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Simultaneous path planning and trajectory optimization for robotic manipulators using discrete mechanics and optimal control

Published online by Cambridge University Press:  18 September 2014

Zeeshan Shareef  and
Ansgar Trächtler
Zeeshan Shareef*
Affiliation:
Department of Control Engineering and Mechatronics, Heinz Nixdorf Institute, University of Paderborn, Fürstenallee 11, 33102 Paderborn, Germany. E-mail: [email protected]
Ansgar Trächtler
Affiliation:
Department of Control Engineering and Mechatronics, Heinz Nixdorf Institute, University of Paderborn, Fürstenallee 11, 33102 Paderborn, Germany. E-mail: [email protected]
*
*Corresponding author. E-mail: [email protected]
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Summary

In robotics, path planning and trajectory optimization are usually performed separately to optimize the path from the given starting point to the ending point in the presence of obstacles. In this paper, path planning and trajectory optimization for robotic manipulators are solved simultaneously by a newly developed methodology called Discrete Mechanics and Optimal Control (DMOC). In DMOC, the Lagrange–d'Alembert equation is discretized directly unlike the conventional variational optimization method in which first the Euler–Lagrange equations are derived and then discretization takes place. In this newly developed method, the constraints for optimization of a desired objective function are the forced discrete Euler–Lagrange equations. In this paper, DMOC is used for simultaneous path planning and trajectory optimization for robotic manipulators in the presence of known static obstacles. Two numerical examples, applied on a DELTA parallel robot, are discussed to show the applicability of this new methodology. The optimal results obtained from DMOC are compared with the other state-of-the-art techniques. The difficulties and problems associated in using the DMOC for Parallel Kinematic Machine (PKM) are also discussed in this paper.

Keywords

Robotic manipulatorsPath planningOptimizationDiscrete mechanicsConstraints
Type
Articles
Information
Robotica , Volume 34 , Issue 6 , June 2016 , pp. 1322 - 1334
Copyright
Copyright © Cambridge University Press 2014 

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References

1.Latombe, J., Robot Motion Planning (Kluwer Academic Publishers, Boston/Dordrecht/London, 1991).Google Scholar
2.Nagata, F. and Watanabe, K., Controller Design for Industrial Robots and Machine Tools, Cambridge, United Kingdom (Woodhead Publishing, 2013).Google Scholar
3.Bohlin, R. and Kavraki, E. E., “Path Planning Using Lazy PRM,” Proceedings of International Conference on Robotics and Automation, vol. 1, San Francisco, USA (Apr. 24–28, 2000) pp. 521–528.Google Scholar
4.Warren, C. W., “Global Path Planning Using Artificial Potential Fields,” Proceedings of International Conference on Robotics and Automation, vol. 1, Scottsdale, USA (May 14–19, 1989) pp. 316–321.Google Scholar
5.Conkur, C. A., “Path planning using potential fields for highly redundant manipulators,” Robot. Auton. Syst. 52 (2–3), 209–228 (2005).Google Scholar
6.Kindel, R., Hsu, D., Latombe, J.-C. and Rock, S., “Kinodynamic Motion Planning Amidst Moving Obstacles,” Proceedings of International Conference on Robotics and Automation, vol. 1, San Francisco, USA (Apr. 24–28, 2000) pp. 537–543.Google Scholar
7.van den Berg, J. and Overmars, M., “Kinodynamic Motion Planning on Roadmaps in Dynamic Environments,” Proceedings of International Conference on Intelligent Robots and Systems, San Diego, USA (Oct. 29–Nov. 2, 2007) pp. 4253–4258.Google Scholar
8.Belghith, K., Kabanza, F., Hartman, L. and Nkambou, R., “Anytime Dynamic Path-Planning with Flexible Probabilistic Roadmaps,” Proceedings of International Conference on Robotics and Automation, Orlando, USA (May 15–19, 2006) pp. 2372–2377.Google Scholar
9.Poty, A., Melchior, P. and Oustaloup, A., “Dynamic Path Planning for Mobile Robots using Fractional Potential Field,” Proceedings of 1st International Symposium on Control, Communications and Signal Processing, Hammamet, Tunisia (Mar. 21–24, 2004) pp. 557–561.CrossRefGoogle Scholar
10.Zhang, Q., Yue, S., Yin, Q. and Zha, Y., “Dynamic obstacle-avoiding path planning for robots based on modified potential field method,” Intell. Comput. Theories Technol. 7996, 332–342 (2013).Google Scholar
11.Warren, C. W., “Fast Path Planning using Modified A* Method,” Proceedings of International Conference on Robotics and Automation, Atlanta, USA (May 2–6, 1993) pp. 662–667.Google Scholar
12.Stentz, A., “Optimal and Efficient Path Planning for Partially-known Environments,” Proceedings of International Conference on Robotics and Automation, vol. 4, San Diego, USA (May 8–13, 1994) pp. 3310–3317.Google Scholar
13.Wang, G., Guo, L., Duan, H., Liu, L. and Wang, H., “A bat algorithm with mutation for UCAV path planning,” Sci. World J. 2012, 1–15 (2012).Google Scholar
14.Saska, K., Macas, M., Preucil, L. and Lhotska, L., “Robot Path Planning using Particle Swarm Optimization of Ferguson Splines,” Proceedings of International Conference on Emerging Technologies and Factory Automation, Prague, Czech Republic (Sep. 20–22, 2006) pp. 833–839.CrossRefGoogle Scholar
15.Wang, L., Liu, Y., Deng, H. and Xu, Y., “Obstacle-Avoidance Path Planning for Soccer Robots Using Particle Swarm Optimization,” Proceedings of International Conference on Robotics and Biomimetics, Kunming, P. R. China (Dec. 17–20, 2006) pp. 1233–1238.Google Scholar
16.Cong, Y. Z. and Ponnambalam, S. G., “Mobile Robot Path Planning using Ant Colony Optimization,” Proceedings of International Conference on Advanced Intelligent Mechatronics, Singapore (Jul. 14–17, 2009) pp. 851–856.Google Scholar
17.Hsiao, Y., Chuang, C. and Chien, C., “Ant Colony Optimization for Best Path Planning,” Proceedings of International Symposium on Communications and Information Technology, vol. 1, Sapporo, Japan (Oct. 26–29, 2004) pp. 109–113.Google Scholar
18.Bhattacharjee, P., Rakshit, P., Goswami, I., Konar, A. and Nagar, A. K., “Multi-Robot Path-Planning using Artificial Bee Colony Optimization Algorithm,” Proceedings of 3rd World Congress on Nature and Biologically Inspired Computing, Salamanca, Spain (Oct. 19–21, 2011) pp. 219–224.Google Scholar
19.liu, C., Gao, Z. and Zhao, W., “A New Path Planning Method Based on Firefly Algorithm,” Proceedings of 5th International Joint Conference on Computational Sciences and Optimization, Harbin, P. R. China (Jun. 23–26, 2012) pp. 775–778.Google Scholar
20.Sontag, E. D. and Sussmann, H. J., “Time-optimal Control of Manipulators,” Proceedings of International Conference on Robotics and Automation, vol. 3, San Francisco, USA (Apr. 7–10, 1986) pp. 1692–1697.Google Scholar
21.Bobrow, J. E., Dubowsky, S. and Gibson, S. J., “Time-optimal control of robotic manipulators along specified paths,” Int. J. Robot. Res. 4 (3), 3–17 (1985).Google Scholar
22.Shin, K. G. and McKay, N. D., “Minimum-time control of robotic manipulators with geometric path constraints,” IEEE Trans. Autom. Control 30 (6), 531–541 (1985).Google Scholar
23.Rajan, V., “Minimum Time Trajectory Planning,” Proceedings of International Conference on Robotics and Automation, vol. 2, St. Louis, USA (Mar. 25–28, 1985) pp. 759–764.Google Scholar
24.Pfeiffer, F. and Johanni, R., “A Concept for Manipulator Trajectory Planning,” Proceedings of International Conference on Robotics and Automation, vol. 3, San Francisco, USA (Apr. 7–10, 1986) pp. 1399–1405.Google Scholar
25.Slotine, J.-J. E. and Yang, H. S., “Improving the efficiency of time-optimal path-following algorithms,” IEEE Trans. Robot. Autom. 5 (1), 118–124 (1989).Google Scholar
26.Shiller, Z. and Lu, H. H., “Computation of path constrained time optimal motions with dynamic singularities,” J. Dyn. Syst. Meas. Control 114 (1), 34–40 (1992).CrossRefGoogle Scholar
27.Tarkiainen, M. and Shiller, Z., “Time Optimal Motions of Manipulators with Actuator Dynamics,” Proceedings of International Conference on Robotics and Automation, vol. 2, Atlanta, USA (May 2–6, 1993) pp. 725–730.Google Scholar
28.Bellman, R., Dynamic Programming (Princeton University Press, Princeton, N.J., 1957).Google Scholar
29.Vukobratovic, M. and Kircanski, M., “A method for optimal synthesis of manipulation robot trajectories,” J. Dyn. Syst. Meas. Control 104 (2), 188–193 (1982).Google Scholar
30.Shin, G. K. and McKay, N. D., “A dynamic programming approach to trajectory planning of robotic manipulators,” IEEE Trans. Autom. Control 31 (6), 491–500 (1986).Google Scholar
31.Singh, S. and Leu, M. C., “Optimal trajectory generation for robotic manipulators using dynamic programming,” J. Dyn. Sys., Meas., Control 109 (2), 88–96 (1987).Google Scholar
32.Shareef, Z. and Trächtler, A., “Joint Selection Criterion for Optimal Trajectory Planning for Robotic Manipulators using Dynamic Programming,” Proceedings of 19th IFAC World Congress, Cape Town, South Africa (Aug. 24–29, 2014) pp. 6025–6031.Google Scholar
33.Yen, V., “An inverse dynamic-based dynamic programming method for optimal point-to-point trajectory planning of robotic manipulators,” Int. J. Syst. Sci. 26 (2), 181–195 (1995).Google Scholar
34.Field, G. and Stepanenko, Y., “Iterative Dynamic Programming: An Approach to Minimum Energy Trajectory Planning for Robotic Manipulators,” Proceedings of International Conference on Robotics and Automation, vol. 3, Minneapolis, USA (Apr. 22–28, 1996) pp. 2755–2760.Google Scholar
35.Junge, O., Marsden, J. E. and Ober-Blöbaum, S., “Discrete Mechanics and Optimal Control,” Proceedings of 16th IFAC World Congress, Prague, Czech Republic (Jul. 4–8, 2005) pp. 744–749.Google Scholar
36.Cadzow, J. A., “Discrete calculus of variations,” Int. J. Control 11, 393–407 (1970).Google Scholar
37.Jordan, B. W. and Polak, E., “Theory of a class of discrete optimal control systems,” J. Electr. Control 17, 697–711 (1964).Google Scholar
38.Hwang, C. L. and Fan, L. T., “A discrete version of Pontryagins maximum principle,” Oper. Res. 15, 139–146 (1967).Google Scholar
39.Marsden, J. E. and West, M., “Discrete mechanics and variational integrators,” Acta Numer. 10, 357–514 (2001).CrossRefGoogle Scholar
40.Ober-Blöbaum, S., Junge, O. and Marsden, J. E., “Discrete mechanics and optimal control: an analysis,” Control Optim. Calculus Variations 17, 322–352 (2011).Google Scholar
41.Pekarek, D., Ames, A. D. and Marsden, J. E., “Discrete Mechanics and Optimal Control Applied to the Compass Gait Biped,” Proceedings of Conference on Decision and Control, New Orleans, USA (Dec. 12–14, 2007) pp. 5376–5382.Google Scholar
42.McLachlan, R. I. and Marsland, S., “Discrete Mechanics and Optimal Control for Image Registration,” ANZIAM J. 48, 1–15 (2007).CrossRefGoogle Scholar
43.Timmermann, J., Khattab, S., Ober-Blöbaum, S. and Trächtler, A., “Discrete Mechanics and Optimal Control and its Application to a Double Pendulum on a Cart,” Proceedings of the 18th IFAC World Congress, Milan, Italy (Aug. 28–Sep. 2, 2011) pp. 10199–10206.Google Scholar
44.Ober-Blöbaum, S. and Timmermann, J., “Optimal Control for a Pitcher's Motion Modeled as Constrained Mechanical System,” Proceedings of International Conference on Multibody Systems, Nonlinear Dynamics, and Control, San Diego, USA (Aug. 30–Sep. 2, 2009) pp. 597–606.Google Scholar
45.Junge, O., Marsden, J. E. and Ober-Blöbaum, S., “Optimal Reconfiguration of Formation Flying Spacecraft – A Decentralized Approach,” Proceedings of the 45th Conference on Decision and Control, San Diego, USA (Dec. 13–15, 2006) pp. 5210–5215.Google Scholar
46.Junge, O. and Ober-Blöbaum, S., “Optimal Reconfiguration of Formation Flying Satellites,” Proceedings of the 44th Conference on Decision and Control and European Control Conference (Dec. 12–15, 2005) pp. 66–71.Google Scholar
47.Codourey, A., “Dynamic Modelling and Mass Matrix Evaluation of the DELTA Parallel Robot for Axes Decoupling Control,” Proceedings of the International Conference on Intelligent Robots and Systems, vol. 3, Osaka, Japan (Nov. 4–8, 1996) pp. 1211–1218.Google Scholar
48.Zhang, C. and Song, S., “An efficient method for inverse dynamics of manipulators based on the virtual work principle,” J. Robot. Syst. 10 (5), 605–627 (1993).Google Scholar
49.Shareef, Z. and Trächtler, A., “Optimal Trajectory Planning for Robotic Manipulators using Discrete Mechanics and Optimal Control (DMOC),” Proceedings of Multi-Conference on Systems and Control (2014). [Under review]Google Scholar
50.Stamper, R. E., A Three Degree of Freedom Parallel Manipulator with Only Translational Degrees of Freedom Ph.D. Thesis (College Park, MD: Institute of System Research, University of Maryland, 1997).Google Scholar