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Input-to-State Stable Coupled Oscillator Networks for Closed-form Model-based Control in Latent Space
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Proposes Coupled Oscillator Networks (CON), a Lagrangian, globally ISS-stable coupled damped-oscillator latent dynamics model with an approximate closed-form integrator (CFA-CON), used for learning latent dynamics from pixels and potential-shaping control in latent space.
Problem
Latent-space control of physical systems requires latent dynamics that have physical structure, conserve stability, are low-dimensional, and have an invertible input-to-forcing mapping; existing latent models (MLPs, NODEs, RNNs, LNNs) lack one or more of these properties.
Approach
Couple damped harmonic oscillators via elastic springs, damping and a tanh neuron-like term, transform to W-coordinates so the coupling force is symmetric and derives a potential energy, prove global asymptotic/ISS stability with a Lyapunov function, learn an input-to-forcing map and forcing decoder, and use potential shaping (feedforward compensation + saturated PID) for latent control.
Key results
- CON-M reduces RMSE on PCC-NS-3 by 6% relative to the closest baseline (MECH-NODE) and matches SoA across actuated soft-robot datasets, with far fewer parameters than NODE.
- Latent P-satI-D+FF controller on a simulated two-segment soft robot achieves 26% lower RMSE (2.12 vs 2.88 rad/m) than a P-satI-D baseline on MECH-NODE latent model.
- CFA-CON is about 30% more accurate than Euler at half its speed and gives 2x training speedup with small accuracy loss; CFA-UDCON is 2.4x faster than CFA-CON with no accuracy loss for underdamped networks.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use CON instead of NODE for latent dynamics | latent dynamics models · M-SP+F | RMSE: 0.0264 ± 0.0010 -> 0.0303 ± 0.0053 | Hurt |
| use CON instead of NODE for latent dynamics | latent dynamics models · S-P+F | RMSE: 0.1260 ± 0.0013 -> 0.1303 ± 0.0064 | Hurt |
| use CFA-CON instead of CON-M (ablation) | CON latent dynamics · D-P+F | RMSE: 0.1323 ± 0.0018 -> 0.1307 ± 0.0012 | Helped |
| use CFA-CON instead of CON-M (ablation) | CON latent dynamics · CS | RMSE: 0.1063 ± 0.0027 -> 0.1462 ± 0.0211 | Hurt |
| use CON-M instead of MECH-NODE | latent dynamics models · PCC-NS-3 | RMSE: 0.1900 ± 0.0024 -> 0.1785 ± 0.0023 | Helped |
| add potential-force compensation feedforward to P-satI-D (ablation) | CON latent-space controller · PCC-NS-2 control | RMSE: 4.08 rad/m -> 2.12 rad/m | Helped |
| use P-satI-D+FF with CON instead of P-satI-D with MECH-NODE | latent-space controller · PCC-NS-2 control | RMSE: 2.88 rad/m -> 2.12 rad/m; 26 % lower RMSE | Helped |
| replace numerical integration with CFA-CON (ablation) | CON integration · CON integration benchmark | RMSE: 0.010 ± 0.003 -> 0.007 ± 0.002; 30 % more accurate than the Euler integrator at half of the speed | Mixed |
| specialize CFA-CON for underdamped oscillators (ablation) | CFA-CON integration · Underdamped CON integration benchmark | simulation time to real time: 36500x -> 41900x | Helped |
Surprising findings
- NODE slightly outperforms CON on the unactuated M-SP+F and S-P+F datasets despite CON having about two orders of magnitude fewer parameters.
- The closed-form approximation performs worse when nonlinear coupled forces dominate, and control performance was only explored when latent dimension equals input dimension.
Limitations
- Assumes Markovian, fully observable systems with an isolated globally asymptotically stable equilibrium; not suited to nonholonomic, partially observable, or multi-equilibrium systems.
- Closed-form approximation degrades when nonlinear coupled dynamics dominate.
- Control only shown for latent dimension equal to input dimension; nz > m or nz < m not investigated.
Experimental setup
Datasets: M-SP+F S-P+F D-P+F CS PCC-NS-2 PCC-NS-3 reaction-diffusion
Benchmarks: M-SP+F S-P+F D-P+F CS PCC-NS-2 PCC-NS-3
Metrics: RMSE number of trainable parameters simulation-time to real-time factor
Baselines: RNN GRU coRNN NODE MECH-NODE Dopri5 Tsit5
Builds on: coupled oscillatory RNN (coRNN) Neural ODE beta-VAE potential shaping
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