An orienting platform is a mechanism which allows rotation of a spatial object without translational motion of that object. In this work, we study a parallel platform with one passive nonholonomic spherical joint and two series of spherical, actuated prismatic and universal joints (the platform is also known in literature as an (nS)-2SPU wrist). To solve the control and motion planning problems, an analytic approach is used. The design of practical stabilization and tracking algorithm is based on transverse functions and a method for motion planning respecting mechanical singularities is derived from endogenous configuration space approach. It is shown that the system is controllable and locally equivalent to the chained form system. Then, the stabilization, tracking, and motion planning algorithms are proposed. Results are verified with computer simulations. A combination of the open-loop motion planning algorithm and the closed-loop tracking provide a tool for designing a motion planning algorithm respecting mechanical singularities and robust to input disturbances.