Interconnection and damping assignment passivity-based control scheme has been used to stabilize many physical systems such as underactuated mechanical systems through total energy shaping. In this method, some partial differential equations (PDEs) related to kinetic and potential energy shaping shall be solved analytically. Finding a suitable desired inertia matrix as the solution of nonlinear PDEs relevant to kinetic energy shaping is a challenging problem. In this paper, a systematic approach to solving this matching equation for systems with one degree of underactuation is proposed. A special structure for desired inertia matrix is proposed to simplify the solution of the corresponding PDE. It is shown that the proposed method is more general than that of some reported methods in the literature. In order to derive a suitable desired inertia matrix, a necessary condition is also derived. The proposed method is applied to three examples, including pendubot, VTOL aircraft, and 2D SpiderCrane.

互连和阻尼分配基于无源控制方案已被用于通过总能量整形来稳定许多物理系统，例如欠驱动机械系统。在该方法中，一些与动能和势能整形相关的偏微分方程 (PDE) 将被解析求解。寻找合适的所需惯性矩阵作为与动能整形相关的非线性偏微分方程的解是一个具有挑战性的问题。在本文中，提出了一种系统的方法来求解具有一个欠驱动度的系统的匹配方程。为了简化相应偏微分方程的求解，提出了一种特殊的期望惯量矩阵结构。结果表明，所提出的方法比文献中报道的一些方法更通用。为了推导出合适的期望惯量矩阵，还推导出一个必要条件。所提出的方法应用于三个例子，包括pendubot、VTOL飞机和2D SpiderCrane。