External Magnetic Field Node#

Module: mime.nodes.actuation.external_magnetic_field Stability: experimental Algorithm ID: MIME-NODE-001 Version: 1.0.0 Verification Mode: Mode 2 (Independent)

Summary#

Generates a rotating uniform magnetic field B(t) = B_0 * [cos(omegat), sin(omegat), 0], modelling a Helmholtz coil pair driven in quadrature or a distant rotating permanent magnet.

Governing Equations#

$$ \mathbf{B}(t) = B_0 \begin{pmatrix} \cos(2\pi f t) \ \sin(2\pi f t) \ 0 \end{pmatrix} $$

where $B_0$ is the field magnitude [T] and $f$ is the rotation frequency [Hz].

For a coil array, the general form is $\mathbf{B}(\mathbf{p}) = \mathcal{B}(\mathbf{p}) \mathbf{I}$ (Appendix C), but the current implementation uses the uniform-field approximation (valid near workspace centre).

Discretization#

Analytical — no discretisation. The field is evaluated exactly at each timestep.

Implementation Mapping#

Equation Term

Implementation

Notes

$B_0 \cos(\omega t)$

mime.nodes.actuation.external_magnetic_field.ExternalMagneticFieldNode.update

jnp.cos(omega * t)

$B_0 \sin(\omega t)$

mime.nodes.actuation.external_magnetic_field.ExternalMagneticFieldNode.update

jnp.sin(omega * t)

mT to T conversion

mime.nodes.actuation.external_magnetic_field.ExternalMagneticFieldNode.update

strength_mt * 1e-3

Assumptions and Simplifications#

  1. Uniform field over the workspace (valid near Helmholtz coil centre)

  2. No eddy currents or shielding from biological tissue

  3. Coil inductance delay negligible (quasi-static field)

  4. Field rotation in xy-plane only

Validated Physical Regimes#

Parameter

Verified Range

Notes

frequency_hz

0–200

Typical microrobot actuation range

field_strength_mt

0–100

Below saturation for most soft-magnetic materials

Known Limitations and Failure Modes#

  1. Uniform field approximation invalid far from workspace centre

  2. No spatial gradient in uniform mode (gradient = 0) — only torque actuation, no gradient force

  3. 2D rotation only — no out-of-plane field components

Stability Conditions#

Unconditionally stable — analytical evaluation with no numerical integration.

State Variables#

Field

Shape

Units

Description

field_vector

(3,)

T

Current B field

field_gradient

(3,3)

T/m

Spatial gradient (zero in uniform mode)

sim_time

()

s

Accumulated simulation time

Parameters#

Parameter

Type

Default

Units

Description

frequency_hz

float

10.0

Hz

Commandable rotation frequency

field_strength_mt

float

10.0

mT

Commandable field magnitude

Boundary Inputs#

Field

Shape

Default

Coupling Type

Description

frequency_hz

()

10.0

replacive

Rotation frequency from ControlPolicy

field_strength_mt

()

10.0

replacive

Field magnitude from ControlPolicy

Boundary Fluxes (outputs)#

Field

Shape

Units

Description

field_vector

(3,)

T

B field for MagneticResponseNode

field_gradient

(3,3)

T/m

dB/dx for MagneticResponseNode

MIME-Specific Sections#

Clinical Relevance#

The external magnetic field is the primary actuation mechanism for helical microrobots navigating CSF. Field frequency and strength are the main control inputs for the ControlPolicy.

References#

  • [@Abbott2009] Abbott, J.J. et al. (2009). How Should Microrobots Swim? — Analysis of magnetic actuation strategies for microrobots.

Verification Evidence#

  • MIME-VER-005: Full chain force-velocity consistency

  • MIME-VER-006: 1000-step chain stability

  • Unit tests: tests/nodes/test_external_magnetic_field.py (15 tests)

Changelog#

Version

Date

Change

1.0.0

2026-03-20

Initial implementation — uniform rotating field