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A deep inverse-mapping model for a flapping robotic wing
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Proposes a Seq2Seq model with a novel Adaptive Spectrum Layer (FFT magnitude/phase gating) to learn the inverse mapping from desired aerodynamic forces to flapping-wing kinematics, using a new measured wing dataset and an open-source dataset.
Problem
Inverting the forward wing-kinematics-to-aerodynamic-force mapping of a flapping wing is nontrivial and too computationally costly for real-time control, so the paper learns the inverse (force-to-kinematics) mapping directly from data.
Approach
A GRU-based Seq2Seq model with attention, augmented by an Adaptive Spectrum Layer that applies rFFT, represents each frequency bin by magnitude and phase (sin/cos), gates the bins with a learned weight vector, and returns to the time domain via IFFT with a skip connection.
Key results
- On the authors' dataset, Seq2Seq+ASL reached median MAE 0.1216 rad versus 0.1430 for Transformer and 0.2271 for AutoFormer.
- On the open-source dataset, Seq2Seq+ASL reached median MAE 0.0908 rad versus 0.1021 for Transformer (about 11% better).
- Seq2Seq+ASL inference took 2.00 ms versus 19.53 ms for Transformer, with roughly 200k vs 50k parameters.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add the Adaptive Spectrum Layer to Seq2Seq (ablation) | Seq2Seq · Our dataset test | MAE: 0.1471 -> 0.1323 | Helped |
| Add the Adaptive Spectrum Layer to Seq2Seq (ablation) | Seq2Seq · Our dataset test | Median MAE: 0.1236 -> 0.1216 | Helped |
| Add the Adaptive Spectrum Layer to Seq2Seq (ablation) | Seq2Seq · Open Source dataset test | MAE: 0.1206 -> 0.1130 | Hurt |
| Add the Adaptive Spectrum Layer to Seq2Seq (ablation) | Seq2Seq · Open Source dataset test | Median MAE: 0.1058 -> 0.0908 | Helped |
| Use Seq2Seq+ASL instead of Transformer | Flapping-wing inverse mapping · Our dataset test | MAE: 0.1323 vs. 0.1476; 0.1216 vs. 0.1430 | Mixed |
| Use Seq2Seq+ASL instead of Transformer | Flapping-wing inverse mapping · Open Source dataset test | MAE: 0.1130 vs. 0.1123; 0.0908 vs. 0.1021 | Mixed |
| Use Seq2Seq+ASL instead of Transformer | Flapping-wing inverse mapping · Inference time | inference time: 2.00\pm0.13 \textrm{ms} vs. 19.53 \pm 5 \textrm{ms} | Helped |
| Encode phase with sine/cosine rather than an angle (ablation) | ASL · Flapping-wing inverse mapping | MAE | No clear effect |
| Vary the low-pass filter cutoff frequency among 20, 100, and 210Hz (ablation) | ASL · Flapping-wing inverse mapping | MAE: 20, 100, and 210Hz | Mixed |
| Add ASL gating (ablation) | ASL · Flapping-wing inverse mapping | MAE | No clear effect |
| Use a per-frequency layer to combine frequency bins (ablation) | ASL · Flapping-wing inverse mapping | MAE | No clear effect |
| Learn a new complex representation from the FFT vector (ablation) | ASL · Flapping-wing inverse mapping | MAE | No clear effect |
Surprising findings
- On the open-source dataset, Seq2Seq+ASL and Transformer were statistically indistinguishable in mean MAE (Wilcoxon p=0.58), and the Transformer had the better mean MAE (0.1123 vs 0.1130).
- Seq2Seq+ASL produced more outlier events than the Transformer, so its advantage appears mainly in median rather than mean error.
Limitations
- Demonstrated only on a bench-top single-wing device, not an onboard flapping robot.
- Does not generalize to arbitrary wing forms or Reynolds numbers given the limited datasets.
- Single-axis wing hinge may restrict the range of achievable wing motions.
Experimental setup
Datasets: Our measured flapping-wing dataset (153 events) Open-source flapping-wing dataset (Bayiz and Cheng, 548 events)
Benchmarks: Our flapping-wing dataset test split Open-source flapping-wing dataset test split
Metrics: MAE (radians), mean MAE, median inference time (ms)
Baselines: Transformer AutoFormer FedFormer NLinear Linear Seq2Seq
Builds on: Seq2Seq Bahdanau attention FedFormer Autoformer
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