Remote Magnetic Levitation Using Reduced Attitude Control and Parametric Field Models

TL;DR

Problem: enabling remote magnetic levitation and control of a centimeter-scale rigid body in air using an electromagnetic navigation system. Approach: develop a compact parametric field model based on a multipole expansion and a point-dipole representation, and implement a nonlinear reduced-attitude controller plus translational LQR with integral action. Contributions: remote levitation demonstration on OctoMag, levitator-agnostic field modeling, and stabilization/trajectory tracking over a five-DoF controllable pose subspace. Findings: high-bandwidth feedback enables rapid, accurate trajectory tracking at large rotational excursions, with observed yaw drift and hardware-delay considerations discussed. Impact: advances the feasibility of clinically relevant, contactless magnetic manipulation and informs design trade-offs for future eMNS hardware.

Abstract

Electromagnetic navigation systems (eMNS) are increasingly used in minimally invasive procedures such as endovascular interventions and targeted drug delivery due to their ability to generate fast and precise magnetic fields. In this paper, we utilize the OctoMag eMNS to achieve remote levitation and control of a rigid body across large air gaps which showcases the dynamic capabilities of clinical eMNS. A compact parametric analytical model maps coil currents to the forces and torques acting on the levitating object, eliminating the need for computationally expensive simulations or lookup tables and leading to a levitator agnostic modeling approach. Translational motion is stabilized using linear quadratic regulators. A nonlinear time-invariant controller is used to regulate the reduced attitude accounting for the inherent uncontrollability of rotations about the dipole axis and stabilizing the full five degrees of freedom controllable pose subspace. We analyze key design limitations and evaluate the approach through trajectory tracking experiments. This work demonstrates the dynamic capabilities and potential of feedback control in electromagnetic navigation, which is likely to open up new medical applications.

Figures (5)

• Figure 1: A freely levitating object in OctoMag eMNS. The levitator's main body (bronze colored) was fabricated in a 3D printer using the Colorfabb BronzeFill® filament whose high density leads to high inertia with smaller dimensions to achieve more movement space in the OctoMag's small workspace. The dipole is made up of two N52 NdFeB disc permanent magents ($\varnothing \qty{5}{\milli\meter} \times \qty{10}{\milli\meter}$ each) that are symmetrically attached to the center of the levitator. Overall mass of the levitator is 32.4 and its principal moments of inertia are $(I_{xx}, I_{yy}, I_{zz}) = (6. 21, 5.63, 1.14)\times 10^{-6}\qty{}{\kilo\gram\meter\squared}$. A motion capture system detects three reflective markers attached to the levitator for pose estimation. The fourth marker, colored in white, is not reflective and just attached for achieving a symmetric weight distribution.
• Figure 2: Geometric notations and key components of the eMNS and the levitator. (a) The levitator’s body frame $\left\{ \mathrm{B} \right\}$ and the inertial frame $\left\{ \mathrm{V} \right\}$. The OctoMag consists of eight solenoidal coils (approx. $\varnothing \qty{103}{\milli\meter} \times \qty{170}{\milli\meter}$), each with a soft iron core, arranged in a spherical combination. The origin of $\left\{ \mathrm{V} \right\}$ is located at the center of this sphere. Full pose of the levitator is provided as the rotation matrix $\mathbf{R} \in \mathrm{\mathsf{SO}}(3)$ from $\left\{ \mathrm{B} \right\}$ to $\left\{ \mathrm{V} \right\}$, and the position vector $\boldsymbol{p} \in \mathbb{R}^{3}$ of $\left\{ \mathrm{B} \right\}$ expressed in $\left\{ \mathrm{V} \right\}$. (b) The top and cross-section views of the levitator illustrating the placement of permanent magnets. Three reflective markers from \ref{['fig:octomag_with_steady_levitator']} are shown in gray, while the non-reflective marker is not shown. The origin of $\left\{ \mathrm{B} \right\}$ is approximately at the center of mass of the levitator. The two axially magnetized disc magnets used in the levitator are illustrated with a dot pattern. The net magnetic dipole moment of the assumed ideal point dipole in $\left\{ \mathrm{B} \right\}$ is ${}^{\mathrm{B}}\boldsymbol{\tilde{m}} = -|\boldsymbol{\tilde{m}}| {}^{\mathrm{B}}\boldsymbol{e}_{z}$, since dipole moment is oriented from the magnet stack's overall south-pole ($\texttt{S}$) to the north-pole ($\texttt{N}$) by convention.
• Figure 3: Block diagram of the full levitation pipeline. All state variables with a $\hat{}$ in the diagrams are states estimates from motion capture feedback and finite differences. Solid lines between blocks represent signals with linear gains and operations while dashed lines represent flow of information through non-linear function blocks. The solid black vertical bars represent column vector concatenation blocks.
• Figure 4: Trajectory tracking experiments. Orientation is represented as XYZ intrinsic Euler angles: roll $\phi$, pitch $\theta$, and yaw $\psi$; therefore, in this case $\psi$ is the uncontrollable rotation mode. Left: Reference and actual pose of the levitator for a trajectory with large angles. The object tracks roll and pitch angles as high as $\pm\qty{45}{\degree}$ followed by figure eight maneuvers in the $xy$-plane without integrators. Gyroscopic coupling from imperfections in the levitator leads to large drifts in $\psi$. Right: Reference and actual pose for the figure eight trajectory with integrators enabled. Integrators remove the steady state error in $z$-setpoint tracking compared to the previous case. With minimal roll and pitch changes, gyroscopic coupling effects are negligible, so $\psi$ shows less drift.
• Figure 5: 3D plot of reference and actual positions while tracking the $xy$ figure eight trajectory with integrators enabled.