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Ising Machines for Model Predictive Path Integral-Based Optimal Control

Lorin Werthen-Brabants, Pieter Simoens

TL;DR

The paper addresses the computational burden of real-time MPC, particularly Model Predictive Path Integral (MPPI), by proposing an Ising-machine implementation. It reformulates MPPI as sampling from a Boltzmann distribution over binary control sequences via a QUBO, employing a binary expansion and Gibbs sampling on p-bits. Comparative experiments on a nonlinear bicycle model show Ising-MPPI achieves competitive trajectory tracking with insights into convergence and sampling requirements, relative to linearized and reference MPPI baselines. The work highlights potential speedups and energy efficiency gains with specialized hardware for robotics and autonomous systems, while outlining substantial avenues for hardware validation and model-extension.

Abstract

We present a sampling-based Model Predictive Control (MPC) method that implements Model Predictive Path Integral (MPPI) as an \emph{Ising machine}, suitable for novel forms of probabilistic computing. By expressing the control problem as a Quadratic Unconstrained Binary Optimization (QUBO) problem, we map MPC onto an energy landscape suitable for Gibbs sampling from an Ising model. This formulation enables efficient exploration of (near-)optimal control trajectories. We demonstrate that the approach achieves accurate trajectory tracking compared to a reference MPPI implementation, highlighting the potential of Ising-based MPPI for real-time control in robotics and autonomous systems.

Ising Machines for Model Predictive Path Integral-Based Optimal Control

TL;DR

The paper addresses the computational burden of real-time MPC, particularly Model Predictive Path Integral (MPPI), by proposing an Ising-machine implementation. It reformulates MPPI as sampling from a Boltzmann distribution over binary control sequences via a QUBO, employing a binary expansion and Gibbs sampling on p-bits. Comparative experiments on a nonlinear bicycle model show Ising-MPPI achieves competitive trajectory tracking with insights into convergence and sampling requirements, relative to linearized and reference MPPI baselines. The work highlights potential speedups and energy efficiency gains with specialized hardware for robotics and autonomous systems, while outlining substantial avenues for hardware validation and model-extension.

Abstract

We present a sampling-based Model Predictive Control (MPC) method that implements Model Predictive Path Integral (MPPI) as an \emph{Ising machine}, suitable for novel forms of probabilistic computing. By expressing the control problem as a Quadratic Unconstrained Binary Optimization (QUBO) problem, we map MPC onto an energy landscape suitable for Gibbs sampling from an Ising model. This formulation enables efficient exploration of (near-)optimal control trajectories. We demonstrate that the approach achieves accurate trajectory tracking compared to a reference MPPI implementation, highlighting the potential of Ising-based MPPI for real-time control in robotics and autonomous systems.

Paper Structure

This paper contains 17 sections, 18 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Ising-based Model Predictive Path Integral Control
  3. Results
  4. Overall Performance
  5. Impact of Number of MPPI Samples on Trajectory Quality
  6. Limitations
  7. Conclusion
  8. MPC Formulation
  9. Linear MPC Formulation
  10. Finite set MPC as a QUBO formulation
  11. Binary Expansion Matrix
  12. Interaction terms and bias
  13. Symmetrization
  14. Non-Ising Linear MPPI formulation
  15. Random Trajectory Generation
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Comparison of Ising-MPPI vs Non-Ising Linear MPPI samples across different iteration limits. The $x$-axis represents the number of samples $S$ (log scale), the $y$-axis shows the MSE (log scale). Different shades of blue and orange denote a differing amount of iterations $M$ for the proposed finite-set Ising-MPPI and Non-Ising Linear MPPI respectively. The shaded areas represent the standard deviation. The red dashed line denotes the reference MPPI implementation.
  • Figure A-1: Examples of trajectories and their Ising-MPPI solutions. Note that the $N=8$ last points in the reference trajectory are not solved due to no additional reference points being available. Each trajectory starts at $(0, 0)$. The light red streaks visible are the predicted trajectories at each point.