Closed-loop control of a monolithically 3D nano-printed electromagnetic lens scanner with an integrated Hall sensor

1. Problem Statement

Three-dimensional (3D) nano-printing via two-photon polymerization (2PP) allows for the monolithic fabrication of complex micro-optical and mechanical systems with high alignment accuracy. However, the photopolymer materials used (e.g., IP-S) exhibit intrinsic viscoelasticity and temperature-dependent stiffness.

• Viscoelasticity: Causes creep (time-dependent deformation under constant load) and hysteresis, degrading position accuracy in quasi-static operations.
• Thermal Drift: Joule heating from electromagnetic coils reduces the Young's modulus of the polymer, causing significant drift in actuator response during continuous operation.
• Limitation: These factors make precise, repeatable open-loop control impractical for high-precision applications like endomicroscopes or miniscopes.
• Challenge: Implementing closed-loop control in these monolithic systems is difficult because integrating sensors without disrupting the 3D structure or requiring complex post-processing (like buried conductors for capacitive sensing) is challenging.

2. Methodology

The authors developed a closed-loop control system for a 3D nano-printed lens scanner using a commercial 3-axis Hall sensor for integrated position tracking.

Device Architecture

• Actuator: A monolithically 3D-printed lens scanner featuring ortho-planar linear-motion springs, a movable shuttle with an aspherical microlens, and a ring-shaped NdFeB micro-magnet.
• Actuation: Two electromagnetic coils arranged in an anti-Helmholtz-like configuration drive the scanner bidirectionally. This creates a high magnetic field gradient to generate axial force while minimizing torque.
• Sensing: A compact 3-axis Hall sensor (MLX90394, 2 x 1.5 x 0.35 mm³) is mounted on a flexible PCB and sandwiched between the two coils, as close to the magnet as possible, without increasing the device footprint.

Control Strategy & Calibration

The system relies on separating the magnetic field contributions to isolate the magnet's position. The total measured field ($\vec{B}$) is modeled as:
$$\vec{B} = \vec{B}_{offset} + \vec{B}_{coil}(I_{coil}) + \vec{B}_{magnet}(z_{magnet})$$

To recover the magnet position ($z_{magnet}$), a calibration-based subtraction process is employed:

1. Offset Calibration: Measure $\vec{B}_{offset}$ (sensor bias + static environmental fields) at zero current.
2. Coil Calibration: Mechanically block the shuttle to isolate the coil's field contribution ($\frac{d\vec{B}}{dI_{coil}}$) as a function of current.
3. Position Calibration: Use a Laser Doppler Vibrometer (LDV) as a reference to map the corrected magnetic field to displacement ($\frac{d\vec{B}}{dz_{magnet}}$).
4. Real-time Calculation: The controller subtracts the offset and the calculated coil field from the raw sensor reading to estimate the magnet position, which is then fed into a Proportional-Integral (PI) controller.

Note: A PID controller was rejected because the Hall sensor's sampling rate (1.4 kHz) is only ~4x the device's natural frequency (362 Hz); adding a derivative term would cause instability.

3. Key Contributions

• First Integrated Magnetic Self-Sensing: Demonstrated the first closed-loop control of a monolithically 3D nano-printed electromagnetic actuator using integrated magnetic sensing, avoiding complex micro-fabrication steps for sensors.
• Field Separation Technique: Developed a robust calibration method to mathematically separate the large coil-induced magnetic field from the smaller magnet-induced field, enabling accurate position tracking despite the coil field being an order of magnitude larger.
• Practical Implementation: Showed that a commercial off-the-shelf Hall sensor can be integrated into a 3D-printed optical system without compromising the monolithic design or alignment.

4. Key Results

The system was tested over a displacement range of ±75 µm (150 µm total).

• Accuracy & Precision:

  • Mean Absolute Accuracy: 0.86 µm.
  • Precision (Std Dev): 0.49 µm.
  • The error is primarily attributed to residual calibration/model errors and sensor noise.

• Viscoelastic Creep Elimination:

  • Open-loop: Exhibited significant drift (22 µm drift after 10s for a 75 µm step, ~29% error).
  • Closed-loop: Effectively eliminated creep; no significant drift was observed over time.

• Hysteresis Suppression:

  • Open-loop: Mean relative hysteresis of 9.0%.
  • Closed-loop: Reduced to −1.4% (the negative sign is due to the PI controller's integral action).

• Dynamic Response:

  • Settling Time: Reduced from 69.7 ms (open-loop) to 17.6 ms (closed-loop).
  • Overshoot: Reduced from 44.3% to 18.8%.
  • Ringing: Closed-loop control significantly suppressed ringing at the damped natural frequency.

• Thermal Stability:

  • Under continuous operation (1000 cycles of 150 µm displacement), the open-loop system showed a 10% increase in peak-to-peak displacement due to coil self-heating.
  • The closed-loop system reduced this thermal drift by 83%, though a residual drift remained due to temperature-dependent changes in the magnet's remanence and sensor geometry.

5. Significance

This work provides a practical pathway for the widespread adoption of 3D nano-printed optical microsystems in precision applications (e.g., medical imaging, miniscopes). By solving the critical issues of viscoelastic creep, hysteresis, and thermal drift through a non-invasive, integrated sensing approach, the authors enable these devices to achieve sub-micron stability and repeatability. This bridges the gap between the design freedom of 3D printing and the rigorous performance requirements of high-precision optical instrumentation.