Position tracking is a well-studied problem in Euclidean navigation, yet comparatively little attention has been given to navigation on general manifolds, where motion is governed by the underlying geometry of the space. In this work, we formalize the problem of manifold navigation as a velocity-driven tracking task and propose a coordinate-chart-based learning framework in which learning models operate entirely in a Euclidean representation. Predicted trajectories are mapped to the manifold via an immersion function prior to error computation, allowing geometric consistency to be evaluated without explicitly enforcing manifold constraints during learning. Within this framework, we focus on Gated Recurrent Units (GRUs) for sequential position prediction on curved manifolds after comparisons with alternative recurrent architectures. We evaluate our GRU models, trained with pertinent custom loss functions, against both a standard MSE loss function and a naive vector-addition baseline. We evaluate our approach on both analytically (and numerically) defined manifolds, as well as a surface reconstructed from a LiDAR point cloud. Our results show that recurrent models can learn to approximate manifold-constrained dynamics from velocity sequences and consistently outperform Euclidean baselines. Our results demonstrate that certain custom loss functions significantly improve long-horizon prediction accuracy compared to conventional point-wise loss functions.
Figures & tables
Strategy
Notation
Formula for jth term
Standard
Lk
j,j∈[1,k]
Exponential
exp_k
integer(nmaxj/(k+1))
Mixed linear
mixed_p_q Δ=⌈qnmax−p⌉
{j,p+Δ(j−p),j∈[1,p]j∈(p,p+q]
TABLE I : Strategies for Index Selection
Scheme
Indices
L50
1,…, 50
exp_10
1, 1, 2, 2, 4, 5, 8, 12, 17, 24, 35, 50
mixed_7_7
1, 2, 3, 4, 5, 6, 7, 14, 20, 26, 32, 38, 44, 50
TABLE II : The different loss functions and the corresponding indices aggregated.
Plane
Sphere
Torus
Loss Func.
Brownian
AR(1)
Brownian
AR(1)
Brownian
AR(1)
Baseline
– (–)
– (–)
0.00005 (–)
0.00200 (–)
0.00387 (–)
0.00742 (–)
L50
0.00118 (0.00109)
0.00054 (0.00037)
0.00007 (0.00002)
0.00018 (0.00030)
0.00161 (0.00106)
0.00055 (0.00038)
exp_10
0.00030 (0.00048)
0.00009 (0.00006)
0.00009 (0.00012)
0.00013 (0.00010)
0.00054 (0.00033)
0.00025 (0.00015)
mixed_7_7
0.00027 (0.00022)
0.00007 (0.00002)
0.00006(0.00003)∗
0.00013 (0.00012)
0.00029 (0.00018)
0.00046 (0.00018)
TABLE III : Prediction errors (MSE over all 100 timesteps) of GRUs trained with different loss functions. Values shown are averaged over 10 runs, with standard deviation in parentheses. Bold signifies the lowest MSE, while an asterisk next to the result indicates the lowest recurrent-based MSE.
Fig. 1 : Mean squared error at each time step in the test trajectories for GRU models with different loss functions averaged across 10 independently trained models of each respective type against the vector addition baseline model.
Setup
Brownian
AR(1)
Baseline
0.01268 (NA)
0.00225 (NA)
L50
0.00291 (0.00112)
0.00032 (0.00011)
exp_10
0.00276 (0.00102)
0.00032 (0.00014)
mixed_7_7
0.00271 (0.00095)
0.00031 (0.00011)
TABLE IV : Prediction MSE on the reconstructed surface
Fig. 2 : Predicton MSE for Brownian motion trajectories on the torus and the reconstructed surface with the 10th time step of the velocity sequence missing.
Fig. 3 : Mean squared error of predictions for Brownian motion trajectories on the torus and the reconstructed surface for different missing velocity portions.
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Plane
Sphere
Torus
Loss Func.
Model
Brownian
AR(1)
Brownian
AR(1)
Brownian
AR(1)
Baseline
– (–)
– (–)
0.00005 (–)
0.00200 (–)
0.00387 (–)
0.00742 (–)
L10
GRU
0.10376 (0.09999)
0.28024 (0.08625)
0.03291 (0.00406)
0.02627 (0.01133)
0.11175 (0.02301)
0.09435 (0.01313)
LSTM
0.54111 (0.32620)
0.06560 (0.02796)
0.00936 (0.00489)
0.01819 (0.00631)
0.14782 (0.05111)
0.51796 (0.13962)
RNN
2.26389 (6.02657)
0.00985 (0.02976)
0.00150 (0.00171)
0.00811 (0.00604)
0.08622 (0.16793)
0.10722 (0.27902)
L30
GRU
0.00574 (0.00411)
0.00608 (0.00385)
0.00149 (0.00041)
0.00019 (0.00017)
0.00334 (0.00155)
0.01392 (0.00225)
Appendix
TABLE V : Prediction error (MSE) for GRU, LSTM, and RNN with different versions of the standard loss function. The mean values and standard deviation (in parentheses) of the localization MSE are displayed.
Setup
Brownian
AR(1)
Baseline
0.01268 (NA)
0.00225 (NA)
GRU
0.00291 (0.00112)
0.00032 (0.00011)
LSTM
0.00264 (0.00094)
0.00036 (0.00010)
RNN
0.01611 (0.02645)
0.00112 (0.00034)
Appendix
TABLE VI : Prediction error (MSE) of recurrent models on the reconstructed surface with the L50 loss function.
Setup
Brownian
AR(1)
Baseline
0.01268 (NA)
0.00225 (NA)
GRU
0.00271 (0.00095)
0.00031 (0.00011)
LSTM
0.00295 (0.00098)
0.00036 (0.00011)
RNN
0.06543 (0.07541)
0.01220 (0.02937)
Appendix
TABLE VII : Prediction error (MSE) of recurrent models on reconstructed surface with the mixed_7_7 loss function.
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.
Diana Cuervo Espinosa, Mahathi Anand, Angela P. Schoellig
Chair of Robotics and System Intelligence, Technical University of Munich, Germany · Learning Systems and Robotics Lab, Technical University of Munich, Germany
We introduce Manifold Sobolev Informed Neural Optimization (MSINO), a curvature aware training framework for neural networks defined on Riemannian manifolds. The method replaces standard Euclidean derivative supervision with a covariant Sobolev loss that aligns gradients using parallel transport and improves stability via a Laplace Beltrami smoothness regularization term. Building on classical results in Riemannian optimization and Sobolev theory on manifolds, we derive geometry dependent constants that yield (i) a Descent Lemma with a manifold Sobolev smoothness constant, (ii) a Sobolev Polyak Lojasiewicz inequality giving linear convergence guarantees for Riemannian gradient descent and stochastic gradient descent under explicit step size bounds, and (iii) a two step Newton Sobolev method with local quadratic contraction in curvature controlled neighborhoods. Unlike prior Sobolev training in Euclidean space, MSINO provides training time guarantees that explicitly track curvature and transported Jacobians. Applications include surface imaging, physics informed learning settings, and robotics on Lie groups such as SO(3) and SE(3). The framework unifies value and gradient based learning with curvature aware convergence guarantees for neural training on manifolds.
A common way for trajectory planning is to leverage generative models trained on large collections of expert trajectories. At inference time, the model generates executable trajectories by conditioning on task goal constraints. However, trajectory-based methods rely on costly supervision, scale poorly with sequence length, and often generalize poorly to unseen constraints such as novel start-goal pairs. We propose an alternative to learn the underlying state-space manifold and use the geometry of the manifold for trajectory planning. This approach requires only state observations and enables generalization to unseen constraints by con- structing trajectories on the learned manifold of the state space. Experiments on Maze2D and robotic motion-planning benchmarks show that Ariadne constructs feasible paths from state-only supervision and generalizes to unseen start-goal combinations. On high-dimensional dual-arm planning, it remains competitive with trajectory-supervised and classical planners, while requiring no trajectory data for training.
Silong Yong, Anji Liu, Cunxi Dai +5
Carnegie Mellon University · National University of Singapore · DEVCOM Army Research Laboratory +1