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.
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 and R=[r1r2r3] , 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.
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 p^∗ is illustrated for (a) the case where the screw axis intersects the spherical manifold, i.e., when (py∗)2+(pz∗)2<L2 , and (b) the case where the screw axis does not intersect the spherical manifold, i.e., when (py∗)2+(pz∗)2>L2 .
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 case, the lower-diagonal entries εij of U are exactly zero; (d) Components of U^ obtained with regularization. In this case, the lower-diagonal entries εij of U are approximately zero.
Appendix figures & tables1 asset
Supplementary material from the paper’s appendix.
Appendix
Figure 4: Geometric relations between α[xi−1] , α[xi] , u11 , u12 , and u22 .
Robot model identification is commonly performed by least-squares regression on inverse dynamics, but existing formulations measure residuals directly in coordinate force space and therefore depend on the chosen coordinate chart, units, and scaling. This paper proposes a coordinate-independent identification method that weights inverse-dynamics residuals by the dual metric induced by the system Riemannian metric. Using the force--velocity vector--covector duality, the dual metric provides a physically meaningful normalization of generalized forces, pulling coordinate residuals back into the ambient mechanical space and eliminating coordinate-induced bias. The resulting objective remains convex through an affine-metric and Schur-complement reformulation, and is compatible with physical-consistency constraints and geometric regularization. Experiments on an inertia-dominated Crazyflie--pendulum system and a drag-dominated LandSalp robot show improved identification accuracy, especially on shape coordinates, on both downsampled and full datasets.
Yanhao Yang, Ross L. Hatton
Collaborative Robotics and Intelligent Systems (CoRIS) Institute at Oregon State University, Corvallis, OR USA
Dynamic feedback linearization (DFL) is a classical technique for trajectory tracking of unicycle-type mobile robots, but the resulting DFL-based controller becomes singular when the linear velocity vanishes, rendering standard DFL-based controllers unsuitable for stop-and-reverse maneuvers. This paper proposes a quadratic-program (QP)-based optimal control framework that avoids this singularity, while establishing local Lipschitz continuity of the resulting feedback law. Our approach reformulates the DFL constraints as an equality-constrained QP with a slack variable, ensuring feasibility for all states and reference signals, including at points where the robot's velocity vanishes. By introducing slack variables and tunable parameters, we demonstrate that the singular configuration can be avoided for a large class of reference trajectories. The effectiveness of the proposed approach for trajectory tracking is demonstrated through ROS 2-Gazebo simulations on a TurtleBot3 Waffle robot. The code is available at https://gradslab.github.io/DFL_QP_Unicycle/
Hamza Tariq, Usman Ali, Adeel Akhtar
Department of Mechanical & Industrial Engineering at the New Jersey Institute of Technology, Newark, NJ 07102, USA · School of Engineering and Sustainable Development at the De Montfort University, Leicester, LE1 9BH, UK
The Park-Ravani construction generates a twice continuously differentiable, frame-invariant spline on SO(3) by exponentiating cubic canonical-coordinate polynomials. We show that the construction transfers, without changing form, to the group of orthogonal dual tensors, a representation of rigid displacements. The transferred recurrence is stated compactly through the dual extension of the right Jacobian of the exponential map and its first Fréchet derivative. This yields interpolation of prescribed rigid poses and continuity of the body dual twist and its first derivative. Using the higher-order rigid-body kinematics of dual spatial twists, we then prove that the resulting curve has a continuous physical acceleration field, not merely a continuous quantity obtained by formally differentiating the dual part of a twist. We also distinguish algebraic dual transfer from temporal differential prolongation: their simultaneous first-order use takes place in a hyper-dual algebra, and interpolation of arbitrary prolonged nodal data need not be holonomic. A noncommuting three-pose example verifies the recurrence, all knot continuity statements, and dimensional covariance under a change from meters to millimeters. We define and analyze the first-order holonomy defect of a generic hyper-dual interpolant, exhibit an exact counterexample, and remove the defect by cubic or quintic Hermite interpolation in dual logarithmic coordinates.
Daniel Condurache
“Gheorghe Asachi” Technical University of Ia¸i, Romania