We explore global randomized joint-space path planning for articulated robots that are subjected to task-space constraints. This paper describes a representation of constrained motion for joint-space planners and develops two simple and efficient methods for constrained sampling of joint configurations: tangent-space sampling (TS) and first-order retraction (FR). FR is formally proven to provide global sampling for linear task-space transformations. Constrained joint-space planning is important for many real-world problems, which involves redundant manipulators. On the one hand, tasks are designated in workspace coordinates: to rotate doors about fixed axes, to slide drawers along fixed trajectories, or to hold objects level during transport. On the other hand, joint-space planning gives alternative paths that use redundant degrees of freedom (DOFs) to avoid obstacles or satisfy additional goals while performing a task. We demonstrate that our methods are faster and more invariant to parameter choices than the techniques that exist.