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Receding-Constraint Model Predictive Control using a Learned Approximate Control-Invariant Set

Gianni Lunardi, Asia La Rocca, Matteo Saveriano, Andrea Del Prete

TL;DR

This work tackles safety in constrained nonlinear control by moving away from exact control-invariant sets and toward a conservative backward-reachable set $\hat{\mathcal{V}}$ combined with an $N$-step control-invariance notion. The authors propose a Safe Model Predictive Control framework with two components: Safe Task Abortion, which can steer the system to equilibrium via a backup OCP, and Receding-Constraint MPC, which enforces safety through a sliding constraint along the horizon and optional soft termination. They prove that the receding-constraint formulation achieves recursive feasibility under $N$-step invariance, and show via simulation on a 3-joint manipulator that the approach reduces constraint violations while incurring modest tracking-cost increases and acceptable computation times. The results suggest practical safety benefits for model-based and data-driven controllers, with future work aimed at robustifying guarantees, accelerating the safe-abort OCP, and integrating learning to improve the safe set and warm-starts.

Abstract

In recent years, advanced model-based and data-driven control methods are unlocking the potential of complex robotics systems, and we can expect this trend to continue at an exponential rate in the near future. However, ensuring safety with these advanced control methods remains a challenge. A well-known tool to make controllers (either Model Predictive Controllers or Reinforcement Learning policies) safe, is the so-called control-invariant set (a.k.a. safe set). Unfortunately, for nonlinear systems, such a set cannot be exactly computed in general. Numerical algorithms exist for computing approximate control-invariant sets, but classic theoretic control methods break down if the set is not exact. This paper presents our recent efforts to address this issue. We present a novel Model Predictive Control scheme that can guarantee recursive feasibility and/or safety under weaker assumptions than classic methods. In particular, recursive feasibility is guaranteed by making the safe-set constraint move backward over the horizon, and assuming that such set satisfies a condition that is weaker than control invariance. Safety is instead guaranteed under an even weaker assumption on the safe set, triggering a safe task-abortion strategy whenever a risk of constraint violation is detected. We evaluated our approach on a simulated robot manipulator, empirically demonstrating that it leads to less constraint violations than state-of-the-art approaches, while retaining reasonable performance in terms of tracking cost, number of completed tasks, and computation time.

Receding-Constraint Model Predictive Control using a Learned Approximate Control-Invariant Set

TL;DR

This work tackles safety in constrained nonlinear control by moving away from exact control-invariant sets and toward a conservative backward-reachable set combined with an -step control-invariance notion. The authors propose a Safe Model Predictive Control framework with two components: Safe Task Abortion, which can steer the system to equilibrium via a backup OCP, and Receding-Constraint MPC, which enforces safety through a sliding constraint along the horizon and optional soft termination. They prove that the receding-constraint formulation achieves recursive feasibility under -step invariance, and show via simulation on a 3-joint manipulator that the approach reduces constraint violations while incurring modest tracking-cost increases and acceptable computation times. The results suggest practical safety benefits for model-based and data-driven controllers, with future work aimed at robustifying guarantees, accelerating the safe-abort OCP, and integrating learning to improve the safe set and warm-starts.

Abstract

In recent years, advanced model-based and data-driven control methods are unlocking the potential of complex robotics systems, and we can expect this trend to continue at an exponential rate in the near future. However, ensuring safety with these advanced control methods remains a challenge. A well-known tool to make controllers (either Model Predictive Controllers or Reinforcement Learning policies) safe, is the so-called control-invariant set (a.k.a. safe set). Unfortunately, for nonlinear systems, such a set cannot be exactly computed in general. Numerical algorithms exist for computing approximate control-invariant sets, but classic theoretic control methods break down if the set is not exact. This paper presents our recent efforts to address this issue. We present a novel Model Predictive Control scheme that can guarantee recursive feasibility and/or safety under weaker assumptions than classic methods. In particular, recursive feasibility is guaranteed by making the safe-set constraint move backward over the horizon, and assuming that such set satisfies a condition that is weaker than control invariance. Safety is instead guaranteed under an even weaker assumption on the safe set, triggering a safe task-abortion strategy whenever a risk of constraint violation is detected. We evaluated our approach on a simulated robot manipulator, empirically demonstrating that it leads to less constraint violations than state-of-the-art approaches, while retaining reasonable performance in terms of tracking cost, number of completed tasks, and computation time.
Paper Structure (16 sections, 2 theorems, 11 equations, 2 figures, 3 tables, 2 algorithms)

This paper contains 16 sections, 2 theorems, 11 equations, 2 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Problem statement
  5. Model Predictive Control and Recursive Feasibility
  6. Soft Terminal Constraint
  7. Safe Model Predictive Control
  8. Safe Task Abortion
  9. Receding-Constraint MPC
  10. Safe Task Abortion with Receding Constraint
  11. Simulation Results
  12. Simulation Setup
  13. Small Safety Margin Tests
  14. Large Safety Margin Tests
  15. Cost and Computation Time
  16. ...and 1 more sections

Key Result

Lemma 1

Under Assumptions ass:conservative_set to ass:ocp_time, the hard terminal-constraint MPC with safe task abortion described in Alg. alg:terminal guarantees that constraints are never violated.

Figures (2)

  • Figure 1: Example of Receding-Constraint MPC with $N=4$. After the MPC loop 3, the receding constraint slides forward because $x_{4|3} \in \hat{\mathcal{V}}$.
  • Figure 2: Comparison between ST (task failed) and STWA (task aborted). The bottom plot shows the value of the terminal constraint \ref{['eq:terminal_constraint']}. The vertical line highlights the start of the safe-abort trajectory.

Theorems & Definitions (2)

  • Lemma 1
  • Theorem 1