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SafeDiffuser: Safe Planning with Diffusion Probabilistic Models

Wei Xiao, Tsun-Hsuan Wang, Chuang Gan, Daniela Rus

arXiv:2306.00148v1cs.LGcs.ROcs.SYeess.SY

TL;DR

Diffusion models show promise for data-driven trajectory planning, but lack safety guarantees for safety-critical applications. SafeDiffuser embeds finite-time diffusion invariance, using control barrier functions, into denoising. The paper reports guaranteed specification satisfaction across maze, locomotion, and manipulation settings, subject to differentiable specifications.

  • Problem

    Diffusion-based trajectory planning has shown promise, but existing approaches provide no safety guarantees for safety-critical applications.

    Source: §1 · p. 1

  • Method

    SafeDiffuser embeds finite-time diffusion invariance using control barrier functions into the denoising procedure to guarantee specifications.

    Source: §1 · p. 2

  • Results

    The proposed safe diffusers guarantee specification satisfaction in maze tasks, while RoS-diffuser guarantees collision-free locomotion and supports differentiable manipulation specifications.

    Source: §5.1 · p. 8, §5.2 · p. 8, §5.3 · p. 9

  • Takeaways & Limitations

    The method demonstrates safe planning with diffusion models across maze, robot locomotion, and manipulation tasks.

    Source: §1 · p. 2, §7 · p. 12

    Specifications must be continuously differentiable, and they may be unknown for planning tasks.

    Source: §7 · p. 12

Abstract

from arXiv · show

Diffusion model-based approaches have shown promise in data-driven planning, but there are no safety guarantees, thus making it hard to be applied for safety-critical applications. To address these challenges, we propose a new method, called SafeDiffuser, to ensure diffusion probabilistic models satisfy specifications by using a class of control barrier functions. The key idea of our approach is to embed the proposed finite-time diffusion invariance into the denoising diffusion procedure, which enables trustworthy diffusion data generation. Moreover, we demonstrate that our finite-time diffusion invariance method through generative models not only maintains generalization performance but also creates robustness in safe data generation. We test our method on a series of safe planning tasks, including maze path generation, legged robot locomotion, and 3D space manipulation, with results showing the advantages of robustness and guarantees over vanilla diffusion models.

1. Introduction

Diffusion-based trajectory planning can generalize to novel environments, but it lacks safety guarantees needed for safety-critical applications. SafeDiffuser embeds finite-time diffusion invariance into denoising and is demonstrated on maze, locomotion, and manipulation tasks.

1. Introduction

  • Diffusers generate planning trajectories from noise conditioned on the current state and objectives, but provide no safety guarantees.The planning process repeats after applying one control step until the objective is achieved.Source: §1 · p. 1
  • In Fig. 1, SafeDiffuser generates safe trajectories with guarantees while the ordinary diffuser fails along the illustrated path.The figure contrasts SafeDiffuser below with the diffuser above, from J to N.Source: Figure 1 · p. 1, §1 · p. 1
  • SafeDiffuser uses finite-time diffusion invariance and control barrier functions to guarantee specifications within the denoising procedure.The approach combines receding-horizon control with diffusion-based path generation and validates paths through simulation before actuation.Source: §1 · p. 2
  • The method is demonstrated on safe planning tasks involving maze paths, robot locomotion, and manipulation.The proposed contributions include formal guarantees through control-theoretic invariance and three safe-diffuser variants.Source: §1 · p. 2

2. Preliminaries

The preliminaries describe diffusion models as iterative denoising processes and establish notation distinguishing diffusion time from planning time. They also review forward invariance and control barrier functions, which the paper maps to finite-time invariance during diffusion.

2. Preliminaries

  • Diffusion probabilistic models represent data generation as iterative denoising, reversing a forward process that gradually adds noise to clean data.The reverse transitions are often parameterized as Gaussians with time-dependent mean and variance.Source: §2 · p. 2
  • The paper distinguishes diffusion time, which runs from N to 0, from planning time, which runs from 0 to horizon H.A trajectory is a sequence of discretized planning-time states at a given diffusion step.Source: §2 · p. 3
  • Forward invariance means trajectories starting in a set C remain in C, and a CBF condition can ensure this under a suitable controller.The safe set is defined by b(x_t) ≥ 0; the CBF condition uses Lie derivatives and an extended class K function.Source: §2 · p. 3
  • SafeDiffuser maps control-theoretic forward invariance to finite-time diffusion invariance by incorporating CBFs into diffusion time.This differs from their regular use in planning time and is also used to address local traps during diffusion.Source: §2 · p. 3

3. Safe Diffuser

Safe Diffuser seeks to enforce planning specifications during denoising through finite-time diffusion invariance. It proposes three safe diffuser variants to achieve this, including variants intended to address local traps.

3. Safe Diffuser

  • Finite-time diffusion invariance brings initially violated planning constraints to satisfaction across planning states by a finite denoising step; continuous-time safety can use a lower-level CBF controller.Source: §3 · pp. 3–4

3.1. Robust-Safe Diffuser

  • Robust-Safe Diffuser adds an invariance QP step to diffusion dynamics and is finite-time diffusion invariant with almost probability 1.The CBF conditions are imposed across planning states and diffusion steps.Source: Figure 2 · p. 4, §3.1 · pp. 4–5

3.2. Relaxed-Safe Diffuser

  • Relaxed-Safe Diffuser decreases a time-varying relaxation weight toward zero and may add Na diffusion steps; its procedure is finite-time invariant with almost probability 1.Additional steps address the potential adverse effect of enforcing a hard constraint.Source: §3.2 · p. 5

3.3. Time-Varying-Safe Diffuser

  • Time-Varying-Safe Diffuser targets local traps by changing the specification over diffusion time and enforcing it with CBFs; its procedure is finite-time invariant.Source: §3.3 · pp. 5–6

4. Enforcing Invariance in Diffuser

SafeDiffuser enforces finite-time invariance by applying a minimum-deviation QP at each diffusion step. The QP adjusts the diffusion update to satisfy safety conditions while keeping the control close to the diffusion dynamics.

4. Enforcing Invariance in Diffuser

  • A minimum-deviation QP enforces invariance conditions at every diffusion step by adjusting the diffusion update.The resulting control updates the diffusion state at the end of each step.Source: §4 · p. 6
  • Optimizing the condition at diffusion time zero may produce more coherent trajectories.The passage presents this as a possibility, not a guaranteed outcome.Source: §4 · p. 6
  • The robust-safe and time-varying-safe variants choose controls that stay close to the diffusion dynamics while satisfying their invariance conditions.The update is applied within each diffusion step.Source: §4 · p. 6
  • The relaxed-safe QP minimizes relaxation variables alongside the control, with either separate relaxations per specification or one shared variable.After solving the QP, the chosen control updates the diffusion state.Source: §4 · p. 6
  • The implementation selects a QP formulation according to the diffuser variant, then updates the diffusion state with its optimized control.The robust-safe, relaxed-safe, and time-varying-safe branches use different conditions or variables.Source: §4 · p. 7
  • Enforcing invariance has QP complexity O(q^3), where q is the decision-variable dimension; adding specifications adds constraints.The three proposed safe diffusers have similar complexity.Source: §4 · p. 7

5. Experiments

The experiments assess specification satisfaction, planning performance, and computation across maze, locomotion, and manipulation tasks. The proposed safe diffusers are reported to guarantee specifications in these settings, with task-specific qualifications.

5. Experiments

  • The experiments ask whether the method matches its theoretical potential, enforces safety against state-of-the-art approaches, and affects diffusion performance under guarantees.The maze task imposes trajectory constraints on paths generated from randomly sampled initial positions and destinations.Source: §5 · p. 7, §5.1 · p. 7

5.1. Safe Planning in Maze

  • In maze planning, RoS-, ReS-, and TVS-diffusers guarantee even complex differentiable specifications while maximally preserving diffusion-model performance.ReS- and TVS-diffusers also address local traps from specifications; the reported satisfaction scores are not strictly positive.Source: §5.1 · p. 8, Table 1 · p. 8

5.2. Safe Planning for Robot Locomotion

  • RoS-diffuser guarantees collision-free locomotion, while truncation fails on complex specifications and classifier guidance improves satisfaction without guarantees.The evaluation uses MuJoCo robot locomotion and considers avoiding obstacles such as a roof.Source: §5.2 · p. 8

5.3. Safe Planning for Manipulation

  • For manipulation, RoS-diffuser works for all differentiable specifications, and the authors report a possible reward improvement over diffusion models.The task generates joint trajectories as controls conditioned on object locations; the proposed method's computation time is reported as comparable to other methods.Source: §5.3 · p. 9

6. Related Works

Prior work uses diffusion models in image generation, planning, and language, but the cited literature lacks a method for specification guarantees in diffusion models. The paper relates this gap to prior control-barrier and neural-network guarantee approaches.

Diffusion models and planning

  • Diffusion models have been applied to image generation, planning, and language, while the cited literature lacks methods to equip them with specification guarantees.The passage frames such guarantees as especially important for safety-critical applications.Source: §6 · p. 10

Set invariance and CBFs

  • Control barrier functions are widely used to prove set invariance, but prior CBF approaches are usually applied in planning time rather than diffusion time.The passage identifies few studies of CBFs in spaces other than planning time, including the diffusion-time horizon.Source: §6 · p. 10

Guarantees in neural networks

  • Prior neural-network safety approaches include CBF filters, verification-in-the-loop training, and CBFs in neural ODEs, but the authors say none applies to diffusion models.The cited verification approaches cannot ensure coverage of the entire state space; non-trainable CBFs may limit learning performance.Source: §6 · p. 11

7. Conclusions, Discussions and Future Work

The paper concludes that finite-time diffusion invariance can support safe planning and reports demonstrations across robotic planning tasks. It identifies differentiable, potentially unknown specifications and the gap between planning and control as future-work constraints.

Conclusion and future work

  • The authors report demonstrating finite-time diffusion invariance for safe planning across a series of robotic planning tasks.They also identify possible application to guarantees in image generation as future work.Source: §7 · p. 12

Limitations and future work

  • Specifications must be continuously differentiable and may be unknown in planning tasks; the authors identify the planning-control gap as another limitation.Proposed future directions include learning specifications from trajectory histories and studying safe control policies when robot dynamics are known or learned.Source: §7 · p. 12

S1. Proofs

The proofs establish finite-time diffusion invariance under the stated conditions for three theorems. Their reported finite-time endpoints differ across the results.

S1.1. Proof of Theorem 3.2

  • Theorem 3.2's proof concludes finite-time diffusion invariance, with finite time N in the stated case and almost-probability-one invariance in another.The proof invokes Nagumo's theorem and Lyapunov stability arguments.Source: S1.1. Proof of Theorem 3.2 · pp. 15–16

S1.2. Proof of Theorem 3.3

  • Theorem 3.3's proof concludes that the diffusion procedure is finite-time invariant with almost probability 1 under the stated weight choice.The proof sets the weight to zero at j = 0, making condition (10) a hard constraint when j < 0.Source: S1.2. Proof of Theorem 3.3 · p. 16

S1.3. Proof of Theorem 3.4

  • The diffusion procedure p_theta(tau_i-1|tau_i), for i in {0,...,N}, is finite-time diffusion invariant, and the finite time in diffusion invariance is 0.Source: S1.3. Proof of Theorem 3.4 · p. 16

S2. Experiment Details

The experiments apply safety specifications to maze paths, robot locomotion, and manipulation trajectories. In the maze, robust-safe diffusion guarantees safety but may leave points in local traps; relaxed-safe and time-varying-safe variants can address that issue.

S2.1. Safe Planning in Maze

  • Maze models use a 384-step planning horizon and 256 diffusion steps, plus 50 additional diffusion steps for the proposed methods.Training used 2 million steps and took about 10 hours on an Nvidia RTX-3090 GPU.Source: S2.1. Safe Planning in Maze · p. 17
  • Relaxed-safe and time-varying-safe diffusers address local traps observed with the robust-safe diffuser, which guarantees maze-trajectory safety.The maze experiments use ellipse- and super-ellipse-shaped obstacles; truncation becomes difficult for complex specifications.Source: S2.1. Safe Planning in Maze · p. 17

S2.2. Safe Planning for Robot Locomotion

  • Robot locomotion experiments specify roof collision avoidance, including a speed-dependent constraint that requires greater clearance during faster jumps.The speed-dependent constraint accounts for uncertainties or perturbations; the experiments use Walker2D and Hopper models, with no local-trap problem considered.Source: S2.2. Safe Planning for Robot Locomotion · pp. 17–18

S2.3. Safe Planning for Manipulation

  • Manipulation specifications constrain seven joint angles to allowed ranges, with complex constraints also limiting joint speeds estimated from successive planned states.The experiments use pre-trained models and condition joint trajectories on object locations for grasping and placing.Source: S2.3. Safe Planning for Manipulation · p. 19

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