Extending the Cooperative Dual-Task Space in Conformal Geometric Algebra

TL;DR

The paper addresses the need for richer geometric representations in dual-arm manipulation by extending the cooperative dual-task space from the dual-quaternion formulation to conformal geometric algebra ($CGA$). It introduces a cooperative pointpair and derives relative/absolute motor representations and their Jacobians within CGA, enabling direct modeling of geometric primitives and constraints. The CGA-CDTS is integrated into an existing geometric-algebra-based optimal control framework, with residuals defined for reaching primitives and containment relations, and applied to real two-arm robot experiments. The results demonstrate enhanced expressiveness for dual-arm tasks, robust optimization formulations via MPC/iLQR, and practical validation on two Franka Emika robots.

Abstract

In this work, we are presenting an extension of the cooperative dual-task space (CDTS) in conformal geometric algebra. The CDTS was first defined using dual quaternion algebra and is a well established framework for the simplified definition of tasks using two manipulators. By integrating conformal geometric algebra, we aim to further enhance the geometric expressiveness and thus simplify the modeling of various tasks. We show this formulation by first presenting the CDTS and then its extension that is based around a cooperative pointpair. This extension keeps all the benefits of the original formulation that is based on dual quaternions, but adds more tools for geometric modeling of the dual-arm tasks. We also present how this CGA-CDTS can be seamlessly integrated with an optimal control framework in geometric algebra that was derived in previous work. In the experiments, we demonstrate how to model different objectives and constraints using the CGA-CDTS. Using a setup of two Franka Emika robots we then show the effectiveness of our approach using model predictive control in real world experiments.

Figures (6)

• Figure 1: Cooperative dual-task space using conformal geometric algebra. The figure shows the two manipulators as well as their individual, relative and absolute motors. Additionally, it shows the cooperative pointpair.
• Figure 2: Dual-Arm manipulator reaching a plane. The target plane is shown in red. The white Franka Emika robots show the initial configurations, the green ones are the result of individually reaching the plane, and the blue ones cooperatively.
• Figure 3: Two Franka Emika robots reaching for a single point in the cooperative dual-task space. The initial configurations are shown in white and the final ones in gray. The target point is shown in red.
• Figure 4: Two Franka Emika robots reaching a circle in the cooperative dual-task space while trying to maintain the same relative motor.
• Figure 5: Aligning the x-Axis.