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A Predictive and Sampled-Data Barrier Method for Safe and Efficient Quadrotor Control

Ming Gao, Zhanglin Shangguan, Shuo Liu, Liang Wu, Bo Yang, Wei Xiao

TL;DR

This paper proposes a cascaded control framework for quadrotor trajectory tracking with formal safety guarantees, and shows that embedding SdHOCBFs as control-affine constraints into the MPC formulation guarantees both safety and optimality while preserving convexity for real-time implementations.

Abstract

This paper proposes a cascaded control framework for quadrotor trajectory tracking with formal safety guarantees. First, we design a controller consisting of an outer-loop position model predictive control (MPC) and an inner-loop nonlinear attitude control, enabling decoupling of position safety and yaw orientation. Second, since quadrotor safety constraints often involve high relative degree, we adopt high order control barrier functions (HOCBFs) to guarantee safety. To employ HOCBFs in the MPC formulation that has formal guarantees, we extend HOCBFs to sampled-data HOCBF (SdHOCBFs) by introducing compensation terms, ensuring safety over the entire sampling interval. We show that embedding SdHOCBFs as control-affine constraints into the MPC formulation guarantees both safety and optimality while preserving convexity for real-time implementations. Finally, comprehensive simulations are conducted to demonstrate the safety guarantee and high efficiency of the proposed method compared to existing methods.

A Predictive and Sampled-Data Barrier Method for Safe and Efficient Quadrotor Control

TL;DR

This paper proposes a cascaded control framework for quadrotor trajectory tracking with formal safety guarantees, and shows that embedding SdHOCBFs as control-affine constraints into the MPC formulation guarantees both safety and optimality while preserving convexity for real-time implementations.

Abstract

This paper proposes a cascaded control framework for quadrotor trajectory tracking with formal safety guarantees. First, we design a controller consisting of an outer-loop position model predictive control (MPC) and an inner-loop nonlinear attitude control, enabling decoupling of position safety and yaw orientation. Second, since quadrotor safety constraints often involve high relative degree, we adopt high order control barrier functions (HOCBFs) to guarantee safety. To employ HOCBFs in the MPC formulation that has formal guarantees, we extend HOCBFs to sampled-data HOCBF (SdHOCBFs) by introducing compensation terms, ensuring safety over the entire sampling interval. We show that embedding SdHOCBFs as control-affine constraints into the MPC formulation guarantees both safety and optimality while preserving convexity for real-time implementations. Finally, comprehensive simulations are conducted to demonstrate the safety guarantee and high efficiency of the proposed method compared to existing methods.

Paper Structure

This paper contains 15 sections, 3 theorems, 26 equations, 3 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES AND PROBLEM FORMULATION
  3. Quadrotor Model
  4. Safety via High Order Control Barrier Functions
  5. Problem Formulation
  6. METHODOLOGY
  7. Outer-loop MPC Controller
  8. Model
  9. HOCBFs for Sampled Data System
  10. MPC Formulation
  11. Inner-loop Controller
  12. SIMULATIONS AND DISCUSSIONS
  13. Circle with two cylinders
  14. Circle with a narrow gap
  15. CONCLUSIONS

Key Result

Theorem 1

xiaohocbf Given an HOCBF $h(\bm{x})$ from Def. def: hocbfdef with the sets $\mathcal{C}_i, i \in \lbrace 0, \cdots, \rho-1 \rbrace$, if $\bm{x}(t_0) \in \mathcal{C}_0 \cap \cdots \cap \mathcal{C}_{\rho-1}$, then any Lipschitz continuous controller $\bm{u}(t)$ satisfying eq: HOCBF$\forall t > t_0$ re

Figures (3)

  • Figure 1: Diagram of the proposed cascaded controller for safe quadrotor control. MPC-SdHOCBF takes position states to generate $\bm{s}_v$, which is converted into thrust $f_z$ and angular states. The attitude controller then uses the angular states to compute torques $\bm{\tau}$.
  • Figure 2: Trajectories, computation time, HOCBF values and velocities from different controllers. (a) Quadrotor's trajectories generated by different controllers. The black dash line, two cyan circles, red triangle and black arrow denote the reference trajectory, cylindrical obstacles, the start point the direction of trajectories, respectively. (b) Computation time to slove MPC optimization of different controllers. (c) the velocities of quadrotor by different controllers. (d) the value of $h(\bm{x})=x^2+(y-2)^2-1$. $h(\bm{x}) \geq 0$ imply the safety of quadrotor.
  • Figure 3: Trajectories generated by different controllers for the circle with narrow gap. The black arrows indicate the direction along the trajectories. (a) Trajectories generated by SdHOCBF with $p=5,6,7,8,9$. (b) Trajectories generated by HOCBF with $p=5,6,7,8,9$. (c) Trajectories generated by DCBF with $\lambda=0.1,0.2,0.4,0.6,0.8,1$. When $\lambda=1$, DCBF is equivalent to MPC-DC. (d) Trajectories generated by DHOCBF with $\lambda=0.1,0.2,0.4,0.6,0.8$.

Theorems & Definitions (9)

  • Definition 1: HOCBF xiaohocbf
  • Theorem 1
  • Definition 2: SdHOCBF
  • Theorem 2
  • proof
  • Remark 1
  • Theorem 3
  • proof
  • Remark 2