High‑Precision Multi‑Degree‑of‑Freedom Motion Measurement Using PDMS Cross‑Coupled Diffraction Gratings

Background

Multi‑degrees‑of‑freedom (MDOF) motion parameters are critical for precise positioning and attitude control in large‑scale systems, including aircraft, robotic arms, and precision manufacturing. High‑precision sensing of displacement, pitch, and deflection angles is therefore essential across aerospace, UAV, and optical alignment applications.

Traditional MDOF measurement techniques rely on arrays of high‑performance sensors that must operate synchronously at high speed and with sub‑micrometer accuracy. Recent advances in micro‑nanofabrication and nanomaterials have enabled single‑chip implementations that reduce size, cost, and complexity while maintaining or improving accuracy.

Prior work has explored three‑dimensional sensor arrays and angle‑dependent grating techniques. However, these approaches often suffer from assembly‑induced errors and computational complexity. In contrast, the method presented here uses a single orthogonal PDMS grating pair fabricated via oxygen‑plasma processing, enabling a compact, low‑cost, and highly accurate MDOF measurement platform.

Experimental

Polydimethylsiloxane (PDMS) Preparation

The Sylgard 184 PDMS (10:1) mixture was spin‑coated onto silicon wafers and cured at <80 °C for 2 h. By controlling the spin speed, 600‑µm‑thick membranes were obtained.

Double Orthogonal Grating Preparation

PDMS films (3 × 3 cm²) were pre‑strained 1.5× in the X‑direction on a homemade translation stage. Wrinkled SiO₂ layers were then formed on the oxygen‑plasma‑treated, pre‑strained PDMS (30 sccm O₂, 40 s) using an IoN Wave 10 chamber. After relaxation, a uniform nanograting array appeared on one side. The process was repeated on the opposite side with a 90° rotation to create orthogonal gratings on both faces.

Fabrication process and morphology characterizations of PDMS double optical grating. a Fabricating double optical grating. b The optical images of grating. c Atomic force microscopy image of the grating. d The uniformity of periodicity for the samples

Test Platform Building

The four‑degree‑of‑freedom displacement‑angle sensor system consists of a He‑Ne laser (680 nm), a rotating platform (0.1° accuracy) and a manual 3‑D adjustment frame (2 µm precision), a specimen holder, a screen, a 480 × 640‑pixel CMOS camera, and a PC running MATLAB. The crossed gratings produce a 2‑D spot array; the camera captures the array in real time, and image‑processing algorithms extract the spot centers to calculate X‑ and Y‑displacements and incident angles.

The principle and test system for the MODF motion parameter. a system diagram. b System setup. c Testing principle of displacement and angle

Analysis and Discussion

Orthogonal PDMS Grating Characterization

Oxygen‑plasma treatment introduces hydrophilic SiO₂ layers and hydroxyl groups on the PDMS surface. When the pre‑strain exceeds a critical value, relaxing the strain forms a regular nanograting array. AFM measurements confirm a uniform period of (2 ± 0.05) µm across the sample.

Diffraction Grating for Position and Angle Motion Parameter Characterization

When a laser passes through the crossed gratings, Fraunhofer diffraction produces a 2‑D spot matrix. The spot positions shift with both source displacement and incident angle, enabling simultaneous measurement of all four degrees of freedom.

The fundamental relation is:

Here, λ is the laser wavelength, d the grating period, α the incidence angle, φ the diffraction angle, and m the diffraction order.

For first‑order spots the spacing equations become:

These expressions allow conversion from measured spot spacings to incident angles.

Orthogonal Diffraction Grating‑Based Multi‑Degree‑of‑Freedom Motion Parameter Detection and Characterization

With one grating on each side of the PDMS, the laser beam is diffracted into orthogonal 1‑D spot patterns that combine into a 2‑D array. Movement of the source translates the zero‑order spot; rotation of the source changes the inter‑spot spacing.

Image‑processing extracts spot centers using Gaussian fitting:

From the fitted coefficients the spot center is:

By comparing successive images, absolute and relative displacements (Δx, Δy) and angle changes (Δθₓ, Δθᵧ) are calculated. The system achieved a displacement sensitivity of 5.4 pixels/µm (0.18 µm resolution) and an angular sensitivity of 2.3 pixels/° (0.0075 rad resolution). Using a higher‑pixel CCD or optical optimization could push these limits to sub‑nanometer displacement and micro‑radian angle precision.

Hovering Aircraft Rotor Motion Parameter Information Characterization

To validate the method in a real‑world scenario, a four‑rotor UAV (Typhoon Q500) carried a laser pointer at its center. The crossed grating system recorded the 2‑D spot array while the UAV hovered. Data acquired every 0.02 s revealed a maximum translational displacement of 2.1 mm (X) and 2.3 mm (Y), and a maximum angular deviation of 1° in both axes. These high‑frequency, high‑resolution measurements can be fed back to the flight‑control loop within 20 ms, dramatically improving attitude stability.

Conclusions

We demonstrated a straightforward, low‑cost fabrication process for orthogonal 2‑µm period PDMS gratings on both sides of a substrate. Using the Fraunhofer diffraction principle and a 480 × 640‑pixel CCD, the system measures displacement to 0.18 µm and deflection angles to 0.0075 rad in real time. With higher‑resolution imaging, sub‑nanometer and micro‑radian precision is attainable. This platform delivers rapid, accurate four‑degree‑of‑freedom data for hovering UAVs, enabling closed‑loop stabilization and precise control for autonomous flight.