# Constraint-Aware Diffusion Models for Trajectory Optimization

arXiv: https://arxiv.org/abs/2406.00990 · HTML: https://mlsanity.com/paper/2406.00990/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion and flow policies for robotics, one of 14 directions in Diffusion models.
- Builds on 22 of its 26 references that are in the map.
- First posted on arXiv 3 Jun 2024.

## 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, 22 of 22)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [Planning with Diffusion for Flexible Behavior Synthesis](https://mlsanity.com/paper/2205.09991/) (2022)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [Is Conditional Generative Modeling all you need for Decision-Making?](https://mlsanity.com/paper/2211.15657/) (2022)
- [DIFUSCO: Graph-based Diffusion Solvers for Combinatorial Optimization](https://mlsanity.com/paper/2302.08224/) (2023)
- [Design-Bench: Benchmarks for Data-Driven Offline Model-Based Optimization](https://mlsanity.com/paper/2202.08450/) (2022)
- [Diffusion Models Beat GANs on Topology Optimization](https://mlsanity.com/paper/2208.09591/) (2022)
- [Diffusion Models for Black-Box Optimization](https://mlsanity.com/paper/2306.07180/) (2023)
- [Compositional Diffusion-Based Continuous Constraint Solvers](https://mlsanity.com/paper/2309.00966/) (2023)
- [DiffDock-PP: Rigid Protein-Protein Docking with Diffusion Models](https://mlsanity.com/paper/2304.03889/) (2023)
- [Trajectory Generation, Control, and Safety with Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2306.15512/) (2023)
- [Denoising Heat-inspired Diffusion with Insulators for Collision Free Motion Planning](https://mlsanity.com/paper/2310.12609/) (2023)

## Built on by (most cited first, 11 of 11)

- [Global Search for Optimal Low Thrust Spacecraft Trajectories using Diffusion Models and the Indirect Method](https://mlsanity.com/paper/2501.07005/) (2025)
- [Generative AI Agents in Autonomous Machines: A Safety Perspective](https://mlsanity.com/paper/2410.15489/) (2024)
- [Dynamics-aware Diffusion Models for Planning and Control](https://mlsanity.com/paper/2504.00236/) (2025)
- [How Well do Diffusion Policies Learn Kinematic Constraint Manifolds?](https://mlsanity.com/paper/2510.01404/) (2025)
- [Grounding Generative Policies in Physics: Optimization-Guided Diffusion for Robot Control](https://mlsanity.com/paper/2606.24208/) (2026)
- [Toward Generalist Neural Motion Planners for Robotic Manipulators: Challenges and Opportunities](https://mlsanity.com/paper/2603.24318/) (2026)
- [GLENS: Global Search via Learning from Solver Iterates with Diffusion Models](https://mlsanity.com/paper/2606.00366/) (2026)
- [Alternators With Noise Models](https://mlsanity.com/paper/2505.12544/) (2025)
- [Training-Free Imitation Learning with Closed-Form Diffusion Policies](https://mlsanity.com/paper/2606.01238/) (2026)
- [Complexity Bounds and Approaches to Learning Projected Gradient Descent Solver Iterates](https://mlsanity.com/paper/2607.22467/) (2026)
- [When Do Learned Diffusion Proposals Help Constraint Solving? A Controlled Study on Continuous Algebraic Systems](https://mlsanity.com/paper/2607.27169/) (2026)

## Abstract (arXiv)

The diffusion model has shown success in generating high-quality and diverse solutions to trajectory optimization problems. However, diffusion models with neural networks inevitably make prediction errors, which leads to constraint violations such as unmet goals or collisions. This paper presents a novel constraint-aware diffusion model for trajectory optimization. We introduce a novel hybrid loss function for training that minimizes the constraint violation of diffusion samples compared to the groundtruth while recovering the original data distribution. Our model is demonstrated on tabletop manipulation and two-car reach-avoid problems, outperforming traditional diffusion models in minimizing constraint violations while generating samples close to locally optimal solutions.