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In many applications, the manipulations require only part of the degrees of freedom (DOFs) of the end-effector, or some DOFs are more important than the rest. We name these applications prioritized manipulations. The end-effector's DOFs are divided into those which are critical and must be controlled as precisely as possible, and those which have loose specifications, so their tracking performance can be traded off to achieve other needs. In this paper, for the class of general constrained rigid multibody systems (including passive joints and multiple closed kinematic loops), we derive a formulation for partitioning the task space into major and secondary task directions, and finding the velocity and static force mappings that precisely accomplish the major task and optimize some secondary goals such as reliability enhancement, obstacle and singularity avoidance, fault tolerance, or joint limit avoidance. The major task and secondary goals need to be specified in term of velocities/forces. In addition, a framework is developed to handle two kinds of common actuator failures, torque failure and position failure, by reconfiguring the differential kinematics and static force models. The techniques are tested on a 6-DOF parallel robot. Experimental results illustrate that the approach is practical and yields good performance.