In this paper, we develop a control algorithm for the point-to-point motion control of a visco-elastic servo to achieve both precision positioning and fast move while satisfying various physical constraints. The visco-elastic servo is modeled as two lumped masses connected by a spring and a damper The control input is the actuation applied on one mass and the goal is to maneuver the motion of the other To achieve fast and precision point-to-point move in the presence of physical constraints, we employ a dual-module control structure, i.e., a control module for trajectory tracking and a planning module for real-time trajectory generation. While the control module uses a full state feedback controller for trajectory tracking, the planning module generates a reference trajectory that conforms to the 4th order dynamics of the system, minimizes the convergence time and satisfies all the hard constraints on spring displacement, mass velocity and input saturation, which is the key focus of this paper We propose an analytical solution to this problem by applying a state and input transformation to the desired trajectory dynamics, and developing an integrated phase portrait method to generate a bang-bang-like trajectory that meets all the design objectives. The analytical nature of the proposed method renders it computationally efficient and easy to implement. The effectiveness of the proposed method is tested via simulation.