Toolbox including optimization techniques for estimation of Globally Asymptotically Stable Dynamical Systems focused on Linear Parameter Varying formulation with GMM-based mixing function and different Lyapunov candidate functions as proposed in [1], where a non-linear DS formulated as:

is learned from demonstrations in a decoupled manner. Where the GMM parameters used to parametrize are estimated via the physically-consistent GMM approach proposed in [1] and provided in phys-gmm and the DS parameters are estimated via semi-definite optimization problem that ensures global asymptotic stability of the system via constraints derived from either a:

• QLF (Quadratic Lyapunov Function):
• P-QLF(Parametrized QLF):

This allows us to accurately encode highly non-linear, non-monotic trajectories as the ones below:

while ensuring global asymptotic stability.

• [Necessary] phys-gmm: This package needs the physically-consisent GMM (PC-GMM) fitting proposed in [1]. Please download it, install its dependencies and place it in your MATLAB workspace path.

• [Optional]: For comparison purposes, this toolbox also includes demo scripts for DS learning with SEDS [2].

  • To run the SEDS learning demo script, download SEDS implementation from

  $ git clone https://bitbucket.org/khansari/seds SEDS 
  

  and place it in the thirdparty/seds folder.

Each demo_learn_*.m script includes self-explanatory code-block instructions to learn DS with:

• demo_learn_lpvds.m: The proposed LPV-DS approach [1] allowing to test different mixing function estimation approaches and DS parameter constraint optimization variants.
• demo_learn_seds.m: The se-DS approach [2] allowing to test different estimation approaches for the intial GMM and different object functions for the DS optimization.

In each of these scripts you can load the datasets shown below or any motion from the LASA Handwriting dataset. Also with the demo_drawData_DS.m you can draw your own 2D datasets on a GUI and

The demo_incremental_lpvDS.m script shows an implementation of the incremental learning framework proposed in [1] using the 2D datasets used in the paper or by drawing your own 2D datasets!

These examples + more datasets are provided in ./datasets folder. Following we show some notably challenging motions that cannot be accurately encoded with SEDS [3] (1st Column) or a PC-GMM-based LPV-DS [1] with a simple Quadradtic Lyapunov Function (QLF) (2nd Column), yet can be correctly encoded with the proposed PC-GMM-based LPV-DS with a parametrized QLF (P-QLF) [1] (3rd Column) yielding comparable (in some cases BETTER) results than the global diffeomorphic matching approach [3] (4th Column), which is the state-of-the-art approach known to outperform SEDS. To reproduce result for the latter, please contact the original authors of that paper.

• 2D S-shape Dataset

• 2D Multi-Behavior (Single Target) Dataset

• 2D Messy Snake Dataset

• 2D SharpC-shape from LASA Handwriting Dataset

• 2D N-shape from LASA Handwriting Dataset

• 2D Hee-shape from LASA Handwriting Dataset

• 2D Snake-shape from LASA Handwriting Dataset

The following 3D datasets where collected via kinesthetic teaching of a KUKA LWR 4+ robot and processed using the code provided in the easy-kinesthetic-recording package.

• 3D Sink Motion for "Inspection Line" Task

• 3D Via-point Motion for "Branding Line" Task

• 3D Cshape-top Motion for "Shelf Arranging" Task

• 3D Cshape-botton Motion for "Shelf Arranging" Task

References

[1] Figueroa, N. and Billard, A. (2018) A Physically-Consistent Bayesian Non-Parametric Mixture Model for Dynamical System Learning. In Proceedings of the 2nd Conference on Robot Learning (CoRL). Accepted.
[2] Khansari Zadeh, S. M. and Billard, A. (2011) Learning Stable Non-Linear Dynamical Systems with Gaussian Mixture Models. IEEE Transaction on Robotics, vol. 27, num 5, p. 943-957.
[3] N. Perrin and P. Schlehuber-Caissier, “Fast diffeomorphic matching to learn globally asymptotically stable nonlinear dynamical systems,” Systems & Control Letters (2016).

Contact: Nadia Figueroa (nadia.figueroafernandez AT epfl dot ch)

Acknowledgments This work was supported by the EU project Cogimon H2020-ICT-23-2014.