Curve-Induced Dynamical Systems on Riemannian Manifolds and Lie Groups

TL;DR

This work introduces Curve-induced Dynamical systems on Smooth Manifolds (CDSM), a real-time framework for constructing dynamical systems directly on Riemannian manifolds and Lie groups and demonstrates improved trajectory accuracy, reduced path deviation, and faster generation and query times compared to state-of-the-art methods.

Abstract

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 (14)

• Figure 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.
• Figure 2: Curve on $S^2$ 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.
• Figure 3: (a) Distance field for a curve on $S^2$ 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.
• Figure 4: Spline on $\mathcal{S}_{++}^{2}$, which corresponds to the $\mathcal{S}_{++}^{2}$-spline used in Fig. \ref{['fig:variable_spd_2D_rn2_vectorfield']}, obtained from the covariance. The axes indicate the diagonal ($A_{1,1}$ and $A_{1,2}$) and off-diagonal ($A_{1,2}$) terms of the $2\times 2$ matrices.
• Figure 5: Vector fields of a dynamical system in $\mathbb{R}^2$ 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.

Theorems & Definitions (2)

• Definition 4.1: Closest Point Projection
• Definition 4.2: Dynamical System