The dynamics of guided projectile systems are inherently stochastic in nature. While deterministic control algorithms such as impact point prediction (IPP) may prove effective in many scenarios, the probability of impacting obstacles and constrained areas within an impact zone cannot be accounted for without accurate uncertainty modeling. A stochastic model predictive guidance algorithm is developed, which incorporates nonlinear uncertainty propagation to predict the impact probability density in real-time. Once the impact distribution is characterized, the guidance system aim point is computed as the solution to an optimization problem. The result is a guidance law that can achieve minimum miss distance while avoiding impact area constraints. Furthermore, the acceptable risk of obstacle impact can be quantified and tuned online. Example trajectories and Monte Carlo simulations demonstrate the effectiveness of the proposed stochastic control formulation in comparison to deterministic guidance schemes.

References

References
1.
Ravindra
,
V. C.
,
Bar-Shalom
,
Y.
, and
Willett
,
P.
,
2010
, “
Projectile Identification and Impact Point Prediction
,”
IEEE Trans. Aerosp. Electron. Syst.
,
46
(
4
), pp.
2004
2021
.10.1109/TAES.2010.5595610
2.
Hardiman
,
D. F.
,
Kerce
,
J. C.
, and
Brown
,
G. C.
,
2006
, “
Nonlinear Estimation Techniques for Impact Point Prediction of Ballistic Targets
,” Proceedings of the
SPIE
Conference on Signal and Data Processing of Small Targets, Orlando, FL, May 19, Vol.
6236
.10.1117/12.679557
3.
Kashigawi
,
Y.
,
1968
, “
Prediction of Ballistic Missile Trajectories
,” Stanford Research Institute, Menlo Park, CA, Technical Report No. SRI-H-8976.
4.
Morgan
,
R. W.
,
Tharp
,
H.
, and
Vincent
,
T.
,
2011
, “
Minimum Energy Guidance for Aerodynamically Controlled Missiles
,”
IEEE Trans. Autom. Control
,
56
(
9
), pp.
2026
2037
.10.1109/TAC.2011.2108619
5.
Hainz
,
L.
, and
Costello
,
M.
,
2005
, “
Modified Projectile Linear Theory for Rapid Trajectory Prediction
,”
J. Guid., Control, Dyn.
,
28
(
5
), pp.
1006
1014
.10.2514/1.8027
6.
Slegers
,
N.
,
2008
, “
Predictive Control of a Munition Using Low-Speed Linear Theory
,”
J. Guid., Control, Dyn.
,
31
(
3
), pp.
768
775
.10.2514/1.34329
7.
Fresconi
,
F.
,
Cooper
,
G.
, and
Costello
,
M.
,
2011
, “
Practical Assessment of Real-Time Impact Point Estimators for Smart Weapons
,”
ASCE J. Aerosp. Eng.
,
24
(
1
), pp.
1
11
.10.1061/(ASCE)AS.1943-5525.0000044
8.
Speyer
,
J.
,
1976
, “
An Adaptive Terminal Guidance Scheme Based on an Exponential Cost Criterion With Application to Homing Missile Guidance
,”
IEEE Trans. Autom. Control
,
21
(
3
), pp.
371
375
.10.1109/TAC.1976.1101206
9.
Zarchan
,
P.
,
1979
, “
Complete Statistical Analysis of Nonlinear Missile Guidance Systems-SLAM
,”
J. Guid., Control, Dyn.
,
2
(
1
), pp.
71
78
.10.2514/3.55834
10.
Hull
,
R. A.
,
Schumacher
,
D.
, and
Qu
,
Z.
,
1995
, “
Design and Evaluation of Robust Nonlinear Missile Autopilots From a Performance Perspective
,”
Proceedings of the American Control Conference
, Seattle, WA, Vol.
1
, pp.
189
193
.
11.
Yavin
,
Y.
,
Frangos
,
G.
, and
Fourie
,
J.
,
1992
, “
The Performance of a Projectile Which Uses a Bang-Bang Type Guidance Law – Part 1
,”
Comput. Math. Appl.
,
23
(
1
), pp.
111
118
.10.1016/0898-1221(92)90086-W
12.
Yavin
,
Y.
,
Frangos
,
G.
, and
Fourie
,
J.
,
1992
, “
The Performance of a Projectile Which Uses a Bang-Bang Type Guidance Law – Part 2 – A Semirigid Body Model
,”
Comput. Math. Appl.
,
24
(
4
), pp.
85
92
.10.1016/0898-1221(92)90010-F
13.
Hexner
,
G.
, and
Shima
,
T.
,
2007
, “
Stochastic Optimal Control Guidance Law With Bounded Acceleration
,”
IEEE Trans. Aerosp. Electron. Syst.
,
43
(
1
), pp.
71
78
.10.1109/TAES.2007.357155
14.
Abate
,
A.
,
Prandini
,
M.
,
Lygeros
,
J.
, and
Sastry
,
S.
,
2008
, “
Probabilistic Reachability and Safety for Controlled Discrete Time Stochastic Hybrid Systems
,”
Automatica
,
44
(
11
), pp.
2724
2734
.10.1016/j.automatica.2008.03.027
15.
Summers
,
S.
, and
Lygeros
,
J.
,
2010
, “
Verification of Discrete Time Stochastic Hybrid Systems: A Reach-Avoid Decision Problem
,”
Automatica
,
46
(12)
, pp.
1951
1961
.10.1016/j.automatica.2010.08.006
16.
Summers
,
S.
,
Kamgarpour
,
M.
,
Tomlin
,
C.
, and
Lygeros
,
J.
,
2011
, “
A Stochastic Reach-Avoid Problem With Random Obstacles
,”
14th International Conference on Hybrid Systems: Computation and Control (HSCC’11)
, Chicago, IL, Apr. 12–14.
17.
Thrun
,
S.
,
Burgard
,
W.
, and
Fox
,
D.
,
2006
,
Probabilistic Robotics
,
MIT
,
Cambridge, MA
, p.
496
.
18.
Blondel
,
V.
, and
Tsitsiklis
,
J.
,
2000
, “
A Survey of Computational Complexity Results in Systems and Control
,”
Automatica
,
36
(9)
, pp.
1249
1274
.10.1016/S0005-1098(00)00050-9
19.
Rust
,
J.
,
1997
, “
Using Randomization to Break the Curse of Dimensionality
,” Ph.D. dissertation, Yale University, Department of Economics, New London, CT.
20.
Doucet
,
A.
,
de Freitas
,
N.
, and
Gordon
,
N.
,
2001
,
Sequential Monte Carlo Methods in Practice
,
Springer–Verlag
,
New York
, pp.
6
14
.
21.
Wiener
,
N.
,
1938
, “
The Homogeneous Chaos
,”
Am. J. Math.
,
60
(
4
), pp.
897
936
.10.2307/2371268
22.
Roberts
,
J. B.
, and
Spanos
,
P. D.
,
1990
,
Random Vibration and Statistical Linearization
,
Wiley
,
New York
, pp.
122
176
.
23.
Risken
,
H.
,
1989
,
The Fokker–Planck Equation: Methods of Solution and Applications
,
Springer
,
New York
, pp.
32
62
.
24.
Scott
,
D.
, 1992,
Multivariate Density Estimation: Theory, Visualization, and Practice
,
Wiley
,
New York
, p.
149
.
25.
Ilg
,
M.
,
Rogers
,
J.
, and
Costello
,
M.
,
2011
, “
Projectile Monte-Carlo Trajectory Analysis Using a Graphics Processing Unit
,”
2011 AIAA Atmospheric Flight Mechanics Conference
, Portland, OR, Aug. 7–10.
26.
SECO Corporation,
2012
, “
Carma Devkit – SECO
,” Accessed Dec. 20, http://www.nvidia.com/object/carma-devkit.html
27.
SVTronics, Inc.,
2013
, “
OMAP 5432 uEVM Development Board
,” Accessed Apr. 18, http://www.svtronics.com/index.php?route=product/product&product_id=33
28.
Rogers
,
J.
, and
Costello
,
M.
,
2010
, “
Design of a Roll-Stabilized Mortar Projectile With Reciprocating Canards
,”
J. Guid., Control, and Dyn.
,
33
(
4
), pp.
1026
1034
.10.2514/1.47820
29.
Costello
,
M.
, and
Rogers
,
J.
,
2011
, “
BOOM: A Computer-Aided Engineering Toolbox for Exterior Ballistics of Smart Projectiles
,” U.S. Army Research Laboratory Aberdeen Proving Ground, MD, Technical Report No. ARL-CR-670.
30.
Press
,
W.
,
Teukolsky
,
S.
,
Vetterling
,
W.
, and
Flannery
,
B.
,
2007
,
Numerical Recipes in C: The Art of Scientific Computing
,
Cambridge University
,
New York
, pp.
714
722
.
You do not currently have access to this content.