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Optimal trajectory planning for nonlinear systems: robust and constrained solution

Published online by Cambridge University Press:  15 August 2014

P. Boscariol  and
A. Gasparetto
P. Boscariol*
Affiliation:
Dipartimento di Ingegneria Elettrica, Gestionale e Meccanica University of Udine, Italy Via delle Scienze 208, 33100 Udine
A. Gasparetto
Affiliation:
Dipartimento di Ingegneria Elettrica, Gestionale e Meccanica University of Udine, Italy Via delle Scienze 208, 33100 Udine
*
*Corresponding author. E-mail: paolo.boscariol@uniud.it
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Summary

This paper presents a solution to the problem of generating constrained robust trajectory planning for nonlinear mechatronic systems. By using an indirect variational solution method, the necessary optimality conditions deriving from the Pontryagin's minimum principle are imposed, and lead to a differential Two-Point Boundary Value Problem (TPBVP); numerical solution of the latter is accomplished by means of collocation techniques. The robustness to parametric mismatches is obtained trough the use of sensitivity functions, while a hard constraint on actuator effort is obtained using a smoothing technique. Numerical results shows that the robustness can be greatly improved, and that the inclusion of constraints on actuator effort does not affect it.

Keywords

Motion PlanningRobustnessRobot DynamicsMechatronic SystemsPath Planning
Type
Articles
Information
Robotica , Volume 34 , Issue 6 , June 2016 , pp. 1243 - 1259
Copyright
Copyright © Cambridge University Press 2014 

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References

1.Siciliano, B. and Khatib, O., Springer Handbook of Robotics (Springer, Berlin, 2008).Google Scholar
2.Bischoff, R., Kurth, J., Schreiber, G., Koeppe, R., Albu-Schäffer, A., Beyer, A., Eiberger, O., Haddadin, S., Stemmer, A., Grunwald, G., et al., “The Kuka-dlr Lightweight Robot Arm-a New Reference Platform for Robotics Research and Manufacturing,” Robotics (ISR), 2010 41st International Symposium on and 2010 6th German Conference on Robotics (ROBOTIK) VDE (2010) pp. 1–8.Google Scholar
3.Moberg, S., On Modeling and Control of Flexible Manipulators Ph.D. Thesis (Department of Electrical Engineering, Linköping University, Sweden, 2007).Google Scholar
4.De Luca, A. and Di Giovanni, G., “Rest-to-rest Motion of a Two-link Robot with a Flexible Forearm,” Proceedings. 2001 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 2001, vol. 2, IEEE (2001) pp. 929–935.Google Scholar
5.Benosman, M. and Le Vey, G., “Control of flexible manipulators: A survey,” Robotica 22 (05), 533–545 (2004).Google Scholar
6.Boschetti, G., Richiedei, D. and Trevisani, A., “Delayed reference control for multi-degree-of-freedom elastic systems: Theory and experimentation,” Control Eng. Pract. 19 (9), 1044–1055 (2011).CrossRefGoogle Scholar
7.Ata, A. A., “Optimal trajectory planning of manipulators: A review,” J. Eng. Sci. Technol. 2 (1), 32–54 (2007).Google Scholar
8.Gasparetto, A., Boscariol, P., Lanzutti, A. and Vidoni, R., “Trajectory planning in robotics,” Math. Comput. Sci. 1–11 6 (3), 269–279 (2012).Google Scholar
9.Zanotto, V., Gasparetto, A., Lanzutti, A., Boscariol, P. and Vidoni, R., “Experimental validation of minimum time-jerk algorithms for industrial robots,” J. Intell. Robot. Syst. 64 (2), 197–219 (2011).Google Scholar
10.Caracciolo, R., Richiedei, D. and Trevisani, A., “Experimental validation of a model-based robust controller for multi-body mechanisms with flexible links,” Multibody Syst. Dyn. 20 (2), 129–145 (2008).Google Scholar
11.Islam, S. and Liu, X. P., “Robust sliding mode control for robot manipulators,” IEEE Trans. Ind. Electron. 58 (6), 2444–2453 (2011).Google Scholar
12.Biagiotti, L. and Melchiorri, C., Trajectory Planning for Automatic Machines and Robots (Springer, Berlin, 2008).Google Scholar
13.Boryga, M. and Graboś, A., “Planning of manipulator motion trajectory with higher-degree polynomials use,” Mech. Mach. Theory 44 (7), 1400–1419 (2009).Google Scholar
14.Louembet, C., Cazaurang, F. and Zolghadri, A., “Motion planning for flat systems using positive b-splines: An lmi approach,” Automatica 46 (8), 1305–1309 (2010).Google Scholar
15.Lambrechts, P., Boerlage, M. and Steinbuch, M., “Trajectory planning and feedforward design for electromechanical motion systems,” Control Eng. Pract. 13 (2), 145–157 (2005).Google Scholar
16.Balkcom, D. and Mason, M., “Time optimal trajectories for bounded velocity differential drive vehicles,” Int. J. Robot. Res. 21 (3), 199–217 (2002).CrossRefGoogle Scholar
17.Dahl, O., “Path Constrained Motion Optimization for Rigid and Flexible Joint Robots,” Proceedings of the International Conference on Robotics and Automation, vol. 2, IEEE (1993) pp. 223–229.Google Scholar
18.Korayem, M., Nohooji, H. R. and Nikoobin, A., “Optimal motion generating of nonholonomic manipulators with elastic revolute joints in generalized point-to-point task,” Int. J. Adv. Des. Manuf. Technol. 3 (2), 1–9 (2010).Google Scholar
19.Abe, A., “Trajectory planning for residual vibration suppression of a two-link rigid-flexible manipulator considering large deformation,” Mech. Mach. Theory 44 (9), 1627–1639 (2009).Google Scholar
20.Korayem, M. H., Nikoobin, A. and Azimirad, V., “Trajectory optimization of flexible link manipulators in point-to-point motion,” Robotica 27 (6), 825 (2009).Google Scholar
21.Korayem, M., Haghpanahi, M., Rahimi, H. and Nikoobin, A., “Finite Element Method and Optimal Control Theory for Path Planning of Elastic Manipulators,” In: New Advances in Intelligent Decision Technologies (Springer, 2009) pp. 117–126.Google Scholar
22.Boscariol, P. and Gasparetto, A., “Model-based trajectory planning for flexible-link mechanisms with bounded jerk,” Robot. Comput.-Integr. Manuf. 29, 90–99 (2013).Google Scholar
23.Ackermann, J., Bartlett, A., Kaesbauer, D., Sienel, W. and Steinhauser, R., Robust Control (Springer, 1993).Google Scholar
24.Karkoub, M., Balas, G., Tamma, K. and Donath, M., “Robust control of flexible manipulators via μsynthesis,” Control Eng. Pract. 8 (7), 725–734 (2000).Google Scholar
25.Tomei, P., “Robust adaptive friction compensation for tracking control of robot manipulators,” IEEE Trans. Autom. Control 45 (11), 2164–2169 (2000).Google Scholar
26.Gallardo, D., Colomina, O., Flórez, F. and Rizo, R., “A genetic algorithm for robust motion planning,” Tasks Methods Appl. Artif. Intell. 115–121 (1998).Google Scholar
27.Shin, K. G. and McKay, N. D., “Robust trajectory planning for robotic manipulators under payload uncertainties,” IEEE Trans. Autom. Control 32 (12), 1044–1054 (1987).Google Scholar
28.Houska, B., Robust Optimization of Dynamic Systems Ph.D. Thesis (Katholieke Universiteit Leuven, 2011).(ISBN: 978-94-6018-394-2), 2011.Google Scholar
29.Singh, T., Optimal Reference Shaping for Dynamical Systems: Theory and Applications (CRC PressI Llc, Boca-Raton, Florida, 2010).Google Scholar
30.Hindle, T. A. and Singh, T., “Desensitized Minimum Power/Jerk Control Profiles for Rest-to-rest Maneuvers,” Proceedings of the 2000 American Control Conference, 2000, vol. 5, IEEE (2000) pp. 3064–3068.Google Scholar
31.Kased, R. and Singh, T., “Rest-to-rest Motion of an Experimental Flexible Structure Subject to Friction: Linear Programming Approach,” Proceedings of the AIAA Guidance, Navigation and Control Conference, San Francisco, CA (2005).Google Scholar
32.Boschetti, G., Caracciolo, R., Richiedei, D. and Trevisani, A., “Moving the suspended load of an overhead crane along a pre-specified path: A non-time based approach,” Robot. Comput.-Integr. Manuf. 30 (3), 256–264 (2014).Google Scholar
33.Pontryagin, L. and Gamkrelidze, R., The Mathematical Theory of Optimal Processes, vol. 4 (CRC, 1986).Google Scholar
34.Shampine, L., Gladwell, I. and Thompson, S., Solving ODEs with MATLAB (Cambridge University Press, Cambridge, 2003).Google Scholar
35.Holsapple, R. W., “A modified simple shooting method for solving two-point boundary value problems,” Master of Science Thesis (Texas Tech University, 2012).Google Scholar
36.Stein, W., et al., “Sage: Open source mathematical software,” (2008).Google Scholar
37.Jenks, R. D., Sutor, R. S. and Morrison, S. C., AXIOM: the Scientific Computation System, vol. 6 (Springer-Verlag, New York, 1992).Google Scholar
38.Moberg, S. and Öhr, J., “Robust Control of a Flexible Manipulator Arm: A Benchmark Problem,” Proceedings of the 16th IFAC World Congress, Prague, Czech Republic (2005).Google Scholar
39.Axelsson, P., “Evaluation of Six Different Sensor Fusion Methods for an Industrial Robot Using Experimental Data,” Proceedings of the 10th International IFAC Symposium on Robot Control, Dubrovnik, Croatia (2012).Google Scholar
40.Axelsson, P., Norrlöf, M., Wernholt, E. and Gustafsson, F., “Extended Kalman Filter Applied to Industrial Manipulators,” Proceedings of Reglermöte 2010, Lund, Sweden (2010).Google Scholar
41.Moberg, S., Öhr, J. and Gunnarsson, S., “A Benchmark Problem for Robust Control of a Multivariable Nonlinear Flexible Manipulator,” Proceedings of the 17th IFAC World Congress (2008).Google Scholar
42.Moberg, S., Wernholt, E., Hanssen, S. and Brogårdh, T., “Modeling and Parameter Estimation of Robot Manipulators using Extended Flexible Joint Models,” J. Dyn. Syst. Meas. Control 136 (3), 031005 (2014).Google Scholar
43.Spong, M. W., “Modeling and control of elastic joint robots,” J. Dyn. Syst. Meas. Control 109 (4), 310–318 (1987).Google Scholar
44.Shampine, L. F., Kierzenka, J. and Reichelt, M. W., “Solving boundary value problems for ordinary differential equations in matlab with bvp4c,” Tutorial Notes (2000).Google Scholar
45.Avvakumov, S. and Kiselev, Y. N., “Boundary value problem for ordinary differential equations with applications to optimal control,” Spectral Evol. Probl. 2000 10 (2000).Google Scholar