geodex: A Library for Motion Planning on Riemannian Manifolds

Phone Thiha Kyaw, Ben Wei, Sepehr Samavi, Miguel Angel Rogel Garcia, Jonathan Kelly · N/A · 2026

Summary

Abstract Summary

Planning motions that respect the intrinsic geometry of a robot's configuration space, including its curvature and a configuration-dependent notion of cost, yields shorter, lower-energy, and more natural trajectories than planning under the ambient flat metric. Existing libraries for optimization on manifolds provide rich geometric primitives but do not plan around obstacles. While general-purpose motion planning libraries support many state spaces and custom distance functions, they do not yet treat a configuration-dependent Riemannian metric as the geometry that drives distance, interpolation, and geodesics. We present geodex, an open-source C++20 library with Python bindings. The library exposes the manifold, its Riemannian metric, the retraction, and the sampler as independent, interchangeable components through a single sampling-based motion planning interface. The same planner runs unchanged on canonical spaces $\mathbb{R}^n$, $\mathbb{T}^n$, $\mathbb{S}^n$, matrix Lie groups such as $SO(2)$, $SE(2)$, $SO(3)$, and $SE(3)$, products of these spaces, and articulated-robot configuration spaces, each equipped with a user-defined Riemannian metric. We make geodex publicly available with documentation, tests, and a reproducible benchmark suite.

Key Points

Planning motions that respect the intrinsic geometry of a robot's configuration space, including its curvature and a configuration-dependent notion of cost, yields shorter, lower-energy, and more natural trajectories than planning under the ambient flat metric.
Existing libraries for optimization on manifolds provide rich geometric primitives but do not plan around obstacles.
While general-purpose motion planning libraries support many state spaces and custom distance functions, they do not yet treat a configuration-dependent Riemannian metric as the geometry that drives distance, interpolation, and geodesics.
We present geodex, an open-source C++20 library with Python bindings
The library exposes the manifold, its Riemannian metric, the retraction, and the sampler as independent, interchangeable components through a single sampling-based motion planning interface.

Design Goals and System Overview

Motion planning algorithms typically rely on several subroutines, such as distance functions or a cost functional, local steering methods, and nearest-neighbor queries to find collision-free paths. On a Riemannian manifold, these components must all be consistent with the underlying geometry.

In most planning libraries, however, they are implemented separately and assume a fixed metric, most often the ambient Euclidean one, so adapting a planner to a new geometry means reimplementing each component by hand.

Design Goals and System Overview — geodex: A Library for Motion Planning on Riemannian Manifolds — Fig. 3 : Low-discrepancy sampling on manifolds. Pseudo-random points on the unit square clump and leave gaps (left), while a Halton sequence of the same size spreads evenly (center). The measure-preserving from_unit_cube map then carries these same points onto the sphere, shown here as the equal-area map from the unit square to $\mathbb{S}^{2}$ (right), so the points that are uniform in the square remain uniform on the manifold.

Architecture

The design goals of Section III demand a single planner that runs under any Riemannian metric at close to the speed of Euclidean planning. geodex achieves this by expressing every manifold in terms of one base topology and three interchangeable policies (metric, retraction, and sampler) that a single generic planner consumes through one compile-time interface.

Evaluation

We first apply geodex to a downstream manipulation task. We perform shelving with a Franka arm using the MoveIt 2 plugin of geodex , and evaluate plans using two different metrics without touching the underlying planner or the rest of the planning stack. Under the Euclidean metric, all directions of motion have the same cost, so the planner returns the shortest motion in joint space.

Instead, we could also measure distance differently and plan the same task under a different metric.

Evaluation — geodex: A Library for Motion Planning on Riemannian Manifolds — Fig. 5 : Map (roughly $1,600~\mathrm{m}^{2}$) used for nonholonomic navigation on $\mathrm{SE}(2)$. White areas are traversable, and gray indicate obstacles. We also illustrate areas traversed by the robot in the real-robot experiments (Sec. V-B2 ). The narrow areas are inside a cluttered office space, and the far areas a publicly accessible atrium space in a university building.

Sources and Demonstrations

Figures are reproduced from Kyaw et al. See the full paper for experimental details and the project page, when the paper links one, for demonstrations and videos.