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Safe Returning FaSTrack with Robust Control Lyapunov-Value Functions

Zheng Gong, Boyang Li, Sylvia Herbert

TL;DR

The paper tackles safe real-time navigation in unknown environments by integrating robust control Lyapunov-value functions with the FaSTrack framework. It offline-computes a robust relative value function (R-CLVF) between the tracker and planner to guarantee exponential convergence back to the planning model even under unexpected disturbances, and introduces a safe returning mechanism that can deliberately jump the planner forward to accelerate progress. The SR-F framework thus combines disturbance rejection with speed-up opportunities in open regions, validated on a 10D quadrotor where SR-F outperforms FaSTrack by approximately 20% while preserving safety. This approach offers practical benefits for high-dimensional robotic systems operating under unmodeled disturbances, providing formal safety guarantees and improved navigation efficiency in uncertain environments.

Abstract

Real-time navigation in a priori unknown environment remains a challenging task, especially when an unexpected (unmodeled) disturbance occurs. In this paper, we propose the framework Safe Returning Fast and Safe Tracking (SR-F) that merges concepts from 1) Robust Control Lyapunov-Value Functions (R-CLVF), and 2) the Fast and Safe Tracking (FaSTrack) framework. The SR-F computes an R-CLVF offline between a model of the true system and a simplified planning model. Online, a planning algorithm is used to generate a trajectory in the simplified planning space, and the R-CLVF is used to provide a tracking controller that exponentially stabilizes to the planning model. When an unexpected disturbance occurs, the proposed SR-F algorithm provides a means for the true system to recover to the planning model. We take advantage of this mechanism to induce an artificial disturbance by ``jumping'' the planning model in open environments, forcing faster navigation. Therefore, this algorithm can both reject unexpected true disturbances and accelerate navigation speed. We validate our framework using a 10D quadrotor system and show that SR-F is empirically 20\% faster than the original FaSTrack while maintaining safety.

Safe Returning FaSTrack with Robust Control Lyapunov-Value Functions

TL;DR

The paper tackles safe real-time navigation in unknown environments by integrating robust control Lyapunov-value functions with the FaSTrack framework. It offline-computes a robust relative value function (R-CLVF) between the tracker and planner to guarantee exponential convergence back to the planning model even under unexpected disturbances, and introduces a safe returning mechanism that can deliberately jump the planner forward to accelerate progress. The SR-F framework thus combines disturbance rejection with speed-up opportunities in open regions, validated on a 10D quadrotor where SR-F outperforms FaSTrack by approximately 20% while preserving safety. This approach offers practical benefits for high-dimensional robotic systems operating under unmodeled disturbances, providing formal safety guarantees and improved navigation efficiency in uncertain environments.

Abstract

Real-time navigation in a priori unknown environment remains a challenging task, especially when an unexpected (unmodeled) disturbance occurs. In this paper, we propose the framework Safe Returning Fast and Safe Tracking (SR-F) that merges concepts from 1) Robust Control Lyapunov-Value Functions (R-CLVF), and 2) the Fast and Safe Tracking (FaSTrack) framework. The SR-F computes an R-CLVF offline between a model of the true system and a simplified planning model. Online, a planning algorithm is used to generate a trajectory in the simplified planning space, and the R-CLVF is used to provide a tracking controller that exponentially stabilizes to the planning model. When an unexpected disturbance occurs, the proposed SR-F algorithm provides a means for the true system to recover to the planning model. We take advantage of this mechanism to induce an artificial disturbance by ``jumping'' the planning model in open environments, forcing faster navigation. Therefore, this algorithm can both reject unexpected true disturbances and accelerate navigation speed. We validate our framework using a 10D quadrotor system and show that SR-F is empirically 20\% faster than the original FaSTrack while maintaining safety.
Paper Structure (18 sections, 2 theorems, 16 equations, 4 figures, 1 table, 2 algorithms)

This paper contains 18 sections, 2 theorems, 16 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Models
  4. Tracker Model
  5. Planner Model
  6. Relative Dynamics
  7. HJ Reachability and Fastrack
  8. HJ reachability (Offline)
  9. Online Execution
  10. R-CLVF
  11. Safe Returning with Unexpected Disturbance
  12. SR-F Algorithm
  13. Sensing Block
  14. Planning Block and the Safe Returning Function
  15. Tracking Block
  16. ...and 3 more sections

Key Result

Theorem 1

The relative state can be exponentially stabilized to the TEB from $\mathcal{D}_\gamma \setminus \mathcal{B}$, if the R-CLVF exists in $\mathcal{D}_\gamma$. where $k > 0$ and $t \leq s \leq 0$.

Figures (4)

  • Figure 1: Online simulations for an 8D quadrotor tracking a 2D planning model paired with Rapidly-Exploring Random Trees. Results using SR-F (ours), M-F fridovich2018planning, and classic MPC are shown. Top left: the entire trajectory using SR-F. The quadrotor starts at the origin and navigates to a goal (red plus sign). The obstacle (red) is augmented by the tracking error bound. The system's trajectory shown in cyan (fast) and blue (slow). A position disturbance (labeled "real dstb") is applied on the quadrotor, pushing it (blue dashed line) close to the obstacle. Top right: zoomed-in views. On the left shows the SR-F algorithm under the real disturbance. The pink region indicates the safe resetting region for the planner (sTEB), which indicates where the planning model may restart to ensure safe convergence. The right shows how the SR-F speeds up navigation in the cyan regions of the trajectory. The algorithm selects the furthest point in the sTEB to reset the planning model, forcing faster navigation while maintaining safety. Bottom right: both M-F and MPC crash after the unexpected disturbance.
  • Figure 2: Comparison of relative trajectory using FaSTrack (left) and SR-F (right). The red dotted line denotes an unexpected disturbance that causes the relative state to leave the TEB. The FaSTrack can only guarantee the relative state stays in the current level set, while the SR-F can stabilize the relative state back to the TEB.
  • Figure 3: Online flowchart for SR-F. The online algorithm contains three main blocks: the sensing block, the planning block, and the tracking block. The sensing block senses the environment, determines the sTEB and augments the obstacle. The planning block takes in the current tracker state, and does a series of logical judgment. A raw path and next planning state is obtained from the planning block. The tracking block takes in the next plan state, determines the optimal controller, and updates the tracker state.
  • Figure 4: 10D-3D simulation using SR-F. The tracker's trajectory switches between cyan and blue, indicating that the tracker jumps (cyan) when obstacle-free, and tracks a RRT path when not obstacle-free. The planner's position is the green star in the translucent blue box (representing sTEB). Both systems start on the left and navigate to a goal on the right. The three light grey rectangles are obstacles, and once sensed by the quadrotor they turn red. When the quadrotor is passing near an obstacle, it experiences an unexpected disturbance to its position (black dashed line), mimicking a sudden wind gust. The green dashed line shows the change of the planner's position after replanning in Alg. \ref{['algo: safe correction']}.

Theorems & Definitions (7)

  • Remark 1
  • Definition 1
  • Theorem 1
  • Remark 2
  • Theorem 2
  • proof
  • Remark 3