For underactuated robots working in complex environments, an important objective is to drive all variables (particularly for unactuated end-effectors) to move along the specific path and restrict positions/velocities to avoid obstacles, rather than using only point-to-point control. Unfortunately, most path planning methods are only suitable to fully actuated systems or depend on linearized models. The main motivations of our work are to directly fulfill motion constraints and achieve path following for both actuated and unactuated states (e.g., payload swing of cranes) when lacking effective control inputs. To this end, this article presents a new time-optimal trajectory planning-based motion control method for general underactuated robots. By constructing auxiliary signals (in Cartesian space) to express all actuated/unactuated variables (in joint space), their position/velocity constraints are converted into some convex/nonconvex inequalities related to a to-be-optimized path parameter and its derivatives. Then, an optimization algorithm is constructed to solve the available path parameter and derive a group of time-optimal trajectories for actuated states. As we know, this is the first study to ensure path following and necessary full-state constraints for actuated/unactuated states. Then, a tradeoff among path-constrained motions, time optimization, and state constraints is achieved together. This article takes the rotary crane as an example and provides detailed analysis of calculating desired trajectories based on the proposed planning frame, whose effectiveness is also verified through hardware experiments.