SSMRc: Spectral Submanifold Reduction for Control of High-Dimensional Robots

This repository contains the code accompanying our recent ICRA paper: Data-Driven Spectral Submanifold Reduction for Nonlinear Optimal Control of High-Dimensional Robots. We learn low-dimensional, faithful control dynamics of high degree-of-freedom (DOF) mechanical systems on attracting, invariant manifolds called Spectral Submanifolds ¹. This work extends recent work on learning autonomous dynamics of nonlinearizable systems to the control setting.²

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Dependencies

• SSMLearn: Repo has a stand-alone version installed in ext/ folder. This could potentially be deprecated, so we recommend users to install the latest version.
• Soft Robot Control: Required to generate data from SOFA simulation of soft robots.

Getting Started

• To get started, in Matlab run install.m.
• Go to the ROM/ folder and run generateSSMmodel.mlx to generate a discrete and continuous-time SSMRc model based on our dataset dataDecayObs, which can be downloaded here.
• The generated model must then be imported into the examples/hardware/SSMmodels folder of the soft-robot-control repo. Follow the instructions on the corresponding repo to run the controller.

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Generating your own model

Step 1: Collect Training Data

• To parameterize the autonomous manifold, we must first collect decaying data trajectories from either a simulation environment (e.g., SOFA for soft robot applications) or from hardware experiments.
  • To use the consolidateSSMTrainData.m function (which consolidates the individual training data into appropriate matlab cell format), the individual training data must be saved as .mat files. See hw_train.mlx for working example.
• Once the individual raw trajectory files are consolidated into the appropriate cell format (i.e., cell column 1 := time, cell column 2 := decay trajectory), the transients from the dataset is truncated to ensure the training trajectories are near the invariant manifold. See ROM/generateSSMmodel.mlx, ROM/mainObsCDS.mlx, or ROM/mainObsDDS.mlx for an example.
  • Discrete-time model: ROM/mainObsCDS.mlx
  • Continuous-time model: ROM/mainObsDDS.mlx
  • Both: ROM/generateSSMmodel.mlx
• Below is a visual depiction of this process on our soft robot example.

Raw Data

Training Data

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Step 2: Parameterize Manifold and Reduced Dynamics

• Examples for learning the autonomous dynamics is located in ROM/generateSSMmodel.mlx, ROM/mainObsCDS.mlx, or ROM/mainObsDDS.mlx.
• PCA is used to approximate the spectral subspace and determines the dimension of the invariant manifold.
• The polynomial order of the approximation is chosen as a hyperparameter as a tradeoff to increase the accuracy of the parameterization and mitigate overfitting.

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Step 3: Learning the Reduced actuation matrix

• To learn the effect of actuation, we must correlate the effect of actuation on the reduced dynamics. Thus, we must collect state transition data under the influence of control inputs as shown in the figure above. Again, this can be collected from either a simulator or experiments.
• Code for regressing the reduced order actuation matrix is located in ROM/generateSSMmodel.mlx.

Footnotes

1. Haller, George, and Sten Ponsioen. "Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction." Nonlinear dynamics 86.3 (2016): 1493-1534. ↩

2. Cenedese, Mattia, et al. "Data-driven modeling and prediction of non-linearizable dynamics via spectral submanifolds." Nature communications 13.1 (2022): 1-13. ↩