Learning from Demonstrations Over Riemannian Manifolds Using Neural ODEs

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. I. MOTIVATION Traditional methods for robot motion planning are usually hard coded for specific tasks and require extensive coding expertise. This makes it difficult to adapt to new tasks, and significant time and effort is required to identify relevant goal locations or sequential waypoints. Learning from demonstra- tions seeks to enhance generalization by collecting demon- strations from human experts and directly converting them to robot motion in an adaptive manner without requiring significant programming experience [1], [2]. However, they are usually limited to generating motion in the Euclidean task space. On the other hand, robot orientations, which evolve over curved manifolds, cannot be represented accurately with a purely Euclidean framework, and any extension to capturing full end-effector data requires careful enforcement of geometric constraints [3]. Recently, a Riemannian perspective to LfD has emerged [4]. Here, demonstrations are encoded into a curved latent space, i.e. a Riemannian manifold, where full end-effector poses can be described naturally. Then, robot motion may be learned directly via geodesics, i.e., the shortest paths between two arbitrary points. However, finding geodesics is a challenging problem, as it requires solving complex, second order differential equations. Several numerical relaxations, including iterative schemes [5] and graph-based methods [4] are common. Nevertheless, they are either computationally intensive, or rely on discrete approximations making it tedious to *This work is supported in part by Robotics Institute Germany, funded by BMFTR grant 16ME0997K 1Chair of Robotics and System Intelligence, Technical University of Munich, Germany diana.cuervo@tum.de 2Learning Systems and Robotics Lab, Technical University of Munich, Germany {mahathi.anand, angela.schoellig}@tum.de Position and Orientation Data Input Phase 1: Learning Riemannian Manifolds from Demonstrations using Variational Autoencoders Phase 2: Proxy-Geodesic Learning for Computation of Robot Motion Encoded Demonstrations Riemannian Metric Reconstructed Input Encoder Decoder Low-Dimensional Latent Space (Riemannian Manifold) Neural ODE Proxy Geodesic Learning Geodesic Path Robot Motion Fig. 1: The architecture for learning from demonstrations over Riemannian manifolds. interpolate motion between two unseen data points. In this work, we propose an adaptive motion generation pipeline over Riemannian manifolds by utilizing variational autoencoders for encoding demonstrations into manifolds and neural ordinary differential equations (NODE) for geodesic computation. The NODE can quickly generate a motion path between any two locations in the manifold at inference time, making our approach very attractive for adaptive generalization and for hybrid, sequential tasks. II. ARCHITECTURE The proposed Riemannian motion generation framework is decoupled into two phases as described in Fig. 1 – the learning of robot’s spatial constraints via the Riemannian manifold, and solving for the robot’s temporal dynamics via geodesic paths in the learned manifold. The first phase is similar to the one proposed in [4], where a variational autoencoder (VAE) takes the demonstration data x ∈X as input and encodes it into a lower-dimensional latent variable z ∈Z. In particular, the variational autoencoder consists of an encoder with parameters φ : X →Z, responsible for approximating the posterior distribution pφ(z | x), and a decoder with parameters θ : Z →X that approximates the generative distribution pθ(x | z). It is trained under the standard evidence lower bound (ELBO) loss that consists of a regularization term as well as a reconstruction term. The resulting decoder function θ is then used to construct a local Riemannian metric given by M(z) = JT θ (z)Jθ(z), ∀z ∈ Z, which characterizes the length and energy of a curve z : [0, 1] →Z as lz = R 1 0 p ̇z(t)T M(z(t)) ̇z(t)dt, and Ez = 1 2 R 1 0 ̇z(t)T M(z(t)) ̇z(t)dt, respectively. As a result, finding the geodesic, i.e., the shortest path between any two points in the manifold is reduced to finding a path that minimizes the energy (or length), respectively. ICRA2026 Late Breaking Results Poster presented at 2026 IEEE International Conference on Robotics and Automation (ICRA 2026) June 1-5, 2026. Vienna, Austria