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The volume of the orientation workspace is measured based on two invariant principles: It must be invariant with respect to the ground frame and invariant with respect to the orientation description. A method which is based on quaternions and differential geometry is developed for the measurement of the volume correctly. The method is extended for an arbitrary orientation description by means of a mapping theorem proposed for the first time. An example of a serial spherical wrist shows that the volumes that are obtained by the proposed method are consistent with the two invariant principles.