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The static analysis of a parallel wrist, known as the 3-universal-prismatic-universal (UPU) wrist, is presented, and the 3-UPU wrist singularity conditions are interpreted from a statics point of view. An equation (singularity equation) that enables finding all the 3-UPU wrist singular configurations is written in explicit form. Moreover, the use of the Rodrigues parameters to parameterize the 3-UPU wrist platform orientation allows the singularity equation to become a fourth-degree polynomial equation in three unknowns. Finally, a numerical example is presented.