The transfer matrix method (TMM) can be a powerful tool for modeling flexible structures under feedback control. It is particularly well suited to modeling structures composed of serially connected elements. The TMM is capable of modeling continuous elements such as beams or flexible robot links without discretization. The ability to incorporate controller transfer functions into the transfer matrix model of the system makes it a useful approach for control design. A limitation of the traditional formulation of the TMM is that it can only model feedback where the actuators and sensors are strictly collocated. The primary contribution of this paper is an algorithm for modeling noncollocated feedback with the TMM. Two cases of noncollocated sensors are considered (upstream and downstream). The approach is experimentally verified on a flexible robot that has one upstream and one downstream sensor in its feedback loops.

1.
De Luca
,
A.
, and
Siciliano
,
B.
, 1991, “
Closed-Form Dynamic-Model of Planar Multilink Lightweight Robots
,”
IEEE Trans. Syst. Man Cybern.
0018-9472,
21
(
4
), pp.
826
839
.
2.
Kang
,
B. S.
, and
Mills
,
J. K.
, 2002, “
Dynamic Modeling of Structurally-Flexible Planar Parallel Manipulator
,”
Robotica
0263-5747,
20
, pp.
329
339
.
3.
Chalhoub
,
N. G.
, and
Chen
,
L.
, 1998, “
A Structural Flexibility Transformation Matrix for Modelling Open-Kinematic Chains With Revolute and Prismatic Joints
,”
J. Sound Vib.
0022-460X,
218
(
1
), pp.
45
63
.
4.
Book
,
W. J.
, 1984, “
Recursive Lagrangian Dynamics of Flexible Manipulator Arms
,”
Int. J. Robot. Res.
0278-3649,
3
(
3
), pp.
87
101
.
5.
Brüls
,
O.
,
Duysinx
,
P.
, and
Golinval
,
J. -C.
, 2007, “
The Global Modal Parameterization for Nonlinear Model-Order Reduction in Flexible Multibody Dynamics
,”
Int. J. Numer. Methods Eng.
0029-5981,
69
, pp.
948
977
.
6.
Brüls
,
O.
,
Duysinx
,
P.
, and
Golinval
,
J. -C.
, 2005, “
A Unified Finite Element Framework for the Dynamic Analysis of Controlled Flexible Mechanisms
,”
Proceedings of the ECCOMAS Conference on Advances in Computational Multibody Dynamics
.
7.
Hallauer
,
W.
, and
Liu
,
R.
, 1982, “
Beam Bending-Torsion Dynamic Stiffness Method for Calculation of Exact Vibration Modes
,”
J. Sound Vib.
0022-460X,
85
, pp.
105
113
.
8.
Banerjee
,
J. R.
, 1997, “
Dynamic Stiffness Formulation for Structural Elements: A General Approach
,”
Comput. Struct.
0045-7949,
63
(
1
), pp.
101
103
.
9.
Banerjee
,
J. R.
, and
Fisher
,
S. A.
, 1992, “
Coupled Bending-Torsional Dynamic Stiffness Matrix for Axially Loaded Beam Elements
,”
Int. J. Numer. Methods Eng.
0029-5981,
33
, pp.
739
751
.
10.
Yu
,
J. -F.
,
Lien
,
H. -C.
, and
Wang
,
B. P.
, 2004, “
Exact Dynamic Analysis of Space Structures Using Timoshenko Beam Theory
,”
AIAA J.
0001-1452,
42
(
4
), pp.
833
839
.
11.
Yang
,
B.
, 1992, “
Transfer Functions of Constrained/Combined One-Dimensional Continuous Dynamic Systems
,”
J. Sound Vib.
0022-460X,
156
(
3
),
425
443
.
12.
Yang
,
B.
, 2005,
Stress, Strain, and Structural Dynamics
,
Elsevier
,
New York
/
Academic
,
New York
.
13.
Pestel
,
E. C.
, and
Leckie
,
F. A.
, 1963,
Matrix Methods in Elastomechanics
,
McGraw Hill
,
New York
.
14.
Book
,
W.
,
Maizza-Neto
,
O.
, and
Whitney
,
D.
, 1975, “
Feedback Control of Two Beam, Two Joint Systems With Distributed Flexibility
,”
ASME J. Dyn. Syst., Meas., Control
0022-0434,
97
(
4
), pp.
424
431
.
15.
Book
,
W.
, and
Majette
,
M.
, 1983, “
Controller Design for Flexible, Distributed Parameter Mechanical Arms via Combined State Space and Frequency Domain Techniques
,”
ASME J. Dyn. Syst., Meas., Control
0022-0434,
105
(
4
), pp.
245
254
.
16.
George
,
L.
, and
Book
,
W.
, 2003, “
Inertial Vibration Damping Control of a Flexible Base Manipulator
,”
IEEE/ASME Trans. Mechatron.
1083-4435,
8
(
2
), pp.
268
271
.
17.
Esser
,
B.
,
Huston
,
D.
, and
Miller
,
J.
, 2003, “
Aerospace Electronics Weight Reduction Through the Use of Active Mass Damping
,”
Smart Mater. Struct.
0964-1726,
5052
, pp.
433
444
.
18.
Krauss
,
R.
, and
Book
,
W.
, 2004, “
Stability in Active Mass Damping Control of a Flexible Robot
,”
2004 IEEE Aerospace Conference
, Vol.
5
.
19.
Krauss
,
R. W.
, and
Book
,
W. J.
, 2007, “
A Python Software Module for Automated Identification of Systems Modeled With the Transfer Matrix Method
,”
Proceedings of IMECE2007
.
20.
Meirovitch
,
L.
, 1986,
Elements of Vibration Analysis
,
2nd ed.
,
McGraw-Hill
,
New York
.
You do not currently have access to this content.