Due to their high flexibility, low cost, and energy-saving advantages, applying Unmanned Aerial Vehicles (UAVs) in logistics is a promising field to achieve better social and economic benefits. Since UAVs’ energy storage capacity is generally low, it is essential to reduce energy costs to improve their system’s energy efficiency. In this paper, we proposed a novel trajectory planning framework to achieve the optimal trajectory with the minimum amount of energy consumption under the constraints of obstacles in a static environment. Based on UAV dynamics, we first derived the required power functions of multi-rotor UAVs in vertical and horizontal flight. To generate a feasible trajectory, we first adopted the A* algorithm to find a path and developed a safe flight corridor for the UAV to fly across by expanding the waypoints against the environment, and then proposed a time-discretization method to formulate the trajectory generation problem and solve it by the convex optimization algorithm. The optimization results in a static environment with obstacles demonstrated that the proposed method could efficiently and effectively obtain the optimal trajectory with the minimum amount of energy consumption under different allowed mission times and payloads. The framework would promote a variety of logistics UAV applications relevant to trajectory planning.