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.

翻译：在人类环境中，许多任务需要多个运动链之间的协作行为，无论是为搬运大型笨重物体提供额外支撑，还是实现手内操作所需的灵巧性。由于这些复杂系统通常具有极高的自由度数量，协调其运动历来难以建模。本文基于使用共形几何代数定义的几何基元，提出了多臂机器人系统协同任务空间的理论基础推导。基于这些协同几何基元的相似变换，我们推导出复杂机器人系统的抽象表示方法，使其能够以直接对应于单臂系统的方式进行表征。通过推导相关的解析雅可比矩阵和几何雅可比矩阵，我们展示了该方法如何无缝集成于基于操作空间控制的经典控制技术。我们使用双机械臂系统、仿人机器人和多指手在最优控制实验（用于达到期望几何基元）和基于微分运动学控制的遥操作实验中验证了该框架。随后我们讨论了这些几何基元如何将零空间结构自然地嵌入控制器，从而可用于引入次级控制目标。本工作奠定了该协同操作控制框架的理论基础，因此实验以抽象方式呈现，同时为未来潜在应用提供了方向指引。