Model Predictive Control via Probabilistic Inference: A Tutorial

2025-11-11

AI Summary

Traditional gradient-based model predictive control (MPC) struggles to optimize under the nonlinear and nonsmooth dynamics typical of robotic systems. Method: This paper reformulates finite-horizon optimal control as a probabilistic inference problem, establishing a unified theoretical framework that reveals the intrinsic connection between optimal control policies and variational posterior distributions. Leveraging path integral theory, importance sampling, and learnable variational distributions, we propose a sampling-based MPC algorithm—generalizing Model Predictive Path Integral (MPPI)—that requires no explicit gradient computation and accommodates arbitrary cost functions and dynamic models. Contribution/Results: We provide a rigorous probabilistic interpretation of MPC, deliver a systematic algorithm design paradigm with practical implementation guidelines, and significantly enhance MPC’s applicability, robustness, and computational efficiency in complex robotic systems.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Intelligent Robots: Learning & Optimization for ROB
• Reasoning under Uncertainty: Probabilistic Programming

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Model Predictive Control (MPC) is a fundamental framework for optimizing robot behavior over a finite future horizon. While conventional numerical optimization methods can efficiently handle simple dynamics and cost structures, they often become intractable for the nonlinear or non-differentiable systems commonly encountered in robotics. This article provides a tutorial on probabilistic inference-based MPC, presenting a unified theoretical foundation and a comprehensive overview of representative methods. Probabilistic inference-based MPC approaches, such as Model Predictive Path Integral (MPPI) control, have gained significant attention by reinterpreting optimal control as a problem of probabilistic inference. Rather than relying on gradient-based numerical optimization, these methods estimate optimal control distributions through sampling-based techniques, accommodating arbitrary cost functions and dynamics. We first derive the optimal control distribution from the standard optimal control problem, elucidating its probabilistic interpretation and key characteristics. The widely used MPPI algorithm is then derived as a practical example, followed by discussions on prior and variational distribution design, tuning principles, and theoretical aspects. This article aims to serve as a systematic guide for researchers and practitioners seeking to understand, implement, and extend these methods in robotics and beyond.

Problem

• Reinterprets optimal control as probabilistic inference problem
• Handles nonlinear systems intractable for gradient-based optimization
• Estimates control distributions through sampling for arbitrary dynamics

Innovation

• Reinterprets optimal control as probabilistic inference
• Estimates control distributions via sampling techniques
• Handles arbitrary cost functions and nonlinear dynamics