Forward Kinematics Solution of Stewart-Gough using Improved Hybrid Strategy (Neural Network and 3rd-order Newton-Raphson)
Authors
Abstract
Many efforts have been done in recent years to decrease the required time for analysis of FKP (Forward Kinematics
Problem) of parallel robots.This paper starts with developing kinematics of a parallel robot and finishes with a suggested
algorithm to solve forward kinematics of robots. In this paper, by combining the artificial neural networks and a 3rd-order
numerical algorithm, an improved hybrid strategy is proposed in order to increase the accuracy and speed of forward kinematics
analysis of parallel manipulators. First, an approximate solution of the forward kinematics problem is produced by the neural
network. This approximate solution is then considered as the initial guess for the 3rd-order Newton-Raphson numerical
technique. By applying Stewart-Gough parallel manipulator, the efficiency of the proposed method is evaluated. It is shown that
replacing the Newton-Raphson algorithm by the 3rd-order one leads to a reduction of the iterations required to reach the desired
accuracy level and thus a reduction of the FKP analysis time. Finally, Stewart robot is used to simulate the movement of jaw.
This novel algorithm can be applied to any forward kinematics of serial or parallel robots.
Problem) of parallel robots.This paper starts with developing kinematics of a parallel robot and finishes with a suggested
algorithm to solve forward kinematics of robots. In this paper, by combining the artificial neural networks and a 3rd-order
numerical algorithm, an improved hybrid strategy is proposed in order to increase the accuracy and speed of forward kinematics
analysis of parallel manipulators. First, an approximate solution of the forward kinematics problem is produced by the neural
network. This approximate solution is then considered as the initial guess for the 3rd-order Newton-Raphson numerical
technique. By applying Stewart-Gough parallel manipulator, the efficiency of the proposed method is evaluated. It is shown that
replacing the Newton-Raphson algorithm by the 3rd-order one leads to a reduction of the iterations required to reach the desired
accuracy level and thus a reduction of the FKP analysis time. Finally, Stewart robot is used to simulate the movement of jaw.
This novel algorithm can be applied to any forward kinematics of serial or parallel robots.
Keywords
1. Patel, A. J., and Ehmann, K. F., “Calibration of a
Hexapod Machine Tool Using a Redundant Leg”,
International Journal of Machine Tools &
Manufacture, Vol. 40, pp. 489-512, 2000.
2. Enferadi, J., and Akbarzadeh Tootoonchi, A.,
“Accuracy and Stiffness Analysis of a 3-RRP
Spherical Parallel Manipulator”, Robotica, Vol. 29,
pp. 193-209, 2011.
3. Enferadi, J., and Akbarzadeh, A., “A Novel
Approach for Forward Position Analysis of a
Double-Triangle Spherical Parallel Manipulator”,
European Journal of Mechanics- A/Solids, Vol. 29,
No. 3, pp. 348-355, 2010.
4. Kamali, K., and Akbarzadeh, A., “A Novel Method
for Direct Kinematics Solution of Fully Parallel
Manipulators Using Basic Regions Theory”, Journal
of Systems and Control Engineering, Vol. 225,
pp. 683-701. 2011.
5. Gregorio, R. Di., and Parenti-Castelli, V., “Position
Analysis in Analytical Form of the 3-PSP
Mechanism”, Journal of Mechanical Design, Vol.
123, pp. 51-57, 2001.
6. Rezaei, A., Akbarzadeh, A., and Mahmoodi Nia, P.,
“Position, Jacobian and Workspace Analysis of a 3-
PSP Spatial Parallel Manipulator”, Robotics and
Computer-Integrated Manufacturing, Vol. 29,
pp. 158-173, 2013.
7. Zhang, Y., Liu, H., and Wu, X., “Kinematics
Analysis of a Novel Parallel Manipulator”,
Mechanism and Machine Theory, Vol. 44, pp. 1648-
1657, 2009.
8. Merlet, J. P., “Direct Kinematics of Parallel
Manipulators”. Robotics and Automation, IEEE
Transactions on, Vol. 9, No. 6, pp. 842-846, 1993.
9. Der-Ming, Ku., “Direct Displacement Analysis of a
Stewart Platform Mechanism”, Mechanism and
Machine Theory, Vol. 34, No. 3, pp. 453-465, 1999.
10. Sadjadian, H., and Taghirad, H. D., “Numerical
Methods for Computing the Forward Kinematics of a
Redundant Parallel Manipulator”, Proceedings of the
IEEE Conference on Mechatronics and Robotics,
Aachen, Germany, 2004.
11. Liu, K., Fitzgerald, J. M., and Lewis., F. L.,
“Kinematic Analysis of a Stewart Platform
Manipulator”, IEEE Transactions on Industrial
Electronics, Vol. 40, No. 2, pp. 282-293, 1993.
12. Li, Y., and Xu, Q., “Kinematic Analysis of a 3-PRS
Parallel Manipulator”, Robotics and Computer-
Integrated Manufacturing, Vol. 23, No. 4, pp.395-
408, 2007.
13. Cheng, H. H., Lee, J. J., and Penkar, R., “Kinematic
Analysis of a Hybrid Serial-and-parallel-driven
Redundant Industrial Manipulator”, International
Journal of Robotics and Automation, Vol. 10, No. 4,
pp. 159-166, 1995.
14. Parikh, P. J., and Lam, S. S., “A Hybrid Strategy to
Solve the Forward Kinematics Problem in Parallel
Manipulators”, Robotics, IEEE Transactions on,
Vol. 21, No. 1, pp. 18-25, 2005.
15. Laosiritaworn, W., and Chotchaithanakorn, N.,
“Artificial Neural Networks Parameters Optimization
with Design of Experiments: An Application in
Ferromagnetic Materials Modeling”, Chiang Mai
Journal of Science, Vol. 36, No. 1, pp. 83-91, 2009.
16. Sekar, B. D., Dong, M. C., Shi, J., and Hu, X. Y.,
“Fused Hierarchical Neural Networks for
Cardiovascular Disease Diagnosis”, Sensors Journal,
IEEE, Vol. 12, No. 3, pp. 644-650, 2012.
17. Boudreau, R., Levesque, G., and Darenfed, S.,
“Parallel Manipulator Kinematics Learning Using
Holographic Neural Network Models”, Robotics and
Computer-Integrated Manufacturing, Vol. 14. No. 1,
pp. 37-44, 1998.
18. Sadjadian, H., Taghirad, H. D., and Fatehi, A.,
“Neural Networks Approaches for Computing the
Forward Kinematics of a Redundant Parallel
Manipulator”, International Journal of Computational
Intelligence, Vol. 2, No. 1, pp. 40-47, 2005.
19. Kang, R., Chanal, H., Bonnemains, T., Pateloup, S.,
Branson D. T., and Ray, P., “Learning the Forward
Kinematics Behavior of a Hybrid Robot Employing
Artificial Neural Networks”, Robotica, Vol. 30, No.
5, pp. 847-855, 2012.
20. Raghavan, M., “The Stewart Platform of General
Geometry Has 40 Configurations”, ASME Journal of
Mechanical Design, Vol. 115, No. 2, pp. 277-282,
1993.
21.Husty, M. L., “An Algorithm for Solving the Direct
Kinematics of General Stewart-Gough Platforms”,
Mechanism and Machine Theory, Vol. 31, No. 4,
pp. 365-380, 1996.
22. Innocenti, C., “Forward Kinematics in Polynomial
Form of the General Stewart Platform”, Journal of
Mechanical Design, Vol. 123, No. 2, pp. 254-260,
2001.
23. Dhingra, A., Almadi, A., and Kohli, D., “A Gröbner-
Sylvester Hybrid Method for Closed-Form
Displacement Analysis of Mechanisms”, 25th
Biennial Mechanisms Conference, Atlanta GA,
Paper: DETC98/MECH-5969, 1998.
24. Kardan, I., and Akbarzadeh, A., “An Improved
Hybrid Method for Forward Kinematics Analysis of
Parallel Robots”, Advanced Robotics, Vol. 29, No. 6,
pp. 401-411, 2015.
25.Darvishi, M. T., and Barati., A. “A Third-order
Newton-type Method to Solve Systems of Nonlinear
Equations”, Applied Mathematics and Computation,
Vol. 187, No. 2, pp. 630-635, 2007.
26.Hagan, M. T., and Demuth, H. B., “Neural Networks
for Control”, Preceeding of 1999 American Control
Conference, Vol. 3, No. 5, pp. 1642-1656, 1999.
27. Hagan, M. T., Demuth, H. B., and Jesus, O. De., “An
Introduction to the Use of Neural Networks in
Control Systems”, International Journal of Robust
and Nonlinear Control, Vol. 12, No. 11, pp. 959-985,
2002.
Hexapod Machine Tool Using a Redundant Leg”,
International Journal of Machine Tools &
Manufacture, Vol. 40, pp. 489-512, 2000.
2. Enferadi, J., and Akbarzadeh Tootoonchi, A.,
“Accuracy and Stiffness Analysis of a 3-RRP
Spherical Parallel Manipulator”, Robotica, Vol. 29,
pp. 193-209, 2011.
3. Enferadi, J., and Akbarzadeh, A., “A Novel
Approach for Forward Position Analysis of a
Double-Triangle Spherical Parallel Manipulator”,
European Journal of Mechanics- A/Solids, Vol. 29,
No. 3, pp. 348-355, 2010.
4. Kamali, K., and Akbarzadeh, A., “A Novel Method
for Direct Kinematics Solution of Fully Parallel
Manipulators Using Basic Regions Theory”, Journal
of Systems and Control Engineering, Vol. 225,
pp. 683-701. 2011.
5. Gregorio, R. Di., and Parenti-Castelli, V., “Position
Analysis in Analytical Form of the 3-PSP
Mechanism”, Journal of Mechanical Design, Vol.
123, pp. 51-57, 2001.
6. Rezaei, A., Akbarzadeh, A., and Mahmoodi Nia, P.,
“Position, Jacobian and Workspace Analysis of a 3-
PSP Spatial Parallel Manipulator”, Robotics and
Computer-Integrated Manufacturing, Vol. 29,
pp. 158-173, 2013.
7. Zhang, Y., Liu, H., and Wu, X., “Kinematics
Analysis of a Novel Parallel Manipulator”,
Mechanism and Machine Theory, Vol. 44, pp. 1648-
1657, 2009.
8. Merlet, J. P., “Direct Kinematics of Parallel
Manipulators”. Robotics and Automation, IEEE
Transactions on, Vol. 9, No. 6, pp. 842-846, 1993.
9. Der-Ming, Ku., “Direct Displacement Analysis of a
Stewart Platform Mechanism”, Mechanism and
Machine Theory, Vol. 34, No. 3, pp. 453-465, 1999.
10. Sadjadian, H., and Taghirad, H. D., “Numerical
Methods for Computing the Forward Kinematics of a
Redundant Parallel Manipulator”, Proceedings of the
IEEE Conference on Mechatronics and Robotics,
Aachen, Germany, 2004.
11. Liu, K., Fitzgerald, J. M., and Lewis., F. L.,
“Kinematic Analysis of a Stewart Platform
Manipulator”, IEEE Transactions on Industrial
Electronics, Vol. 40, No. 2, pp. 282-293, 1993.
12. Li, Y., and Xu, Q., “Kinematic Analysis of a 3-PRS
Parallel Manipulator”, Robotics and Computer-
Integrated Manufacturing, Vol. 23, No. 4, pp.395-
408, 2007.
13. Cheng, H. H., Lee, J. J., and Penkar, R., “Kinematic
Analysis of a Hybrid Serial-and-parallel-driven
Redundant Industrial Manipulator”, International
Journal of Robotics and Automation, Vol. 10, No. 4,
pp. 159-166, 1995.
14. Parikh, P. J., and Lam, S. S., “A Hybrid Strategy to
Solve the Forward Kinematics Problem in Parallel
Manipulators”, Robotics, IEEE Transactions on,
Vol. 21, No. 1, pp. 18-25, 2005.
15. Laosiritaworn, W., and Chotchaithanakorn, N.,
“Artificial Neural Networks Parameters Optimization
with Design of Experiments: An Application in
Ferromagnetic Materials Modeling”, Chiang Mai
Journal of Science, Vol. 36, No. 1, pp. 83-91, 2009.
16. Sekar, B. D., Dong, M. C., Shi, J., and Hu, X. Y.,
“Fused Hierarchical Neural Networks for
Cardiovascular Disease Diagnosis”, Sensors Journal,
IEEE, Vol. 12, No. 3, pp. 644-650, 2012.
17. Boudreau, R., Levesque, G., and Darenfed, S.,
“Parallel Manipulator Kinematics Learning Using
Holographic Neural Network Models”, Robotics and
Computer-Integrated Manufacturing, Vol. 14. No. 1,
pp. 37-44, 1998.
18. Sadjadian, H., Taghirad, H. D., and Fatehi, A.,
“Neural Networks Approaches for Computing the
Forward Kinematics of a Redundant Parallel
Manipulator”, International Journal of Computational
Intelligence, Vol. 2, No. 1, pp. 40-47, 2005.
19. Kang, R., Chanal, H., Bonnemains, T., Pateloup, S.,
Branson D. T., and Ray, P., “Learning the Forward
Kinematics Behavior of a Hybrid Robot Employing
Artificial Neural Networks”, Robotica, Vol. 30, No.
5, pp. 847-855, 2012.
20. Raghavan, M., “The Stewart Platform of General
Geometry Has 40 Configurations”, ASME Journal of
Mechanical Design, Vol. 115, No. 2, pp. 277-282,
1993.
21.Husty, M. L., “An Algorithm for Solving the Direct
Kinematics of General Stewart-Gough Platforms”,
Mechanism and Machine Theory, Vol. 31, No. 4,
pp. 365-380, 1996.
22. Innocenti, C., “Forward Kinematics in Polynomial
Form of the General Stewart Platform”, Journal of
Mechanical Design, Vol. 123, No. 2, pp. 254-260,
2001.
23. Dhingra, A., Almadi, A., and Kohli, D., “A Gröbner-
Sylvester Hybrid Method for Closed-Form
Displacement Analysis of Mechanisms”, 25th
Biennial Mechanisms Conference, Atlanta GA,
Paper: DETC98/MECH-5969, 1998.
24. Kardan, I., and Akbarzadeh, A., “An Improved
Hybrid Method for Forward Kinematics Analysis of
Parallel Robots”, Advanced Robotics, Vol. 29, No. 6,
pp. 401-411, 2015.
25.Darvishi, M. T., and Barati., A. “A Third-order
Newton-type Method to Solve Systems of Nonlinear
Equations”, Applied Mathematics and Computation,
Vol. 187, No. 2, pp. 630-635, 2007.
26.Hagan, M. T., and Demuth, H. B., “Neural Networks
for Control”, Preceeding of 1999 American Control
Conference, Vol. 3, No. 5, pp. 1642-1656, 1999.
27. Hagan, M. T., Demuth, H. B., and Jesus, O. De., “An
Introduction to the Use of Neural Networks in
Control Systems”, International Journal of Robust
and Nonlinear Control, Vol. 12, No. 11, pp. 959-985,
2002.
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