This paper proposes a new formulation for modeling 3-DOF lifting vehicles in Cartesian coordinates. We address the problem of describing the lift force as orthogonal to velocity vector of the vehicle, which is not well-defined in traditional models. In spherical coordinates there is a singularity in vertical flight whereas in Cartesian coordinates, using a reference direction to define the lift vector also introduces a singularity. The proposed approach consists of a moving frame carried along the trajectory by parallel transport, with its first axis remaining co-linear to the velocity vector. The other two vectors are orthogonal to the velocity vector at all time and are used to express the lift force in a well-defined manner. The advantages of this approach are illustrated in the simulation of a simplified vehicle performing a looping.