Numerical search for local (partial) differential flatness

Differential flatness is a property of certain systems that greatly simplifies the generation of optimal and dynamically feasible trajectories. Using a differentially flat model, there is no need to integrate the system dynamics to retrieve the states and the constraints of the optimization problem are simpler. Recently, the concept of partial differential flatness has been introduced covering a broader class of systems. In particular, it allows to reduce the need for integration by limiting it to a subset of the states. However, finding an analytical expression for the (partial) differential flatness requires the manipulation of the equations of motion in a very specific manner such that a series of properties are fulfilled. In general, finding such analytical model is not straightforward nor compatible with algorithmic models. In order to tackle this problem, in this paper we present a numerical method to find a (partially) differentially flat model of a system around a collection of states and inputs trajectories. We present results on three underactuated nonlinear systems (cart-pole, planar ballbot and a 3D quadrotor). As use case examples, we show online trajectory re-planning tasks. The validity of the trajectories obtained with the locally flat models is verified by forward integrating the original equations of motion together with an optimal stabilizer.