Robotics Motion Learning: Training Linked Robot Arms with Kuramoto Models

Table of Links

Abstract and 1. Introduction

2. Some recent trends in theoretical ML

   2.1 Deep Learning via continuous-time controlled dynamical system

   2.2 Probabilistic modeling and inference in DL

   2.3 Deep Learning in non-Euclidean spaces

   2.4 Physics Informed ML

3. Kuramoto model

   3.1 Kuramoto models from the geometric point of view

   3.2 Hyperbolic geometry of Kuramoto ensembles

   3.3 Kuramoto models with several globally coupled sub-ensembles

4. Kuramoto models on higher-dimensional manifolds

   4.1 Non-Abelian Kuramoto models on Lie groups

   4.2 Kuramoto models on spheres

   4.3 Kuramoto models on spheres with several globally coupled sub-ensembles

   4.4 Kuramoto models as gradient flows

   4.5 Consensus algorithms on other manifolds

5. Directional statistics and swarms on manifolds for probabilistic modeling and inference on Riemannian manifolds

   5.1 Statistical models over circles and tori

   5.2 Statistical models over spheres

   5.3 Statistical models over hyperbolic spaces

   5.4 Statistical models over orthogonal groups, Grassmannians, homogeneous spaces

6. Swarms on manifolds for DL

   6.1 Training swarms on manifolds for supervised ML

   6.2 Swarms on manifolds and directional statistics in RL

   6.3 Swarms on manifolds and directional statistics for unsupervised ML

   6.4 Statistical models for the latent space

   6.5 Kuramoto models for learning (coupled) actions of Lie groups

   6.6 Grassmannian shallow and deep learning

   6.7 Ensembles of coupled oscillators in ML: Beyond Kuramoto models

7. Examples

   7.1 Wahba’s problem

   7.2 Linked robot’s arm (planar rotations)

   7.3 Linked robot’s arm (spatial rotations)

   7.4 Embedding multilayer complex networks (Learning coupled actions of Lorentz groups)

8. Conclusion and References

7.2 Linked robot’s arm (planar rotations)

Suppose that we have a set of observations of the linked robot’s arm with m − 1 joints. Mathematically, this is formalized as a problem of learning m coupled actions of the group SO(2). Recent study [48] experimented with three types of normalizing flows on the torus with applications to this problem.

7.2.1 Deterministic policy for learning coupled planar rotations

7.2.2 Stochastic policy for learning coupled planar rotations

The simplicity of such stochastic policy implies limited representative power. Indeed, this model can learn only distributions whose marginals are symmetric and unimodal. This can be sufficient if the motions are not very complicated.

This approach can naturally be applied to the sequential (temporal) data. Therefore, the Kuramoto model can be trained to predict and imitate motions of the linked robot’s arm in real time.