Due to high dimensionality and non-convexity, real-time optimal control using full-order dynamics models for legged robots is challenging. Therefore, Nonlinear Model Predictive Control (NMPC) approaches are often limited to reduced-order models or local approximations. Sampling-based MPC has shown potential in nonconvex even discontinuous problems, but often yields suboptimal solutions with high variance, which limits its applications in high-dimensional locomotion. This work introduces DIAL-MPC (Diffusion-Inspired Annealing for Legged MPC), a sampling-based MPC framework with a novel diffusion-style annealing process. Such a process is supported by the theoretical landscape analysis of Model Predictive Path Integral Control (MPPI) and the connection between MPPI and single-step diffusion. Algorithmically, DIALMPC iteratively refines solutions online and achieves both global coverage and local convergence. In quadrupedal torquelevel control tasks, DIAL-MPC reduces the tracking error of standard MPPI by 13.4 times and outperforms reinforcement learning (RL) policies by 50 % in challenging climbing tasks without any training. In particular, DIAL-MPC enables precise real-world quadrupedal jumping with payload. To the best of our knowledge, DIAL-MPC is the first training-free method that optimizes over full-order legged dynamics in real-time.

Recent advances in diffusion models have demonstrated their strong capabilities in generating high-fidelity samples from complex distributions through an iterative refinement process. Despite the empirical success of diffusion models in motion planning and control, the model-free nature of these methods does not leverage readily available model information and limits their generalization to new scenarios beyond the training data (e.g., new robots with different dynamics). In this work, we introduce Model-Based Diffusion (MBD), an optimization approach using the diffusion process to solve trajectory optimization (TO) problems without data. The key idea is to explicitly compute the score function by leveraging the model information in TO problems, which is why we refer to our approach as model-based diffusion. Moreover, although MBD does not require external data, it can be naturally integrated with data of diverse qualities to steer the diffusion process. We also reveal that MBD has interesting connections to sampling-based optimization. Empirical evaluations show that MBD outperforms state-of-the-art reinforcement learning and sampling-based TO methods in challenging contact-rich tasks. Additionally, MBD’s ability to integrate with data enhances its versatility and practical applicability, even with imperfect and infeasible data (e.g., partial-state demonstrations for high-dimensional humanoids), beyond the scope of standard diffusion models. Videos and codes are available in the supplementary materials.