Cooperative Task Spaces for Multi-Arm Manipulation Control based on Similarity Transformations

TL;DR

The paper develops a geometric framework for cooperative multi-arm manipulation by representing multi-chain coordination with cooperative geometric primitives embedded in conformal geometric algebra and transformed via similarity transformations. It derives analytic and geometric Jacobians, and shows how to integrate these into impedance control, optimal control, and teleoperation in a way that generalizes single-arm operational space control while naturally embedding a geometric nullspace. Key contributions include the cooperative task space modeling, explicit similarity transformations for point/line/plane/circle/sphere primitives, and a unified treatment of kinematics, dynamics, and control within this algebraic setting. The results demonstrate abstract yet versatile experiments across line, circle, plane, and sphere primitives, highlighting robustness, redundancy management, and potential for human-robot collaboration and scalable multi-arm systems. The framework offers a principled path toward geometrically consistent, distributed, and cooperative manipulation with controlled secondary objectives and reduced dimensionality, pointing to broad future applications and real-world deployment possibilities.

Abstract

Many tasks in human environments require collaborative behavior between multiple kinematic chains, either to provide additional support for carrying big and bulky objects or to enable the dexterity that is required for in-hand manipulation. Since these complex systems often have a very high number of degrees of freedom coordinating their movements is notoriously difficult to model. In this article, we present the derivation of the theoretical foundations for cooperative task spaces of multi-arm robotic systems based on geometric primitives defined using conformal geometric algebra. Based on the similarity transformations of these cooperative geometric primitives, we derive an abstraction of complex robotic systems that enables representing these systems in a way that directly corresponds to single-arm systems. By deriving the associated analytic and geometric Jacobian matrices, we then show the straightforward integration of our approach into classical control techniques rooted in operational space control. We demonstrate this using bimanual manipulators, humanoids and multi-fingered hands in optimal control experiments for reaching desired geometric primitives and in teleoperation experiments using differential kinematics control. We then discuss how the geometric primitives naturally embed nullspace structures into the controllers that can be exploited for introducing secondary control objectives. This work, represents the theoretical foundations of this cooperative manipulation control framework, and thus the experiments are presented in an abstract way, while giving pointers towards potential future applications.

Figures (22)

• Figure 1: Overview of various daily-life tasks and how they can be modeled using the cooperative geometric primitives. Note that the geometric primitives do not approximate the shape of the objects, but instead model the cooperative behavior of the kinematic chains. The corresponding reference points are shown in red. Left: Human activities with an overlay of the proposed geometric primitive to capture the task constraints. Right: Corresponding robotic system with the cooperative geometric primitive achieving the same behavior.
• Figure 2: Forward kinematics function for cooperative systems using similarity transformations.
• Figure 3: Cooperative geometric primitives for multiple kinematic chains. The red points indicate the reference points on the robots that are considered for the cooperative geometric primitive, which is then shown in green.
• Figure 4: Interpolation of different geometric primitives using similarity transformations.
• Figure 5: Three Kuka IIWA14 robots arranged in a circle. The green circle shows the cooperative geometric primitive for the current joint configuration.