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Safety-Critical Controller Synthesis with Reduced-Order Models

Max H. Cohen, Noel Csomay-Shanklin, William D. Compton, Tamas G. Molnar, Aaron D. Ames

TL;DR

This paper formalizes the connection between full and ROMs by defining projection mappings that relate the states and inputs of these models and leverage simulation functions to establish conditions under which safety guarantees may be transferred from a ROM to its corresponding full-order model.

Abstract

Reduced-order models (ROMs) provide lower dimensional representations of complex systems, capturing their salient features while simplifying control design. Building on previous work, this paper presents an overarching framework for the integration of ROMs and control barrier functions, enabling the use of simplified models to construct safety-critical controllers while providing safety guarantees for complex full-order models. To achieve this, we formalize the connection between full and ROMs by defining projection mappings that relate the states and inputs of these models and leverage simulation functions to establish conditions under which safety guarantees may be transferred from a ROM to its corresponding full-order model. The efficacy of our framework is illustrated through simulation results on a drone and hardware demonstrations on ARCHER, a 3D hopping robot.

Safety-Critical Controller Synthesis with Reduced-Order Models

TL;DR

This paper formalizes the connection between full and ROMs by defining projection mappings that relate the states and inputs of these models and leverage simulation functions to establish conditions under which safety guarantees may be transferred from a ROM to its corresponding full-order model.

Abstract

Reduced-order models (ROMs) provide lower dimensional representations of complex systems, capturing their salient features while simplifying control design. Building on previous work, this paper presents an overarching framework for the integration of ROMs and control barrier functions, enabling the use of simplified models to construct safety-critical controllers while providing safety guarantees for complex full-order models. To achieve this, we formalize the connection between full and ROMs by defining projection mappings that relate the states and inputs of these models and leverage simulation functions to establish conditions under which safety guarantees may be transferred from a ROM to its corresponding full-order model. The efficacy of our framework is illustrated through simulation results on a drone and hardware demonstrations on ARCHER, a 3D hopping robot.

Paper Structure

This paper contains 6 sections, 4 theorems, 34 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Reduced-Order Models
  4. Safety-Critical Control with ROMs
  5. Hardware Demonstrations
  6. Conclusions

Key Result

Theorem 1

A closed set $\mathcal{S}\subset\mathbb{R}^n$ is forward invariant for eq:dyn if and only if $\bf(\mathbf{x})\in\mathcal{T}_{\mathcal{S}}(\mathbf{x})$ for all $\mathbf{x}\in\partial\mathcal{S}$, where $\mathcal{T}_{\mathcal{S}}(\mathbf{x})$ denotes the contingent coneSee Blanchini for a precise defi

Figures (4)

  • Figure 2: Overview: We project high-dimensional control systems onto reduced-order spaces, design safety-critical controllers for a reduced-order representation of the original system, and then relate the inputs of this reduced-order system back to the full-order system. Our theoretical developments are illustrated through their application to ARCHER, a 3D hopping robot, a video of which is available at https://vimeo.com/1010060590?share=copy.
  • Figure 3: (Left) Evolution of the quadrotor's position for different choices of $\alpha$ in \ref{['eq:issf-rom']}. (Right) Difference between quadrotor's planar velocity $\mathbf{v}=\bm{\psi}(\mathbf{x})$ and desired velocity $\bm{\kappa}(\bm{\pi}(\mathbf{x}))$ generated by a single integrator. The velocity error remains bounded per Lemma \ref{['lemma:simulation']}, although the error is too large when $\alpha=2$, leading to safety violations. For each of these simulations we took $\varepsilon=20$ and omitted the $\sigma$ term since $\nabla h = L_{\mathbf{g}}h$. Varying $\varepsilon$ had minimal effect on trajectories.
  • Figure 4: (Top) Evolution of ARCHER's position where the yellow cubes denote the obstacles. (Bottom) Evolution of the ROM's CBF $h$ along the trajectory of the full-order system, which remains positive for all time.
  • Figure 5: Commanded velocities output by the reduced-order safety filter $\bm{\kappa}(\bm{\pi}(\mathbf{x}))=(\dot{x}_{\rm{safe}}, \dot{y}_{\rm{safe}})$ and the velocities of the full-order system $(\dot{x},\dot{y})$.

Theorems & Definitions (10)

  • Theorem 1
  • Example 1
  • Definition 1
  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Example 2