Optimal control in general, and flatness-based control in particular, of robotic arms necessitate to compute the first and second time derivatives of the joint torques/forces required to achieve a desired motion. In view of the required computational efficiency, recursive $O(n)$ -algorithms were proposed to this end. Aiming at compact yet efficient formulations, a Lie group formulation was recently proposed, making use of body-fixed and hybrid representation of twists and wrenches. In this letter, a formulation is introduced using the spatial representation. The second-order inverse dynamics algorithm is accompanied by a fourth-order forward and inverse kinematics algorithm. An advantage of all Lie group formulations is that they can be parameterized in terms of vectorial quantities that are readily available. The method is demonstrated for the 7 DOF Franka Emika Panda robot.

中文翻译：

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使用扭曲空间表示的串行机械手高阶运动学和逆动力学的 O(n) 算法

机械臂的一般最佳控制，尤其是基于平面度的控制，需要计算实现所需运动所需的关节扭矩/力的一阶和二阶导数。鉴于所需的计算效率，为此提出了递归$O(n)$ -算法。针对紧凑而高效的公式，最近提出了李群公式，利用扭动和扳手的体固定和混合表示。在这封信中，介绍了使用空间表示的公式。二阶逆动力学算法伴随着四阶正逆运动学算法。所有李群公式的一个优点是它们可以根据容易获得的矢量量进行参数化。