Dynamic Electromagnetic Navigation

TL;DR

This work demonstrates dynamic electromagnetic navigation (eMNS) by stabilizing a 3D inverted pendulum on a magnetically driven arm using an eight-coil OctoMag system. It develops a Lagrangian-based dynamic model, identifies key parameters via multisine excitation, and designs a cascaded controller with state feedback, online calibration compensation, and a norm-optimal Iterative Learning Control for repetitive trajectory tracking. The study characterizes electrical dynamics across coil scales, showing substantial bandwidth benefits that persist in clinical-scale hardware, and analyzes magnetic-field gradient effects as a fundamental limitation. Overall, the approach enables high-bandwidth, disturbance-rejecting control in dynamic magnetic navigation with potential applications in eMNS-guided cardiovascular interventions and beyond.

Abstract

Magnetic navigation offers wireless control over magnetic objects, which has important medical applications, such as targeted drug delivery and minimally invasive surgery. Magnetic navigation systems are categorized into systems using permanent magnets and systems based on electromagnets. Electromagnetic Navigation Systems (eMNSs) are believed to have a superior actuation bandwidth, facilitating trajectory tracking and disturbance rejection. This greatly expands the range of potential medical applications and includes even dynamic environments as encountered in cardiovascular interventions. To showcase the dynamic capabilities of eMNSs, we successfully stabilize a (non-magnetic) inverted pendulum on the tip of a magnetically driven arm. Our approach employs a model-based framework that leverages Lagrangian mechanics to capture the interaction between the mechanical dynamics and the magnetic field. Using system identification, we estimate unknown parameters, the actuation bandwidth, and characterize the system's nonlinearity. To explore the limits of electromagnetic navigation and evaluate its scalability, we characterize the electrical system dynamics and perform reference measurements on a clinical-scale eMNS, affirming that the proposed dynamic control methodologies effectively translate to larger coil configurations. A state-feedback controller stabilizes the inherently unstable pendulum, and an iterative learning control scheme enables accurate tracking of non-equilibrium trajectories. Furthermore, to understand structural limitations of our control strategy, we analyze the influence of magnetic field gradients on the motion of the system. To our knowledge, this is the first demonstration to stabilize a 3D inverted pendulum through electromagnetic navigation.

Figures (5)

• Figure 1: 3D inverted pendulum balanced using the OctoMag electromagnetic navigation system. The system includes an arm driven by the external magnetic field, and a non-magnetic pendulum, free to rotate spherically. Both arms are connected through a (non-magnetic) spherical joint.
• Figure 2: The figure illustrates the parametrization of orientations, describing rotations with respect to the inertial frame of the motion capture system. Both the actuator and the inverted pendulum possess two kinematic degrees of freedom. These are parametrized by the generalized coordinates $\alpha$ and $\beta$ for the actuator, and $\varphi$ and $\theta$ for the inverted pendulum. The control input to the system is the orientation of the magnetic field vector $\bm{b}$, parametrized by the angles $u_\alpha, u_\beta$. Rotations described by $u_\alpha, \alpha,$ and $\varphi$ are with respect to the inertial $\bm{e}_y$ axis, whereas $u_\beta, \beta,$ and $\theta$ correspond to the inertial $\bm{e}_x$ axis. We demonstrate pendulum stabilization of lengths between $L=\unit[20-40]{cm}$, with an actuator length of $\ell=\unit[22]{cm}$. The distance between the pivot point and the magnet center is denoted by $\ell_\mathrm{m}$. The pendulum, actuator, joint, and magnet masses are denoted by $M$, $m$, $m_\mathrm{j}$, and $m_\mathrm{m}$, respectively.
• Figure 3: Bode diagram obtained from system identification experiments of the actuator dynamics (blue). For the actuator dynamics, similar results are observed for the $\beta$ direction. The standard deviation $\hat{\sigma}_{\alpha, \mathrm{nl}}$ captures the uncertainty of the estimate. We map $\hat{\sigma}_{\alpha, \mathrm{nl}}$ to values between $(0,1]$, used as weights (cyan) during curve fitting to obtain a parametric transfer function estimate (blue line). Furthermore, the dynamics of the closed-loop current control system for both the OctoMag coil (red) and the clinically ready Navion eMNS coil (green) are depicted; their respective bandwidths ([-3]dB gain reduction) indicated by dashed lines. Although the bandwidth advantage is slightly reduced for the larger Navion coil, it still remains an order of magnitude higher than expected disturbances in potential clinical applications.
• Figure 4: Block diagram of the system illustrating the cascaded control structure. The prefilters, $F_\alpha, F_\beta$, are designed to reduce steady-state errors by scaling the setpoints appropriately. Setpoints, $\alpha_\mathrm{SP}, \beta_\mathrm{SP}$, are mapped to the state vector using $\mathrm{\Psi}_{x, \alpha}: \alpha_\mathrm{SP} \mapsto \alpha_\mathrm{SP}0\dot{\alpha}_\mathrm{SP}0 ^\top$, and similarly for $\beta_\mathrm{SP}$. The state feedback controllers, $\bm{\mathrm{K}}_\alpha, \bm{\mathrm{K}}_\beta$, operate at [100]Hz determining the magnetic field orientation, $u_\alpha, u_\beta$, which is converted to the magnetic field vector using the allocation $\mathrm{\Psi}_b$. The magnetic field vector is then converted into electrical currents using the actuation matrix, $\bm{\mathcal{A}}^\dag$. The electrical currents are controlled by eight independent PI controllers (drivers). Full state information is derived using finite-difference differentiation. An Iterative Learning Control (ILC) scheme is included calculating feedforward correction signals, $u_\alpha^n, u_\beta^n$, to counteract any repetitive error during the tracking of periodic reference trajectories.
• Figure 5: Experimental results tracking a circular and a figure eight trajectory, without learning and with activated learning control scheme (top plots). Iteration 0 corresponds to no learning. The ILC scheme is able to compensate for all repetitive disturbances. Due to the inherent instability of the inverted pendulum non-repetitve disturbances persist. The bottom plot shows the torque contribution (in body-fixed frame $\mathcal{B}$) for each of the ten magnets arising from the magnetic- and its gradient field, indicating that the magnetic field varies signifcantly over the magnetic volume, being the main source of the repetitve errors. Similar results are obtained for $\prescript{}{\mathcal{B}}{\tau_\beta}$.