Suitable representations of dynamical systems can simplify their analysis and control. On this line of thought, this article aims to answer the following question: Can a transformation of the generalized coordinates under which the actuators directly perform work on a subset of the configuration variables be found? We not only show that the answer to this question is yes but also provide necessary and sufficient conditions. More specifically, we look for a representation of the configuration space such that the right-hand side of the dynamics in Euler–Lagrange form becomes $[\boldsymbol{I}\; \boldsymbol{O}]^{T}\boldsymbol{u}$ , being $\boldsymbol{u}$ the system input. We identify a class of systems, called collocated , for which this problem is solvable. Under mild conditions on the input matrix, a simple test is presented to verify whether a system is collocated or not. By exploiting power invariance, we provide necessary and sufficient conditions that a change of coordinates decouples the input channels if and only if the dynamics is collocated. In addition, we use the collocated form to derive novel controllers for damped underactuated mechanical systems. To demonstrate the theoretical findings, we consider several Lagrangian systems with a focus on continuum soft robots.

中文翻译：

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通过坐标变换拉格朗日系统的输入解耦：一般表征及其在软机器人中的应用

动力系统的适当表示可以简化其分析和控制。沿着这个思路，本文旨在回答以下问题：能否找到执行器直接对配置变量的子集执行工作的广义坐标的变换？我们不仅证明这个问题的答案是可以，但也要提供充分必要条件。更具体地说，我们寻找配置空间的表示，使得欧拉-拉格朗日形式的动力学右侧变为$[\boldsymbol{I}\; \boldsymbol{O}]^{T}\boldsymbol{u}$ ， 存在$\boldsymbol{u}$系统输入。我们确定一类系统，称为collocated ，这个问题是可以解决的。在输入矩阵的温和条件下，提出了一个简单的测试来验证系统是否并置。通过利用幂不变性，我们提供了必要且充分的条件，即当且仅当动态配置时，坐标的变化才能解耦输入通道。此外，我们使用并置形式导出用于阻尼欠驱动机械系统的新型控制器。为了证明理论结果，我们考虑了几个重点关注连续软机器人的拉格朗日系统。