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Variational integrators for optimal control of foldable drones

L. Colombo, J. Giribet, D. Martín de Diego

TL;DR

The paper addresses attitude control and trajectory planning for foldable UAVs whose inertia changes with internal configuration. It develops structure-preserving, discrete-time variational integrators on Lie groups by discretizing reduced Lagrangians and external forces, yielding discrete Euler–Poincaré/Lie–Poisson dynamics with control. It then formulates and analyzes discrete-time optimal-control problems for controlled inertia rigid bodies and foldable drones, using trapezoidal discretization and Cayley retractions to obtain implementable update rules and solvability guarantees. Simulation results on a foldable quadrotor model demonstrate stable attitude stabilization and accurate tracking under challenging initial conditions, highlighting the approach's potential for reconfigurable UAVs and energy-efficient control.

Abstract

Numerical methods that preserves geometric invariants of the system such as energy, momentum and symplectic form, are called geometric integrators. These include variational integrators as an important subclass of geometric integrators. The general idea for those variational integrators is to discretize Hamilton's principle rather than the equations of motion and as a consequence these methods preserves some of the invariants of the original system (symplecticity, symmetry, good behavior of energy,...). In this paper, we construct variational integrators for control-dependent Lagrangian systems on Lie groups. These integrators are derived via a discrete-time variational principle for discrete-time control-dependent reduced Lagrangians. We employ the variational integrator into optimal control problems for path planning of foldable unmanned aerial vehicles (UAVs). Simulation are shown to validate the performance of the geometric integrator.

Variational integrators for optimal control of foldable drones

TL;DR

The paper addresses attitude control and trajectory planning for foldable UAVs whose inertia changes with internal configuration. It develops structure-preserving, discrete-time variational integrators on Lie groups by discretizing reduced Lagrangians and external forces, yielding discrete Euler–Poincaré/Lie–Poisson dynamics with control. It then formulates and analyzes discrete-time optimal-control problems for controlled inertia rigid bodies and foldable drones, using trapezoidal discretization and Cayley retractions to obtain implementable update rules and solvability guarantees. Simulation results on a foldable quadrotor model demonstrate stable attitude stabilization and accurate tracking under challenging initial conditions, highlighting the approach's potential for reconfigurable UAVs and energy-efficient control.

Abstract

Numerical methods that preserves geometric invariants of the system such as energy, momentum and symplectic form, are called geometric integrators. These include variational integrators as an important subclass of geometric integrators. The general idea for those variational integrators is to discretize Hamilton's principle rather than the equations of motion and as a consequence these methods preserves some of the invariants of the original system (symplecticity, symmetry, good behavior of energy,...). In this paper, we construct variational integrators for control-dependent Lagrangian systems on Lie groups. These integrators are derived via a discrete-time variational principle for discrete-time control-dependent reduced Lagrangians. We employ the variational integrator into optimal control problems for path planning of foldable unmanned aerial vehicles (UAVs). Simulation are shown to validate the performance of the geometric integrator.

Paper Structure

This paper contains 12 sections, 1 theorem, 66 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Dynamics of mechanical systems on Lie groups
  3. Optimal control for controlled inertia rigid bodies
  4. Necessary conditions for optimallity of controlled inertia rigid bodies
  5. Optimal control problem
  6. Necessary conditions for optimality
  7. Optimal control problem for foldable drones
  8. Variational integrators on Lie groups for control-dependent systems
  9. Variational integrators on Lie groups
  10. Variational integrators for the optimal control of controlled inertia rigid bodies
  11. Variational integrators on Lie groups for the optimal control of foldable drones
  12. Simulation results

Key Result

Theorem 5.1

Let $\ell:\mathfrak{g}\times U\to\mathbb{R}$ be a controlled reduced Lagrangian function and $f:G\times\mathfrak{g}\times U\times\bar{U}\to G\times\mathfrak{g}^{*}$ external forces. For a curve $g(t)$ consider the curve on the Lie algebra $\mathfrak{g}$ given by $\omega(t)=g^{-1}\dot{g}$, then Lagra holds for all variations of $g(t)$ constrained to $\delta\omega=\dot{\eta}+ad_{\omega}\eta$, where

Figures (5)

  • Figure 1: Foldable quadrotor model. In solid line, the vehicle is in "X" configuration. In the dashed line, the vehicle is in "H" configuration. $u$ denotes the (constant) internal control and $\phi$, $\theta$, $\psi$ denotes roll, pitch and yaw for the quadrotor motion.
  • Figure 1: Retraction map.
  • Figure 1: Vehicle attitude (roll, pitch, and yaw) during stabilization and tracking phases.
  • Figure 2: Tracking error for roll, pitch, and yaw angles over time.
  • Figure 3: Angle $u$ (internal control - see Figure \ref{['drone']}) of the quadcopter's arms during the control maneuvers.

Theorems & Definitions (2)

  • Example 3.1
  • Theorem 5.1