This dissertation is concerned with the position and vibration control of flexible articulated space robots consisting of a rigid platform, two flexible arms, and a rigid end-effector carrying a payload, all components being serially connected through revolute joints. The mission is to carry a payload over a prescribed trajectory in the inertial space, while suppressing the elastic vibration of the arms and the rigid-body perturbations. The equations of motion governing the robot dynamics are derived by means of Lagrangian mechanics and they include actuator dynamics. Based on the assumption that the elastic deformations and the rigid-body perturbations are small relative to the nominal trajectory-following rigid-body motions, a perturbation approach is adopted to separate the equations into nonlinear rigid-body equations and linear perturbation equations. The nominal trajectory is planned to conserve the limited actuator resources and keep the platform attitude stationary, by eliminating the inherent kinematic redundancy of the manipulator. By assuming perfect sensing, i.e., all the states are completely accessible, two kinds of controls are designed in discrete-time. First, a feedforward control is designed to minimize the persistent disturbance resulting from the nominal motions. Next, a feedback control is synthesized based on the Linear Quadratic Regulator (LQR) theory with a prescribed degree of stability to make the system stable and further enhance the disturbance-rejection performances. These controls are subsequently applied to the case in which only the sensor outputs are available, and they are noisy. A finite number of sensors is assumed. A Kalman filter is designed to estimate the state on the assumption of zeromean Gaussian white plant and measurement noise. In the real situation, controls are applied to the original plant rather than the linearized model, so that the Linear Quadratic Gaussian (LQG) control combined with robustness recovery methods is tested on the plant. Due to difficulties in implementing a Kalman filter, a Maximum Likelihood Estimator (MLE) is proposed. A numerical example illustrates the approach.