Suitable representations of dynamical systems can simplify their analysis and control. On this line of thought, this paper considers the input decoupling problem for input-affine Lagrangian dynamics, namely the problem of finding a transformation of the generalized coordinates that decouples the input channels. We identify a class of systems for which this problem is solvable. Such systems are called collocated because the decoupling variables correspond to the coordinates on which the actuators directly perform work. 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, it is proven that a change of coordinates decouples the input channels if and only if the dynamics is collocated. We illustrate the theoretical results by considering several Lagrangian systems, focusing on underactuated mechanical systems, for which novel controllers that exploit input decoupling are designed.

翻译：动力系统的合适表示可简化其分析与控制。基于这一思路，本文研究了输入仿射拉格朗日动力学中的输入解耦问题，即寻找能够解耦输入通道的广义坐标变换。我们识别出一类可解此类问题的系统，这类系统被称为共位系统，因为解耦变量对应于执行器直接做功的坐标。在输入矩阵满足温和条件的前提下，我们提出了一种简单的判别方法，用于验证系统是否为共位系统。通过利用功率不变性，我们证明：当且仅当动力系统具有共位性时，坐标变换方能解耦输入通道。为说明理论结果，我们考虑多个拉格朗日系统，重点研究欠驱动机械系统，并设计了利用输入解耦特性的新型控制器。