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Lossless optimal transient control for rigid bodies in 3D space

Riccardo Zanella, Federico Califano, Antonio Franchi, Stefano Stramigioli

Abstract

In this letter, we propose a control scheme for rigid bodies designed to optimise transient behaviors. The search space for the optimal control input is parameterized to yield a passive, specifically lossless, nonlinear feedback controller. As a result, it can be combined with other stabilizing controllers without compromising the stability of the closed-loop system. The controller commands torques generating fictitious gyroscopic effects characteristics of 3D rotational rigid body motions, and as such does not inject nor extract kinetic energy from the system. We validate the controller in simulation using a model predictive control (MPC) scheme, successfully combining stability and performance in a stabilization task with obstacle avoidance constraints.

Lossless optimal transient control for rigid bodies in 3D space

Abstract

In this letter, we propose a control scheme for rigid bodies designed to optimise transient behaviors. The search space for the optimal control input is parameterized to yield a passive, specifically lossless, nonlinear feedback controller. As a result, it can be combined with other stabilizing controllers without compromising the stability of the closed-loop system. The controller commands torques generating fictitious gyroscopic effects characteristics of 3D rotational rigid body motions, and as such does not inject nor extract kinetic energy from the system. We validate the controller in simulation using a model predictive control (MPC) scheme, successfully combining stability and performance in a stabilization task with obstacle avoidance constraints.

Paper Structure

This paper contains 12 sections, 2 theorems, 24 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Passivity
  4. Rigid Body Model
  5. Lossless control and Optimisation
  6. Case studies and Simulations
  7. Discrete model and software setup
  8. Nominal controller
  9. Task description
  10. Simulation with nonlinear MPC
  11. Results and discussion
  12. Conclusions

Key Result

Proposition 1

Consider system (eq:dyn1-eq:dyn2) with control (eq:gravitycomp) and any torque controller in the form for any (possibly state-dependent) term $\bm{a}\in\mathbb{R}^3$. Then, the closed-loop system is passive with respect to its kinetic energy as storage function and input/output pairs $(\bm{f}', \bm{v})$ and $(\bm{\tau}' ,\bm{\omega})$.

Figures (4)

  • Figure 1: Free dynamics ($\bm{\tau}'=\bm{a}=\bm{0}$)
  • Figure 2: Lossless optimal control without torque control ($\bm{\tau}'=\bm{0}$)
  • Figure 3: Nominal attitude control alone ($\bm{a}=\bm{0}$)
  • Figure 4: Lossless optimal transient control and nominal attitude control

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • proof
  • Proposition 2
  • proof
  • Remark 1: Closed-loop inertia shaping