Differential flatness and control of protocentric aerial manipulators with any number of arms and mixed rigid-/elastic-joints

We introduce a particularly relevant class of aerial manipulators that we name protocentric. These robots are formed by an underactuated aerial vehicle, a planar- Vertical Take-Off and Landing (PVTOL), equipped with any number of different parallel manipulator arms with the only property that all the first joints are attached at the Center of Mass (CoM) of the PVTOL, while the center of actuation of the PVTOL can be anywhere. We prove that protocentric aerial manipulators (PAMs) are differentially flat systems regardless the number of joints of each arm and their kinematic and dynamic parameters. The set of flat outputs is constituted by the CoM of the PVTOL and the absolute orientation angles of all the links. The relative degree of each output is equal to four. More amazingly, we prove that PAMs are differentially flat even in the case that any number of the joints are elastic, no matter the internal distribution between elastic and rigid joints. The set of flat outputs is the same but in this case the total relative degree grows quadratically with the number of elastic joints. We validate the theory by simulating object grasping and transportation tasks with unknown mass and parameters and using a controller based on dynamic feedback linearization.