GPD: Guided Polynomial Diffusion for Motion Planning

TL;DR

GPD addresses the bottleneck of slow diffusion-based motion planning by performing diffusion in the Bernstein coefficient space, parameterizing trajectories as Bernstein polynomials with the transform $\boldsymbol{\tau} = \boldsymbol{\alpha} \cdot \boldsymbol{B}$. This enables effective gradient-guided updates $\nabla_{\boldsymbol{\alpha}_t} J$ via $\boldsymbol{q}_t = \boldsymbol{\alpha}_t \cdot \boldsymbol{B}$ and a preconditioning with $\boldsymbol{B}^T$, yielding faster convergence and smoother trajectories. A stitching algorithm then exploits the diffusion priors’ diversity to assemble fully collision-free trajectories from segments across multiple samples using a local planner like RRT-Connect. Empirically, GPD achieves state-of-the-art performance in speed and success on robotic manipulators and demonstrates applicability to reactive navigation tasks, including indoor and urban driving, at near real-time rates. The approach balances a compact diffusion model, smooth trajectory priors, and an inference-time stitching mechanism to deliver robust, fast motion planning with strong generalization.

Abstract

Diffusion-based motion planners are becoming popular due to their well-established performance improvements, stemming from sample diversity and the ease of incorporating new constraints directly during inference. However, a primary limitation of the diffusion process is the requirement for a substantial number of denoising steps, especially when the denoising process is coupled with gradient-based guidance. In this paper, we introduce, diffusion in the parametric space of trajectories, where the parameters are represented as Bernstein coefficients. We show that this representation greatly improves the effectiveness of the cost function guidance and the inference speed. We also introduce a novel stitching algorithm that leverages the diversity in diffusion-generated trajectories to produce collision-free trajectories with just a single cost function-guided model. We demonstrate that our approaches outperform current SOTA diffusion-based motion planners for manipulators and provide an ablation study on key components.

Figures (8)

• Figure 1: Guided Polynomial Diffusion (GPD) denoises in the Bernstein space and produces smooth trajectories that converge rapidly to the prior (2nd row). In contrast, waypoint-diffusion models take a much longer time to produce trajectories that resemble the prior, as shown in the first row. (a) initial diffusion steps where GPD maintains smoothness while waypoint diffusers don't; (b) shows intermediate steps of diffusion where GPD already resembles the prior; (c) denotes the final diffusion steps.
• Figure 2: Architecture: GPD uses a parameterization module alongside a guided diffusion model. The diffusion model denoises the control points $\boldsymbol{\alpha}$ sampled from a Gaussian distribution and is guided by a cost function $J$. The parameterization module uses a Bernstein transform to convert a set of polynomial coefficients or control points to a trajectory in the waypoint space to compute the cost function, which is used to guide the denoising process. The right-most image illustrates the control points and trajectory generated by GPD.
• Figure 3: Guidance: GPD exhibits more effective guidance as compared to waypoint space diffusion. Row 1 shows guidance applied directly to the waypoints, where we see that just moving the individual waypoints out of collision might not lead to a final collision-free trajectory. Row 2 shows the enhanced effect of guidance in GPD for the same cost function, where the entire "string" of trajectory is moved out of collision, even when a single waypoint is in collision.
• Figure 4: Stitching: (a) Diverse batch of generated trajectories. (b) trajectory with the lowest cost. (c) pair of trajectories that can be stitched. (d) shows the final collision-free trajectory by stitching two different trajectories. The stitch is shown in yellow.
• Figure 5: Out-of-distribution scenes: GPD generalises to unseen object-in-hand scene without any additional training just by modifying our cost function directly at inference