Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T08:17:32.343Z Has data issue: false hasContentIssue false
  • English
  • Français

Characteristics analysis and stabilization of a planar 2R underactuated manipulator

Published online by Cambridge University Press:  09 July 2014

Guang-Ping He ,
Zhi-Lü Wang ,
Jie Zhang  and
Zhi-Yong Geng
Guang-Ping He*
Affiliation:
Department of Mechanical and Electrical Engineering, North China University of Technology, Beijing 100144, P. R. China State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Aerospace Engineering, Peking University, Beijing 100871, P. R. China
Zhi-Lü Wang
Affiliation:
Department of Mechanical and Electrical Engineering, North China University of Technology, Beijing 100144, P. R. China
Jie Zhang
Affiliation:
Department of Mechanical and Electrical Engineering, North China University of Technology, Beijing 100144, P. R. China
Zhi-Yong Geng
Affiliation:
State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Aerospace Engineering, Peking University, Beijing 100871, P. R. China
*
*Corresponding author. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Summary

The weightless planar 2R underactuated manipulators with passive last joint are considered in this paper for investigating a feasible method to stabilize the system, which is a second-order nonholonomic-constraint mechanical system with drifts. The characteristics including the controllability of the linear approximation model, the minimum phase property, the Small Time Local Controllability (STLC), the differential flatness, and the exactly nilpotentizable properties, are analyzed. Unfortunately, these negative characteristics indicate that the simplest underactuated mechanical system is difficult to design a stable closed-loop control system. In this paper, nilpotent approximation and iterative steering methods are utilized to solve the problem. A globally effective nilpotent approximation model is developed and the parameterized polynomial input is adopted to stabilize the system to its non-singularity equilibrium configuration. In accordance with this scheme, it is shown that designing a stable closed-loop control system for the underactuated mechanical system can be ascribed to solving a set of nonlinear algebraic equations. If the nonlinear algebraic equations are solvable, then the controller is asymptotically stable. Some numerical simulations demonstrate the effectiveness of the presented approach.

Keywords

UnderactuationManipulatorsNilpotent approximationIterative steeringControl
Type
Articles
Information
Robotica , Volume 34 , Issue 3 , March 2016 , pp. 584 - 600
Copyright
Copyright © Cambridge University Press 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Murray, R. M., “Nilpotent bases for a class of non-integrable distributions with applications to trajectory generation for nonholonomic systems,” Math. Control, Signals Syst. 7, 58–75 (1994).CrossRefGoogle Scholar
2.Murray, R. M. and Sastry, S. S., “Nonholonomic motion planning: Steering using sinusoids,” IEEE Trans. Autom. Control 38 (5), 700–716 (1993).CrossRefGoogle Scholar
3.El-Hawwary, M. I., Elshafei, A. L., Emara, H. M. and Fattah, H. A. A., “Adaptive fuzzy control of the inverted pendulum problem,” IEEE Trans. Control Syst. Technol. 14 (6), 1135–1144 (2006).CrossRefGoogle Scholar
4.Mita, T., Hyon, S. H. and Nam, T. K., “Analysis time optimal control solution for a two-link planar Acrobot with initial angular momentum,” IEEE Trans. Robot. Autom. 13 (3), 361–366 (2001).CrossRefGoogle Scholar
5.Bhattacharya, S. and Agrawal, S. K., “Spherical rolling robot: A design and motion planning studies,” IEEE Trans. Robot. Autom. 16 (6), 835–839 (2000).CrossRefGoogle Scholar
6.Li, Z. and Canny, J., “Motion of two rigid bodies with rolling constraint,” IEEE Trans. Robot. Autom. 6 (1), 62–71 (1990).CrossRefGoogle Scholar
7.De Luca, A., Mattone, R. and Oriolo, G., “Stabilization of an underactuated planar 2R manipulator,” Int. J. Robust Nonlinear Control 10, 181–198 (2000).3.0.CO;2-X>CrossRefGoogle Scholar
8.Oriolo, G. and Nakamura, Y., “Free-Joint Manipulators: Motion Control Under Second-Order Nonholonomic Constraints,” Proceedings of the IEEE/RSJ International Workshop on Intelligent Robots and Systems, Osaka, Japan (Nov. 3–5, 1991) pp. 1248–1253.Google Scholar
9.De Luca, A. and Oriolo, G., “Motion Planning and Trajectory Control of an Underactuated Three-Link Robot via Dynamic Feedback Linearization,” Proceedings of the IEEE International Conference on Robotics and Automation, San Francisco, USA (Apr. 24–28, 2000) pp. 2789–2795.Google Scholar
10.De Luca, A., Iannitti, S., Mattone, R., and Oriolo, G., “Control Problems in Underactuated Manipulators,” Proceedings of the IEEE/ASME International Conference on Advanced Intelligent Mechatronics Proceedings, Como, Italy (Jul. 6–12, 2002) pp. 855–861.Google Scholar
11.Arai, H., Tanie, K. and Shiroma, N., “Nonholonomic control of a three-DOF planar underactuated manipulator,” IEEE Trans. Robot. Autom. 14 (5), 681–695 (1998).CrossRefGoogle Scholar
12.Shiroma, N., Arai, H. and Tanie, K., “Nonholonomic motion planning for coupled planar rigid bodes with passive revolute joints,” Int. J. Robot. Res. 21 (5–6), 563–574 (2002).CrossRefGoogle Scholar
13.De Luca, A., Mattone, R. and Oriolo, G., “Control of Underactuated Mechanical Systems: Application to the Planar 2R Robot,” Proceedings of the Conference on Decision and Control, Kobs, Japan (Dec. 11–13, 1996) pp. 1455–1460.CrossRefGoogle Scholar
14.De Luca, A., Mattone, R. and Oriolo, G., “Stabilization of Underactuated Robots: Theory and Experiments for a Planar 2R Manipulator,” Proceedings of the IEEE International Conference on Robotics and Automation, Albuquerque, USA (Apr. 20–25, 1997) pp. 3274–3280.CrossRefGoogle Scholar
15.De Luca, A., Iannitti, S. and Oriolo, G., “Stabilization of a PR Planar Underactuated Robot,” Proceedings of the IEEE International Conference on Robotics and Automation, Seoul, South Korea (2001) pp. 2090–2095.Google Scholar
16.Hermes, H., “Nilpotent and high-order approximations of vector field systems,” SIAM Rev. 33 (2), 238–264 (1991).CrossRefGoogle Scholar
17.Lucibello, P. and Oriolo, G., “Robust stabilization via iterative state steering with an application to chained-form systems,” Automatica 37, 71–79 (2001).CrossRefGoogle Scholar
18.Arai, H., Tanie, K. and Shiroma, N., “Time-scaling control of an underactuated manipulator,” J. Robot. Syst. 15 (9), 525–536 (1998).3.0.CO;2-M>CrossRefGoogle Scholar
19.Hong, K.-S., “An open-loop control for underactuated manipulators using oscillatory inputs: Steering capability of an unactuated joint,” IEEE Trans. Control Syst. Technol. 10 (3), 469–480 (2002).CrossRefGoogle Scholar
20.Rosas-Flores, J. A., Alvarez-Gallegos, J. and Castro-Linares, R., “Trajectory planning and control of an underactuated planar 2R Manipulator,” Proceedings of the IEEE International Conference on Control Applications, Mexico City, Mexico (Sep. 5–7, 2001) pp. 548–552.Google Scholar
21.Mahindrakar, A. D., Rao, S. and Banavar, R. N., “Point-to point control of a 2R planar horizontal underactuated manipulator,” Mech. Mach. Theory 41, 838–844 (2006).CrossRefGoogle Scholar
22.De Luca, A. and Oriolo, G., “Stabilization of the Acrobot Via Iterative State Steering,” Proceedings of the IEEE International Conference on Robotics and Automation, Leuven, Belgium (May 16–20, 1998) pp. 3581–3587.Google Scholar
23.Oriolo, G. and Vendittelli, M., “A framework for the stabilization of general nonholonomic systems with an application to the plate-ball mechanism,” IEEE Trans. Robot. 21 (2), 162–175 (2005).CrossRefGoogle Scholar
24.Vendittelli, M., Oriolo, G., and Laumond, J.-P., “Steering Nonholonomic Systems via Nilpotent Approximations: The General Two-Trailer System,” Proceedings of the IEEE International Conference on Robotics and Automation, Detroit, USA (May 10–15, 1999) pp. 823–829.Google Scholar
25.Vendittelli, M. and Oriolo, G., “Stabilization of the general two-trailer system,” Proceedings of the IEEE International Conference on Robotics and Automation, San Francisco, USA (Apr. 24–28, 2000) pp. 1817–1823.Google Scholar
26.Lafferriere, G. and Sussmann, H. J., “Motion Planning for Controllable Systems Without Drift,” Proceedings of International Conference on Robotics and Automation, Sacramento, USA (Apr. 9–11, 1991) pp. 1148–1153.Google Scholar
27.Mullhaupt, P., Srinivasan, B., and Bonvin, D., “Analysis of exclusively kinetic two-link underactuated mechanical systems,” Automatica 38, 1565–1573 (2002).CrossRefGoogle Scholar
28.Sussmann, H. J., “A General theorem on local controllability,” SIAM J. Control Optim. 25 (1), 158–194 (1987).CrossRefGoogle Scholar
29.Rathinam, M. and Murray, R. M., “Configuration flatness of Lagrangian systems underactuated by one control,” SIAM J. Control Optim. 36 (1), 164–179 (1998).CrossRefGoogle Scholar
30.Murray, R. M., Rathinam, M. and Sluis, W., “Differential flatness of mechanical control systems: A catalog of prototype systems,” Proceedings of the ASME International Mechanical Engineering Congress and Expo, San Francisco, USA (Nov. 12–17, 1995) pp. 1–9.Google Scholar
31.Fliess, M., Levine, J., Martin, P. and Rouchon, P., “Flatness and defect of nonlinear systems: Introductory theory and examples,” Int. J. Control 61 (6), 1327–1361 (1995).CrossRefGoogle Scholar
32.Hermes, H., “Nilpotent approximations of control systems and distributions,” SIAM J. Control Optim. 24 (4), 731–736 (1986).CrossRefGoogle Scholar
33.Bellaïche, A., Laumond, J.-P. and Chyba, M., “Canonical Nilpotent Approximation of Control Systems: Application to Nonholonomic Motion Planning,” Proceedings of the Conference on Decision and Control, San Antonio, USA (Dec. 15–17, 1993) pp. 2694–2699.CrossRefGoogle Scholar
34.Hermes, H., Lundell, A. and Sullivan, D., “Nilpotent bases for distributions and control systems,” J. Diff. Eqns. 55, 385–400 (1984).CrossRefGoogle Scholar
35.Vendittelli, M., Oriolo, G., Jean, F. and Laumond, J.-P., “Nonhomogeneous nilpotent approximations for nonholonomic systems with singularities,” IEEE Trans. Autom. Control 49 (2), 261–266 (2004).CrossRefGoogle Scholar
36.Murray, R. M., Li, Z. and Sastry, S. S., A Mathematical Introduction to Robotic Manipulation (CRC Press, London, 1994).Google Scholar
37.Paul, B., Kinematics and Dynamics of Planar Machinery (Prentice Hall, New Jersey, 1979).Google Scholar