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System-level Safety Guard: Safe Tracking Control through Uncertain Neural Network Dynamics Models

Xiao Li, Yutong Li, Anouck Girard, Ilya Kolmanovsky

TL;DR

The paper addresses safety verification for NN-driven control in discrete-time systems by treating NN dynamics as predictive models subject to bounded intrinsic and extrinsic uncertainties. It develops a MILP-based robust tracking framework that propagates uncertainty through NN layers, enforces input feasibility, and imposes a safety constraint so that the NN-derived reachable set stays within a safe region. Key contributions include a complete MILP encoding of input feasibility, NN nonlinearity, and safety constraints, theoretical guarantees of safety under bounded disturbances, and demonstrations in obstacle avoidance for an omnidirectional robot and a vehicle with set-theoretic localization. This approach enables safe deployment of NN-based control in safety-critical robotics with tractable optimization times suitable for real-time planning.

Abstract

The Neural Network (NN), as a black-box function approximator, has been considered in many control and robotics applications. However, difficulties in verifying the overall system safety in the presence of uncertainties hinder the deployment of NN modules in safety-critical systems. In this paper, we leverage the NNs as predictive models for trajectory tracking of unknown dynamical systems. We consider controller design in the presence of both intrinsic uncertainty and uncertainties from other system modules. In this setting, we formulate the constrained trajectory tracking problem and show that it can be solved using Mixed-integer Linear Programming (MILP). The proposed MILP-based approach is empirically demonstrated in robot navigation and obstacle avoidance through simulations. The demonstration videos are available at https://xiaolisean.github.io/publication/2023-11-01-L4DC2024.

System-level Safety Guard: Safe Tracking Control through Uncertain Neural Network Dynamics Models

TL;DR

The paper addresses safety verification for NN-driven control in discrete-time systems by treating NN dynamics as predictive models subject to bounded intrinsic and extrinsic uncertainties. It develops a MILP-based robust tracking framework that propagates uncertainty through NN layers, enforces input feasibility, and imposes a safety constraint so that the NN-derived reachable set stays within a safe region. Key contributions include a complete MILP encoding of input feasibility, NN nonlinearity, and safety constraints, theoretical guarantees of safety under bounded disturbances, and demonstrations in obstacle avoidance for an omnidirectional robot and a vehicle with set-theoretic localization. This approach enables safe deployment of NN-based control in safety-critical robotics with tractable optimization times suitable for real-time planning.

Abstract

The Neural Network (NN), as a black-box function approximator, has been considered in many control and robotics applications. However, difficulties in verifying the overall system safety in the presence of uncertainties hinder the deployment of NN modules in safety-critical systems. In this paper, we leverage the NNs as predictive models for trajectory tracking of unknown dynamical systems. We consider controller design in the presence of both intrinsic uncertainty and uncertainties from other system modules. In this setting, we formulate the constrained trajectory tracking problem and show that it can be solved using Mixed-integer Linear Programming (MILP). The proposed MILP-based approach is empirically demonstrated in robot navigation and obstacle avoidance through simulations. The demonstration videos are available at https://xiaolisean.github.io/publication/2023-11-01-L4DC2024.
Paper Structure (16 sections, 4 theorems, 53 equations, 3 figures)

This paper contains 16 sections, 4 theorems, 53 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Model Preliminaries
  4. Robust Constrained Tracking Problem
  5. Mixed-integer Linear Programming
  6. Constraints Embedding Input Feasibility
  7. Constraints Encoding NN Structural Non-Linearity
  8. Constraints Enforcing System Safety
  9. Safe Tracking Control using MILP
  10. Obstacle Avoidance and Reachability-Guided RRT
  11. Vehicle Navigation and Set-theoretical Localization
  12. Conclusion and Future Work
  13. Formal Safety Guarantee
  14. Input Constraints
  15. NN Structural Constraints
  16. ...and 1 more sections

Key Result

proposition 1

Given state measurement $y_k$ and a decision variable $\tilde{u}_k$, assume that the unknown actual quantities $x_k$ and $u_k$ obey eq:y_bound, eq:u_bound with assumptions in eq:y_bound_cube, eq:u_bound_cube, eq:x_feasible_set, eq:u_feasible_set. Let the decision variables $\tilde{u}_k \in \mathbb{R where $\delta^a_j$ denotes the $j$th element in column vector $\delta^a$, $a_{0,m:n}$ represents th

Figures (3)

  • Figure 1: Trajectory tracking through an NN-learned predictive model: The controller leverages an NN-learned model (with prediction errors) for predicting the unknown dynamics and tracks the reference trajectories. The NN predictions depend on the states $x_k$ before the sensing module and the control $u_k$ out of the actuator module, which are not directly accessible by the controller in this pipeline, and are uncertain quantities due to sensing noises and actuator disturbances.
  • Figure 2: Schematic of obstacle avoidance using an omnidirectional robot: (Left) The reachability-guided RRT algorithm expands the tree from the start $x_0$ to the goal $x_g$ over the safe state space $\mathcal{X}_s$. Then, we use the Dijkstra planning algorithm to find a reference path (black lines with red dots) from $x_0$ to $x_g$ that has the shortest distance defined by $\ell_1$ norm. (Middle) We use the proposed method in \ref{['eq:mip']} to track the reference states (red dots). Our method can guarantee that the robot motion is collision-free under uncertainties, i.e., the unknown actual states in blue asterisks are in the safe set $\mathcal{X}_s$, even with most of the reference points located near the obstacles.
  • Figure 3: Schematic of navigating a vehicle through a maze with tracking controller \ref{['eq:mip']} and with a set-theoretic localization algorithm. The zoom-in view, at the top-left, demonstrates that the set-theoretic localization algorithm provides estimates of the vehicle body and orientation that are guaranteed to contain the actual ones. The tracking controller \ref{['eq:mip']} leverages this information, together with an NN-learned vehicle dynamics model, to avoid obstacles.

Theorems & Definitions (8)

  • Remark
  • proposition 1
  • proposition 2
  • proposition 3
  • proposition 4
  • proof
  • proof
  • proof