Robot end-effector tracking of a specific spatial path to fulfill certain requirements is a topic of great interest. First, considering both dynamic constraints and kinematic constraints, the planned velocity has characteristics of bounded high-order derivatives to ensure smooth motion. Second, to meet high-efficiency requirements, the feedrate of the end-effector should be planned within the best possible time. Although some studies have shown that they obtained the time-optimal solution by considering the second-order constraints, this is still an open problem for third-order constraints. Therefore, this article proposes an approximate time-optimal feedrate (ATOF) planning algorithm that considers third-order constraints. This algorithm is primarily based on the bidirectional scanning method, which includes two steps. First, we calculate the second-order time-optimal velocity (TOV2) curve under the third-order maximum velocity curve. Then, considering TOV2, we propose an ATOF algorithm that includes single-direction speed planning and speed curve smoothing. It is more practical for not only considering the Coulomb viscous friction and torque rate but also avoiding difficult calculations at the singular point and solves discontinuous joint-acceleration under second-order constraints. Finally, we design two sets of experiments to verify the higher calculation efficiency and effectiveness of the proposed ATOF algorithm.