Fig. 2: Spectral Submanifold Reduction (SSMR) for Model Learning and Control.
From: Discovering dominant dynamics for nonlinear continuum robot control
a Collect decaying and controlled trajectories of the continuum robot. The decaying trajectories are truncated to remove initial transients, ensuring the data is close to the slow, attracting SSM of preselected dimension. Critically, the decaying trajectories will characterize the structure of the autonomous SSM while controlled trajectories will calibrate the internal dynamics of this invariant manifold as it smoothly deforms under control. b Use SSMLearn36 to extract the dominant dynamics of the continuum robot on the autonomous SSM. Optionally, we can reparameterize the SSM to obtain a model with observables more favorable for control. Lastly, the calibration step learns the effect of control parallel to the autonomous manifold. c The SSMR-based, control-oriented model is used in a model predictive control scheme where planning and control is conducted in the reduced coordinates that parameterize the SSM.
