On the Collocated Form with Input Decoupling of Lagrangian Systems

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.