Abstract : In this paper we present the dynamic Lagrangian modeling, system analysis, and nonlinear control of a robot constituted by a planar-vtol (PVTOL) underactuated aerial vehicle equipped with a rigid-or an elastic-joint arm, which constitutes an aerial manipulator. For the design of the aerial manipulator, we first consider generic offsets between the center of mass (CoM) of the PVTOL, and the attachment point of the joint-arm. Later we consider a model in which these two points are the coinciding. It turns out to be that the choice of this attachment point is significantly affecting the capabilities of the platform. Furthermore, in both cases we consider the rigid-and elastic-joint arm configurations. For each of the resulting four cases we formally assess the presence of exact linearizing and differentially flat outputs and the possibility of using the dynamic feedback linearization (DFL) controller. Later we formalize an optimal control problem exploiting the differential flatness property of the systems, which is applied, as an illustrative example, to the aerial throwing task. Finally we provide extensive and realistic simulation results for comparisons between different robot models in different robotic tasks such as aerial grasping and aerial throwing, and a discussion on the applicability of computationally simpler controllers for the coinciding-point models to generic-point ones. Further exhaustive simulations on the trajectory tracking and the high-speed arm swinging capabilities are provided in a technical attachment.