Permanent Magnet Response Node#

Module: mime.nodes.robot.permanent_magnet_response Stability: experimental Algorithm ID: MIME-NODE-008 Version: 1.0.0 Verification Mode: Mode 2 (Independent)

Summary#

Computes magnetic torque and force on a permanent-magnet microrobot body from an externally applied field. The magnetic moment is fixed in the body frame (rigid permanent magnet), rotated to the lab frame via the robot quaternion. No susceptibility tensor, no saturation, no volume multiplier.

Governing Equations#

Fixed moment in body frame: $$ \mathbf{m}{\text{body}} = n \cdot m{\text{single}} \cdot \hat{\mathbf{a}} $$

where $\hat{\mathbf{a}}$ is the normalised moment axis and $n$ is the number of magnets.

Rotate to lab frame: $$ \mathbf{m}{\text{lab}} = R(\mathbf{q}) , \mathbf{m}{\text{body}} $$

Magnetic torque (lab frame): $$ \mathbf{T} = \mathbf{m}_{\text{lab}} \times \mathbf{B} $$

Magnetic force (lab frame): $$ \mathbf{F} = (\nabla \mathbf{B}) , \mathbf{m}_{\text{lab}} $$

Discretization#

Analytical — no discretisation. Single-evaluation algebraic computation.

Implementation Mapping#

Equation Term	Implementation	Notes
$\mathbf{m}{\text{body}} = n \cdot m{\text{single}} \cdot \hat{\mathbf{a}}$	`mime.nodes.robot.permanent_magnet_response.PermanentMagnetResponseNode.update`	Fixed moment from constructor params
$\mathbf{m}{\text{lab}} = R(\mathbf{q}) \mathbf{m}{\text{body}}$	`mime.nodes.robot.permanent_magnet_response.PermanentMagnetResponseNode.update`	`rotate_vector(q, m_body)`
$\mathbf{T} = \mathbf{m}_{\text{lab}} \times \mathbf{B}$	`mime.nodes.robot.permanent_magnet_response.PermanentMagnetResponseNode.update`	`jnp.cross(m_lab, B_lab)`
$\mathbf{F} = (\nabla \mathbf{B}) \mathbf{m}_{\text{lab}}$	`mime.nodes.robot.permanent_magnet_response.PermanentMagnetResponseNode.update`	`grad_B @ m_lab`

Assumptions and Simplifications#

Rigid permanent moment (no demagnetization)
Moment axis fixed in body frame
No temperature dependence of moment

Validated Physical Regimes#

Parameter	Verified Range	Notes
field_strength	0–0.1 T	NdFeB permanent magnet in mT-range field

Known Limitations and Failure Modes#

No demagnetization effects
No hysteresis
No temperature-dependent moment

State Variables#

Field	Shape	Units	Description
magnetization	(3,)	A*m^2	Current magnetic moment (lab frame)
magnetic_torque	(3,)	N.m	Magnetic torque (lab frame)
magnetic_force	(3,)	N	Magnetic force (lab frame)

Parameters#

Parameter	Type	Default	Units	Description
n_magnets	int	1	-	Number of permanent magnets
m_single	float	1.07e-3	A*m^2	Magnetic moment per magnet
moment_axis	tuple	(0,1,0)	-	Moment direction in body frame

Boundary Inputs#

Field	Shape	Default	Coupling Type	Description
field_vector	(3,)	zeros	replacive	B field from ExternalMagneticFieldNode
field_gradient	(3,3)	zeros	replacive	dB/dx from ExternalMagneticFieldNode
orientation	(4,)	[1,0,0,0]	replacive	Quaternion from RigidBodyNode

Boundary Fluxes (outputs)#

Field	Shape	Units	Description
magnetic_torque	(3,)	N.m	To RigidBodyNode (additive)
magnetic_force	(3,)	N	To RigidBodyNode (additive)

MIME-Specific Sections#

Biocompatibility Context#

Default material: NdFeB N45. Biocompatibility NOT assessed — manufacturer must perform ISO 10993 evaluation. Typically encapsulated in biocompatible resin or parylene coating.

Clinical Relevance#

Computes the torque and force on an untethered magnetic robot (UMR) containing permanent magnets driven by an external rotating field. The step-out phenomenon occurs when the field rotation frequency exceeds the maximum torque capacity of the permanent magnets.

References#

[@deBoer2025] de Boer, M.C.J. et al. (2025). Wireless mechanical and hybrid thrombus fragmentation of ex vivo endovascular thrombosis model in the iliac artery.

Verification Evidence#

Unit tests: tests/nodes/test_permanent_magnet_response.py

Changelog#

Version	Date	Change
1.0.0	2026-03-22	Initial implementation — permanent magnet model