# Trajectory Generation, Control, and Safety with Denoising Diffusion Probabilistic Models

arXiv: https://arxiv.org/abs/2306.15512 · HTML: https://mlsanity.com/paper/2306.15512/

## Key facts

- Cited by 13 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 23 of its 33 references that are in the map.
- First posted on arXiv 27 Jun 2023.

## 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, 23 of 23)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [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)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [Soft Actor-Critic Algorithms and Applications](https://mlsanity.com/paper/1812.05905/) (2018)
- [Planning with Diffusion for Flexible Behavior Synthesis](https://mlsanity.com/paper/2205.09991/) (2022)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Model-Based Reinforcement Learning for Atari](https://mlsanity.com/paper/1903.00374/) (2019)
- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)
- [Deep Reinforcement Learning: An Overview](https://mlsanity.com/paper/1701.07274/) (2017)
- [End-to-End Safe Reinforcement Learning through Barrier Functions for Safety-Critical Continuous Control Tasks](https://mlsanity.com/paper/1903.08792/) (2019)
- [Control Barrier Function Based Quadratic Programs for Safety Critical Systems](https://mlsanity.com/paper/1609.06408/) (2016)
- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [Safety-Critical Model Predictive Control with Discrete-Time Control Barrier Function](https://mlsanity.com/paper/2007.11718/) (2020)
- [Joint Synthesis of Safety Certificate and Safe Control Policy using Constrained Reinforcement Learning](https://mlsanity.com/paper/2111.07695/) (2021)
- [Unsupervised Representation Learning in Deep Reinforcement Learning: A Review](https://mlsanity.com/paper/2208.14226/) (2022)
- [Safe Policy Synthesis in Multi-Agent POMDPs via Discrete-Time Barrier Functions](https://mlsanity.com/paper/1903.07823/) (2019)
- [Safety-Critical Kinematic Control of Robotic Systems](https://mlsanity.com/paper/2009.09100/) (2020)

## Built on by (most cited first, 13 of 13)

- [DiffuSolve: Diffusion-based Solver for Non-convex Trajectory Optimization](https://mlsanity.com/paper/2403.05571/) (2024)
- [Diffusion Model for Planning: A Systematic Literature Review](https://mlsanity.com/paper/2408.10266/) (2024)
- [LTLDoG: Satisfying Temporally-Extended Symbolic Constraints for Safe Diffusion-based Planning](https://mlsanity.com/paper/2405.04235/) (2024)
- [DDAT: Diffusion Policies Enforcing Dynamically Admissible Robot Trajectories](https://mlsanity.com/paper/2502.15043/) (2025)
- [Amortized Global Search for Efficient Preliminary Trajectory Design with Deep Generative Models](https://mlsanity.com/paper/2308.03960/) (2023)
- [Constraint-Aware Diffusion Models for Trajectory Optimization](https://mlsanity.com/paper/2406.00990/) (2024)
- [Constraint-Aware Diffusion Guidance for Robotics: Real-Time Obstacle Avoidance for Autonomous Racing](https://mlsanity.com/paper/2505.13131/) (2025)
- [Aligning Diffusion Model with Problem Constraints for Trajectory Optimization](https://mlsanity.com/paper/2504.00342/) (2025)
- [Zero-Shot Trajectory Planning for Signal Temporal Logic Tasks](https://mlsanity.com/paper/2501.13457/) (2025)
- [SAD-Flower: Flow Matching for Safe, Admissible, and Dynamically Consistent Planning](https://mlsanity.com/paper/2511.05355/) (2025)
- [Constraint-Aware Flow Matching: Decision Aligned End-to-End Training for Constrained Sampling](https://mlsanity.com/paper/2605.12754/) (2026)
- [Denoising Diffusion Planner: Learning Complex Paths from Low-Quality Demonstrations](https://mlsanity.com/paper/2410.21497/) (2024)
- [Diffusion-Based Forecasting for Uncertainty-Aware Model Predictive Control](https://mlsanity.com/paper/2503.15095/) (2025)

## Abstract (arXiv)

We present a framework for safety-critical optimal control of physical systems based on denoising diffusion probabilistic models (DDPMs). The technology of control barrier functions (CBFs), encoding desired safety constraints, is used in combination with DDPMs to plan actions by iteratively denoising trajectories through a CBF-based guided sampling procedure. At the same time, the generated trajectories are also guided to maximize a future cumulative reward representing a specific task to be optimally executed. The proposed scheme can be seen as an offline and model-based reinforcement learning algorithm resembling in its functionalities a model-predictive control optimization scheme with receding horizon in which the selected actions lead to optimal and safe trajectories.