Forward Kinematics Analysis of a 3-PRS Parallel Manipulator
In this article the homotopy continuation method (HCM) to solve the forward kinematic problem of the 3-PRS parallel manipulator is used. Since there are many difficulties in solving the system of nonlinear equations in kinematics of manipulators, the numerical solutions like Newton-Raphson are inevitably used. When dealing with any numerical solution, there are two troublesome problems. One is that good initial guesses are not easy to detect and another is related to whether the used method will converge to useful solutions. Results of this paper reveal that the homotopy continuation method can alleviate the drawbacks of traditional numerical techniques.
[1] B. Dasgupta and T. S. Mruthyunjaya, 2000, "The stewart platform
manipulator: a review, "Mech. Mach Theory, Vol. 35, pp. 15-40.
[2] M. Griffis and J. Duffy, 1989, "A forward displacement analysis of a
class of Stewart Platforms, "Journal of Robotic Systems, Vol. 6, No. 6,
pp. 703-720.
[3] L. W. Tsai, G. C. Walsh, and R. E. Stamper, 1996, "Kinematics of a
novel three DOF translational platform, "Proceedings of IEEE
international conference on robotics and automation, Minneapolis,
Minnesota, pp. 3446-51.
[4] R. Di. Gregorio and V. Parenti-Castelli, 1998, "A translational 3-DOF
parallel manipulator, "In: J. Lenarcic, M. L. Husty, (Eds), Advances in
robot kinematics: analysis and control. Dordrecht: Kluwer Academic
Publishers, pp. 49-58.
[5] Y. Li and Q. Xu, 2006, "Kinematic analysis and design of a new 3-DOF
translational parallel manipulator, "ASME J. Mech. Des., Vol. 128, pp.
729-737.
[6] X. J. Liu, X. Tang, and j. Wang, 2005, "HANA: a novel spatial parallel
manipulator with one rotational and two translational degrees of
freedom, "Robotica, Vol. 23, No. 2, pp. 257-70.
[7] K. M. Lee and S. Arjunan, 1991, "A three-degrees-of-freedom
micromotion in parallel actuated manipulator, "IEEE Trans Robot
Automat, Vol. 7, No. 5, pp. 634-41.
[8] D. Liu, R. Che, Z. Li, and Luo, X., 2003, "Research on the theory and
the virtual prototype of 3-DOF parallel-link coordinating-measuring
machine, "IEEE Trans Instrum Meas, Vol. 52, No. 1, pp. 119-25.
[9] J. A. Carretero, R. P. Podhorodeski, M. A. Nahon, and C. M. Gosselin, ,
2000, "Kinematic analysis and optimization of a new three degree-of
freedom spatial parallel manipulator, "ASME J Mech Des., Vol. 122,
No. 1, pp. 17-24.
[10] M. S. Tsai, T. N. Shiau, Y. J. Tsai, and T. H. Chang, 2003, "Direct
kinematic analysis of a 3-PRS parallel mechanism, "Mech. Mach
Theory, Vol. 38, No. 1, pp. 71-83.
[11] I. A. Bonev and J. Ryu, 2000, "A new method for solving the direct
kinematics of general 6-6 Stewart Platforms, "Mech. Mach Theory, Vol.
35, pp. 423-436.
[12] J. H. He, 2000, "A coupling method of a homotopy technique and a
perturbation technique for nonlinear problems, "International journal of
non-linear mechanics 35, 37-43.
[13] J. H. He, 1999, "Variational iteration method-a kind of non-linear
analytical technique: some examples, "International journal of non-linear
mechanics, Vol. 34, pp. 699-708.
[14] T. M. Wu, 2006, "The inverse kinematics problem of spatial 4P3R
robot manipulator by the homotopy continuation method with an
adjustable auxiliary homotopy function, "Nonlinear Analysis, Vol. 64,
pp. 2373-2380.
[15] T. M. Wu, 2005, "A study of convergence on the Newton- homotopy
continuation method, "Applied Mathematics and Computation, Vol. 168,
pp. 1169-1174.
[16] T. M. Wu, 2006, "Solving the nonlinear equations by the Newtonhomotopy
continuation method with adjustable auxiliary homotopy
function, "Applied Mathematics and Computation, Vol. 173, pp. 383-
388.
[17] D. D. Ganji, G. A. Afrouzi, and R. A. Talarposhti, 2007, "Application of
variational iteration method and homotopy-perturbation, "Physics
Letters A, Vol. 368, pp. 450-457.
[18] A. P. Morgan, 1981, "A method for computing all Solutions to Systems
of Polynomial Equations, "GM Research Publication, GMR 3651.
[19] A. P. Morgan and C. W. Wampler, 1990, "Solving a planar four-bar
design problem using continuation, "ASME J. Mech. Des., Vol. 112, pp.
544 -550.
[20] C. B. Garcia and W. I. Zangwill, 1981, "Pathways to solutions, fixed
points, and equilibria, "Prentice-Hall Book Company, Inc., Englewood
Cliffs, New Jersey.
[21] E. L. Allgower and K. Georg, 1990, "Numerical Continuation Methods,
An Introduction, "Springer, NewYork.
[22] S. M. Varedi, H. M. Daniali, and D. D. Ganji, 2008, "Kinematics of an
offset 3-UPU translational parallel manipulator by the homotopy
continuation method, "Nonlinear Analysis: Real World Applications,
doi:10.1016/j.nonrwa.2008.02.014.
[23] Y. Li. and Q. Xu, 2007, "Kinematic analysis of a 3-PRS parallel
manipulator, "Robotics and Computer-Integrated Manufacturing, Vol.
23, pp. 395-408.
[1] B. Dasgupta and T. S. Mruthyunjaya, 2000, "The stewart platform
manipulator: a review, "Mech. Mach Theory, Vol. 35, pp. 15-40.
[2] M. Griffis and J. Duffy, 1989, "A forward displacement analysis of a
class of Stewart Platforms, "Journal of Robotic Systems, Vol. 6, No. 6,
pp. 703-720.
[3] L. W. Tsai, G. C. Walsh, and R. E. Stamper, 1996, "Kinematics of a
novel three DOF translational platform, "Proceedings of IEEE
international conference on robotics and automation, Minneapolis,
Minnesota, pp. 3446-51.
[4] R. Di. Gregorio and V. Parenti-Castelli, 1998, "A translational 3-DOF
parallel manipulator, "In: J. Lenarcic, M. L. Husty, (Eds), Advances in
robot kinematics: analysis and control. Dordrecht: Kluwer Academic
Publishers, pp. 49-58.
[5] Y. Li and Q. Xu, 2006, "Kinematic analysis and design of a new 3-DOF
translational parallel manipulator, "ASME J. Mech. Des., Vol. 128, pp.
729-737.
[6] X. J. Liu, X. Tang, and j. Wang, 2005, "HANA: a novel spatial parallel
manipulator with one rotational and two translational degrees of
freedom, "Robotica, Vol. 23, No. 2, pp. 257-70.
[7] K. M. Lee and S. Arjunan, 1991, "A three-degrees-of-freedom
micromotion in parallel actuated manipulator, "IEEE Trans Robot
Automat, Vol. 7, No. 5, pp. 634-41.
[8] D. Liu, R. Che, Z. Li, and Luo, X., 2003, "Research on the theory and
the virtual prototype of 3-DOF parallel-link coordinating-measuring
machine, "IEEE Trans Instrum Meas, Vol. 52, No. 1, pp. 119-25.
[9] J. A. Carretero, R. P. Podhorodeski, M. A. Nahon, and C. M. Gosselin, ,
2000, "Kinematic analysis and optimization of a new three degree-of
freedom spatial parallel manipulator, "ASME J Mech Des., Vol. 122,
No. 1, pp. 17-24.
[10] M. S. Tsai, T. N. Shiau, Y. J. Tsai, and T. H. Chang, 2003, "Direct
kinematic analysis of a 3-PRS parallel mechanism, "Mech. Mach
Theory, Vol. 38, No. 1, pp. 71-83.
[11] I. A. Bonev and J. Ryu, 2000, "A new method for solving the direct
kinematics of general 6-6 Stewart Platforms, "Mech. Mach Theory, Vol.
35, pp. 423-436.
[12] J. H. He, 2000, "A coupling method of a homotopy technique and a
perturbation technique for nonlinear problems, "International journal of
non-linear mechanics 35, 37-43.
[13] J. H. He, 1999, "Variational iteration method-a kind of non-linear
analytical technique: some examples, "International journal of non-linear
mechanics, Vol. 34, pp. 699-708.
[14] T. M. Wu, 2006, "The inverse kinematics problem of spatial 4P3R
robot manipulator by the homotopy continuation method with an
adjustable auxiliary homotopy function, "Nonlinear Analysis, Vol. 64,
pp. 2373-2380.
[15] T. M. Wu, 2005, "A study of convergence on the Newton- homotopy
continuation method, "Applied Mathematics and Computation, Vol. 168,
pp. 1169-1174.
[16] T. M. Wu, 2006, "Solving the nonlinear equations by the Newtonhomotopy
continuation method with adjustable auxiliary homotopy
function, "Applied Mathematics and Computation, Vol. 173, pp. 383-
388.
[17] D. D. Ganji, G. A. Afrouzi, and R. A. Talarposhti, 2007, "Application of
variational iteration method and homotopy-perturbation, "Physics
Letters A, Vol. 368, pp. 450-457.
[18] A. P. Morgan, 1981, "A method for computing all Solutions to Systems
of Polynomial Equations, "GM Research Publication, GMR 3651.
[19] A. P. Morgan and C. W. Wampler, 1990, "Solving a planar four-bar
design problem using continuation, "ASME J. Mech. Des., Vol. 112, pp.
544 -550.
[20] C. B. Garcia and W. I. Zangwill, 1981, "Pathways to solutions, fixed
points, and equilibria, "Prentice-Hall Book Company, Inc., Englewood
Cliffs, New Jersey.
[21] E. L. Allgower and K. Georg, 1990, "Numerical Continuation Methods,
An Introduction, "Springer, NewYork.
[22] S. M. Varedi, H. M. Daniali, and D. D. Ganji, 2008, "Kinematics of an
offset 3-UPU translational parallel manipulator by the homotopy
continuation method, "Nonlinear Analysis: Real World Applications,
doi:10.1016/j.nonrwa.2008.02.014.
[23] Y. Li. and Q. Xu, 2007, "Kinematic analysis of a 3-PRS parallel
manipulator, "Robotics and Computer-Integrated Manufacturing, Vol.
23, pp. 395-408.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:62283", author = "Ghasem Abbasnejad and Soheil Zarkandi and Misagh Imani", title = "Forward Kinematics Analysis of a 3-PRS Parallel Manipulator", abstract = "In this article the homotopy continuation method (HCM) to solve the forward kinematic problem of the 3-PRS parallel manipulator is used. Since there are many difficulties in solving the system of nonlinear equations in kinematics of manipulators, the numerical solutions like Newton-Raphson are inevitably used. When dealing with any numerical solution, there are two troublesome problems. One is that good initial guesses are not easy to detect and another is related to whether the used method will converge to useful solutions. Results of this paper reveal that the homotopy continuation method can alleviate the drawbacks of traditional numerical techniques.
", keywords = "Forward kinematics, Homotopy continuationmethod, Parallel manipulators, Rotation matrix", volume = "4", number = "1", pages = "91-7", }
