Input Decoupling of Lagrangian Systems via Coordinate Transformation: General Characterization and Its Application to Soft Robotics

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.