Input constraints are active in robot trajectory planning when a mobile robot traverses mobility challenges such as steep hills that limit the acceleration of the robot due to the torque constraints of the motor or engine or in manipulator lifting tasks when the load is sufficiently heavy that the torque constraints of the robot's motor prevent it from statically supporting the load in regions of the robot's workspace. This paper presents a general methodology for solving these planning tasks using a minimum-time cost function and applies it to the problem of a multiple degrees-of-freedom (DOF) manipulator lifting a heavy load. Planning for these types of problems requires use of the robot's dynamic model. Here, we plan using sampling-based model predictive optimization (SBMPO), which is only practical if the planning can be done quickly. Hence, substantial attention is given to efficient computations by: (1) using the dynamic model without integrating it, (2) using optimal control theory to develop an “optimistic A* estimate of cost-to-goal,” which is in this case a rigorous lower bound on the minimum time from a current state to a goal state, and (3) using prior experience to speed up the computation of a new trajectory. The methodology is experimentally verified for heavy lifting using a two-link manipulator.