This paper presents a tutorial and survey on probabilistic inference-based model predictive control (PI-MPC) for robotics. PI-MPC reformulates finite-horizon optimal control as inference over an optimal control distribution expressed as a Boltzmann distribution weighted by a control prior, and generates actions through variational inference. In the tutorial part, we derive this formulation and explain action generation via variational inference, highlighting Model Predictive Path Integral (MPPI) control as a representative algorithm with a closed-form sampling update. In the survey part, we organize existing PI-MPC research around key design dimensions, including prior design, multi-modality, constraint handling, scalability, hardware acceleration, and theoretical analysis. This paper provides a unified conceptual perspective on PI-MPC and a practical entry point for robotics researchers and practitioners.

翻译：本文针对机器人学中基于概率推理的模型预测控制（PI-MPC）提供一份教程与综述。PI-MPC将有限时域最优控制问题重新表述为对最优控制分布的概率推理，该分布表示为由控制先验加权的玻尔兹曼分布，并通过变分推理生成控制动作。在教程部分，我们推导了该公式，并解释了通过变分推理生成动作的过程，重点介绍了作为代表性算法的模型预测路径积分（MPPI）控制，其具有闭式采样更新规则。在综述部分，我们围绕关键设计维度对现有PI-MPC研究进行了梳理，包括先验设计、多模态性、约束处理、可扩展性、硬件加速和理论分析。本文为PI-MPC提供了统一的概念视角，并为机器人学研究者与实践者提供了实用的入门指南。