This thesis focuses on the safe control of space manipulators using a rigorous geometric model and control approach based on Lie groups. We begin with a comprehensive literature review of in-orbit robotic servicing missions and common Guidance, Navigation, and Control (GNC) methodologies. We also survey our chosen toolset for safe control of space manipulators, geometric mechanics (e.g. dynamic reduction and control on Lie groups) for formulating the dynamics and kinematics of rigid multi-body systems. To enable the design of a safe model-based control of space manipulators, we frst develop a combined form of Euler-Poincaré equations on Lie groups (for spacecraft) and Euler-Lagrange dynamical equations (for the equivalent fxed-base manipulator), called the Lagrange-Poincaré Equations (LPE), to establish a formalism for spacecraft-manipulator dynamics that is inherently free of parametrization singularities. In this model, we derive a block-diagonalized inertia matrix and closedform solutions for Coriolis and coupling efects, which efectively separate external (locked-arm rigid system) and internal (arm’s motion) dynamics. Symmetry-breaking external wrenches and control forces on each degree of freedom are also efciently incorporated to enable the development of control logics that are resilient against such actions. We extend this LPE by integrating Euler-Lagrange equations, Euler-Poincaré equations, and Kepler’s equations for orbital dynamics into a singularity-free formulation of in-orbit disturbed dynamics of space manipulators relative to a non-inertial orbital frame. This model captures how the disturbances from orbital motion critically afect the safety of the operation of a space manipulator. Leveraging the unique geometric characteristics of the model, we design a full-pose workspace controller that applies feedback linearization on the corresponding Lie group of the relative poses between the end-efector and the target. We propose an intrinsic feedforward-feedback PID control law on Lie groups for comprehensive full-pose regulation, incorporating measures to avoid kinematic and dynamic singularities and workspace boundaries. Additionally, we develop the following safety-assurance measures on top of the controller:​ (i) a manipulability optimization algorithm, (ii) a collision-avoidance method, and (iii) a full formulation and proof of the internal/external stability and boundedness of null-space states of the system. Finally, we demonstrate the robustness and features of the controller through a comprehensive series of experiments in Simulink, showcasing its resilience to inertial uncertainties and measurement noise.