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 isyes, 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 [IO]tu, being u the system input. We identify a class of systems, calledcollocated, 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.

Performing precise, repetitive motions is essential in many robotic and automation systems. Iterative learning control (ILC) allows determining the necessary control command by using a very rough system model to speed up the process. Functional iterative learning control is a novel technique that promises to solve several limitations of classic ILC. It operates by merging the input space into a large functional space, resulting in an over-determined control task in the iteration domain. In this way, it can deal with systems having more outputs than inputs and accelerate the learning process without resorting to model discretizations. However, the framework lacks so far a validation in experiments. This paper aims to provide such experimental validation in the context of robotics. To this end, we designed and built a one-link flexible arm that is actuated by a stepper motor, which makes the development of an accurate model more challenging and the validation closer to the industrial practice. We provide multiple experimental results across several conditions, proving the feasibility of the method in practice.