The recently developed bio-inspired virtual motion camouflage (VMC) method can be used to rapidly solve nonlinear constrained optimal trajectory problems. However, the optimality of VMC solution is affected by the dimension reduced search space. Compared with the VMC method, the B-spline augmented VMC (BVMC) method studied in this paper improves the optimality of the solution, while the computational cost will not be significantly increased. Two simulation examples, the Snell's river problem and a robotic minimum time obstacle avoidance problem, are used to show the advantages of the algorithm.

References

References
1.
Nusyirwan
, I
. F.
, and
Bil
,
C.
,
2007
, “
Effect of Uncertainties on UAV Optimisation Using Evolutionary Programming
,”
IEEE Information, Decision, and Control Conference
, Feb. 11–14,
Adelaide, Australia
, pp.
219
223
.
2.
Kaneshige
,
J.
, and
Krishnakumar
,
K.
,
2007
, “
Artificial Immune System Approach for Air Combat Maneuvering
,”
Intelligent Computing: Theory and Applications
,
K. L.
Priddy
and
E.
Ertin
, eds., Society of Photo Optical, New York.
3.
Merchan-Cruz
,
E. A.
, and
Morris
,
A. S.
,
2006
, “
Fuzzy-GA-Based Trajectory Planner for Robot Manipulators Sharing a Common Workspace
,”
IEEE Trans. Rob.
,
22
(
4
), pp.
613
624
.10.1109/TRO.2006.878789
4.
Hristu-Varsakelis
,
D.
, and
Shao
,
C.
,
2004
, “
Biologically-Inspired Optimal Control: Learning From Social Insects
,”
Int. J. Control
,
77
(
18
), pp.
1549
1566
.10.1080/00207170412331330098
5.
Lawden
,
D. F.
,
1991
, “
Rocket Trajectory Optimization: 1950–1963
,”
J. Guid. Control Dyn.
,
14
(
4
), pp.
705
711
.10.2514/3.20703
6.
Ocampo
,
C.
,
2004
, “
Finite Burn Maneuver Modeling for a Generalized Spacecraft Trajectory Design and Optimization System
,”
Ann. N. Y. Acad. Sci.
,
1017
, pp.
210
233
.10.1196/annals.1311.013
7.
Pontryagin
,
L. S.
,
Boltyanskii
,
V. G.
,
Gamkrelidze
,
R. V.
, and
Mishchenko
,
E. F.
,
1962
,
The Mathematical Theory of Optimal Processes
,
Wiley-Interscience
,
New York
.
8.
Betts
,
J.
,
2001
,
Practical Methods for Optimal Control Using Nonlinear Programming
,
SIAM
,
Philadelphia
.
9.
Fahroo
,
F.
, and
Ross
, I
. M.
,
2001
, “
Costate Estimation by a Legendre Pseudospectral Method
,”
J. Guid. Control Dyn.
,
24
(
2
), pp.
270
275
.10.2514/2.4709
10.
Hager
,
W.
,
2000
, “
Runge-Kutta Methods in Optimal Control and the Transformed Adjoint System
,”
Numer. Math.
,
87
(
2
), pp.
247
282
.10.1007/s002110000178
11.
Milam
,
M.
,
Mushambi
,
K.
, and
Murray
,
R.
,
2000
, “
A Computational Approach to Real-Time Trajectory Generation for Constrained Mechanical Systems
,”
IEEE Conference on Decision and Control
, Dec. 12–15,
Sydney, Australia
, pp.
845
851
.
12.
Dai
,
R.
, “
B-Splines Based Optimal Control Solution
,”
2010 AIAA Guidance, Navigation, and Control Conference
, Aug. 2–5,
Toronto, ON
, Paper No. AIAA-2010-7888.
13.
Jackiewicz
,
Z.
, and
Welfert
,
B. D.
,
2003
, “
Stability of Gauss-Radau Pseudospectral Approximations of the One-Dimensional Wave Equation
,”
J. Sci. Comput.
,
18
(
2
), pp.
287
313
.10.1023/A:1021121008091
14.
Kameswaran
,
S.
, and
Biegler
,
L. T.
,
2008
, “
Convergence Rates for Direct Transcription of Optimal Control Problems Using Collocation at Radau Points
,”
Comput. Optim. Appl.
,
41
(
1
), pp.
81
126
.10.1007/s10589-007-9098-9
15.
Jacobson
,
D. H.
, and
Lele
,
M. M.
,
1969
, “
A Transformation Technique for Optimal Control Problems With a State Variable Inequality Constraint
,”
IEEE Trans. Autom. Control
,
14
(
5
), pp.
457
464
.10.1109/TAC.1969.1099283
16.
Mehra
,
R. K.
, and
Davis
,
R. E.
,
1972
, “
A Generalized Gradient Method for Optimal Control Problems With Inequality Constraints and Singular Arcs
,”
IEEE Trans. Autom. Control
,
17
(
1
), pp.
69
79
.10.1109/TAC.1972.1099881
17.
Betts
,
J.
,
1998
, “
Survey of Numerical Methods for Trajectory Optimization
,”
J. Guid. Control Dyn.
,
21
(
2
), pp.
193
207
.10.2514/2.4231
18.
Goerzen
,
C.
,
Kong
,
Z.
, and
Mettler
,
B.
,
2010
, “
A Survey of Motion Planning Algorithms From the Perspective of Autonomous UAV Guidance
,”
J. Intell. Robotic Syst.
,
57
, pp.
65
100
.10.1007/s10846-009-9383-1
19.
Xu
,
Y.
,
2007
, “
Motion Camouflage and Constrained Suboptimal Trajectory Control
,”
2007 AIAA Guidance, Control, and Dynamics Conference
, Aug. 20–23,
Hilton Head, SC
.
20.
Xu
,
Y.
,
2010
, “
Analytical Solutions to Formation Flying System Trajectory Guidance Via the Virtual Motion Camouflage Approach
,”
J. Guid. Control Dyn.
,
33
(
5
), pp.
1376
1386
.10.2514/1.48691
21.
Kumar
,
R. R.
, and
Seywald
H.
,
1996
, “
Should Controls be Eliminated While Solving Optimal Control Problems Via Direct Methods?
,”
J. Guid. Control Dyn.
,
19
(
2
), pp.
418
423
.10.2514/3.21634
22.
Piegl
,
L.
, and
Tiller
,
W.
,
1997
,
The NURBS Book: Second Edition
,
Springer-Verlag
,
New York
.
23.
Benson
,
D. A.
,
Huntington
,
G. T.
,
Thorvaldsen
,
T. P.
, and
Rao
,
A. V.
,
2006
, “
Direct Trajectory Optimization and Costate Estimation Via an Orthogonal Collocation Method
,”
J. Guid. Control Dyn.
,
29
(
6
), pp.
1435
1440
.10.2514/1.20478
24.
Rao
,
C. V.
,
Wright
,
S. J.
, and
Rawlings
,
J. B.
,
1998
, “
Application of Interior-Point Methods to Model Predictive Control
,”
J. Optim. Theory Appl.
,
99
(
3
), pp.
723
757
.10.1023/A:1021711402723
25.
Xu
,
Y.
, and
Basset
,
G.
,
2010
, “
Real-Time Optimal Coherent Phantom Track Generation Via the Virtual Motion Camouflage Approach
,”
2010 AIAA Guidance, Navigation, and Control Conference
, Aug. 2–5,
Toronto, ON, Canada
, Paper No. AIAA-2010-7714.
26.
Hartl
,
R. F.
,
Sethi
,
S. P.
, and
Vickson
,
R. G.
,
1995
, “
A Survey of the Maximum Principles for Optimal Control Problems With State Constraints
,”
SIAM Rev.
,
37
(
2
), pp.
181
218
.10.1137/1037043
27.
Srinivasan
,
M. V.
, and
Davey
,
M.
,
1995
, “
Strategies for Active Camouflage Motion
,”
Proc. R. Soc. London
,
259
(
1354
), pp.
19
25
.10.1098/rspb.1995.0004
28.
Gong
,
Q.
,
Kang
,
W.
, and
Ross
, I
. M.
,
2006
, “
A Pseudospectral Method for the Optimal Control of Constrained Feedback Linearizable Systems
,”
IEEE Trans. Autom. Control
,
51
(
7
), pp.
1115
1129
.10.1109/TAC.2006.878570
29.
Gong
,
Q.
,
Ross
, I
. M.
,
Kang
,
W.
, and
Fahroo
,
F.
,
2008
, “
Connections Between the Covector Mapping Theorem and Convergence of Pseudospectral Methods for Optimal Control
,”
Comput. Optim. Appl.
,
41
(
3
), pp.
307
335
.10.1007/s10589-007-9102-4
30.
Ferrante
,
A.
, and
Ntogramatzidis
,
L.
,
2006
, “
A Unified Approach to the Finite-Horizon LQ Regulator—Part I: The Continuous Time
,”
45th IEEE Conference on Decision and Control
, Dec. 13–15,
San Diego, CA
, pp.
5651
5656
.
31.
Bryson
,
A. E.
, Jr.
, and
Ho
,
Y. C.
,
1975
,
Applied Optimal Control: Optimization, Estimation, and Control
,
Taylor & Francis
,
Washington, DC
.
32.
Laumond
,
J. P.
,
Sekhavat
,
S.
, and
Lamiraux
,
F.
,
1998
, Guidelines in Nonholonomic Motion Planning for Mobile Robots (Lecture Notes in Control and Information Sciences), LNCIS 229, Springer-Verlag, New York, pp.
1
44
.
You do not currently have access to this content.