In this paper, we propose a novel approach to the control of closed kinematic chains (CKCs). This method is based on a recently developed singularly perturbed model for CKCs. Conventionally, the dynamics of CKCs are described by differential-algebraic equations (DAEs). Our approach transfers the control of the original DAE system to the control of an artificially created singularly perturbed system in which the slow dynamics corresponds to the original DAE when the perturbation parameter tends to zero. Compared to control schemes that rely on solving nonlinear algebraic constraint equations, the proposed method uses an ordinary differential equation (ODE) solver to obtain the dependent coordinates, hence, eliminates the need for Newton-type iterations and is amenable to real-time implementation. The composite Lyapunov function method is used to show that the closed-loop system, when controlled by typical open kinematic chain schemes, achieves asymptotic trajectory tracking. Simulations and experimental results on a parallel robot, the Rice planar Delta robot, are also presented to illustrate the efficacy of our method.

1.
Merlet
,
J. P.
, 2000,
Parallel Robots
,
Kluwer Academic Publishers
,
Dordrecht
.
2.
Tsai
,
L. W.
, 1999,
Robot Analysis: The Mechanics of Serial and Parallel Manipulators
,
Wiley
,
New York
.
3.
Merlet
,
J. P.
, 1988, “
Force-Feedback Control of Parallel Manipulators
,”
IEEE Int. Conf. on Robotics and Automation
, Philadelphia,
IEEE
,
New York
, pp.
1484
1489
.
4.
Nguyen
,
C. C.
,
Pooran
,
F. J.
, and
Premack
,
T.
, 1988, “
Trajectory Control of Robot Manipulators With Closed-Kinematic Chain Mechanism
,”
Proceedings of the 20th Southeastern Symposium on System Theory, 1988
, March,
IEEE
,
New York
, pp.
454
458
.
5.
Walker
,
M. W.
, 1990, “
Adaptive Control of Manipulators Containing Closed Kinematic Loops
,”
IEEE Trans. Rob. Autom.
1042-296X,
6
(
1
), pp.
10
19
.
6.
Ghorbel
,
F. H.
,
Chételat
,
O.
,
Gunawardana
,
R.
, and
Longchamp
,
R.
, 2000, “
Modeling and Set Point Control of Closed-Chain Mechanisms: Theory and Experiment
,”
IEEE Trans. Control Syst. Technol.
1063-6536,
8
(
5
), pp.
801
815
.
7.
Chiacchio
,
P.
,
Pierrot
,
F.
,
Sciavicco
,
L.
, and
Siciliano
,
B.
, 1993, “
Robust Design of Independent Joint Controllers With Experimentation on a High-Speed Parallel Robot
,”
IEEE Trans. Indus. Electronics.
,
40
(
4
), pp.
393
403
.
8.
Li
,
D.
, and
Salcudean
,
S. E.
, 1997, “
Modeling, Simulation, and Control of a Hydraulic Stewart Platform
,”
Proceedings of 1997 IEEE International Conference on Robotics and Automation, ICRA 1997
, Albuquerque, April,
IEEE
,
New York
, pp.
3360
3366
.
9.
Su
,
Y. X.
,
Duan
,
B. Y.
,
Zheng
,
C. H.
,
Zhang
,
Y. F.
,
Chen
,
G. D.
, and
Mi
,
J. W.
, 2004, “
Disturbance-Rejection High-Precision Motion Control of a Stewart Platform
,”
IEEE Trans. Control Syst. Technol.
1063-6536,
12
(
3
), pp.
364
374
.
10.
Cheng
,
Y.
,
Ren
,
G.
, and
Dai
,
S.
, 2003, “
Vibration Control of Gough-Stewart Platform on Flexible Suspension
,”
IEEE Trans. Rob. Autom.
1042-296X,
19
(
3
), pp.
489
493
.
11.
Geng
,
Z.
, and
Hayes
,
L. S.
, 1991, “
Dynamic Control of a Parallel Link Manipulator Using CMAC Neural Network
,” in
Proceedings of 1991 IEEE International Symposium on Intelligent Control
, Arlington, VA, August,
IEEE
,
New York
, pp.
411
416
.
12.
Begon
,
P.
,
Pierrot
,
F.
, and
Dauchez
,
P.
, 1995, “
Fuzzy Sliding Mode Control of a Fast Parallel Robot
,”
IEEE Int. Conf. on Robotics and Automation
, Nagoya, May,
IEEE
,
New York
, pp.
1178
1183
.
13.
Ghorbel
,
F. H.
,
Gunawardana
,
R.
, and
Dabney
,
J. B.
, 2001, “
Experimental Validation of a Reduced Model Based Tracking Control of Parallel Robots
,”
Proceedings of the 2001 IEEE International Conference on Control Applications
, Mexico City, September,
IEEE
,
New York
, pp.
375
382
.
14.
Choi
,
H. B.
,
Company
,
O.
,
Pierrot
,
F.
,
Konno
,
A.
,
Shibukawa
,
T.
, and
Uchiyama
,
M.
, 2003, “
Design and Control of a Novel 4-dofs Parallel Robot H4
,”
Proceedings of 2003 IEEE Int. Conf. on Robot. and Automat.
, Taipei, September,
IEEE
,
New York
, pp.
1185
1190
.
15.
Aghili
,
F.
, 2003, “
Inverse and Direct Dynamics of Constrained Multibody System Based on Orthogonal Decomposition of Generalized Force
,”
Proceedings of 2003 IEEE International Conference on Robotics and Automation, ICRA 2003
, Taipei, September,
IEEE
,
New York
, pp.
4035
4041
.
16.
Kim
,
N. I.
, and
Lee
,
C. W.
, 1998, “
High Speed Tracking Control of Stewart Platform Manipulato via Enhanced Sliding Model Control
,”
Proceedings of the 1998 IEEE International Conference on Robotics and Automation, ICRA 1998
, Leuven, Belgium, May,
IEEE
,
New York
, pp.
2716
2721
.
17.
Kim
,
D. H.
,
Kang
,
J. Y.
, and
Lee
,
K. I.
, 2000, “
Robust Tracking Control Design for a 6 DOF Parallel Manipulator
,”
J. Rob. Syst.
0741-2223,
17
(
10
), pp.
527
547
.
18.
Aguilar-Ibanez
,
C.
,
Martinez-Garcia
,
J. C.
, and
Salazar-Cruz
,
S.
, 2000, “
Illustrating a Robust Nonlinear Tracking Control Methodology With a Closed-Kinematic Chain
,”
Proceedings of American Control Conference
, Chicago, June,
IEEE
,
New York
, pp.
2506
2510
.
19.
Nguyen
,
C. C.
,
Antrazi
,
S. S.
,
Zhou
,
Z. L.
, and
Campbell
,
J. C. E.
, 1993, “
Adaptive Control of a Stewart Platform-Based Manipulator
,”
J. Rob. Syst.
0741-2223,
10
(
5
), pp.
657
687
.
20.
Honegger
,
M.
,
Codourey
,
A.
, and
Burdet
,
E.
, 1997, “
Adaptive Control of the Hexaglide, a 6-DOF Parallel Manipulator
,”
Proceeding of 1997 IEEE Int. Conf. on Robotics and Automation
, Albuquerque, April,
IEEE
,
New York
, pp.
543
548
.
21.
Fasse
,
E. D.
, and
Gosselin
,
C. M.
, 1999, “
Spatio-Geometric Impedance Control of Gough-Stewart Plat-Forms
,”
IEEE Trans. Rob. Autom.
1042-296X,
15
(
2
), pp.
281
288
.
22.
Caccavale
,
F.
,
Siciliano
,
B.
, and
Villani
,
L.
, 2003, “
The Tricept Robot: Dynamics and Impedance Control
,”
IEEE/ASME Trans. Mechatron.
1083-4435,
8
(
2
), pp.
263
268
.
23.
Kock
,
S.
, 2000, “
Control of a Fast Parallel Robot With a Redundant Chain and Gearboxes: Experimental Results
,”
Proceeding of 2000 IEEE International Conference on Robotics and Automation
, San Francisco, April,
IEEE
,
New York
, pp.
1924
1929
.
24.
Liu
,
G.
, and
Li
,
Z.
, 2002, “
A Unified Geometric Approach to Modeling and Control of Constrained Mechanical System
,”
IEEE Trans. Rob. Autom.
1042-296X,
18
(
4
), pp.
574
587
.
25.
Dabney
,
J.
,
Ghorbel
,
F. H.
, and
Wang
,
Z.
, 2002, “
Modeling Closed Kinematic Chains via Singular Perturbations
,”
Proceeding of the American Control Conference
, Anchorage, AK, May,
IEEE
,
New York
, pp.
4104
4110
.
26.
Wang
,
Z.
,
Ghorbel
,
F. H.
, and
Dabney
,
J.
, 2004, “
On the Domain and Error Characterization in the Singular Perturbation Modeling of Closed Kinematic Chains
,”
Proceeding of the 2004 American Control Conference
, Boston, June,
IEEE
,
New York
, pp.
493
498
.
27.
Slotine
,
J. J. E.
, and
Li
,
W.
, 1987, “
On the Adaptive Control of Robot Manipulators
,”
Int. J. Robot. Res.
0278-3649,
6
(
3
), pp.
49
59
.
28.
Saberi
,
A.
, and
Khalil
,
H.
, 1984, “
Quadratic-Type Lyapunov Functions for Singularly Perturbed Systems
,”
IEEE Trans. Autom. Control
0018-9286,
29
(
6
), pp.
542
550
.
29.
Yun
,
X.
, and
Sarkar
,
N.
, 1998, “
Unified Formulation of Robotic Systems With Holonomic and Noholonomic Constraints
,”
IEEE Trans. Rob. Autom.
1042-296X,
14
(
4
), pp.
640
649
.
30.
Wang
,
Z.
, 2004, “
Modeling and Control of Closed Kinematic Chains: A Singular Perturbation Approach
,” Ph.D. dissertation, Rice University, Houston.
31.
Ghorbel
,
F. H.
,
Srinivasant
,
B.
, and
Spong
,
M. W.
, 1993, “
On the Positive Definiteness and Uniform Boundedness of the Inertia Matrix of Robot Manipulators
,” in
Proceedings of 32nd Conference on Decision and Control, December
, San Antonio, December, IEEE, New York, pp.
1103
1108
.
32.
Vidyasagar
,
M.
, 1993,
Nonlinear Systems Analysis
,
2nd ed.
,
Prentice Hall
,
Englewood Cliffs, NJ
.
You do not currently have access to this content.