In this paper, we formulate and explore the characteristics of iterative learning in ballistic control problems, where the projectile experiences a constant gravitational force and a fluid drag force that is quadratic in speed. Three scenarios are considered in the spatial learning process, where the shooting speed, shooting angle, or their combination, are, respectively, the manipulated variables. The viewed endpoint displacement is the controlled variable. Under the framework of iterative learning control, ballistic learning convergence is derived in the presence of process uncertainties. In the end, an illustrative example is provided to verify the validity of the proposed ballistic learning control schemes.

References

1.
Arimoto
,
S.
,
Kawamura
,
S.
, and
Miyazaki
,
F.
,
1984
, “
Bettering Operation of Robots by Learning
,”
J. Rob. Syst.
,
1
(
2
), pp.
123
140
.10.1002/rob.4620010203
2.
Zhang
,
S.
, and
Zhu
,
N.
,
2001
, “
Simulation and Research of Inertia Control of End Control Cannonball
,”
Proceedings of the International Conference on Modeling and Simulation in Distributed Applications
,
Changsha
,
Hunan, China
, Sept. 25–27, pp.
273
276
.
3.
Swetz
,
F. J.
,
1989
, “
An Historical Example of Mathematical Modelling: The Trajectory of a Cannonball
,”
Int. J. Math. Educ. Sci. Technol.
,
20
(
5
), pp.
731
741
.10.1080/0020739890200511
4.
Moore
,
K. L.
,
1999
, “
Iterative Learning Control—An Expository Overview
,”
Applied and Computational Control, Signals, and Circuits
, Vol.
1
,
Birkhaeuser Boston
,
Cambridge, MA
, pp.
151
214
.
5.
Longman
,
R. W.
,
2000
, “
Iterative Learning Control and Repetitive Control for Engineering Practice
,”
Int. J. Control
,
73
(
10
), pp.
930
954
.10.1080/002071700405905
6.
Bien
,
Z.
, and
Xu
,
J.-X.
,
1998
,
Iterative Learning Control—Analysis, Design, Integration and Applications
,
Kluwer Academic Press
,
Boston
.
7.
Xu
,
J.-X.
, and
Tan
,
Y.
,
2003
,
Linear and Nonlinear Iterative Learning Control
,
Springer-Verlag
,
Berlin
.
8.
Bristow
,
D. A.
,
Tharayil
,
M.
, and
Allyne
,
A. G.
,
2006
, “
A Survey of Iterative Learning
,”
IEEE Control Syst. Mag.
,
26
(
3
), pp.
96
114
.10.1109/MCS.2006.1636313
9.
Parker
,
G. W.
,
1977
, “
Projectile Motion With Air Resistance Quadratic in the Speed
,”
Am. J. Phys.
,
45
(
7
), pp.
606
610
.10.1119/1.10812
10.
Tan
,
A.
, and
Miller
,
G.
,
1981
, “
Kinematics of the Free Throw in Basketball
,”
Am. J. Phys.
,
49
(
6
), pp.
542
544
.10.1119/1.12668
11.
Xu
,
J.-X.
, and
Huang
,
D.
,
2008
, “
Initial State Iterative Learning for Final State Control in Motion Systems
,”
Automatica
,
44
, pp.
3162
3169
.10.1016/j.automatica.2008.05.017
12.
Xu
,
J.-X.
,
Chen
,
Y. Q.
,
Lee
,
T. H.
, and
Yamamoto
,
S.
,
1999
, “
Terminal Iterative Learning Control With an Application to RTPCVD Thickness Control
,”
Automatica
,
35
(
9
), pp.
1535
1542
.10.1016/S0005-1098(99)00076-X
13.
Chen
,
Y. Q.
,
Wen
,
C. Y.
,
Gong
,
Z.
, and
Sun
,
M. X.
,
1999
, “
An Iterative Learning Controller With Initial State Learning
,”
IEEE Trans. Autom. Control
,
44
(
2
), pp.
371
376
.10.1109/9.746269
14.
Holsapple
,
R.
,
Venkataraman
,
R.
, and
Doman
,
D.
,
2003
, “
A Modified Simple Shooting Method for Solving Two-Point Boundary-Value Problems
,”
Proceedings of Aerospace Conference
, pp.
2783
2790
.
15.
Keller
,
H. B.
,
1968
,
Numerical Methods for Two-Point Boundary-Value Problems
,
Blaisdell
,
Waltham, MA
.
16.
Keller
,
H. B.
,
1976
, “
Numerical Solution of Two Point Boundary Value Problems
,”
CBMS-NSF Regional Conference Series in Applied Mathematics
.
17.
Salomon
,
R.
,
1998
, “
Evolutionary Algorithms and Gradient Search: Similarities and Differences
,”
IEEE Trans. Evol. Comput.
,
2
(
2
), pp.
45
55
.10.1109/4235.728207
18.
Batchelor
,
G. K.
,
1967
,
An Introduction to Fluid Dynamics
,
Cambridge University Press
,
London
.
19.
McPhee
,
J. J.
, and
Andrews
,
G. C.
,
1988
, “
Effect of Sidespin and Wind on Projectile Trajectory, With Particular Application to Golf
,”
Am. J. Phys.
,
56
(
10
), pp.
933
939
.10.1119/1.15363
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