In this brief the dynamic behavior of a parametrically forced manipulator, or pendulum, system with PD control is examined. For an excitation of sufficient amplitude or frequency a Hopf bifurcation to a steady-state limit cycle is shown to result, appearing as a precursor to instability. The parameter space is mapped in order to illustrate regions where control failure will likely occur, even in the strongly damped case. For weakly damped systems, the Hopf bifurcation can additionally exhibit a dependence on initial conditions. The resulting case of competing point and periodic attractors is discussed.

1.
Ravishankar
,
A. S.
, and
Ghosal
,
A.
,
1999
, “
Nonlinear Dynamics and Chaotic Motions in Feedback Controlled Two- and Three-Degree-of-Freedom Robots
,”
Int. J. Robot. Res.
,
18
, No.
1
, pp.
93
108
.
2.
Lankalapalli, S., and Ghosal, A., 1996, “Possible Chaotic Motions in a Feedback Controlled 2R Robot,” Proceedings of the 1996 IEEE International Conference On Robotics and Automation, Minneapolis, Vol. 2, pp. 1241–1245.
3.
Mahout, V., Lopez, P., Carcasses, J. P., and Mira, C., 1993, “Complex Behaviors of a Two Revolute Joints Robot: Harmonic, Subharmonic, Higher Harmonic, Fractional Harmonic, Chaotic Responses,” IEEE International Conference On Systems, Man, and Cybernetics, Le Touquet, France, pp. 201–205.
4.
Spong, M. W., and Vidyasagar, M., 1989, Robot Dynamics and Control, Wiley, New York, NY.
5.
Arimoto, S., 1996, Control Theory of Nonlinear Systems: A Passivity Based and Circuit-Theoretical Approach, Oxford University Press.
6.
Kelly
,
R.
,
1997
, “
PD Control with Desired Gravity Compensation of Robotic Manipulators: A Review
,”
Int. J. Robot. Res.
,
16
, No.
5
, pp.
660
672
.
7.
Takegaki
,
M.
, and
Arimoto
,
S.
,
1981
, “
A New Feedback Method for Dynamic Control of Manipulators
,”
ASME J. Dyn. Syst., Meas., Control
,
103
, pp.
119
125
.
8.
Blizter
,
L.
,
1965
, “
Inverted Pendulum
,”
Am. J. Phys.
,
33
, pp.
1076
1078
.
9.
Capecchi
,
D.
,
1995
, “
Geometric Aspects of the Parametrically Driven Pendulum
,”
Nonlinear Dyn.
,
7
, pp.
231
247
.
10.
Leven
,
R. W.
, and
Koch
,
B. P.
,
1981
, “
Chaotic Behavior of a Parametrically Excited Damped Pendulum
,”
Phys. Lett. A
,
86
, No.
2
, pp.
71
74
.
11.
McLaughlin
,
J.
,
1981
, “
Period Doubling Bifurcations and Chaotic Motion for a Parametrically Forced Pendulum
,”
J. Stat. Phys.
,
24
, No.
2
, pp.
375
388
.
12.
Arimoto, S., 1989, “Control” Robotics Science, M. Brady ed., MIT Press, pp. 349–377.
13.
McLachlan, N. W., 1947, Theory and Application of Mathieu Functions, Oxford Press.
14.
Turyn
,
L.
,
1993
, “
The Damped Mathieu Equation
,”
Q. Appl. Math.
,
51
, No.
2
, pp.
389
398
.
15.
Taylor
,
J. H.
, and
Narendra
,
K.
,
1969
, “
Stability Regions for the Damped Mathieu Equation
,”
SIAM (Soc. Ind. Appl. Math.) J. Appl. Math.
,
17
, No.
2
, pp.
343
352
.
16.
Gunderson
,
H.
,
Rigas
,
H.
, and
VanVleck
,
F. S.
,
1974
, “
A Technique for Determining Stability Regions for the Damped Mathieu Equation
,”
SIAM (Soc. Ind. Appl. Math.) J. Appl. Math.
,
26
, No.
2
, pp.
345
349
.
17.
Leiber
,
T.
, and
Risken
,
H.
,
1988
, “
Stability of Parametrically Excited Dissipative Systems
,”
Phys. Lett. A
,
129
, No.
4
, pp.
214
218
.
18.
Kuznetsov, Y., 1998, Elements of Applied Bifurcation Theory, Springer Verlag, pp. 86–103.
19.
Hsu
,
C. S.
,
1975
, “
Limit Cycle Oscillations of Parametrically Excited Second-Order Nonlinear Systems
,”
ASME J. Appl. Mech.
,
42
, pp.
176
182
.
You do not currently have access to this content.