cs.ROApr 11, 2026

A Coordinate-Invariant Local Representation of Motion and Force Trajectories for Identification and Generalization Across Coordinate Systems

Authors: Arno Verduyn, Erwin Aertbeliën, Maxim Vochten, Joris De Schutter

Organizations: Department of Mechanical Engineering and Flanders Make at KU Leuven, 3001 Leuven, Belgium. · Department of Mechanics, Royal Military Academy, 1000 Brussels, Belgium.

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Abstract

Identifying the trajectories of rigid bodies and of interaction forces is essential for a wide range of tasks in robotics, biomechanics, and related domains. These tasks include trajectory segmentation, recognition, and prediction. For these tasks, a key challenge lies in achieving consistent results when the trajectory is expressed in different coordinate systems. A way to address this challenge is to utilize trajectory models that can generalize across coordinate systems. The focus of this paper is on such trajectory models obtained by transforming the trajectory into a coordinate-invariant representation. However, coordinate-invariant representations often suffer from sensitivity to measurement noise and the manifestation of singularities in the representation, where the representation is not uniquely defined. This paper aims to address this limitation by introducing the novel Dual-Upper-Triangular Invariant Representation (DUTIR), with improved robustness to singularities, along with its computational algorithm. The proposed representation is formulated at a level of abstraction that makes it applicable to both rigid-body trajectories and interaction-force trajectories, hence making it a versatile tool for robotics, biomechanics, and related domains.

Figures & tables

Figure 1: Geometric interpretation of p\boldsymbol{p} and R=[r1 r2 r3]\boldsymbol{R}=[\boldsymbol{r}_{1}~\boldsymbol{r}_{2}~\boldsymbol{r}_{3}] , which are the components of the screw-transformation matrix S\boldsymbol{S} in the SUSU -decomposition of Ξ=[ξ[xi−1] ξ[xi] ξ[xi+1]]\boldsymbol{\Xi}=[\boldsymbol{\xi}[x_{i-1}]~\boldsymbol{\xi}[x_{i}]~\boldsymbol{\xi}[x_{i+1}]] . The screw axes of ξ[xi−1]\boldsymbol{\xi}[x_{i-1}] and ξ[xi]\boldsymbol{\xi}[x_{i}] are shown as solid black lines. The common normal of the two screw axes is shown as a dotted black line.
Figure 2: Two-dimensional illustration of the regularization of p∗\boldsymbol{p}^{*} , showing the coordinate system of Ξ\boldsymbol{\Xi} (green), the spherical manifold defined by LL (black circle), the screw axis of ξ[xi−1]\boldsymbol{\xi}[x_{i-1}] (black line), and the functional coordinate system defined by p∗\boldsymbol{p}^{*} , r1\boldsymbol{r}_{1} , and r2\boldsymbol{r}_{2} (red). The solution for p^∗\hat{\boldsymbol{p}}^{*} is illustrated for (a) the case where the screw axis intersects the spherical manifold, i.e., when (py∗)2+(pz∗)2<L2(p^{*}_{y})^{2}+(p^{*}_{z})^{2}<L^{2} , and (b) the case where the screw axis does not intersect the spherical manifold, i.e., when (py∗)2+(pz∗)2>L2(p^{*}_{y})^{2}+(p^{*}_{z})^{2}>L^{2} .
Figure 3: Numerical example of the SUSU -decomposition applied to rigid-body trajectory data. (a) Visualization of the trajectory for two different choices (red and blue) of the coordinate system attached to the body; (b) Twist coordinates extracted from the trajectory via numerical differentiation using the logarithmic map [ 10 ] ; (c) Components of the invariant representation U\boldsymbol{U} obtained without regularization. In this case, the lower-diagonal entries εij\varepsilon_{ij} of U\boldsymbol{U} are exactly zero; (d) Components of U^\hat{\boldsymbol{U}} obtained with regularization. In this case, the lower-diagonal entries εij\varepsilon_{ij} of U\boldsymbol{U} are approximately zero.
Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure 4: Geometric relations between α[xi−1]\boldsymbol{\alpha}[x_{i-1}] , α[xi]\boldsymbol{\alpha}[x_{i}] , u11u_{11} , u12u_{12} , and u22u_{22} .
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