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This paper presents an efficient kinematical solution to a multi-robot system with serial and parallel mechanisms. JL-I is a reconfigurable robot featuring active spherical joints formed by serial and parallel mechanisms endowing the robotic system with the ability of changing shapes in three dimensions. The active joint here can combine the advantages of the high rigidity of a parallel mechanism and the extended workspace of a serial mechanism. However, the kinematic analysis of the serial and parallel mechanism is always the bottleneck in designing a robot and control realization. In order to deal with this problem, the whole kinematical analysis is organized in the sequence from the direct mechanical analysis related to the serial and parallel mechanism over the numerical solutions to the simplified kinematics expression. The latest results obtained demonstrate that the deduced closure-form solution is time efficient and easy to implement while offering a satisfactory motion performance in on-site experiments.