# PAD-TRO: Projection-Augmented Diffusion for Direct Trajectory Optimization

arXiv: https://arxiv.org/abs/2510.04436 · HTML: https://mlsanity.com/paper/2510.04436/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion and flow policies for robotics, one of 14 directions in Diffusion models.
- Builds on 10 of its 23 references that are in the map.
- First posted on arXiv 6 Oct 2025.

## Placed in

- Diffusion models / [Diffusion and flow policies for robotics](https://mlsanity.com/diffusion/diffusion-and-flow-policies-for-robotics/)

## Builds on (most cited first, 10 of 10)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [Model Predictive Path Integral Control using Covariance Variable Importance Sampling](https://mlsanity.com/paper/1509.01149/) (2015)
- [SafeDiffuser: Safe Planning with Diffusion Probabilistic Models](https://mlsanity.com/paper/2306.00148/) (2023)
- [Robust and Efficient Quadrotor Trajectory Generation for Fast Autonomous Flight](https://mlsanity.com/paper/1907.01531/) (2019)
- [Model-Based Diffusion for Trajectory Optimization](https://mlsanity.com/paper/2407.01573/) (2024)
- [Full-Order Sampling-Based MPC for Torque-Level Locomotion Control via Diffusion-Style Annealing](https://mlsanity.com/paper/2409.15610/) (2024)
- [DDAT: Diffusion Policies Enforcing Dynamically Admissible Robot Trajectories](https://mlsanity.com/paper/2502.15043/) (2025)
- [Equality Constrained Diffusion for Direct Trajectory Optimization](https://mlsanity.com/paper/2410.01939/) (2024)

## Abstract (arXiv)

Recently, diffusion models have gained popularity and attention in trajectory optimization due to their capability of modeling multi-modal probability distributions. However, addressing nonlinear equality constraints, i.e, dynamic feasibility, remains a great challenge in diffusion-based trajectory optimization. Recent diffusion-based trajectory optimization frameworks rely on a single-shooting style approach where the denoised control sequence is applied to forward propagate the dynamical system, which cannot explicitly enforce constraints on the states and frequently leads to sub-optimal solutions. In this work, we propose a novel direct trajectory optimization approach via model-based diffusion, which directly generates a sequence of states. To ensure dynamic feasibility, we propose a gradient-free projection mechanism that is incorporated into the reverse diffusion process. Our results show that, compared to a recent state-of-the-art baseline, our approach leads to zero dynamic feasibility error and approximately 4x higher success rate in a quadrotor waypoint navigation scenario involving dense static obstacles.