Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Continuous-Time Control Synthesis for Multiple Quadrotors under Signal Temporal Logic Specifications

Yating Yuan

TL;DR

Simulation results demonstrate that the proposed approach enables formally verifiable multi-agent coordination in constrained environments, with provable tracking guarantees under bounded disturbances.

Abstract

Ensuring continuous-time control of multiple quadrotors in constrained environments under signal temporal logic (STL) specifications is challenging due to nonlinear dynamics, safety constraints, and disturbances. This letter proposes a two-stage framework to address this challenge. First, exponentially decaying tracking error bounds are derived with multidimensional geometric control gains obtained via differential evolution. These bounds are less conservative, while the resulting tracking errors exhibit smaller oscillations and improved transient performance. Second, leveraging the time-varying bounds, a mixed-integer convex programming (MICP) formulation generates piecewise Bézier reference trajectories that satisfy STL and velocity limits, while ensuring inter-agent safety through convex-hull properties. Simulation results demonstrate that the proposed approach enables formally verifiable multi-agent coordination in constrained environments, with provable tracking guarantees under bounded disturbances.

Continuous-Time Control Synthesis for Multiple Quadrotors under Signal Temporal Logic Specifications

TL;DR

Simulation results demonstrate that the proposed approach enables formally verifiable multi-agent coordination in constrained environments, with provable tracking guarantees under bounded disturbances.

Abstract

Ensuring continuous-time control of multiple quadrotors in constrained environments under signal temporal logic (STL) specifications is challenging due to nonlinear dynamics, safety constraints, and disturbances. This letter proposes a two-stage framework to address this challenge. First, exponentially decaying tracking error bounds are derived with multidimensional geometric control gains obtained via differential evolution. These bounds are less conservative, while the resulting tracking errors exhibit smaller oscillations and improved transient performance. Second, leveraging the time-varying bounds, a mixed-integer convex programming (MICP) formulation generates piecewise Bézier reference trajectories that satisfy STL and velocity limits, while ensuring inter-agent safety through convex-hull properties. Simulation results demonstrate that the proposed approach enables formally verifiable multi-agent coordination in constrained environments, with provable tracking guarantees under bounded disturbances.
Paper Structure (32 sections, 4 theorems, 113 equations, 4 figures, 1 table)

This paper contains 32 sections, 4 theorems, 113 equations, 4 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Quadrotor Dynamics Model
  4. Geometric Control
  5. Bézier Curves
  6. Signal Temporal Logic
  7. Problem Statement
  8. Tracking Error Bounds
  9. Time-varying Error Bounds Analysis
  10. Lyapunov Function for Translation-error Dynamics
  11. Lyapunov Function for Rotational-error Dynamics
  12. Time-varying Lyapunov Function Bound (see \ref{['apped:Lyapunov_Bounds']} for details)
  13. Optimization-based Control Gains
  14. Reference Trajectory Construction
  15. Dynamic Constraints
  16. ...and 17 more sections

Key Result

Proposition 1

Supposing $K_R$ has distinct positive diagonal entries, define If $\Psi_K(t) < \psi < h_1$ for some positive $\psi$, then where $g_1=\frac{h_1}{h_2+h^2_3}$ and $g_2=\frac{h_3}{h_1\left(h_1-\psi\right)}$.

Figures (4)

  • Figure 1: The inertial frame $\mathcal{I}$, the body frame $\mathcal{B}$, and desired body axes $b_{d,1}(t)$, $b_{d,2}(t)$, $b_{d,3}(t)$ representing the desired orientation $R_d(t)$.
  • Figure 2: Trajectory tracking visualization for the single case. The reference trajectory (blue solid line) and twenty tracking trajectories (colored dashed lines) are shown. The left panel shows the top view, while the top-right panel presents the side view, with obstacles omitted for clarity of the trajectories. The bottom-right panel quantifies initial point deviations relative to the reference trajectory.
  • Figure 3: Velocity tracking performance and error analysis. (a) presents the desired Bézier velocity (solid blue line) and tracking velocities (dashed colored lines), all satisfying $|v| \leq v_{\mathrm{\max}}$. (b) Tracking errors $e_p(t)$ and $e_v(t)$ for the proposed method (solid) and scalar gain methodserry2024reach (dashed). Ten of twenty color-matched trajectory pairs with identical initial conditions are shown for clarity. Errors converge after $t \geq 4 \mathrm{~s}$. (c) shows that the proposed method attains a lower median error and a narrower interquartile range compared to serry2024reach.
  • Figure 4: Multi-agent task results under STL formula $\varphi_2$. (a) shows desired (solid) and tracking (dashed) trajectories, along with two sets of quadrotor poses: a faded set at early time $t=8.91\mathrm{~s}$ for visualization, and an opaque set at $t=17.42\mathrm{~s}$, where the minimum distance $1.9 \mathrm{~m}$ occurs in (b). (b) shows that all pairwise distances between agents' trajectories exceed $\epsilon_{\text{inter}}=0.2\mathrm{~m}$, confirming safety.

Theorems & Definitions (9)

  • Proposition 1: Adapted from lee2012robust; see also invernizzi2017geometric
  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 2: Initial conditions
  • Remark 1
  • Lemma 1: lee2012robust
  • Lemma 2: See \ref{['apped:prof_lem2']}
  • proof