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MPC Based Linear Equivalence with Control Barrier Functions for VTOL-UAVs

Ali Mohamed Ali, Hashim A. Hashim, Chao Shen

TL;DR

This work tackles safe navigation for VTOL-UAVs under obstacle avoidance by blending a cascaded Dynamic Feedback Linearization (DFL) with Model Predictive Control (MPC) and Control Barrier Functions (CBF). By converting the nonlinear underactuated UAV dynamics into a linear equivalent form via DFL, the authors formulate MPC on a linear system and impose safety through a CBF, yielding a Quadratic Constraint Quadratic Programming (QCQP) problem. They prove closed-loop stability and recursive feasibility within this MPC-CBF-DFL framework and validate performance through numerical simulations, including robustness to noise and comparisons with Euclidean-distance safety constraints. The approach offers reduced computational load relative to nonlinear MPC while maintaining rigorous safety guarantees, enabling real-time, obstacle-avoidant operation for VTOL-UAVs.

Abstract

In this work, we propose a cascaded scheme of linear Model prediction Control (MPC) based on Control Barrier Functions (CBF) with Dynamic Feedback Linearization (DFL) for Vertical Take-off and Landing (VTOL) Unmanned Aerial Vehicles (UAVs). CBF is a tool that allows enforcement of forward invariance of a set using Lyapunov-like functions to ensure safety. The First control synthesis that employed CBF was based on Quadratic Program (QP) that modifies the existing controller to satisfy the safety requirements. However, the CBF-QP-based controllers leading to longer detours and undesirable transient performance. Recent contributions utilize the framework of MPC benefiting from the prediction capabilities and constraints imposed on the state and control inputs. Due to the intrinsic nonlinearities of the dynamics of robotics systems, all the existing MPC-CBF solutions rely on nonlinear MPC formulations or operate on less accurate linear models. In contrast, our novel solution unlocks the benefits of linear MPC-CBF while considering the full underactuated dynamics without any linear approximations. The cascaded scheme converts the problem of safe VTOL-UAV navigation to a Quadratic Constraint Quadratic Programming (QCQP) problem solved efficiently by off-the-shelf solvers. The closed-loop stability and recursive feasibility is proved along with numerical simulations showing the effective and robust solutions. Keywords: Unmanned Aerial Vehicles, Vertical Take-off and Landing, Model Predictive Control, MPC, Nonlinearity, Dynamic Feedback Linearization, Optimal Control.

MPC Based Linear Equivalence with Control Barrier Functions for VTOL-UAVs

TL;DR

This work tackles safe navigation for VTOL-UAVs under obstacle avoidance by blending a cascaded Dynamic Feedback Linearization (DFL) with Model Predictive Control (MPC) and Control Barrier Functions (CBF). By converting the nonlinear underactuated UAV dynamics into a linear equivalent form via DFL, the authors formulate MPC on a linear system and impose safety through a CBF, yielding a Quadratic Constraint Quadratic Programming (QCQP) problem. They prove closed-loop stability and recursive feasibility within this MPC-CBF-DFL framework and validate performance through numerical simulations, including robustness to noise and comparisons with Euclidean-distance safety constraints. The approach offers reduced computational load relative to nonlinear MPC while maintaining rigorous safety guarantees, enabling real-time, obstacle-avoidant operation for VTOL-UAVs.

Abstract

In this work, we propose a cascaded scheme of linear Model prediction Control (MPC) based on Control Barrier Functions (CBF) with Dynamic Feedback Linearization (DFL) for Vertical Take-off and Landing (VTOL) Unmanned Aerial Vehicles (UAVs). CBF is a tool that allows enforcement of forward invariance of a set using Lyapunov-like functions to ensure safety. The First control synthesis that employed CBF was based on Quadratic Program (QP) that modifies the existing controller to satisfy the safety requirements. However, the CBF-QP-based controllers leading to longer detours and undesirable transient performance. Recent contributions utilize the framework of MPC benefiting from the prediction capabilities and constraints imposed on the state and control inputs. Due to the intrinsic nonlinearities of the dynamics of robotics systems, all the existing MPC-CBF solutions rely on nonlinear MPC formulations or operate on less accurate linear models. In contrast, our novel solution unlocks the benefits of linear MPC-CBF while considering the full underactuated dynamics without any linear approximations. The cascaded scheme converts the problem of safe VTOL-UAV navigation to a Quadratic Constraint Quadratic Programming (QCQP) problem solved efficiently by off-the-shelf solvers. The closed-loop stability and recursive feasibility is proved along with numerical simulations showing the effective and robust solutions. Keywords: Unmanned Aerial Vehicles, Vertical Take-off and Landing, Model Predictive Control, MPC, Nonlinearity, Dynamic Feedback Linearization, Optimal Control.
Paper Structure (9 sections, 4 theorems, 22 equations, 3 figures)

This paper contains 9 sections, 4 theorems, 22 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. VTOL-UAV Dynamic Model
  4. Linear Equivalence Model
  5. Feedback Linearization
  6. Dynamic Feedback Linearization (DFL)
  7. Proposed Scheme
  8. Numerical Results
  9. Conclusion

Key Result

Lemma 1

isidori1985nonlinear The relative degree of eq:pr2 at $x_{0}$ is described as $r=[r_{1},\ldots,r_{m}]^{\top}\in\mathbb{R}^{m}$ exists if:

Figures (3)

  • Figure 1: VTOL-UAV safe navigation task where $\{I\}$ is the body fixed frame and $\{B\}$ is the Global frame
  • Figure 2: MPC-CBF-DFL scheme for the VTOL-UAV.
  • Figure 3: Output performance of MPC-CBF-DFL scheme: (a) illustrates the safe navigation of the VTOL-UAV against a spherical Obstacle with different pairs of $N$ and $\gamma$; (b) presents a comparison between MPC-CBF-DF vs. MPC-ED-DF schemes; (c) shows the trajectories of the proposed scheme against a noisy feedback signal corrupted by a Gaussian noise; (d) shows the error signals where dashed-line and solid-line refer to the $x$ and $y$ errors, respectively; (e) depicts the distance between the VTOL-UAV and the obstacle (MPC-CBF-DF vs. MPC-ED-DF schemes)

Theorems & Definitions (6)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Theorem 1
  • proof