Direct kinematics of a 3-PRPS type parallel manipulator


Parallel manipulators with six degrees of freedom and three limbs have a large workspace and less complex singular configurations compared to the parallel manipulators with six degrees of freedom and six limbs. This paper is presented to solve the direct kinematics of a novel 3-PRPS type parallel manipulator with six-degrees-of-freedom, where P, R, and S are prismatic, revolute and spherical kinematic pairs respectively. The considered parallel manipulator is formed by connecting a moving platform with a fixed platform (base) through three closing kinematic chains of a PRPS type in which the prismatic kinematic pairs are active and they are located on a fixed platform and legs. The constant and variable parameters of the considered parallel manipulator characterizing its geometry and kinematics respectively are determined. In the direct kinematics, the positions of the moving platform are determined by the known constant parameters of the links and the given variable parameters of the active kinematic pairs. An analysis of the obtained equations of the direct kinematics showed that the variable parameters of the active prismatic kinematic pairs are set free, and these equations are reduced to a 16th –order polynomial equation with passive kinematic pairs variables. Numerical examples of the considered parallel manipulator’s direct kinematics are presented, and the results showed that the direct kinematics equations have four solutions corresponding to the four assemblies of the parallel manipulator.


[1] Jean-Pierre Merlet, "Parallel robots", (Kluwer Academic Publishers, 2000), 48-62.
[2] K.H. Hunt, "Structural kinematic of in-parallel-actuated robot-arms", ASME J. Mechanisms, Transmissions, and Automat. Des., 105(1983), 705-711.
[3] Xin-Jun Liu Jinsong Wang, "Parallel Kinematics: Type, Kinematics, and Optimal Design", (Springer Verlag Berlin Heidelberg, 2014), ch. 1, pp. 3-30, ch.2, pp.31-77.
[4] Takeda Y., Funubashi., Ichimaru H., "Development of spatial in-parallel actuated manipulators with six degrees of freedom with high motion transmissibility", JSME International Journal Series C 2(1997), 299-308.
[5] Ma O., Angeles J., Optimum architecture design of platform manipulators. Proceeding of the Fifth International Conference on Advanced Robotics (Piza, IEEE, Press, Piscataway, N.J., 1991), 1130-1135.
[6] Wang L.C.T. and Change C.C., "On the dynamic analysis of general parallel manipulators", In Journal Robotics and Automation 2(1997), 81-87.
[7] McAree P.R., Selig J.M., "Constrained robot dynamics II: Parallel machines", Journal of Robotic Systems 16:9(1999), 487-498.
[8] Clearly C., Uebel M., Jacobian Formulation for a Novel 6-DOF Parallel Manipulator (IEEE International Conference on Robotics and Automation, 3, 1994), 2377-2382.
[9] Tsai L.-W., Tahmasebi F., "Synthesis and Analysis of a New Class of Six-Degree-of-Freedom Parallel Manipulators", Journal of Robotic Systems 10:5(1993), 561-580.
[10] Alizade R.I., Tagiyev N.R., and Duffy J., "A Forward and Reverse Displacement Analysis of a 6-DOF In-Parallel Manipulator", Mechanism and Machine Theory 29:1(1994), 115-124.
[11] Collins C.L. Long G.L., "The Singularity Analysis of an in-Parallel Hand Controller for Force-Reflected Teleoperation", IEEE Transaction on Robotics and Automation 11(1995), 661-669.
[12] Mimuza N., Funabashi Y., A New Analytical System Applying 6 DOF Parallel Link Manipulator For Evaluating Motion Sensation (IEEE International Conference on Robotics and Automation, 1995), 227-333.
[13] Ebert U., Gosselin C.M., Kinematic Study of a New Type of Spatial Parallel Platform Mechanism (ASME Design Engineering Technical Conference, Atlanta, 13-16 September, 1998), 170-177.
[14] Byun Y.K., Cho H.S., "Analysis of a Novel 6-DOF 3-PPSP Parallel Manipulator", The International Journal of Robotics Research 16:6(1997), 859-872.
[15] Baigunchekov Zh., Mustafa A., Kairov R., Kassinov A., "Structural synthesis and geometry of 3-PRRS and 3-PRPS type parallel manipulators", Herald of the Kazakh-British Technical University 16(2019), 67-71.
[16] Khali D., Lee S-H, Tsai K.-Y., Sandor G.N., "Manipulator Configurations Based on Rotary-Liner (R-L) Actuators and Their Direct and Inverse Kinematics", Journal of Mechanisms, Transmissions, and Automation in Design 110(1988), 397-404.
[17] Zhumadil Baigunchekov, Myrzabai Izmambetov, Inverse Kinematics of Six - DOF Three – Limbed Parallel Manipulator (Advances in Robot Design and Intelligent Control, Proceedings of the 25th Conference on Robotics in Aple-Adria-Danube Region – RAAD 2016, Springer, 2016), 171-178.
[18] Zhumadil Baigunchekov, Myrzabai Izmambetov, Geometry and Direct Kinematics of Six-DOF Three-Limbed Parallel Manipulators (ROMANSY 21 - Robot Design, Dynamics and Control. Proceedings of the 21st CISM – IFToMM Symposium.
Springer, 2016), 39-46.
[19] Zh. Baigunchekov, М.А. Laribi, A. Mustafa, R. Kaiyrov, B.Amanov, A.Kassinov, "Inverse kinematics of 3-PRPS type parallel manipulator", Herald of the Kazakh-British Technical University 16(2019), 82–86.
[20] Lung-Wen Tsai, "Robot Analysis: The Mechanics of Serial and Parallel Manipulators", (John Wiley & Sons, Inc., 1999), ch.3, pp.116-164.
How to Cite
BAIGUNCHEKOV, Zh. Zh.; KASSINOV, A. N.. Direct kinematics of a 3-PRPS type parallel manipulator. Journal of Mathematics, Mechanics and Computer Science, [S.l.], v. 108, n. 4, p. 38-46, dec. 2020. ISSN 2617-4871. Available at: <>. Date accessed: 20 jan. 2021. doi:
Keywords parallel manipulator, moving and fixed platforms, direct kinematics