Technical Briefs

Iterative Learning in Ballistic Control: Formulation of Spatial Learning Processes for Endpoint Control

[+] Author and Article Information
Jian-Xin Xu

e-mail: elexujx @nus.edu.sg

Deqing Huang

e-mail: elehd@nus.edu.sg
Department of Electrical and Computer Engineering,
National University of Singapore,
Singapore 117576

Contributed by the Dynamic Systems Division of ASME for publication in the Journal of Dynamic Systems, Measurement, and Control. Manuscript received November 23, 2010; final manuscript received May 8, 2012; published online November 7, 2012. Assoc. Editor: Warren E. Dixon.

J. Dyn. Sys., Meas., Control 135(2), 024501 (Nov 07, 2012) (11 pages) Paper No: DS-10-1344; doi: 10.1115/1.4007236 History: Received November 23, 2010; Revised May 08, 2012

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.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Arimoto, S., Kawamura, S., and Miyazaki, F., 1984, “Bettering Operation of Robots by Learning,” J. Rob. Syst., 1(2), pp. 123–140. [CrossRef]
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.
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. [CrossRef]
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.
Longman, R. W., 2000, “Iterative Learning Control and Repetitive Control for Engineering Practice,” Int. J. Control, 73(10), pp. 930–954. [CrossRef]
Bien, Z., and Xu, J.-X., 1998, Iterative Learning Control—Analysis, Design, Integration and Applications, Kluwer Academic Press, Boston.
Xu, J.-X., and Tan, Y., 2003, Linear and Nonlinear Iterative Learning Control, Springer-Verlag, Berlin.
Bristow, D. A., Tharayil, M., and Allyne, A. G., 2006, “A Survey of Iterative Learning,” IEEE Control Syst. Mag., 26(3), pp. 96–114. [CrossRef]
Parker, G. W., 1977, “Projectile Motion With Air Resistance Quadratic in the Speed,” Am. J. Phys., 45(7), pp. 606–610. [CrossRef]
Tan, A., and Miller, G., 1981, “Kinematics of the Free Throw in Basketball,” Am. J. Phys., 49(6), pp. 542–544. [CrossRef]
Xu, J.-X., and Huang, D., 2008, “Initial State Iterative Learning for Final State Control in Motion Systems,” Automatica, 44, pp. 3162–3169. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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.
Keller, H. B., 1968, Numerical Methods for Two-Point Boundary-Value Problems, Blaisdell, Waltham, MA.
Keller, H. B., 1976, “Numerical Solution of Two Point Boundary Value Problems,” CBMS-NSF Regional Conference Series in Applied Mathematics.
Salomon, R., 1998, “Evolutionary Algorithms and Gradient Search: Similarities and Differences,” IEEE Trans. Evol. Comput., 2(2), pp. 45–55. [CrossRef]
Batchelor, G. K., 1967, An Introduction to Fluid Dynamics, Cambridge University Press, London.
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. [CrossRef]


Grahic Jump Location
Fig. 1

The sketch of ballistic control problem under the effect of air resistance

Grahic Jump Location
Fig. 2

In ballistic control with shooting angle learning, there is a unique critical shooting angle ϕc ∈ (0,π/2-θ) such that the projectile can achieve the furthest flight distance. (a) The motion trajectories of projectile when the shooting angle φ is less than φc, (b) the motion trajectories of projectile when the shooting angle φ is greater than φc. Due to the effect of air resistance, φc is usually less than π/4 − θ. In practice, case (b) with a larger incident angle is preferred when an obstacle exists in between P and O.

Grahic Jump Location
Fig. 3

Variation of the parameter b versus flight height y in ballistic control process. When the flight height of projectile is increased from 0 m to 10 km, the magnitude of b is reduced by 2/3.

Grahic Jump Location
Fig. 4

The output error profile versus iteration number. Within 13 iterations, the output error in dramatically decreased from 2517 m to 0.54 m.

Grahic Jump Location
Fig. 5

The profile of initial shooting speed signal versus iteration number. The desired shooting speed is v0,d = 807.47 m/s.

Grahic Jump Location
Fig. 6

The motion trajectory of projectile with given shooting angle φ = π/6 and the learned initial speed v0 = 807.47 m/s. It can be seen that the target located at zd = 2 × 104/cos(θ) is attacked accurately.

Grahic Jump Location
Fig. 7

The profile of learning gain versus iteration number. Within 12 iterations, it changes from 0.74 to 81.35.

Grahic Jump Location
Fig. 8

Gradient variation profile versus parametric perturbation and input change, where 9057 ≤ dz/dφ ≤ 7.76 × 104 and the process parameters are m = 4.7 kg, θ = π/12 rad, v0 = 900 m/s

Grahic Jump Location
Fig. 9

The profile of output error versus iteration number. Within five iterations, the output error in dramatically decreased from 2313 m to 0.11 m.

Grahic Jump Location
Fig. 10

The profile of input signal versus iteration number. Due to the parametric perturbation, the desired projection angle is 0.3328 rad.

Grahic Jump Location
Fig. 11

The motion trajectory of projectile with given initial speed v0 = 900 m/s and the learned projection angle φd = 0.3328 rad

Grahic Jump Location
Fig. 12

The variation of learning gain versus iteration number. Within four iterations, it increases from 1.289 × 10−5 to 4.349 × 10−5 and then decreases to 2.899 × 10−5.

Grahic Jump Location
Fig. 13

Variation of the performance index J. Within 34 iterations, J is decreased from 5.348 × 104 to 0.0078. Accordingly, the output error is dramatically decreased from 2313 m to 0.88 m.

Grahic Jump Location
Fig. 14

The profile of input components versus iteration number. The desired shooting angle 0.2654 rad and shooting speed v0 = 978.0 m/s are achieved within 34 iterations.

Grahic Jump Location
Fig. 15

The output error profile versus iteration number under wind effect. After learning 12 iterations, the output error is decreased by 99.9%, from the initial discrepancy 2517 m to the final 2.44 m in average.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In