In this work, we propose a geometric framework for analyzing mechanical manipulation, for instance, by a robotic agent. Under the assumption of conservative forces and quasi-static manipulation, we use energy methods to derive a metric. In the first part of the paper, we review how quasi-static mechanical manipulation tasks can be naturally described via the so-called force-space, i.e. the cotangent bundle of the configuration space, and its Lagrangian submanifolds. Then, via a second order analysis, we derive the control Hessian of total energy. As this is not necessarily positive-definite, from an optimal control perspective, we propose the use of the squared-Hessian, also motivated by insights derived from both mechanics (Gauss' Principle) and biology (Separation Principle). In the second part of the paper, we apply such methods to the problem of an elastically-driven, inverted pendulum. Despite its apparent simplicity, this example is representative of an important class of robotic manipulation problems for which we show how a smooth elastic potential can be derived by regularizing mechanical contact. We then show how graph theory can be used to connect each numerical solution to `nearby' ones, with weights derived from the very metric introduced in the first part of the paper.

翻译：本文提出了一种用于分析机械操作（例如机器人操作）的几何框架。在保守力与准静态操作的假设下，我们利用能量方法推导出一种度规。论文第一部分回顾了准静态机械操作任务如何自然地通过所谓“力空间”（即构型空间的余切丛）及其拉格朗日子流形进行描述。随后，通过二阶分析，我们推导出总能量的控制海森矩阵。由于该矩阵不一定正定，我们从最优控制角度出发，提出使用平方海森矩阵，该方法的动机同时源于力学（高斯原理）与生物学（分离原理）的洞见。论文第二部分将该方法应用于弹性驱动倒立摆问题。尽管该示例看似简单，但它代表了一类重要的机器人操作问题，我们展示了如何通过正则化机械接触导出平滑的弹性势能。最后，我们证明图论可用于将每个数值解连接至“邻近”解，其权重由论文第一部分引入的度规推导得出。