Extending the Cooperative Dual-Task Space in Conformal Geometric Algebra

Tobias Löw and Sylvain Calinon

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.

Preprint: https://arxiv.org/pdf/2310.19093.pdf

Appendix: appendix_cdts.pdf

Code Repository: https://gitlab.com/gafro

Experiments

Reaching a Point

This task shows how a single point is reached by using the cooperative pointpair. The target is defined by an Aruco marker. It can be seen that the manipulator that is closer to the point performs the reaching task while the other remains in its initial configuration.

Balancing a Plate

In this task the robot balances a plate. The task definition constrains the relative distance and orientation of the end-effectors as well as the absolute orientation of the plate. Due to the way the robots are setup, the workspace for dual arm manipulation is very limited, which reduces the range of perturbations that can be applied.