In this work, we propose a geometric framework for analyzing mechanical manipulation, for example, by a robotic agent. Under the assumption of conservative forces and quasi-static manipulation, we use energy methods to derive a metric. We first review and show that the natural geometric setting is represented by the cotangent bundle and its Lagrangian submanifolds. These are standard concepts in geometric mechanics but usually presented within dynamical frameworks. We review the basic definitions from a static mechanics perspective and show how Lagrangian submanifolds are naturally derived from a first order analysis. Then, via a second order analysis, we derive the Hessian of total energy. As this is not necessarily positive-definite from a control perspective, we propose the use of the squared-Hessian for optimality measures, motivated by insights {derived from both mechanics (Gauss's Principle) and biology (Separation Principle)}. We conclude by showing how such methods can be applied, for example, to the simple case of an elastically driven pendulum. The example is simple enough to allow for analytical solution. However, an extension is further derived and numerically solved, which is more realistically connected with actual robotic manipulation problems.

翻译：在本文中，我们提出了一种用于分析机械操作（例如由机器人代理执行的操作）的几何框架。在保守力和准静态操作的假设下，我们利用能量方法推导出一个度量。我们首先回顾并指出，自然的几何背景由余切丛及其拉格朗日子流形表示。这些是几何力学中的标准概念，但通常是在动力学框架中呈现的。我们从静力学角度回顾基本定义，并展示如何通过一阶分析自然推导出拉格朗日子流形。然后，通过二阶分析，我们推导出总能量的海森矩阵。由于从控制角度来看该矩阵未必是正定的，我们提出使用平方海森矩阵作为最优性度量，其动机源于力学（高斯原理）和生物学（分离原理）的见解。最后，我们展示了此类方法如何应用于简单案例（例如弹性驱动摆）。该示例足够简单，可以解析求解。然而，我们还进一步推导并数值求解了一个扩展案例，该案例与实际机器人操作问题具有更真实的联系。