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Optimal Potential Shaping on SE(3) via Neural ODEs on Lie Groups

Yannik P. Wotte, Federico Califano, Stefano Stramigioli

TL;DR

This work rephrase dynamic systems as so-called neural ordinary differential equations (neural ODEs) as well as formulate the optimization problem on Lie groups, and presents a gradient descent optimization algorithm to tackle the optimization numerically.

Abstract

This work presents a novel approach for the optimization of dynamic systems on finite-dimensional Lie groups. We rephrase dynamic systems as so-called neural ordinary differential equations (neural ODEs), and formulate the optimization problem on Lie groups. A gradient descent optimization algorithm is presented to tackle the optimization numerically. Our algorithm is scalable, and applicable to any finite dimensional Lie group, including matrix Lie groups. By representing the system at the Lie algebra level, we reduce the computational cost of the gradient computation. In an extensive example, optimal potential energy shaping for control of a rigid body is treated. The optimal control problem is phrased as an optimization of a neural ODE on the Lie group SE(3), and the controller is iteratively optimized. The final controller is validated on a state-regulation task.

Optimal Potential Shaping on SE(3) via Neural ODEs on Lie Groups

TL;DR

This work rephrase dynamic systems as so-called neural ordinary differential equations (neural ODEs) as well as formulate the optimization problem on Lie groups, and presents a gradient descent optimization algorithm to tackle the optimization numerically.

Abstract

This work presents a novel approach for the optimization of dynamic systems on finite-dimensional Lie groups. We rephrase dynamic systems as so-called neural ordinary differential equations (neural ODEs), and formulate the optimization problem on Lie groups. A gradient descent optimization algorithm is presented to tackle the optimization numerically. Our algorithm is scalable, and applicable to any finite dimensional Lie group, including matrix Lie groups. By representing the system at the Lie algebra level, we reduce the computational cost of the gradient computation. In an extensive example, optimal potential energy shaping for control of a rigid body is treated. The optimal control problem is phrased as an optimization of a neural ODE on the Lie group SE(3), and the controller is iteratively optimized. The final controller is validated on a state-regulation task.
Paper Structure (50 sections, 7 theorems, 120 equations, 11 figures, 3 tables, 1 algorithm)

This paper contains 50 sections, 7 theorems, 120 equations, 11 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Neural ODEs and relation to existing works
  3. Notation
  4. Main Result
  5. Lie groups
  6. Optimization problem
  7. Optimization algorithm
  8. Technical Tools
  9. Atlas and minimal Atlas on Lie groups
  10. Vectors on Lie groups
  11. Lie group integrators
  12. Gradients on Lie groups
  13. Composition of Lie groups
  14. The cases $SO(3)$ and $SE(3)$
  15. The matrix Lie groups SO(3) and SE(3)
  16. ...and 35 more sections

Key Result

Theorem 2.1

Given are the dynamics eq:dyn_Lie and the cost eq:single_trajectory_cost. Denote by $\tilde{f}_\theta(g,t) := g^{-1} f_\theta(g,t) \in \mathfrak{g}$. Then the parameter gradient $\frac{\partial}{\partial\theta} C^T_{f_\theta}(g_0)$ of the cost is given by the integral equation where the state $g(t) \in G$ and adjoint state $\lambda_g(t) \in \mathbb{R}^n$ are the solutions of the system of equatio

Figures (11)

  • Figure 1: Overview of the main contribution and structure of the article. Given a parameterized dynamical system on a Lie group, the generalized adjoint method on Lie groups lets us compute the parameter gradient of a cost-functional over system trajectories by solving a set of differential equations. This parameter gradient can then be used to iteratively update parameters by gradient descent. In practice, we sample multiple initial conditions and approximate the parameter gradient of the expected cost $C^T_{f_\theta}(\theta) := \mathbb{E}_{g_0\sim\mathbb{P}} C_{f_\theta}^T(g_0,\theta)$.
  • Figure 2: Commutative diagram of a generic Lie group $G$. Boxes represent sets, while arrows represent functions between sets. Relevant variables in a given set are indicated in red.
  • Figure 3: Commutative diagram highlighting how the natural embedding $\hookrightarrow:SE(3)\rightarrow\mathbb{R}^{4\times 4}$ can be used to restrict a general matrix function to $SE(3)$.
  • Figure 4: Visualization of the training progress of the quadratic controller characterized by $V_{Q,\theta}$ and $B_{Q,\theta}$. All figures show data averaged over 2048 sample trajectories at the given epoch, with initial conditions sampled from $\mathbb{P}(\Gamma_0)$.
  • Figure 5: Visualization of the performance of the quadratic controller characterized by $V_{Q,\theta}$ and $B_{Q,\theta}$, over 100 trajectories of rigid bodies with initial conditions sampled from $\mathbb{P}(\Gamma_0)$.
  • ...and 6 more figures

Theorems & Definitions (29)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.1: Generalized Adjoint Method on Matrix Lie Groups
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1: Atlas and Charts
  • Definition 3.2: Minimal Atlas
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 19 more