geodex: A Library for Motion Planning on Riemannian Manifolds

We present Riemannian Informed Trees (RIT*), a planning framework that replaces Euclidean primitives in batch-informed search with their Riemannian counterparts. RIT* constructs a tighter, cost-consistent informed set, performs a nearest-neighbour search under an anisotropic distance metric, and evaluates edge costs efficiently via a cascading scheme. We further introduce a Collision-Adaptive Metric Refinement (CARM), which learns an obstacle-proximity cost field online from collision feedback, reducing the reliance on prior metric design in practical settings. Experiments across environments from 2-D to 14-D show that RIT* is competitive in low-dimensional and spatially constant-metric settings and produces substantially lower-cost solutions when the metric varies spatially in high-dimensional configuration spaces. Performance gains scale with anisotropy and dimension, reaching up to 13.0% improvement in median initial cost over BIT* in the 3-D anisotropic benchmark, up to 9.0% in median final cost over BIT* in 6-DOF manipulation, and 24.8-63.5% in a 14-DOF bimanual planning problem, where Euclidean-informed baselines degrade. Videos and code can be found here: https://muhayyuddin.github.io/ritstar/

In constrained motion planning problems, task and loop-closure constraints restrict a robot's motion to a curved, lower-dimensional submanifold of its configuration space. Planners measure path length with a metric, which sets the cost of moving in each direction. Under the Euclidean metric, this cost is the same everywhere, whereas under a general Riemannian metric, such as the kinetic-energy metric, the cost can vary with direction and configuration. Existing methods often describe the submanifold either implicitly, as a constraint level set, or explicitly, through a parameterization. The implicit representation is typically combined with the Euclidean metric of the configuration space, and the explicit representation with the parameter domain, so the path length that a planner minimizes depends on the representation. Instead, we measure path length with the induced metric, which the submanifold inherits from a Riemannian metric on the configuration space. The implicit and explicit representations yield the same induced metric, expressed in different coordinates, and hence the same geometry. This result holds for any Riemannian metric on the configuration space, not only the Euclidean one. The choice of metric is therefore independent of the choice of representation. Using this result, we extend planning under a Riemannian metric from unconstrained spaces to constraint submanifolds by applying the induced metric in both a sampling-based planner and a trajectory optimizer. For an explicit representation, the induced metric also accounts for the distortion that the parameterization introduces. In experiments on a bimanual manipulation setup with two Franka arms under end-effector task constraints, we compare the Euclidean and kinetic-energy metrics.

Learning from demonstratins (LfD) is usually performed over Euclidean spaces, while the robot state, e.g. orientation, naturally evolves over curved spaces. Therefore, to ensure natural, complex motion generation, we investigate learning from demonstrations over Riemannian manifolds that are capable of encoding both position and orientation data. Here, geodesic paths provide for natural motion between two arbitrary points within the manifold. We propose to numerically estimate geodesics via neural ordinary differential equations, mitigating large computational overhead of existing approaches. Finally, these geodesics can be decoded back into the original task space before deploying on the robot. In this extended abstract, we discuss the architecture of our framework, provide some initial insights from our simulation experiments, including comparison to other geodesic computation mechanisms, and discuss the challenges and prospects for future work.