After its introduction, impedance control has been utilized as a primary control scheme for robotic manipulation tasks that involve interaction with unknown environments. While impedance control has been extensively studied, the geometric structure of SE(3) for the robotic manipulator itself and its use in formulating a robotic task has not been adequately addressed. In this paper, we propose a differential geometric approach to impedance control. Given a left-invariant error metric in SE(3), the corresponding error vectors in position and velocity are first derived. We then propose the impedance control schemes that adequately account for the geometric structure of the manipulator in SE(3) based on a left-invariant potential function. The closed-loop stabilities for the proposed control schemes are verified using Lyapunov function-based analysis. The proposed control design clearly outperformed a conventional impedance control approach when tracking challenging trajectory profiles.

翻译：自阻抗控制被提出以来，它已成为机器人操作任务中涉及未知环境交互的主要控制方案。尽管阻抗控制已得到广泛研究，但机器人操作臂本身的SE(3)几何结构及其在任务公式化中的应用尚未得到充分探讨。本文提出了一种微分几何方法来实现阻抗控制。给定SE(3)中的左不变误差度量，首先推导出相应的位置和速度误差向量。随后，我们基于左不变势函数提出了能够充分考虑操作臂在SE(3)中几何结构的阻抗控制方案。利用基于李雅普诺夫函数的分析验证了所提控制方案的闭环稳定性。在跟踪具有挑战性的轨迹轮廓时，所提控制设计明显优于传统阻抗控制方法。