PegasusFlow: Hierarchical Diffusion Planning

• PegasusFlow is a robot planning framework that bypasses expert demonstrations by sampling trajectory score gradients from environmental interactions.
• It employs a hierarchical rolling-denoising process over spline-parameterized trajectories, leveraging WBFO and AVWBFO for lower-dimensional optimization.
• The framework integrates asynchronous IsaacGym simulations with structured noise scheduling to boost sample efficiency in various robotic tasks.

PegasusFlow is a robot planning framework for diffusion-based trajectory generation that is designed to bypass the usual reliance on expert demonstrations by sampling trajectory score gradients directly from environmental interaction in parallel simulation (YE et al., 10 Sep 2025). In the formulation reported for PegasusFlow, planning is treated as a hierarchical rolling-denoising process over spline-parameterized action trajectories, with the optimizer operating in a low-dimensional basis-function space and the rollout substrate implemented as an asynchronous IsaacGym-based simulation architecture (YE et al., 10 Sep 2025). The same source also defines PegasusFlow not as a generic workflow engine, but as a planning and data-generation framework for robot diffusion planner flow matching, centered on Weighted Basis Function Optimization (WBFO), its action-value variant AVWBFO, Latin Hypercube Sampling (LHS), and Hierarchical Ramp Noise Scheduling (HRNS) (YE et al., 10 Sep 2025).

1. Problem formulation and conceptual basis

PegasusFlow is motivated by a practical limitation in robot diffusion planning: most diffusion planners are trained by imitation learning from expert trajectories, which the paper characterizes as both practically inconvenient and conceptually indirect for specialized robots where data is scarce (YE et al., 10 Sep 2025). The stated alternative is to obtain trajectory score gradients directly from online environmental interaction rather than expert data, and to do so through parallel rollout-based denoising (YE et al., 10 Sep 2025).

The planning problem is posed as a finite-horizon optimal control problem in which the robot state evolves according to

$x_{k+1} = f(x_k, u_k, t_k),$

with optimization over a control sequence on a receding horizon (YE et al., 10 Sep 2025). The paper connects this setting to sampling-based MPC and diffusion-style sampling through an MPPI-style noisy rollout update and the score interpretation

$u^+ = u + \Sigma \nabla \log p_1(u),$

where $$p_0(u) \propto \exp\!\left(-\frac{J(u)}{\lambda}\right)$$ and $p_1(\cdot)$ is a smoothed version of $p_0$ induced by Gaussian rollout noise (YE et al., 10 Sep 2025).

Within that interpretation, “rolling” refers to the MPC receding-horizon shift and “denoising” refers to iterative refinement from noisy trajectory samples toward high-reward or low-cost trajectories (YE et al., 10 Sep 2025). The framework therefore sits at the intersection of trajectory optimization, sampling-based MPC, score-based generative modeling, and RL, with environmental reward supplying the supervision signal and an optional pre-trained RL policy supplying an initialization prior (YE et al., 10 Sep 2025).

2. Hierarchical rolling-denoising framework

The framework is described as hierarchical in three senses: trajectory representation, temporal optimization, and simulation organization (YE et al., 10 Sep 2025). First, trajectories are parameterized at two resolutions. A dense action sequence $\boldsymbol{u} \in \mathbb{R}^{D \times T}$ is represented through $K$ spline control nodes, with $K \ll T$, via

$\boldsymbol{u} = \mathbf{P}\mathbf{\Phi},$

where $\mathbf{P} \in \mathbb{R}^{D \times K}$ is the control-node matrix and $u^+ = u + \Sigma \nabla \log p_1(u),$0 is the basis weight matrix (YE et al., 10 Sep 2025). The inverse reconstruction uses the pseudoinverse,

$u^+ = u + \Sigma \nabla \log p_1(u),$1

At the continuous level, the same representation is written as

$u^+ = u + \Sigma \nabla \log p_1(u),$2

and the implementation uses Catmull-Rom splines, chosen because they interpolate the control nodes exactly (YE et al., 10 Sep 2025). The spline segment description is also given as

$u^+ = u + \Sigma \nabla \log p_1(u),$3

with $u^+ = u + \Sigma \nabla \log p_1(u),$4 (YE et al., 10 Sep 2025).

Second, the optimization is hierarchical across time. Near-future parts of the horizon receive more focused exploration, while farther-horizon parts retain broader exploratory noise under HRNS (YE et al., 10 Sep 2025). Third, the simulator is organized into “main” environments that carry the actual receding-horizon execution state forward and many “rollout” sub-environments cloned from each main environment for parallel score sampling (YE et al., 10 Sep 2025).

The resulting operational cycle is fixed in the source: initialize a trajectory prior, clone main environment states into rollout environments, add scheduled noise using LHS and HRNS, decode nodes to dense trajectories, roll out all candidates in parallel, compute reward-weighted node updates, repeat denoising for $u^+ = u + \Sigma \nabla \log p_1(u),$5 iterations, execute the first action, shift the trajectory forward, append terminal actions using the RL warm-start policy, and iterate (YE et al., 10 Sep 2025). The paper emphasizes that this fully specifies the planning and sample-collection pipeline, while the downstream flow-matching loss for a learned diffusion policy is not fully specified (YE et al., 10 Sep 2025).

3. Weighted Basis Function Optimization

WBFO is presented as the main algorithmic novelty of PegasusFlow (YE et al., 10 Sep 2025). Its purpose is to improve sample efficiency relative to MPPI by optimizing in a lower-dimensional spline basis and by assigning node-specific weights according to where rewards occur in time (YE et al., 10 Sep 2025). In contrast to MPPI’s trajectory-level global weighting, WBFO projects reward information back to spline nodes through the basis functions that connect each node to each timestep (YE et al., 10 Sep 2025).

The paper states the optimization problem as

$u^+ = u + \Sigma \nabla \log p_1(u),$6

For $u^+ = u + \Sigma \nabla \log p_1(u),$7 rollout trajectories with step-wise rewards $u^+ = u + \Sigma \nabla \log p_1(u),$8, the discounted accumulated rewards are defined as

$u^+ = u + \Sigma \nabla \log p_1(u),$9

For pure trajectory optimization, WBFO uses $$p_0(u) \propto \exp\!\left(-\frac{J(u)}{\lambda}\right)$$0, which reduces to timestep-local reward, while AVWBFO uses $$p_0(u) \propto \exp\!\left(-\frac{J(u)}{\lambda}\right)$$1, and in experiments $$p_0(u) \propto \exp\!\left(-\frac{J(u)}{\lambda}\right)$$2 (YE et al., 10 Sep 2025).

The reward-to-node projection is then

$$p_0(u) \propto \exp\!\left(-\frac{J(u)}{\lambda}\right)$$3

so that each sampled trajectory receives a reward score for each spline node (YE et al., 10 Sep 2025). The paper then applies node-level normalization,

$$p_0(u) \propto \exp\!\left(-\frac{J(u)}{\lambda}\right)$$4

followed by node-specific softmax weights,

$$p_0(u) \propto \exp\!\left(-\frac{J(u)}{\lambda}\right)$$5

The updated node is the weighted average

$$p_0(u) \propto \exp\!\left(-\frac{J(u)}{\lambda}\right)$$6

This basis-centric update is the essential WBFO denoising step (YE et al., 10 Sep 2025). The source attributes four effects to this design: local optimality, temporal coherence, lower-dimensional search, and better sample efficiency because each rollout contributes finer-grained learning signal than a single trajectory-level scalar weight (YE et al., 10 Sep 2025).

A central distinction in the paper is between WBFO and AVWBFO. WBFO is described as appropriate for trajectory optimization when local step-wise reward is sufficient, whereas AVWBFO is intended for integrated dynamic control where future consequences matter (YE et al., 10 Sep 2025). The inverted pendulum result is used as the paper’s main supporting example: plain WBFO underperforms in integrated control, while AVWBFO slightly outperforms MPPI, which the authors interpret as validating discounted reward or action-value style updates in dynamic systems (YE et al., 10 Sep 2025).

4. Noise design, rollout architecture, and warm start

PegasusFlow combines its optimizer with two structured exploration mechanisms: Latin Hypercube Sampling and Hierarchical Ramp Noise Scheduling (YE et al., 10 Sep 2025). LHS is used instead of independent Monte Carlo Gaussian samples in order to stratify each dimension into equiprobable intervals and provide more uniform coverage of the noise space with fewer rollout samples (YE et al., 10 Sep 2025). HRNS modulates noise magnitude across the planning horizon according to temporal distance from the current step, producing focused local improvement near the present and broader exploratory perturbation farther ahead (YE et al., 10 Sep 2025).

The rollout substrate is an IsaacGym-based asynchronous parallel simulator with total environment count

$$p_0(u) \propto \exp\!\left(-\frac{J(u)}{\lambda}\right)$$7

where $$p_0(u) \propto \exp\!\left(-\frac{J(u)}{\lambda}\right)$$8 is the number of main environments and $$p_0(u) \propto \exp\!\left(-\frac{J(u)}{\lambda}\right)$$9 is the number of rollout environments attached to each main environment (YE et al., 10 Sep 2025). The main environments hold the true receding-horizon state, while rollout environments are repeatedly synchronized to those states before each denoising phase and then execute many noisy trajectory branches in parallel (YE et al., 10 Sep 2025).

The paper identifies this architecture as a practical strength because it supports hundreds of sub-environments for rollout sampling, uses NVIDIA Warp for accelerated Signed Distance Field and raycasting queries, and provides branch-consistent exploration from identical starting states (YE et al., 10 Sep 2025). The latter property is especially important for direct score or policy-improvement estimation, since each rollout batch probes the local task-induced trajectory distribution around the same planning point (YE et al., 10 Sep 2025).

RL warm start is reported as an important practical component, though not the core contribution (YE et al., 10 Sep 2025). A pre-trained RL policy initializes the horizon action sequence and provides tail actions after the receding-horizon shift (YE et al., 10 Sep 2025). The paper’s interpretation is that this stabilizes optimization and supplies a dynamically sensible prior, particularly in obstacle-rich locomotion and navigation tasks (YE et al., 10 Sep 2025). It also notes a limitation: PegasusFlow bypasses expert demonstrations for score sampling, but its best empirical results still benefit substantially from a learned RL prior (YE et al., 10 Sep 2025).

5. Empirical results and reported task performance

The experimental program spans 2D Navigation, Inverted Pendulum, Franka arm collision avoidance planning, and several legged navigation tasks including hexapod timber piles navigation, quadruped walking with hundreds of rollout environments, hexapod confined-space navigation, and ElSpider Air barrier navigation (YE et al., 10 Sep 2025). Reported metrics include final cost after optimization, completion rate or success rate, mean steps to completion, and accumulated or mean reward (YE et al., 10 Sep 2025).

For 2D Navigation, the setup uses 16 nodes, 64 dense samples, 10 optimization iterations, exponential noise decay with initial noise $p_1(\cdot)$0 and decay $p_1(\cdot)$1, and 5 trials (YE et al., 10 Sep 2025). The key finding is that WBFO is much more sample-efficient than MPPI, especially with $p_1(\cdot)$2 samples (YE et al., 10 Sep 2025). At 10 samples, WBFO achieves final cost $p_1(\cdot)$3, while MPPI gets $p_1(\cdot)$4 (YE et al., 10 Sep 2025).

For Franka collision avoidance planning, the setup uses 30 Franka arms, 64 samples for all methods, a goal behind a wall gap, and a horizon limit of 150 steps (YE et al., 10 Sep 2025). The source reports AVWBFO + MC at 93.3% completion, MPPI + MC at 70%, AVWBFO + LHS at 100%, and MPPI + LHS at 90% (YE et al., 10 Sep 2025). The best method is AVWBFO + LHS with 100% completion and $p_1(\cdot)$5 steps to completion (YE et al., 10 Sep 2025).

For ElSpider Air barrier navigation, the setup includes 20 robots, a start inside a square barrier, barrier height $p_1(\cdot)$6, width $p_1(\cdot)$7, a goal outside the barrier, a 300-step budget equal to 6 s at 0.02 s per step, and 128 samples (YE et al., 10 Sep 2025). The reported methods are Vanilla RL, AVWBFO with RL, AVWBFO without RL, MPPI with RL, and MPPI without RL (YE et al., 10 Sep 2025). The corresponding success rates are 0%, 100%, 75%, 100%, and 40%; the completion speeds are $p_1(\cdot)$8 steps, $p_1(\cdot)$9 steps, $p_0$0 steps, and $p_0$1 steps for the non-vanilla methods in the order given above; and the mean rewards are 0.024, 0.014, 0.016, 0.011, and $p_0$2 for the five methods in the order reported in the source (YE et al., 10 Sep 2025). The paper highlights that AVWBFO with RL warm start is 18% faster than the next-best method, namely warm-started MPPI (YE et al., 10 Sep 2025).

Task	Best reported result	Comparative note
2D Navigation	WBFO final cost $p_0$3 at 10 samples	MPPI: $p_0$4
Franka collision avoidance	AVWBFO + LHS: 100% completion, $p_0$5 steps	MPPI + LHS: 90%
ElSpider Air barrier navigation	AVWBFO w RL: 100% success, $p_0$6 steps	18% faster than MPPI w RL

A plausible implication is that PegasusFlow’s empirical advantage depends on the joint effect of basis-function optimization, structured noise, large-scale synchronized rollouts, and a strong prior when warm start is available. The source itself frames these gains as evidence that AVWBFO, especially with an RL warm start, significantly outperforms baselines in difficult locomotion navigation tasks (YE et al., 10 Sep 2025).

6. Limitations, scope, and naming ambiguity

The paper’s limitations are explicit. PegasusFlow is simulation-heavy and its practical advantage depends on access to GPU-accelerated simulators such as IsaacGym (YE et al., 10 Sep 2025). The paper does not provide a full downstream diffusion-training recipe, network architecture, or loss for the final diffusion policy, even though it repeatedly states that sampled trajectory score gradients can be used for flow matching training of a diffusion policy (YE et al., 10 Sep 2025). It also does not present a formal convergence theorem for WBFO, and it leaves reward design as a critical dependency because the supervision signal is environmental reward (YE et al., 10 Sep 2025).

A further nuance is methodological rather than nominal. In the hardest locomotion task, warm start materially improves outcomes for both AVWBFO and MPPI, so PegasusFlow’s strongest reported results are not purely prior-free in the sense of dispensing with all learned initialization (YE et al., 10 Sep 2025). The paper’s sharper claim is narrower: it bypasses expert demonstrations for score sampling and planner improvement, not necessarily all auxiliary learned priors (YE et al., 10 Sep 2025).

The term “PegasusFlow” is also ambiguous across adjacent literatures. A 2026 poster on rapid development of sensor-driven applications over Pegasus workflows does not use the term “PegasusFlow” explicitly; instead, it presents a pattern-based, AI-assisted methodology layered on top of Pegasus workflows for edge-to-core applications (Thareja et al., 4 May 2026). That work is therefore about Pegasus-based workflow engineering rather than the robot diffusion planner framework named PegasusFlow (Thareja et al., 4 May 2026). Two additional systems named “Pegasus” are also distinct: a dataplane deep-learning inference framework for programmable switches (Zhang et al., 6 Jun 2025) and a hybrid-kinetic particle-in-cell code for astrophysical plasma dynamics (Kunz et al., 2013). This suggests that “PegasusFlow” should be reserved, in the most precise usage, for the hierarchical rolling-denoising robot planning framework defined in 2025 (YE et al., 10 Sep 2025).

Within that scope, PegasusFlow is best characterized as a parallel score-sampling planner that turns receding-horizon trajectory optimization into a denoising process over spline-based action representations, with WBFO and AVWBFO supplying the optimizer and environmental interaction supplying the score signal (YE et al., 10 Sep 2025).