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Increasing the degrees of freedom (DoFs) of robotic systems makes them more versatile and flexible. This usually renders the system kinematically redundant: the main manipulation or interaction task does not fully determine its joint maneuvers. Additional constraints or objectives are required to solve the underdetermined control and planning problems. The state-of-the-art approaches arrange tasks in a hierarchy and decouple lower priority tasks from higher priority tasks on velocity or torque level using projectors. We develop an approach to redundancy resolution and decoupling on position level by determining subspaces of the configurations space independent of the primary task. We call them orthogonal foliations because they are, in a certain sense, orthogonal to the task self-motion manifolds. The approach provides a better insight into the topological properties of robot kinematics and control problems, allowing a global view. A condition for the existence of orthogonal foliations is derived. If the condition is not satisfied, we will still find approximate solutions by numerical optimization. Coordinates can be defined on these orthogonal foliations and can be used as additional task variables for control. We show in simulations that we can control the system without the need for projectors using these coordinates, and we validate the approach experimentally on a seven-DoF robot.