)$. The OctoMag eMNS departs from this paradigm, instead computing the mechanical wrench (torque $\tau$, and where needed, force $f$) required by the task, and allocating coil currents to achieve this wrench directly:

$$\tau = \mathbf{m} \times \mathbf{B}(p), \qquad f = (\mathbf{m}\cdot\nabla)\mathbf{B}(p)$$

The wrench objective is formulated as a minimum-norm least-squares problem:

$$\min_{\mathbf{I}} \left\| \begin{pmatrix} \tau_{\rm des} \ f_{\rm des} \end{pmatrix} - A(p, \mathbf{m})\,\mathbf{I} \right\|^2$$

$$A(p, \mathbf{m}) = \left[ \mathcal{S}(\mathbf{m})F_B(p)\;\; F_G(p) \right], \qquad \mathcal{S}(\mathbf{m}): 3\times3\ \mathrm{skew}$$

This motion-centric approach enables precise, task-specific control actions and removes inefficiencies inherent in field-alignment strategies (Zughaibi et al., 23 Nov 2025).

3. Energy-Optimal Current Allocation Strategies

Given a torque $W \in \mathbb{R}^3$ to be delivered to the magnetic tool, the system solves a constrained minimum-norm problem:

$$\min_{\mathbf{I}} \|\mathbf{I}\|^2\quad\text{subject to}\quad A(p, \mathbf{m})\,\mathbf{I} = W$$

$\mathbf{I}^* = A(p, \mathbf{m})^\dagger W$

$$\forall i,\, -I_{\max} \le I_i \le I_{\max} \quad (I_{\max} = 16\,\mathrm{A})$$

A Moore–Penrose pseudoinverse ($A^\dagger = A^\top \,(A\,A^\top)^{-1}$) yields this allocation in closed form at 200 Hz. Box constraints are enforced by clamping. Experiments demonstrate that this allocation requires coil currents of only 0.1–0.2 A (versus 8–14 A for field-alignment), achieving the mechanical objective with up to two orders of magnitude less power (Zughaibi et al., 23 Nov 2025). No iterative quadratic programming is required, ensuring real-time applicability.

4. High-Bandwidth Feedback and Real-Time Control Loop

A dynamic feedback architecture closes the control loop at 200 Hz (current loop at 26.4 Hz), consisting of:

1. Measurement of position $p$ and orientation $(\alpha, \beta)$ via motion capture.
2. Calculation of state error for each pendulum axis: $x^{\rm SP}_{\alpha}-x_{\alpha}$, $x^{\rm SP}_{\beta}-x_{\beta}$.
3. Computation of desired torque using a decoupled LQRI (Linear Quadratic Regulator with Integral action):

$$\tau_c = K\,(x_{\rm SP}-x) + k_I \int (x_{\rm SP} - x)$$

1. Assembly of system matrix $A(p, \mathbf{m})$.
2. Solution for coil currents via pseudoinverse, enforcing bounds.
3. Actuation: currents sent to drivers, producing near-instantaneous field.
4. Repetition at 200 Hz.

Closed-loop stability is established by the LQRI design for linearized pendulum dynamics, with actuation bandwidth greatly exceeding system resonance (<3 Hz), ensuring robust stabilization (Zughaibi et al., 23 Nov 2025).

5. Quantitative Workspace Expansion and Performance Metrics

Field-alignment control with digital feedback balances a 3D pendulum in the nominal center ($\pm$5 cm) using 8–14 A. The motion-centric, minimum-norm torque approach reduces this to 0.1–0.2 A. On the Navion platform, similar controllers maintain stable balancing at up to 50 cm from the coils, each drawing up to 25 A.

Workspace expansion—measured via Feasibility-Margin plots—shows the region of achievable task execution extends roughly 3$\times$ farther in each dimension under torque/force-based control (for the same current limits) compared to field-alignment. Thus, system-level optimization of control, feedback, and current allocation achieves an order-of-magnitude workspace expansion while minimizing power dissipation (Zughaibi et al., 23 Nov 2025).

Control Paradigm	Required Current (A)	Workspace Extension
Field Alignment	8–14	Baseline
Torque/Force + Pseudoinv	0.1–0.2	%%%%21$i$22%%%%

6. Multi-Agent Magnetic Actuation

With eight symmetrically arranged coils and the $1/r^3$ decay of the magnetic field, spatial nonlinearity and coil redundancy enable independent wrench assignment to multiple agents. For $n$ tools (e.g., two identical 3D inverted pendulums separated by 6.5 cm), the multitask system is formulated as:

$$\begin{bmatrix} A(p_1, \mathbf{m}_1)\ A(p_2, \mathbf{m}_2) \end{bmatrix}\mathbf{I} = \begin{pmatrix} W_{c,1}\ W_{c,2} \end{pmatrix}$$

$\mathbf{I}^* = \left[ \begin{smallmatrix}A_1 \ A_2\end{smallmatrix}\right]^\dagger\! \begin{pmatrix}W_{c,1}\W_{c,2}\end{pmatrix}$

Experiments achieve simultaneous stabilization and independent actuation using $\leq$1 A per coil for both synchronous and asynchronous multi-agent trajectories. The feasible workspace for two agents more than doubles under torque/force-based allocation compared to field-alignment (Zughaibi et al., 23 Nov 2025).

7. Nonlinear Magnetic Field Modeling: Machine Learning Approaches

The accuracy of high-order control and calibration critically depends on magnetic field modeling fidelity. Classical linear multipole electromagnet methods (LMEM) exhibit substantial errors in nonlinear regimes (e.g., saturated cores, $>$30 A). Random forest (RF) and artificial neural network (ANN) models, trained on large datasets of sensor measurements (427,210 samples across workspace and excitation space), outperform LMEM on held-out data:

Model	$R^2_{\rm norm}$	RMSE$_{\rm norm}$ (mT)	High-Current RMSE (30–35 A) (mT)
LMEM	0.29	23.90	42
RF	0.74	14.43	22
ANN	0.99	3.01	7

Replacing LMEM with ANN in OctoMag-style eMNS yields worst-case field errors below 10 mT even at high coil currents, translating to sub-millimeter microrobot positioning and sub-degree orientation accuracy. ANN models execute in under 1 ms, enabling embedding in $\sim$1 kHz control loops (Yu et al., 2019).

Gradient-based optimization given a differentiable ANN allows direct feed-forward current computation for desired $\mathbf{B}^*(\mathbf{p})$, with feedback correction as needed. The calibration protocol involves workspace scanning with a Hall sensor array and random excitation, enabling system deployment in new clinical or experimental environments (Yu et al., 2019).

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In summary, the OctoMag eMNS exemplifies state-of-the-art electromagnetic navigation technologies for medical and microrobotic applications. Combining a symmetric eight-coil hardware architecture with motion-centric, minimum-norm control, high-rate feedback, and advanced magnetic modeling—particularly machine learning-based nonlinear field estimation—enables dramatic expansions in manipulation workspace and efficiency, while supporting scalable multi-agent behaviors within clinically constrained environments (Zughaibi et al., 23 Nov 2025, Yu et al., 2019).