Input Shaping is a technique that seeks to reduce residual vibrations through modification of the reference command given to a system. Namely the reference command is convolved with a suitable train of impulses. Input shaping has proven to be successful in reducing the vibrations of a great variety of linear systems. This article seeks to apply input shaping to robotic manipulators of parallel architecture. Such systems have multiple degrees-of-freedom and non-linear dynamics and therefore standard input shaping techniques cannot be readily applied. In order to apply standard input shaping techniques to such systems, this article linearizes the dynamic equations of the system locally and determines the configuration-dependent natural frequencies and damping ratios throughout its workspace. Techniques are developed to derive the dynamic equations directly in linearized form. The method is demonstrated for a sample manipulator with two degrees-of-freedom. A linearized dynamic model is derived and input shaping is locally tuned according to the linearized dynamic model. Simulation results are provided and discussed.

1.
Uchiyama
,
M.
,
1994
, “
Structures and Characteristics of Parallel Manipulators
,”
,
8
(
6
), pp.
545
557
.
2.
Merlet, J.-P., 2000, Parallel Robots, Series in Solid Mechanics and Its Applications, Kluwer Academic Publishers, 1st edition.
3.
Tsai, L.-W., 1999, Robot Analysis—The Mechanics of Serial and Parallel Manipulators, John Wiley & Sons, 1st edition.
4.
Codourey, A., 1991, “Contribution a la commande des robots rapides et pre´cis. Application au robot Delta a entrainement direct,” PhD thesis, E´cole Polytechnique Fe´de´rale de Lausanne, Lausanne.
5.
Yamane, K., Okada, M., Komine, N., and Nakamura, Y., 1998, “Parallel Dynamics Computation and H∞ Acceleration Control of Parallel Manipulators for Acceleration Display,” Proceedings 1998 IEEE International Conference on Robotics and Automation, pp. 2301–2308, Leuven, Belgium, May.
6.
Ginsberg, J. H., 1998, Advanced Engineering Dynamics, Cambridge Univ. Press, Cambridge, UK, second edition.
7.
Zoble, P. B., and Clavel, R., 1993, “On the Static Calibration of the Delta Parallel Robot,” Proceedings of the IASTED International Conference on Robotics and Manufacturing, p. 88, Oxford, UK, September.
8.
Rappole, B. W., Singer, N. C., and Seering, W. P., 1994, “Multiple-mode Impulse Shaping Sequences for Reducing Residual Vibrations,” 23rd Biennial Mechanisms Conference, pp. 11–16.
9.
Jones
,
S.
, and
Ulsoy
,
A. G.
, 1999, “An Approach to Control Input Shaping With Application to Coordinate Measuring Machines,” ASME J. Dyn. Syst., Meas., Control, pp. 242–247 (June).
10.
Seth, N., Rattan, K., and Brandstetter, R., 1993, “Vibration Control of a Coordinate Measuring Machine,” IEEE International Conf. on Control Apps., pp. 368–373.
11.
Singhose
,
W.
,
Singer
,
N.
, and
Seering
,
W.
, 1996, “Improving Repeatability of Coordinate Measuring Machines with Shaped Command Signals,” Precis. Eng., pp. 138–146 (April).
12.
Singhose
,
W.
,
Singer
,
N.
, and
Seering
,
W.
, 1997, “Time-optimal Negative Input Shapers,” ASME J. Dyn. Syst., Meas., Control, pp. 198–205 (June).
13.
deRoover
,
D.
,
Bosgra
,
O. H.
,
Sperling
,
F. B.
, and
Steinbuch
,
M.
,
1996
, “
High-performance Motion Control of a Flexible Mechanical Servomechanism
,”
Selected Topics in Identification, Modeling and Control
,
9
, pp.
69
78
.
14.
deRoover, D., and Sperling, F. B., 1997, “Point-point Control of a High Accuracy Positioning Mechanism,” Proceedings of the American Control Conf., pp. 1350–1354, Albuquerque, NM.
15.
Singer
,
N. C.
, and
Seering
,
W. P.
,
1990
, “
Preshaping Command Inputs to Reduce System Vibration
,”
ASME J. Dyn. Syst., Meas., Control
, pp.
112
(
1
), March
76
82
.
16.
Singhose
,
W.
,
Seering
,
W.
, and
Singer
,
N.
, 1994, “Residual Vibration Reduction Using Vector Diagrams to Generate Shaped Inputs,” ASME J. Mech. Des., pp. 654–659 (June).
17.
Singhose, W. E., Seering, W. P., and Singer, N. C., 1996, “Input Shaping for Vibration Reduction with Specified Insensitivity to Modeling Errors,” Japan-USA Sym. on Flexible Automation.
18.
Book, W. J., Magee, D. P., and Rhim, S., 1999, “Time-delay Command Shaping Filters: Robust and/or Adaptive,” Journal of the Robotics Society of Japan, 17(6), Sept., pp. 7–15.
19.
Tzes
,
A.
, and
Yurkovich
,
S.
,
1993
, “
An Adaptive Input Shaping Control Scheme for Vibration Suppression in Slewing Flexible Structures
,”
IEEE Trans. Control Syst. Technol.
,
1
, pp.
114
121
(June).
20.
Gorinevsky
,
D.
, and
Vukovich
,
G.
,
1998
, “
Nonlinear Input Shaping Control of Flexible Spacecraft Reorientation Maneuver
,”
J. Guid. Control Dyn.
,
21
(
2
), pp.
264
270
.
21.
Kinceler, R., and Meckl, P. H., 1995, “Input Shaping for Nonlinear Systems,” Proceedings of the American Control Conf., pp. 914–918, Seattle, WA.
22.
Meckl, P. H., and Kinceler, R., 1994, “Trajectory Determination for Vibration-free Motions of a Flexible-joint Robot,” Proceedings of the American Control Conf., pp. 2521–2525, Baltimore, MD.
23.
Kozak, K., Voglewede, P. A., Ebert-Uphoff, I., and Singhose, W., 2002, “Concept paper: On the Significance of the Lowest Natural Frequency of a Parallel Manipulator as a Performance Measure for Concurrent Design,” Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, pp. 112–118, Quebec, Canada.
24.
Whittaker, E. T., 1988, A Treatise on the Analytical Dynamics of Particles & Rigid Bodies, Cambridge University Press, 4th edition, 1st edition was 1904.
25.
Sciavicco, L., and Siciliano, B., 2000, Modeling and Control of Robot Manipulators, Springer, 2nd edition.
26.
Guglielmetti, P., 1994, “Model-Based Control of Fast Parallel Robots: a Global Approach in Operational Space,” PhD thesis, E´cole Polytechnique Fe´de´rale de Lausanne, Dept. of Mechanical Engineering, Lausanne.
27.
Winckler, M. J., and Kraus, C., 2000, “Simulation of Hexapod Machine Tools by Using Natural Coordinates,” Year 2000 Parallel Kinematic Machines International Conference, pp. 109–117.
28.
Lanczos, C., 1986, The Variational Principles of Mechanics, Dover Publications, fourth edition.
29.
Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P., 1992, Numerical Recipes in C, Cambridge University Press, 2nd edition.
30.
Codourey, A., and Burdet, E., 1997, “A Body-oriented Method for Finding a Linear Form of the Dynamic Equation of Fully Parallel Robots,” Proceedings 1997 IEEE International Conference on Robotics and Automation, pp. 1612–1618, April.
31.
Ebert-Uphoff I., and Kozak, K., 2002, “Review of the Role of Quasi-coordinates for the Kinematic and Dynamic Modeling of Parallel Manipulators,” Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, pp. 328–338, Quebec, Canada.