Orientation Learning and Adaptation towards Simultaneous Incorporation of Multiple Local Constraints

TL;DR

This paper tackles orientation learning on the non-Euclidean manifold SO(3) by introducing the Angle-Axis Space, a local, bijective representation that facilitates constraint-aware imitation learning. It integrates probabilistic modeling (GMM/GMR) and kernelized trajectory primitives (KMP) with a Gauss-based geodesic weighting to fuse multiple local constraints and basepoints, while a memory-based algorithm ensures smooth, discontinuity-free rotation averages. The framework supports angular-acceleration constraints and incomplete orientation via-points (IOVPs), including single and multiple IOC scenarios, through auxiliary-frame transformations and multi-tangent-space fusion. Simulations and real-world experiments (earphone peg-in-hole, test-tube insertion) validate that the approach achieves accurate via-point tracking, reduced acceleration costs, and robust generalization when multiple local constraints are present. Overall, the method extends Euclidean IL/LfD techniques to non-Euclidean orientation learning, enabling simultaneous incorporation of multiple local constraints with improved smoothness and practicality for robotic tasks.

Abstract

Orientation learning plays a pivotal role in many tasks. However, the rotation group SO(3) is a Riemannian manifold. As a result, the distortion caused by non-Euclidean geometric nature introduces difficulties to the incorporation of local constraints, especially for the simultaneous incorporation of multiple local constraints. To address this issue, we propose the Angle-Axis Space-based orientation representation method to solve several orientation learning problems, including orientation adaptation and minimization of angular acceleration. Specifically, we propose a weighted average mechanism in SO(3) based on the angle-axis representation method. Our main idea is to generate multiple trajectories by considering different local constraints at different basepoints. Then these multiple trajectories are fused to generate a smooth trajectory by our proposed weighted average mechanism, achieving the goal to incorporate multiple local constraints simultaneously. Compared with existing solution, ours can address the distortion issue and make the off-theshelf Euclidean learning algorithm be re-applicable in non-Euclidean space. Simulation and Experimental evaluations validate that our solution can not only adapt orientations towards arbitrary desired via-points and cope with angular acceleration constraints, but also incorporate multiple local constraints simultaneously to achieve extra benefits, e.g., achieving smaller acceleration costs.

Figures (32)

• Figure 1: Problem Illustration. (a) The single-tangent-space learning framework. All date are projected into a single tangent space to encode the demonstration distributions. Due to non-Euclidean structure, the distribution for points that are outside the neighborhood $\mathcal{B}_{r}\left({\boldsymbol{x}}\right)$ is distorted. For example, the starting/ending points of every demonstration lie in a geodesic in the manifold $\mathcal{M}$. However, due to the distortion, this property is not held in the tangent space. Therefore, to incorporate local constraint, this framework has to project all data into the tangent space that extended around the given basepoint. Obviously, this framework is inapplicable when multiple local constraints are required. (b) To address this problem, we propose to generate multiple trajectories (the colored curves in each tangent spaces) by projecting data into multiple tangent spaces, to incorporate each local constraint at different basepoints. Then these multiple trajectories can be fused into a final trajectory (the colored curve in right subfigure. The color changes from light to dark as the weights increase) by using a fusing mechanism, which is challenging to design. The difficulties lies in two aspects. First, how to guarantee that each reproduced trajectory is dominant in the final trajectory at the given local time domain, such that the local constraint is satisfied. Second, how to guarantee the smoothness of the final reproduced trajectory. We propose a Gauss-based weighted averaging mechanism to address the above difficulties.
• Figure 2: The Angle-Axis Space $\boldsymbol{B}_\pi \backslash \mathcal{S}^2_\pi$ is bijective to SO(3).
• Figure 3: Our exponential and logarithm mappings are redefined between the Angle-Axis Space $\boldsymbol{B}_\pi \backslash \mathcal{S}^2_\pi$ and SO(3), while the intermediate set $so\left(3\right)$ is omitted.
• Figure 4: Comparison between the Angle-Axis Space and the quaternion space. Analogous to $\mathcal{S}^1$ and $\mathcal{S}^2$ cases, the Angle-Axis Space is a tangent space of SO(3) around the identity ${\boldsymbol{I}}_3$. While the quaternion is to represent the $\mathcal{S}^n$ in a $(n+1)$-d coordinate system, where $n$ is the dimension of the manifold. The Euclidean distance of the quaternions can not directly characterize the geodesic distance of orientations.
• Figure 5: (a) The bigger variance on the relaxed DoF drives the reproduced orientation to deviate from the desired geodesic (the blud and solid line), due to distance distortion. This results in the failure in guaranteeing the stricting constraints on the other DoFs. (b) The distances relative to the basepoint are preserved.

Theorems & Definitions (1)

• Theorem 1