This paper addresses the problem of generating a continuous and differentiable trajectory on the Lie group of rigid body motions, se(3), for a class of underactuated vehicles modeled as rigid bodies. The three rotational degrees of freedom (DOF) are independently actuated, while only one translational DOF is actuated by a body-fixed thrust vector. This model is applicable to a large set of unmanned vehicles, including fixed-wing and rotorcraft unmanned aerial vehicles (UAVs). The formulation utilizes exponential coordinates to express the underactuation constraint as an intrinsic part of the problem. It provides steps to generate a rest-to-rest trajectory after obtaining conditions that guarantee controllability. An attitude trajectory is selected to satisfy the given initial and final attitude state. The position trajectory generation is subsequently posed as an optimal control problem expressed as a linear quadratic regulator (LQR) in the exponential coordinates corresponding to position. As a result, an optimal position trajectory is obtained which ensures that the trajectory generated is feasible with realistic velocities and with given initial pose and final pose, while satisfying the underactuation constraint. Numerical simulation results are obtained that validate this trajectory generation scheme.