Sage Journals: Discover world-class research

Abstract

A one-link flexible arm with a load at the end is considered. The arm can be rotated around its hub in a horizontal plane. This rotation is actuated by a DC motor. The dynamics of a controlled flexible arm are described by an infinite-dimensional mathematical model, which contains the Bernoulli-Euler beam equation, equations of the actuator motion and boundary con ditions. The control voltage fed to the motor is a linear com bination of arm angular position, integral, rotation speed, arm deformations, tip position, and velocity.

The principal goal of the article is to find linear feedback gains such that the desired arm position is asymptotically sta ble. This problem of stability arises when an elastic manipula tor is designed. We have analytically described the boundaries of the domains of asymptotic stability in the space of feedback gains and constructed these domains in different cases. It is shown that the stability domains for a flexible arm belong to and are less than the corresponding domains for a rigid arm. We have shown that under feedback on tip position and veloc ity, the one-link flexible manipulator is unstable if damping is small.

Get full access to this article

View all access options for this article.

References

Akulenko, L.D. , and Bolotnik, N.N. 1982. On controlled rotation of an elastic rod. Appl. Math. Mechanics. 46(4):465-471.

Akulenko, L.D. , Michailov, S.A. , and Satovskaya, O.L. 1988. Dynamic models of elastic manipulation robots. Preprint of Inst. of Mechanics Problems AS USSR. No. 349 (in Russian).

Bayo, E. , Papadopoulos, P. , Stuble, J. , and Serna, M.A. 1989. Inverse dynamics and kinematics of multi-link elastic robots: an iterative frequency domain approach. Int. J. Robot. Res. 8(6):49-62.

Bellezze, F. , Lanari, L. , and Ulivi, G. 1990. Exact modeling of the flexible slewing link. Proc. IEEE Conf. on Robotics and Automation. Cincinnati: IEEE, pp. 734-739.

Berbyuk, V.E. 1984. On the controlled rotation of a system of two rigid bodies with elastic elements. App. Math. Mechanics. 48(2):164-170.

Berbyuk, V.E. , and Demidyuk, M.V. 1984. Controlled motion of an elastic manipulator with distributed parameters. Mech. Solids 19(2):57-66.

Book, W.J. , Maizza-Netto, O. , and Whitney, D.E. 1975. Feedback control of two beam, two joint systems with distributed flexibility. ASME J. Dynam. Sys. Measurement Control 97(G4):424-431.

Book, W.J. , and Majette, M. 1985. Controller design for flexible, distributed parameter mechanical arms via combined state space and frequency domain techniques. ASME J. Dynam. Sys. Measurement Control 105:245-254.

Buchin, V.A. 1981. Stabilization of an unstable operation regime of a chemical reactor with recycle as an object with distributed parameters by means of concentrated control systems. Fluid Dynamics 16(3):332-344.

10.

Buchin, V.A. 1982. Solution of the problem of a substantial increase in the critical load of a compressed elastic rod using boundary conditions leading to a multipoint boundary-value problem. Soviet Physics Doclady 27(4):355-357.

11.

Cannon, R. H. Jr. , and Schmitz, E. 1984. Initial experiments on the end-point control of a flexible one-link robot . Int. J. Robot. Res. 13(3):62-75.

12.

Cetinkunt, S. , and Wen-Lung, Y. 1991. Closed-loop behavior of a feedback-controlled flexible arm: a comparative study. Int. J. Robot. Res. 10(3):263-275.

13.

Chilikin, M.G. , and Sandler, A.S. 1982. A General Course on Electric Drives (in Russian). Moscow: Energoizdat.

14.

De Luca, A. , and Siciliano, B. 1989. Trajectory of non-linear one-link flexible arm. Int. J. Control 50(5):1699-1715.

15.

De Luca, A. , and Siciliano, B. 1993. Regulation of flexible arms under gravity. IEEE Trans. Robot. Automation 9(4):463-467.

16.

Gibson, J.S. , and Adamian, A. 1991. Approximation theory for linear-quadratic-Gaussian optimal control of flexible structures. SIAM J. Control Optim. 29(1):1-37.

17.

Ladikov, Y.P. 1978. Stabilization of Processes in Solid Media. (in Russian) . Moscow: Nauka.

18.

Lur'ye, A.I. 1961. Analytical Mechanics (in Russian). Moscow: Fizmatgiz.

19.

Naimark, M.A. 1969. Linear Differential Operators., (in Russian). Moscow: Nauka.

20.

Neimark, Y.I. 1978. Dynamic Systems and Controlled Processes (in Russian) . Moscow: Nauka.

21.

Nguyen, P.K. , Ravindran, R. , Carr, R. , and Gossain, D.M. 1982. Structural flexibility of the shuttle remote manipulator system mechanical arm. Proc. of the Guidance and Control Conf., AIAA. New York, pp. 1582-1586.

22.

Shabat, B.V. 1976. Introduction into Complex Analysis, Part I (in Russian) . Moscow: Nauka.

23.

Sirazetdinov, T.K. 1987. Stability of Systems With Distributed Parameters (in Russian) . Moscow: Nauka.

24.

Timoshenko, S. , Young, D.H. , and Weaver, W. 1974. Vibration Problems in Engineering. New York: John Wiley.

25.

Vol'mir, A.S. 1967. Stability of Deformable Systems (in Russian). Moscow: Nauka.

26.

Wang, H. , and Chen, G. 1991. Asymptotic locations of eigen-frequencies of Euler-Bernoulli beam with nonhomogeneous structured and viscous damping coefficients. SIAM J. Control Optim. 29(2):347-367.

27.

Wang, F.-Y. , and Kwan, O. 1993. Influence of rotatory inertia, shear deformation and loading on vibration behaviors of flexible manipulators. J. Sound Vibrations 167(2):1-15.

28.

Wei, B. 1981. On the modeling and control of flexible space structures. Ph.D. thesis, Stanford University, Department of Aeronautics and Astronautics.

Stabilization of Flexible One-Link Arm Position: Stability Domains in the Space of Feedback Gains

Abstract

Get full access to this article

References