Abstract: Different from the parallel manipulators with 6 DOFs, the limbs of the lower-mobility parallel manipulators not only transmit motion and actuation force or moment but also provide constraint force or moment for their end-effectors. However, the latter one is often neglected in the past studies on static force of this class of parallel manipulators. By making use of the theory of differential manifold, it is revealed clearly that the generalized force space of the manipulators’ end-effector consists of both actuation and constraint force sub spaces. On the basis of the above analysis, the overall mappings of gener- alized force between input and output of manipulator are built up by use of the principle of virtual work, which consists of actuation and constraint forces mappings, i.e., two matrices. Through investigating the condition numbers of these matrices, characteristics of actuation and constraint forces transmission of the manipulator can be obtained. Moreover, this method is unified and distinct in physics and mathematics, in which it is easy to deeply understand the essence of the static force of this class of parallel manipulators. At last, the 3-UPU parallel manip- ulator is used as an example to demonstrate the methodology.

Key words: Differential manifold, Lower-mobility, Parallel manipulator, Static force analysis