Curve-Induced Dynamical Systems on Riemannian Manifolds and Lie Groups
Authors: Saray Bakker, Martin Schonger, Tobias Löw, Javier Alonso-Mora, Sylvain Calinon
Organizations: Delft University of Technology, The Netherlands · Idiap Research Institute, Switzerland · École Polytechnique Fédérale de Lausanne (EPFL), Switzerland
Deploying robots in household environments requires safe, adaptable, and interpretable behaviors that respect the geometric structure of tasks. Often represented on Lie groups and Riemannian manifolds, this includes poses on SE(3) or symmetric positive definite matrices encoding stiffness or damping matrices. In this context, dynamical system-based approaches offer a natural framework for generating such behavior, providing stability and convergence while remaining responsive to changes in the environment. We introduce Curve-induced Dynamical systems on Smooth Manifolds (CDSM), a real-time framework for constructing dynamical systems directly on Riemannian manifolds and Lie groups. The proposed approach constructs a nominal curve on the manifold, and generates a dynamical system which combines a tangential component that drives motion along the curve and a normal component that attracts the state toward the curve. We provide a stability analysis of the resulting dynamical system and validate the method quantitatively. On an S2 benchmark, CDSM demonstrates improved trajectory accuracy, reduced path deviation, and faster generation and query times compared to state-of-the-art methods. Finally, we demonstrate the practical applicability of the framework on both a robotic manipulator, where poses on SE(3) and damping matrices on SPD(n) are adapted online, and a mobile manipulator.
Figures & tables
Fig. 1 : We consider a dressing task in which the robot adapts both its pose and the damping matrix of an impedance controller throughout the motion. External perturbations are applied by physically pushing the robot, and we observe that the proposed dsmpman converges back to the desired nominal trajectory, given the tracked configuration of the human arm via a camera. The illustration in the bottom-left corner was generated using Google Gemini.
Fig. 2 : Curve on S2 with an illustration of the curve segment definitions and control points in multiple tangent spaces. The top left shows the segment base points on the manifold, as well as the reference data in light blue. Here, the curve is encoded with only 3 segments for better visualization of the procedure.
Fig. 3 : (a) Distance field for a curve on S2 with examples of projections and geodesic distances on the manifold. (b) A trajectory of the ds converging towards the curve with the tangential (TC) and normal (NC) term of the ds in the tangent space at the query point. The velocities are scaled for the illustration.
Fig. 4 : Spline on S++2 , which corresponds to the S++2 -spline used in Fig. 5 , obtained from the covariance. The axes indicate the diagonal ( A1,1 and A1,2 ) and off-diagonal ( A1,2 ) terms of the 2×2 matrices.
Fig. 5 : Vector fields of a dynamical system in R2 without (blue) and with (pink) variable transformation matrices applied to the velocities, estimated from the reference data covariance. The velocity vectors not only change length but also direction, highlighting the presence of non-zero off-diagonal elements in the transformation matrices. To not affect the nominal behavior on the curve, we interpolate the transformation matrices toward identity as we get closer to the curve. The dashed lines are forward simulations of the unmodified and transformed velocities.
Trajectory distance
Path distance ( x0 randomized)
Success rate
Computation time trajectory [s]
Train/Fitting time [s]
dsmpman (ours)
0.0042 ± 0.0087
0.0339 ± 0.1292
1.0 0
0.0039 ± 0.0006
234 1.26 ± 1 0.16
Lieflows
0.0131 ± 0.0072
0.0869 ± 0.5574
0.81
0.5157 ± 0.0281
1324.85 ± 1 8.47
PUMA
0.0112 ± 0.0044
0.0982 ± 0.2092
1.0 0
0.0562 ± 0.0111
1 204.03 ± 10.63
TABLE I : Quantitative results on the LASA dataset mapped to the S2 -manifold.
Fig. 7 : Reference data, curve, and DS rollout on SO(3) , plotted over time, with kTC=1 , kNC=2 , and kg=8 . Note the damped motion near the goal due to ( 18 ).
Fig. 8 : The effect of an optimization of the phase modulation layer with an existing spatial curve in S2 where constraints ensure that the speed, ∥v(t)∥ , is below the limit vˉ=3.0 , and (γs)′(t)≥0 .
Fig. 9 : Setup of the arm dressing experiment. The RealSense camera is positioned near the viewpoint of this photo.
Fig. 10 : Various snapshots of the scene during the dressing task. Frames A and F show the beginning and end of the first of two repeated runs, respectively. The other frames show different kinds of perturbations: B push laterally, C push up, D twist counter clockwise, E tilt forward.
Fig. 11 : A snapshot of the curve for a tracked steady arm (dark violet) with two trajectories of the robot when not pushed (pink). We also show some perturbations (gray) and the behavior of the robot after the perturbations (light blue). The world frame is aligned with the robot base frame. Due to the human arm not being perfectly still, the curve moved a little throughout the experiment, and hence the illustrated curve is only a snapshot.
Fig. 12 : Robot moving along a static arm with perturbations to position and orientation, repeated twice. The experiment is performed with the damping varying linearly with the phase along the nominal curve from 0.5 to 2. The resulting twist outputs of the DS are plotted in pink, the phase in gray and the twists when using a fixed damping are in blue.
Fig. 13 : Resulting torque values of the twists applied in Fig. 12 over time, with the phase in gray. The torque commands from ( 38 ) with Dc=diag(90,90,90,4,4,4) are shown in pink, while the torque commands with a fixed damping are in blue.
Fig. 14 : The dressing task executed by a mobile manipulator consisting of a Kinova Gen3Lite mounted on a Clearpath Dingo base. Frames A and F indicate the start and end-pose, respectively.
Dynamical systems studies how states evolve over time, including stability, feedback, chaos, and transitions in physical, biological, and engineered systems.
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 propose an architecture for learning the dynamics of mechanical systems based on discrete forced Euler-Lagrange equations on Lie groups using only position data. By formulating the dynamics directly on manifold-valued configuration spaces, the method naturally respects the geometric structure of the systems and preserves geometric invariants and conservation laws. The reliance on position measurements alone makes the framework applicable in settings where velocity data are unavailable or noisy. The approach extends naturally to multibody systems, accommodates external control inputs, and demonstrates strong performance on both synthetic and real-world datasets.
Martine Dyring Hansen, Marta Ghirardelli, Elena Celledoni +2
Department of Mathematics and Cybernetics, SINTEF Digital Forskningsveien 1, 0373 Oslo, Norway · Department of Mathematical Sciences Norwegian University of Science and Technology (NTNU) Alfred Getz’ vei 1, 7034 Trondheim, Norway · Instituto de Ciencias Matematicas (CSIC) Calle Nicolás Cabrera 13-15, Cantoblanco, 28048 Madrid, Spain
Diffusion Policy (DP) is a powerful Learning from Demonstration (LfD) method for robotic manipulation, yet it suffers from discontinuous and non-smooth trajectories. Spline-based action representations promote smooth motion within individual action chunks, but existing spline-based methods neither guarantee cross-chunk C2 continuity nor account for the group structure of SE(3). We therefore propose LieSpline-DP, a Lie-group B-spline diffusion policy that generates end-effector trajectories directly on SE(3) and couples consecutive plans by sharing their boundary control poses, ensuring C2 continuity throughout the entire planned trajectory. Across three real-robot tasks, LieSpline-DP produces lower trajectory jerk and higher task success rates than the DP baseline. The gains are particularly pronounced in real-world tasks involving liquids and flexible objects: in our real-robot experiments, LieSpline-DP achieved a 100% success rate on both pouring and bucket hooking, whereas the DP baseline achieved only 10% and 30%, respectively.