This work has developed a new robust and reliable O(N) algorithm for solving general inequality/equality constrained minimum-time problems. To our knowledge, no one has ever applied an O(N) algorithm for solving such minimum time problems. Moreover, the algorithm developed here is new and unique and does not suffer the inevitable ill-conditioning problems that pre-existing O(N) methods for inequality-constrained problems do. Herein we demonstrate the new algorithm by solving several cases of a tip path constrained three-link redundant robotic arm problem with torque bounds and joint angle bounds. Results are consistent with Pontryagin’s Maximum Principle. We include a speed/robustness/complexity comparison with a sequential quadratic programming (SQP) code. Here, the O(N) complexity and the significant speed, robustness, and complexity improvements over an SQP code are demonstrated. These numerical results are complemented with a rigorous theoretical convergence proof of the new O(N) algorithm.

1.
Bashein
G.
,
1971
, “
A Simplex Algorithm for On-Line Computation of Time Optimal Controls
,”
IEEE Transactions on Automatic Control
, Vol.
16
, pp.
479
482
.
2.
Bazaraa, M. S., and Shetty, C. M., 1977, Nonlinear Programming, New York: John Wiley and Sons.
3.
Bobrow, J E., et al., 1983, “On the Optimal Control of Robotic Manipulators with Actuator Constraints,” Proceedings of the 1983 American Control Conference, Vol. 2, pp. 782–787.
4.
Bobrow, J. E., Dubowsky, S., and Gibson, J. S., 1985, “Time-Optimal Control of Robotic Manipulators,” International Journal of Robotics Research, Vol. 4, No. 3.
5.
Boltjanski, W. G., 1969, “Mathematic Methods of Optimal Control,” Moskau: Nauka.
6.
Byers, R. M., and Vadali, S. R., 1993, “Quasi-Closed-Form Solution to the Time-Optimal Rigid Spacecraft Reorientation Problem,” Journal of Guidance, Control, and Dynamics, Vol. 16, No. 3.
7.
De Vlieger
J. H.
,
Verbruggen
H. B.
, and
Bruijn
P. M.
,
1982
, “
A Time-Optimal Control Algorithm for Digital Computer Control
,”
Automatica
, Vol.
18
, pp.
239
244
.
8.
Eisler
G. R.
,
Robinett
R. D.
,
Segalman
D. J.
, and
Feddema
J. D.
,
1993
, “
Approximate Optimal Trajectories for Flexible-Link Manipulator Slewing Using Recursive Quadratic Programming
,”
ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL
, Vol.
115
, pp.
405
410
.
9.
Eisler, G. R., Segalman, D. J., and Robinett, R. D., 1990, “Approximate Minimum-Time Trajectories for Two-Link Flexible Manipulators,” Proceedings of the American Control Conference, pp. 870–875.
10.
Geering, H. P., et al., 1986, “Time-Optimal Motions of Robots in Assembly Tasks,” IEEE Transactions on Automatic Control, Vol. AC-31, No. 6.
11.
Gonzaga, et al., 1980, “An Improved Algorithm for Optimization Problems with Functional Inequality Constraints,” IEEE Transactions on Automatic Control, Vol. AC-25, No. 1.
12.
Ho, Y., and Bryson, A. E., 1975, Applied Optimal Control, Hemisphere Publishing, NY.
13.
Hollerbach
J. M.
,
1984
, “
Dynamic Scaling of Manipulator Trajectories
,”
ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL
, Vol.
106
, pp.
102
106
.
14.
Kim
M.
,
1965
, “
On the Minimum Time Control of Linear Sampled-Data Systems
,”
Proceedings of the IEEE
, Vol.
53
, pp.
1263
1264
.
15.
Larson, R. E., and Casti, J. L., 1982, Principles of Dynamic Programming, Part II, Advanced Theory and Applications, Marcel Dekker, NY.
16.
Lewis, F. L., 1986, Optimal Control, Wiley, NY.
17.
Li
F.
, and
Bainum
P.
,
1994
, “
Analytic Time-Optimal Control Synthesis of Fourth-Order System and Manuevers of Flexible Structures
,”
Journal of Guidance, Control, and Dynamics
, Vol.
17
, No.
6
, pp.
1171
1178
.
18.
Li
F.
, et al.,
1995
, “
Three-axis Near-Minimum-Time Maneuvers of RESHAPE: Numerical and Experimental Results
,”
Journal of Guidance, Control, and Dynamics
, Vol.
43
, No.
2
, pp.
161
178
.
19.
Meier, E., and Bryson, A. E., 1990, “Efficient Algorithm for Time-Optimal Control of a Two-Link Manipulator,” Journal of Guidance, Control, and Dynamics, Vol. 13, No. 5.
20.
Pao, L., 1994, “Characteristics of the Time-Optimal Control of Flexible Structures with Damping,” Proceedings of the 1994 IEEE Conference on Control Applications, 1994, pp. 1299–1304.
21.
Polak
E.
, and
Mayne
D.
,
1976
, “
An Algorithm for Optimization Problems with Functional Inequality Constraints
,”
IEEE Transactions on Automatic Control
, Vol.
AC-21
, pp.
184
193
.
22.
Polak
E.
et al.,
1979
, “
Combined Phase I, Phase II Methods of Feasible Directions
,”
Mathematical Programming
, Vol.
17
, No.
1
, pp.
61
74
.
23.
Rasmy
M. E.
, and
Hamza
M. H.
,
1975
, “
Minimum-Effort Time-Optimal Control of Linear Discrete Systems
,”
International Journal of Control
, Vol.
21
, pp.
293
304
.
24.
Ruxton
D. J.
,
1993
, “
Differential Dynamic Programming Applied to Continuous Optimal Control Problems with State Variable Inequality Constraints
,”
Dynamics and Control
, Vol.
3
, pp.
175
185
.
25.
Sahar, Gideon, and Hollerbach, John M., 1985, “Planning of Minimum-Time Trajectories for Robot Arms,” IEEE 1985 International Conference on Robotics and Automation, pp. 751–758.
26.
Seeger, Giudo H., and Paul, Richard P., 1985, “Optimizing Robot Motion Along a Predefined Path,” IEEE 1985 International Conference on Robotics and Automation, pp. 765–770.
27.
Shin, K. G., and McKay, N. D., 1985, “Minimum-Time Control of Robotic Manipulators with Geometric Paths,” IEEE Transactions on Automatic Control, Vol. AC-30, No. 6.
28.
Shin, K. G., and McKay, N. D., 1984, “Open-Loop Minimum-Time Control of Mechanical Manipulators and its Application,” Conference Proceedings of the 1984 American Control Conference, Vol. 3, pp. 1231–1236.
29.
Shin, K. G., and McKay, N. D., 1986, “A Dynamic Programming Approach to Trajectory Planning of Robotic Manipulators,” IEEE Transactions on Automatic Control, Vol. AC-31, No. 6.
30.
Shin, K. G., and McKay, N. D., 1986, “Selection of Near-Minimum Time Geometric Paths for Robotic Manipulators,” IEEE Transactions on Automatic Control, Vol. AC-31, No. 6.
31.
Torng
H. C.
,
1964
, “
Optimization of Discrete Control Systems Through Linear Programming
,”
Journal of the Franklin Institute
, Vol.
278
, pp.
28
44
.
32.
Wie
B.
, and
Sunkel
J.
,
1990
, “
Minimum-Time Pointing Control of a Two-Link Manipulator
,”
Journal of Guidance, Control, and Dynamics
, Vol.
13
, No.
5
, pp.
867
873
.
33.
Wright
S.
,
1993
, “
Interior Point Methods for Optimal Control of Discrete Time Systems
,”
Journal of Optimization Theory and Applications
, Vol.
77
, No.
1
, pp.
161
187
.
34.
Wright, S., 1991, “Structured Interior Point Methods for Optimal Control,” Proceedings of the 30th Conference on Decision and Control, Brighton, England, pp. 1711–1716.
35.
Wu
Chia-Ju
,
1994
, “
A Numerical Approach for Time-Optimal Path-Planning of Kinematically Redundant Manipulators
,”
Robotica
, Vol.
12
, pp.
401
410
.
36.
Wu
Chia-Ju
,
1995
, “
Minimum-Time Control for an Inverted Pendulum under Force Constraints
,”
Journal of Intelligent and Robotic Systems
, Vol.
12
, pp.
127
143
.
37.
Yang
Eugene K.
, and
Hwang
Wei-Shi
,
1992
, “
A Barrier Method for Dynamic Leontief-type Linear Programs
,”
European Journal of Operational Research
, Vol.
60
, pp.
296
305
.
38.
Zadegh
L. A.
, and
Whalen
B. H.
,
1962
, “
On Optimal Control and Linear Programming
,”
IRE Transactions on Automatic Control
, Vol.
7
, pp.
45
46
.
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