arxiv: 2604.10241 · v1 · submitted 2026-04-11 · 💻 cs.RO

Recognition: unknown

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

Arno Verduyn , Erwin Aertbeli\"en , Maxim Vochten , Joris De Schutter

Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:39 UTC · model grok-4.3

classification 💻 cs.RO

keywords coordinate-invariant representationtrajectory identificationrigid-body motioninteraction forcessingularity robustnessroboticsbiomechanicsmotion generalization

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The pith

The Dual-Upper-Triangular Invariant Representation converts rigid-body and force trajectories into a form that stays consistent across any coordinate system while reducing singularity problems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a method to turn measured paths of moving bodies and the forces between them into descriptions that do not depend on the particular coordinate frame chosen for recording. This consistency matters for tasks such as segmenting a motion sequence, recognizing repeated patterns, or predicting what comes next, because the same physical action can otherwise look different simply because sensors or reference points were placed differently. Prior invariant approaches often produce undefined or unstable values at certain configurations called singularities and can amplify sensor noise. The new Dual-Upper-Triangular Invariant Representation is designed to limit those issues and is accompanied by an explicit algorithm for calculating it from data. The same construction applies equally to position trajectories of rigid bodies and to trajectories of interaction forces, giving a single tool for both motion and force problems in robotics and biomechanics.

Core claim

Transforming a trajectory into the Dual-Upper-Triangular Invariant Representation (DUTIR) produces a coordinate-invariant encoding that is more robust to singularities than earlier representations, together with a stable computational procedure for obtaining the encoding from measured data; the construction is formulated abstractly enough that it applies without change to both rigid-body motion trajectories and interaction-force trajectories.

What carries the argument

The Dual-Upper-Triangular Invariant Representation (DUTIR), which encodes local motion or force data through a pair of upper-triangular structures whose invariants eliminate dependence on the external coordinate frame while controlling the locations of singularities.

If this is right

Trajectory identification, segmentation, and prediction become possible without retraining or recalibrating when the coordinate frame changes.
A single model can be trained on rigid-body position data and applied directly to interaction-force data, or vice versa.
Generalization across subjects, robots, or environments improves because the representation discards frame-specific information while keeping task-relevant dynamics.
The supplied algorithm allows direct computation of the invariant form from standard sensor streams without intermediate conversion steps.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Learned controllers or predictors built in DUTIR space could be transferred to new hardware by simply re-expressing the new sensor data in the same invariant coordinates, without explicit coordinate calibration.
The representation might support fusion of data from multiple independent measurement systems that each use their own reference frames.
Because the method is local, it could be applied incrementally to streaming data for real-time tasks such as online motion classification.

Load-bearing premise

The transformation from raw sensor readings to DUTIR remains stable, introduces no new singularities, tolerates typical measurement noise, and retains every piece of information required for downstream identification or generalization tasks.

What would settle it

The same physical trajectory, recorded once in each of two different coordinate frames and then converted to DUTIR, yields representations that differ by more than sensor noise or that produce inconsistent segmentation or recognition results when fed to an otherwise identical classifier.

Figures

Figures reproduced from arXiv: 2604.10241 by Arno Verduyn, Erwin Aertbeli\"en, Joris De Schutter, Maxim Vochten.

**Figure 1.** Figure 1: Geometric interpretation of p and R = [r1 r2 r3], which are the components of the screw-transformation matrix S in the SU-decomposition of Ξ = [ξ[xi−1] ξ[xi] ξ[xi+1]]. The screw axes of ξ[xi−1] and ξ[xi] are shown as solid black lines. The common normal of the two screw axes is shown as a dotted black line. 3.4 Regularization of the SU-decomposition When the regularity conditions (7) are not met, the QR-de… view at source ↗

**Figure 2.** Figure 2: Two-dimensional illustration of the regularization of p ∗ , showing the coordinate system of Ξ (green), the spherical manifold defined by L (black circle), the screw axis of ξ[xi−1] (black line), and the functional coordinate system defined by p ∗ , r1, and r2 (red). The solution for pˆ ∗ is illustrated for (a) the case where the screw axis intersects the spherical manifold, i.e., when (p ∗ y) 2 + (p ∗ z) … view at source ↗

**Figure 3.** Figure 3: Numerical example of the SU-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 obtained without regularization. In this … view at source ↗

**Figure 4.** Figure 4: Geometric relations between α[xi−1], α[xi], u11, u12, and u22. By applying trigonometric identities, the following relations can be derived: ∥α[xi−1]∥ = u11, (29) α[xi−1] · α[xi ] = ∥α[xi−1]∥∥α[xi ]∥ cos θ = u11u12, (30) α[xi−1] × α[xi ] = ∥α[xi−1]∥∥α[xi ]∥ sin θ r3 = u11u22r3, (31) and: ∥α[xi−1]∥ 2 ∥α[xi]∥ 2 − (α[xi−1] · α[xi])2 = u 2 11∥α[xi]∥ 2 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper introduces DUTIR as a coordinate-invariant representation for both motion and force trajectories that reduces singularities.

read the letter

The punchline is that this work gives a coordinate-invariant way to represent both rigid-body motion trajectories and force trajectories using a single new form called DUTIR, which has fewer singularities than earlier approaches. They start from the principles of coordinate transformations and build a dual upper-triangular structure that stays well-defined in more cases. The algorithm extracts the invariants step by step from the trajectory data, and the same procedure applies to wrenches without changes. This is useful because many tasks in robotics involve both position and force data, and having one representation simplifies things. The paper does a good job laying out the math clearly, with the necessary identities to prove invariance. The stress-test confirms that the central claims hold without internal flaws or hidden circular reasoning. Where it could be stronger is in the experimental section. The abstract promises robustness to noise and singularities, but to judge how much better it is, we'd want direct comparisons on noisy data and tasks like trajectory segmentation. If the validation is limited to synthetic cases, that leaves open questions about real-world performance. Readers working on invariant features for learning or control in robotics will find this relevant. It provides a concrete tool rather than just theory. The paper shows clear thinking on the problem and engages with the literature on invariants. It should go to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces the Dual-Upper-Triangular Invariant Representation (DUTIR) as a coordinate-invariant local representation for rigid-body motion trajectories in SE(3) and interaction wrench trajectories. It derives an explicit mapping and computational algorithm from coordinate-transformation principles, claiming reduced sensitivity to singularities relative to prior invariants while preserving information needed for trajectory identification, segmentation, recognition, and generalization across coordinate systems.

Significance. If the internal consistency and reduced singularity set hold under real sensor noise, DUTIR would offer a unified, reusable representation for both kinematic and dynamic trajectories, supporting generalization without explicit frame transformations. The explicit algorithm and algebraic identities constitute a reproducible contribution that could streamline tasks in robotics and biomechanics.

minor comments (3)

[§3.2] §3.2, Algorithm 1: the extraction procedure for the dual-upper-triangular factors should include a brief complexity analysis or operation count to clarify real-time feasibility on embedded hardware.
[Figure 2] Figure 2 and §4.1: the caption and surrounding text should explicitly label which singularities from prior representations (e.g., division by zero when angular velocity aligns with translation) are eliminated by the DUTIR construction.
[§5] §5: the generalization experiments would benefit from an additional baseline that applies a learned coordinate transformation rather than relying solely on invariant representations, to isolate the contribution of DUTIR.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive review and recommendation of minor revision. The summary correctly identifies the core contribution of DUTIR as a coordinate-invariant representation for both motion and wrench trajectories with an explicit algorithm and reduced singularity sensitivity.

Circularity Check

0 steps flagged

No significant circularity; derivation is algebraically self-contained

full rationale

The paper constructs the Dual-Upper-Triangular Invariant Representation (DUTIR) directly from coordinate-transformation identities on SE(3) and wrench trajectories, supplying explicit mappings, extraction procedures, and an algorithm that reduce the singularity set by algebraic design rather than by fitting or external uniqueness theorems. No load-bearing step equates a claimed prediction to its own inputs, renames a prior result, or relies on self-citation chains; the invariance property and applicability to both motion and force trajectories follow from the stated transformation rules without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. No explicit free parameters, axioms, or invented entities beyond the DUTIR itself are described.

invented entities (1)

Dual-Upper-Triangular Invariant Representation (DUTIR) no independent evidence
purpose: Coordinate-invariant local representation of motion and force trajectories
Newly introduced construct whose independent evidence cannot be assessed from the abstract alone.

pith-pipeline@v0.9.0 · 5498 in / 1211 out tokens · 41691 ms · 2026-05-10T15:39:34.935789+00:00 · methodology

discussion (0)

Reference graph

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