A Unifying Variational Framework for Gaussian Process Motion Planning

TL;DR

This work tackles motion planning under uncertainty in high-dimensional robotic systems by introducing vGPMP, a variational Gaussian-process framework that unifies probabilistic-inference-based and optimization-based planners. By representing trajectories with Gaussian-process priors and optimizing a variational ELBO over inducing variables, the method supports hard, soft, and inequality constraints while yielding uncertainty through interval and Monte Carlo estimates. The approach hinges on sparse GPs and a flexible likelihood design to encode joint limits, collision avoidance, and task-specific objectives, enabling end-to-end training and efficient computation. Empirical results across multiple robots and environments show competitive success rates and improved safety via higher clearance, with a demonstrable real-robot execution, highlighting the practical impact of integrating uncertainty into motion planning.

Abstract

To control how a robot moves, motion planning algorithms must compute paths in high-dimensional state spaces while accounting for physical constraints related to motors and joints, generating smooth and stable motions, avoiding obstacles, and preventing collisions. A motion planning algorithm must therefore balance competing demands, and should ideally incorporate uncertainty to handle noise, model errors, and facilitate deployment in complex environments. To address these issues, we introduce a framework for robot motion planning based on variational Gaussian processes, which unifies and generalizes various probabilistic-inference-based motion planning algorithms, and connects them with optimization-based planners. Our framework provides a principled and flexible way to incorporate equality-based, inequality-based, and soft motion-planning constraints during end-to-end training, is straightforward to implement, and provides both interval-based and Monte-Carlo-based uncertainty estimates. We conduct experiments using different environments and robots, comparing against baseline approaches based on the feasibility of the planned paths, and obstacle avoidance quality. Results show that our proposed approach yields a good balance between success rates and path quality.

Figures (8)

• Figure 1: Intuitive illustration of representing motion paths with GPs. (\ref{['fig:sparse gp']}) Sparse GP posterior for 1 joint with inducing variables $\v{u} = [ \v{u}_{\f{c}}, \v{u}']$ denoted by ‘$\circ$’, where $\v{u}_{\f{c}} = \left[\v{\theta}_0, \v{\theta}_1\right]$ and $\v{u}'$ represent the $M=4$ inducing locations $\v{z} \in \c{T}$ subject to the learning process, shown in-between $t=0$ and $t=1$. We interpret the inducing points as waypoints through which the random motion plans travel. (\ref{['fig:motion plans']}) Candidate end-effector trajectories for the Franka Panda robot and associated uncertainty, which is illustrated as the orange shaded region.
• Figure 2: A sampled trajectory for grasping a can. Frames 1-2, 2-3, 3-4 depict respectively the end-effector alignment with the final pose, the approach stage, and the grasping stage.
• Figure 3: Illustration of smoothness properties of different kernels, including the nowhere-differentiable Matérn-$1/2$, once-differentiable Matérn-$3/2$, and twice-differentiable Matérn-$5/2$ kernel.
• Figure 4: Example distribution of collision-free paths generated by vGPMP in the boxes environment, with a random sample shown in green.
• Figure 5: Sampled trajectory executions on the real (bottom) and the simulated (top) robot showing successful transfer using a low-level controller.