In this paper, we present a novel optimization-based framework for autonomous excavator trajectory generation under various objectives, including minimum joint displacement and minimum time. Traditional methods on excavation trajectory generation usually separate the excavation motion into a sequence of fixed phases, resulting in limited trajectory searching space. Our framework explores the space of all possible excavation trajectories represented with waypoints interpolated by a polynomial spline, thereby enabling optimization over a larger searching space. We formulate a generic task specification for excavation by constraining the instantaneous motion of the bucket and further add a target-oriented constraint, i.e. swept volume that indicates the estimated amount of excavated materials. To formulate time related objectives and constraints, we introduce time intervals between waypoints as variables into the optimization framework. We implement the proposed framework and evaluate its performance on a UR5 robotic arm. The experimental results demonstrate that the generated trajectories are able to excavate sufficient mass of soil for different terrain shapes and have 60% shorter minimal length than traditional excavation methods. We further compare our one-stage time optimal trajectory generation with the two-stage method. The result shows that trajectories generated by our one-stage method cost 18% less time on average.