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Although it is common to eliminate the singularity of parallel mechanism by adding the branched chain with actuation redundancy, there is no theory and method for the configuration synthesis of the branched chain with actuation redundancy in parallel mechanism. Branched chains with actuation redundancy are synthesized for eliminating interior singularity of 3-translational and 1- rotational(3T1R) parallel mechanisms. Guided by the discriminance method of hybrid screw group according to Grassmann line geometry, all the possibilities are listed for the occurrence of interior singularities in 3T1R parallel mechanism. Based on the linear relevance of screw system and the principles of eliminating parallel mechanism singularity with actuation redundancy, different types of branched chains with actuation redundancy are synthesized systematically to indicate the layout and the number of the branched chainsinterior with actuation redundancy. A general method is proposed for the configuration synthesis of the branched chains with actuation redundancy of the redundant parallel mechanism, and it builds a solid foundation for the subsequent performance optimization of the redundant actuation parallel mechanism.