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A Constructive Method for Designing Safe Multirate Controllers for Differentially-Flat Systems

Devansh R. Agrawal, Hardik Parwana, Ryan K. Cosner, Ugo Rosolia, Aaron D. Ames, Dimitra Panagou

TL;DR

Using differential flatness, this work provides a constructive means to synthesize the multi-rate controller, thereby removing the need to search for suitable Lyapunov or barrier functions, or to approximately linearize/discretize nonlinear dynamics.

Abstract

We present a multi-rate control architecture that leverages fundamental properties of differential flatness to synthesize controllers for safety-critical nonlinear dynamical systems. We propose a two-layer architecture, where the high-level generates reference trajectories using a linear Model Predictive Controller, and the low-level tracks this reference using a feedback controller. The novelty lies in how we couple these layers, to achieve formal guarantees on recursive feasibility of the MPC problem, and safety of the nonlinear system. Furthermore, using differential flatness, we provide a constructive means to synthesize the multi-rate controller, thereby removing the need to search for suitable Lyapunov or barrier functions, or to approximately linearize/discretize nonlinear dynamics. We show the synthesized controller is a convex optimization problem, making it amenable to real-time implementations. The method is demonstrated experimentally on a ground rover and a quadruped robotic system.

A Constructive Method for Designing Safe Multirate Controllers for Differentially-Flat Systems

TL;DR

Using differential flatness, this work provides a constructive means to synthesize the multi-rate controller, thereby removing the need to search for suitable Lyapunov or barrier functions, or to approximately linearize/discretize nonlinear dynamics.

Abstract

We present a multi-rate control architecture that leverages fundamental properties of differential flatness to synthesize controllers for safety-critical nonlinear dynamical systems. We propose a two-layer architecture, where the high-level generates reference trajectories using a linear Model Predictive Controller, and the low-level tracks this reference using a feedback controller. The novelty lies in how we couple these layers, to achieve formal guarantees on recursive feasibility of the MPC problem, and safety of the nonlinear system. Furthermore, using differential flatness, we provide a constructive means to synthesize the multi-rate controller, thereby removing the need to search for suitable Lyapunov or barrier functions, or to approximately linearize/discretize nonlinear dynamics. We show the synthesized controller is a convex optimization problem, making it amenable to real-time implementations. The method is demonstrated experimentally on a ground rover and a quadruped robotic system.
Paper Structure (10 sections, 5 theorems, 32 equations, 3 figures)

This paper contains 10 sections, 5 theorems, 32 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Differentially-Flat Systems
  4. Input-To-State Stability
  5. CONTROLLER CONSTRUCTION
  6. High-Level Planning
  7. Low-Level Tracking
  8. MAIN RESULT
  9. Experimental Results
  10. CONCLUSIONS

Key Result

Lemma 1

The function $V$eqn:lyapunov_eq, is an ISS-CLF for the flat error system eqn:flat_system_error_dynamics, wrt. the bounded disturbance $w$, $\left\Vert w \right\Vert_\infty \leq \bar{w}$, under the feedback law eqn:riccati_controller, with $\alpha$ and $\iota$ defined as

Figures (3)

  • Figure 1: Snapshots of a quadrupedal robot (left) and ground rover (right) navigating safely from start to goal positions around two rectangular obstacles. The safe set (thick outside line) and the tightened set (thin/dashed lines) are shown. The reference trajectory (red) is solved online using Model Predictive Control, and must lie inside the tightened safe set. A tracking controller ensures the maximum deviation from this reference trajectory is smaller than the tightening. Thus the true path (green) remains within the safe set. Video: https://tinyurl.com/3c58cnjj.
  • Figure 2: Simulation results. (a) shows unicycle (green) and reference (red) trajectories. The reference is discontinuous, since it is recomputed every $T$ seconds. The start of each replanned reference trajectory is marked (red crosses). Black square is magnified in (b). (c) shows Lyapunov function against time, indicating that it remains below $V_{max}$.
  • Figure 3: Experimental Results. The quadruped (a) and the rover (b) navigate around gray obstacles in the environment to reach target location. See Figure \ref{['fig:experiment_scenario']} for snapshots of the robots performing the experiments.

Theorems & Definitions (21)

  • Definition 1
  • Remark 1
  • Remark 2
  • Example 1: Unicycle: Differential flatness
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 3
  • Example 2: Unicycle: High Level Planner
  • Lemma 1
  • ...and 11 more