DDAT

Abstract

The stochastic nature of diffusion models prevents them from generating trajectories exactly satisfying the equations of motion of robots. To alleviate this issue, we introduce DDAT: Diffusion policies for Dynamically Amissible Trajectories. A trajectory is dynamically admissible if each state belongs to the reachable set of its predecessor by the robot's equations of motion. To generate such trajectories, our diffusion policies project their predictions onto a dynamically admissible manifold during both training and inference to align the objective of the denoiser neural network with the dynamical admissibility constraint. Due to the auto-regressive nature of such projections as well as the black-box nature of robot dynamics, exact projections are challenging. We instead sample a polytopic under-approximation of the reachable set onto which we project the predicted successor, before iterating this process with the projected successor. By producing accurate trajectories, this projection eliminates the need for diffusion models to continually replan, enabling one-shot long-horizon trajectory planning. We demonstrate our framework through extensive simulations on a quadcopter and various MuJoCo environments, along with real-world experiments on a Unitree GO1 and GO2.

Projections make trajectories admissible and better

Statewise admissibility error over 500 Hopper trajectories.
The error without projections is much larger than when using
projections at inference or training with projections $\mathcal{P}^\text{ref}$.
All diffusion models generate only state trajectories.

Cumulative admissibility error over 500 Hopper trajectories.
The error without projections is much larger than when using
projections at inference or training with projections $\mathcal{P}^\text{ref}$.
All diffusion models generate only state trajectories.

Ratios of open-loop Hopper trajectories having fallen at a given timestep.
The trajectories without projections and those with projections at inference
start to fall significantly earlier than those training with projections $\mathcal{P}^\text{ref}$.
All diffusion models generate only state trajectories.

Statewise admissibility error over 400 Walker trajectories.
The error without projections is much larger than when using projections at inference.
Our model trained with projections $\mathcal{P}^\text{SA}$ has no error.
All diffusion models generate states and actions.

Cumulative admissibility error over 400 Walker trajectories.
The error without projections is much larger than when using projections at inference.
Our model trained with projections $\mathcal{P}^\text{SA}$ has no error.
All diffusion models generate states and actions.

Ratios of open-loop Walker trajectories having fallen at a given timestep.
The trajectories without projections and those with projections at inference
start to fall significantly earlier than those training with projections $\mathcal{P}^\text{SA}$.
All diffusion models generate states and actions.

Projections should only occur at low noise levels

All diffusion models generate state-action trajectories with projections starting from either the beginning of inference, i.e., projecting at all noise levels, or starting mid-inference, or project only once after inference.

Statewise admissibility error over 500 Hopper trajectories.
The admissibility error is similar between models no matter when projections start,
because they all perform a projection at the end of inference.

Ratios of open-loop Hopper trajectories having fallen at a given timestep.
The trajectories projecting at high noise levels fall significantly earlier than those projecting at small and zero noise levels.

Quadcopter trajectories tasked with slaloming between obstacles following the dashed line.
Trajectories generated with projections at high noise levels are incapable of performing the task,
while trajectories with projections at small and zero noise levels succeed.

Generating quadcopter trajectories

Objective: slalom between obstacles to reach the target.

Model generating state trajectories without projections.
The sampled trajectory would succeed if it was feasible,
but inverse dynamics shows how far off the trajectory actually is.

Model generating state trajectories with projections only at inference.
The sampled trajectory is close to feasible, but not sufficiently to
prevent the inverse dynamics from crashing.

Model generating state trajectories trained with reference projections.
The sampled trajectory is very close to the inverse dynamics and both succeed.

Model generating state trajectories trained with reference projections.
The sampled trajectory slaloms between the obstacles and reaches the target.

Model generating states and actions without projections.
The sampled trajectory would succeed if it was feasible,
but the open-loop shows how far off the trajectory actually is.

Model generating states and actions with projections only at inference.
The sampled trajectory flies through the obstacle,
while the open-loop diverges.

Model generating states and actions trained with SA-projections.
The sampled trajectory is admissible as it matches its
open-loop realisation and both succeed.

Model generating states and actions trained with reference projections.
The sampled trajectory slaloms between the obstacles and reaches the target.

BibTeX

@inproceedings{bouvier2025ddat, title = {DDAT: Diffusion Policies Enforcing Dynamically Admissible Robot Trajectories}, author = {Bouvier, Jean-Baptiste and Ryu, Kanghyun and Nagpal, Kartik and Liao, Qiayuan and Sreenath, Koushil and Mehr, Negar}, booktitle = {under review}, year = {2025} }

DDAT: Diffusion Policies Enforcing Dynamically Admissible Robot Trajectories

Abstract

Auto-regressive trajectory projection by sampling convex under-approximations of reachable sets of black-box dynamics $s_{t+1} = f(s_t, a_t)$.

Unitree GO1 zero-shot hardware deployment of an open-loop trajectory generated by a our DDAT model.

Projections make trajectories admissible and better

Statewise admissibility error over 500 Hopper trajectories.
The error without projections is much larger than when using
projections at inference or training with projections $\mathcal{P}^\text{ref}$.
All diffusion models generate only state trajectories.

Cumulative admissibility error over 500 Hopper trajectories.
The error without projections is much larger than when using
projections at inference or training with projections $\mathcal{P}^\text{ref}$.
All diffusion models generate only state trajectories.

Statewise admissibility error over 400 Walker trajectories.
The error without projections is much larger than when using projections at inference.
Our model trained with projections $\mathcal{P}^\text{SA}$ has no error.
All diffusion models generate states and actions.

Cumulative admissibility error over 400 Walker trajectories.
The error without projections is much larger than when using projections at inference.
Our model trained with projections $\mathcal{P}^\text{SA}$ has no error.
All diffusion models generate states and actions.

Ratios of open-loop Walker trajectories having fallen at a given timestep.
The trajectories without projections and those with projections at inference
start to fall significantly earlier than those training with projections $\mathcal{P}^\text{SA}$.
All diffusion models generate states and actions.

Unitree GO2 open-loop trajectories tasked with going straight.
Both models without projections deviate from walking straight,
whereas our DDAT model follows prefectly the command.

Projections should only occur at low noise levels

Statewise admissibility error over 500 Hopper trajectories.
The admissibility error is similar between models no matter when projections start,
because they all perform a projection at the end of inference.

Ratios of open-loop Hopper trajectories having fallen at a given timestep.
The trajectories projecting at high noise levels fall significantly earlier than those projecting at small and zero noise levels.

Quadcopter trajectories tasked with slaloming between obstacles following the dashed line.
Trajectories generated with projections at high noise levels are incapable of performing the task,
while trajectories with projections at small and zero noise levels succeed.

Generating quadcopter trajectories

Model generating state trajectories without projections.
The sampled trajectory would succeed if it was feasible,
but inverse dynamics shows how far off the trajectory actually is.

Model generating state trajectories with projections only at inference.
The sampled trajectory is close to feasible, but not sufficiently to
prevent the inverse dynamics from crashing.

Model generating state trajectories trained with reference projections.
The sampled trajectory is very close to the inverse dynamics and both succeed.

Model generating state trajectories trained with reference projections.
The sampled trajectory slaloms between the obstacles and reaches the target.

Model generating states and actions without projections.
The sampled trajectory would succeed if it was feasible,
but the open-loop shows how far off the trajectory actually is.

Model generating states and actions with projections only at inference.
The sampled trajectory flies through the obstacle,
while the open-loop diverges.

Model generating states and actions trained with SA-projections.
The sampled trajectory is admissible as it matches its
open-loop realisation and both succeed.

Model generating states and actions trained with reference projections.
The sampled trajectory slaloms between the obstacles and reaches the target.

MuJoCo Hopper open-loop trajectories.
Without projections the Hopper fall earlier than with our projections.

BibTeX

DDAT: Diffusion Policies Enforcing Dynamically Admissible Robot Trajectories

Abstract

Auto-regressive trajectory projection by sampling convex under-approximations of reachable sets of black-box dynamics $s_{t+1} = f(s_t, a_t)$.

Unitree GO1 zero-shot hardware deployment of an open-loop trajectory generated by a our DDAT model.

Projections make trajectories admissible and better

Statewise admissibility error over 500 Hopper trajectories. The error without projections is much larger than when using projections at inference or training with projections $\mathcal{P}^\text{ref}$. All diffusion models generate only state trajectories.

Cumulative admissibility error over 500 Hopper trajectories. The error without projections is much larger than when using projections at inference or training with projections $\mathcal{P}^\text{ref}$. All diffusion models generate only state trajectories.

Statewise admissibility error over 400 Walker trajectories. The error without projections is much larger than when using projections at inference. Our model trained with projections $\mathcal{P}^\text{SA}$ has no error. All diffusion models generate states and actions.

Cumulative admissibility error over 400 Walker trajectories. The error without projections is much larger than when using projections at inference. Our model trained with projections $\mathcal{P}^\text{SA}$ has no error. All diffusion models generate states and actions.

Ratios of open-loop Walker trajectories having fallen at a given timestep. The trajectories without projections and those with projections at inference start to fall significantly earlier than those training with projections $\mathcal{P}^\text{SA}$. All diffusion models generate states and actions.

Unitree GO2 open-loop trajectories tasked with going straight. Both models without projections deviate from walking straight, whereas our DDAT model follows prefectly the command.

Projections should only occur at low noise levels

Statewise admissibility error over 500 Hopper trajectories. The admissibility error is similar between models no matter when projections start, because they all perform a projection at the end of inference.

Ratios of open-loop Hopper trajectories having fallen at a given timestep. The trajectories projecting at high noise levels fall significantly earlier than those projecting at small and zero noise levels.

Quadcopter trajectories tasked with slaloming between obstacles following the dashed line. Trajectories generated with projections at high noise levels are incapable of performing the task, while trajectories with projections at small and zero noise levels succeed.

Generating quadcopter trajectories

Model generating state trajectories without projections. The sampled trajectory would succeed if it was feasible, but inverse dynamics shows how far off the trajectory actually is.

Model generating state trajectories with projections only at inference. The sampled trajectory is close to feasible, but not sufficiently to prevent the inverse dynamics from crashing.

Model generating state trajectories trained with reference projections. The sampled trajectory is very close to the inverse dynamics and both succeed.

Model generating state trajectories trained with reference projections. The sampled trajectory slaloms between the obstacles and reaches the target.

Model generating states and actions without projections. The sampled trajectory would succeed if it was feasible, but the open-loop shows how far off the trajectory actually is.

Model generating states and actions with projections only at inference. The sampled trajectory flies through the obstacle, while the open-loop diverges.

Model generating states and actions trained with SA-projections. The sampled trajectory is admissible as it matches its open-loop realisation and both succeed.

Model generating states and actions trained with reference projections. The sampled trajectory slaloms between the obstacles and reaches the target.

MuJoCo Hopper open-loop trajectories. Without projections the Hopper fall earlier than with our projections.

BibTeX

Statewise admissibility error over 500 Hopper trajectories.
The error without projections is much larger than when using
projections at inference or training with projections $\mathcal{P}^\text{ref}$.
All diffusion models generate only state trajectories.

Cumulative admissibility error over 500 Hopper trajectories.
The error without projections is much larger than when using
projections at inference or training with projections $\mathcal{P}^\text{ref}$.
All diffusion models generate only state trajectories.

Statewise admissibility error over 400 Walker trajectories.
The error without projections is much larger than when using projections at inference.
Our model trained with projections $\mathcal{P}^\text{SA}$ has no error.
All diffusion models generate states and actions.

Cumulative admissibility error over 400 Walker trajectories.
The error without projections is much larger than when using projections at inference.
Our model trained with projections $\mathcal{P}^\text{SA}$ has no error.
All diffusion models generate states and actions.

Ratios of open-loop Walker trajectories having fallen at a given timestep.
The trajectories without projections and those with projections at inference
start to fall significantly earlier than those training with projections $\mathcal{P}^\text{SA}$.
All diffusion models generate states and actions.

Unitree GO2 open-loop trajectories tasked with going straight.
Both models without projections deviate from walking straight,
whereas our DDAT model follows prefectly the command.

Statewise admissibility error over 500 Hopper trajectories.
The admissibility error is similar between models no matter when projections start,
because they all perform a projection at the end of inference.

Ratios of open-loop Hopper trajectories having fallen at a given timestep.
The trajectories projecting at high noise levels fall significantly earlier than those projecting at small and zero noise levels.

Quadcopter trajectories tasked with slaloming between obstacles following the dashed line.
Trajectories generated with projections at high noise levels are incapable of performing the task,
while trajectories with projections at small and zero noise levels succeed.

Model generating state trajectories without projections.
The sampled trajectory would succeed if it was feasible,
but inverse dynamics shows how far off the trajectory actually is.

Model generating state trajectories with projections only at inference.
The sampled trajectory is close to feasible, but not sufficiently to
prevent the inverse dynamics from crashing.

Model generating state trajectories trained with reference projections.
The sampled trajectory is very close to the inverse dynamics and both succeed.

Model generating state trajectories trained with reference projections.
The sampled trajectory slaloms between the obstacles and reaches the target.

Model generating states and actions without projections.
The sampled trajectory would succeed if it was feasible,
but the open-loop shows how far off the trajectory actually is.

Model generating states and actions with projections only at inference.
The sampled trajectory flies through the obstacle,
while the open-loop diverges.

Model generating states and actions trained with SA-projections.
The sampled trajectory is admissible as it matches its
open-loop realisation and both succeed.

Model generating states and actions trained with reference projections.
The sampled trajectory slaloms between the obstacles and reaches the target.

MuJoCo Hopper open-loop trajectories.
Without projections the Hopper fall earlier than with our projections.