TD-CD-MPPI: Temporal-Difference Constraint-Discounted Model Predictive Path Integral Control

Path Integral methods have demonstrated remarkable capabilities for solving non-linear stochastic optimal control problems through sampling-based optimization. However, their computational complexity grows linearly with the prediction horizon, limiting long-term reasoning, while constraints are merely enforced through handcrafted penalties. In this work, we propose a unified and efficient framework for enabling long-horizon reasoning and constraint enforcement within Model Predictive Path Integral (MPPI) control. First, we introduce a practical method to incorporate a terminal value function, learned offline via temporal-difference learning, to approximate the long-term cost-to-go. This allows for significantly shorter roll-outs while enabling infinite-horizon reasoning, thereby improving computational efficiency and motion performance. Second, we propose a discount modulation strategy that adjusts the return of sampled trajectories based on constraint violations. This provides a more interpretable and effective mechanism for enforcing constraints compared to traditional cost shaping. Our formulation retains the flexibility and sampling efficiency of MPPI while supporting structured integration of long-term objectives and constraint handling. We validate our approach on both simulated and real-world robotic locomotion tasks, demonstrating improved performance, constraint-awareness, and generalization under reduced computational budgets.