Suitable representations of dynamical systems can simplify their analysis and control. On this line of thought, this paper 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? Not only we show that the answer to this question is yes, but we 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 $u$ 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.

翻译：动力系统的合适表示可以简化其分析与控制。基于这一思路，本文旨在回答以下问题：能否找到一种广义坐标变换，使得执行器直接对部分构型变量做功？我们不仅证明了该问题的答案是肯定的，还给出了充分必要条件。具体而言，我们寻求构型空间的一种表示，使得欧拉-拉格朗日形式动力学的右端项变为$[\boldsymbol{I} \; \boldsymbol{O}]^{T}\boldsymbol{u}$，其中$\boldsymbol{u}$为系统输入。我们识别出一类称为共位系统的问题可解类。在输入矩阵的温和条件下，给出了检验系统是否共位的简单判据。利用功率不变性，我们证明了坐标变换解耦输入通道的充要条件是系统动力学具有共位性。此外，我们利用共位形式为阻尼欠驱动机械系统推导了新型控制器。为验证理论结果，我们考虑了几个拉格朗日系统，重点集中于连续体软体机器人。