Research Article A New Method for Initial Parameters...

Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2013 Article ID 486020 7 pageshttpdxdoiorg1011552013486020

Research ArticleA New Method for Initial Parameters Optimization ofGuided Projectiles

Feng Bi-Ming and Nie Wan-Sheng

The Academy of Equipment Beijing 101416 China

Correspondence should be addressed to Feng Bi-Ming fengbimingyahoocomcn

Received 31 March 2013 Accepted 4 November 2013

Academic Editor R Ganguli

Copyright copy 2013 F Bi-Ming and N Wan-Sheng This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A new algorithm was developed for the initial parameters optimization of guided projectiles with multiple constraints Due to therelationship between the analytic guidance logic and state variables of guided projectiles the Radau pseudospectral method wasused to discretize the differential equations including control variables and state variables with multiple constraints into seriesalgebraic equations that were expressed only by state variables The initial parameter optimization problem was transformedto a nonlinear programming problem and the sequential quadratic programming algorithm was used to obtain the optimalcombinations of initial height and range to target for the final velocity of guided projectiles maximumwith constraints Comparingwith the appropriate initial conditions solved by Monte Carlo method and the flight characteristics solved by integrating theoriginal differential equations in the optimal initial parameters computed by the new algorithm the feasibility of new algorithmwas validated

1 Introduction

The initial parameter optimization problem of guided projec-tileswithmultiple constraints is considered In addition to theconstraints on its terminal position coordinates the projectilemust also impact the target from a specified direction withthe required circular error probable (CEP) andmake sure theimpact velocity maximum

Presently several studies are conducted for developingthe guidance logic withmultiple constraints [1 2] and search-ing the launch acceptable region (LAR) of a guided bombwhile the initial conditions are known [3 4] However a fewstudies pay attention to the initial parameters optimizationfor the guided projectiles with multiple constraints Theappropriate parameters are calculated based on differentcombinations of initial conditions (height above sea levelvelocity and path angles) by searchingmethod to find out thelargest LAR but they are not optimal conditions [5] theindirectmethod and directmethod both are called calculativemethod which transforms the parameters optimization tothe optimal control problem Indirect method based onthe Pontryagin maximum principle works out the analytical

optimal control logic to obtain the optimal index for theguided projectiles in known initial conditions for instancemaximum impact velocity or least energy [6 7] directmethod based on pseudospectral method discretizes theequations of motion to obtain the numerical optimal controllaw for making parameters optimal with constraints [8 9]Now when the analytical guidance logic exists how to findout the optimal initial parameters of the guided projectileswithmultiple constraints tomake the final velocitymaximumbecomes a new problem

Due to the relationship between the analytical guidancelogic and state variables the 2-dimensional equations ofmotion with both control variables and state variables aredescribed by the differential equations only with the statevariables Then the new differential equations are discretizedto be series algebraic equations expressed by state variableswhile the pseudospectral method is used [10] So the ini-tial parameters optimization is transformed to a nonlinearprogramming problem which can be solved using a sequen-tial quadratic programming (SQP) algorithm to improvecomputation efficiency and reduce the complexity of initialcondition searching

2 International Journal of Aerospace Engineering

X

Y

V0

120582D

120574D

M

x0

y0

Otarget

Figure 1 Coordinate system and geometry

2 Theoretical Development

21 Guidance Logic with Impact Direction Constraint inLongitudinal Plane For development purposes a simplifiedflat earth coordinate system is defined to be shown in Figure 1The target is at the origin of the coordinate system We willfocus on a fixed target in the following discussion The 119909-axis is pointed to the projectile and the 119910-axis is vertical tothe horizontal plane The line-of-sight (LOS) from the originto the projectile is defined by the elevation angle 120582

119863 where

0 le 120582119863le 1205872 is measured from positive 119909-axis in a clockwise

direction The flight path angle 120579 is the angle between therelative velocity vector and the local horizontal plane andminus1205872 le 120579 le 0 is measured from local horizon in a clockwisedirection as defined in Figure 1 120574

119863(minus1205872 le 120574

119863le 0) is the

azimuth angle of relative velocity vector in vertical plane asdefined in Figure 1

By the definition of the coordinate system shown inFigure 1 the standard 2D equations of motion of the projec-tile in the longitudinal plane can be described as follows

=

minus1205881198812119878119862119909 (120572)

2119872

+ 119892119909ℎ

120579 =

1

119881

(

1205881198812119878119862119910 (120572)

2119872

+ 119892119910ℎ)

= minus119881 cos 120579

119910 = 119881 sin 120579

(1)

As described in (1) the position coordinates are 119909 and 119910The earth-relative velocity is 119881 the mass of projectile is119872the atmospheric density model is 120588 = 120588

0sdot 119890(minus119910119867) the angle of

attack is 120572 and 119892119909ℎ

and 119892119910ℎ

can be defined as follows

119892119909ℎ= minus119892ℎsin 120579 119892

119910ℎ= minus119892ℎcos 120579 (2)

The additional parameters are defined as follows

119892ℎ= 981ms2 gravity acceleration

1205880= 1226 kgm3 standard sea-level density

119867 = 725424m atmospheric scale height

119878 = 0102m2 aerodynamic reference area

In this vertical plane the guidance logic of the projectilewith impact direction constraint may be represented by [6]

120574119863=

119870119866119863120582119863+ 119870119871119863(120582119863+ 120574119863119865)

119879119892

(3)

Here 120574119863119865

is defined as the desired impact direction

22 Differential Equations Expressed by State VariablesBefore the parameter optimization the equations of motionwith control variable are transformed to the differentialequations only including state variables By the discretizationof new differential equations the equation of motion canbe described by some algebraic equations So the initialparameter optimization can be transformed to the nonlinearprogramming problem

As shown in Figure 1 the 120582119863is also easily obtained

120582119863= arctan(

119910

119909

) (4)

So the derivative of 120582119863can be described as

120582119863=

minus119881 cos 120579 sin 120582119863+ 119881 sin 120579 cos 120582

119863

radic1199092+ 1199102

119879119892=

radic1199092+ 1199102

119881 cos 120579 cos 120582119863+ 119881 sin 120579 sin 120582

119863

(5)

Using (5) the guidance logic equation (3) can be repre-sented as follows

120574119863= 119870119866119863


119863

radic1199092+ 1199102

+ 119870119871119863(120582119863+ 120574119863119865)

119881 cos 120579 cos 120582119863+ 119881 sin 120579 sin 120582

119863

radic1199092+ 1199102

(6)

While the equation 120579 = 120574119863satisfies in the vertical plane

the lift coefficient 119862119910(120572) yields from (1)

119862119910 (120572) =

2 ( 120574119863119881 minus 119892

119910ℎ) sdot 119872

1205880sdot 119890(minus119910119867)

1198812119878

(7)

The aerodynamic coefficients 119862120572119910 1198620

119909 and 119862120572

119909come from

[11] and the lift coefficient can be defined as 119862119910(120572) = 119862

120572

119910120572 so

the 120572 is obtained

120572 =

119862119910 (120572)

119862120572

119910

=

1

119862120572

119910

sdot

2 ( 120574119863119881 minus 119892

119910ℎ) sdot 119872

1205880sdot 119890(minus119910119867)

1198812119878

(8)

International Journal of Aerospace Engineering 3

Because the drag coefficient is defined as 119862119909(120572) = 119862

0

119909+

119862120572

1199091205722 the drag force119863 can be readily achieved

119863 =

1205880sdot 119890(minus119910119867)

1198812119878

2

1198620

119909+ 119862120572

119909[

119862119910 (120572)

119862120572

119910

]

2

(9)

By (9) and (6) (1) with control variable 120572 can be trans-formed to differential equations expressed by state variables[119909 119910 119881 120579] it is described as follows

119889119881

119889119905

= minus

1205880sdot 119890(minus119910119867)

1198812119878

2119872

1198620

119909+ 119862120572

119909[

2 ( 120574119863119881 minus 119892

119910ℎ) sdot 119872

119862120572

1199101205880sdot 119890(minus119910119867)

1198812119878

]

2

+ 119892119909ℎ

119889120579

119889119905

= 119870119866119863


119863

radic1199092+ 1199102

+119870119871119863(120582119863+ 120574119863119865)

119881 cos 120579 cos 120582119863+ 119881 sin 120579 sin 120582

119863

radic1199092+ 1199102

119889119909

119889119905

= minus119881 cos 120579

119889119910

119889119905

= 119881 sin 120579(10)

If the initial state x0is known the only final state x

119891is

readily obtained by integrating (10) So the initial parameteroptimization problem can be described as follows

Cost function

min 120593 (x0 x119891) (11)

Subject to

the dynamic constraints

x = f (x) (12)

the inequality constraints

|120572| =

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119862120572

119910

sdot

2 ( 120574119863119881 minus 119892

119910ℎ) sdot 119872

1205880sdot 119890(minus119910119867)

1198812119878

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120572LIMIT

10038161003816100381610038161003816119899119910

10038161003816100381610038161003816=

10038161003816100381610038161003816120574119863119881 minus 119892

119910ℎ

10038161003816100381610038161003816

119892ℎ

le 119899119910 LIMIT

10038161003816100381610038161003816119909119891

10038161003816100381610038161003816le CEP 119909

0gt 0 119910

0gt 0 119881

119891gt 0

(13)

the boundary conditions

1205790= 120579initial 119881

0= 119881initial 119910

119891= 119910terminal

120579119891= 120574119863119865

(14)

23 Discretization of Differential Equations For instance thecontinuous differential equations are discretized into thealgebraic equations by the Radau pseudospectral methodThe parameter optimization problem is described as follows

Cost functional

min 119869 asymp Φ (X(1)1 1199050X(119870)119873119870+1

119905119870) (15)

Equation constraint

equation of motion

119873119896+1

sum

119895=1

119863(119896)

119894119895X(119896)119895minus

119905119896minus 119905119896minus1

2

f (X(119896)119894 120591(119896)

119894 119905119896minus1 119905119896) = 0

(119894 = 1 119873119896)

(16)

equation of boundary condition

120601 (X(1)1 1199050X(119870)119873119870+1

119905119870) = 0 (17)

Inequation constraint

angle of attack and overloading constraint

C(119896) (X(119896)119894 120591119894 119905119896minus1 119905119896) le 0 (119894 = 1 119873

119896) (18)

inequation of boundary condition

120601 (X(1)1 1199050X(119870)119873119870+1

119905119870) le 0 (19)

Then the initial parameter optimization problem of theequations of motion that satisfied the guidance logic is trans-formed to the nonlinear programming problem of algebraicequations expressed by state variables The optimal initialparameter can be computed by SQP

3 Results and Discussions

Assuming that the initial path angle and velocity exist theoptimal initial height and range to the target are solved to getthe maximum impact velocityThe problem can be describedas follows

Cost function

min minus119881119891 (20)

Subject to


x = f (x) (21)


[

[

[

[

[

[

minus30∘

minus60

0

0

minus10m

]

]

]

]

]

]

le

[

[

[

[

[

[

120572

119899119910

1199100

1199090

119909119891

]

]

]

]

]

]

le

[

[

[

[

[

[

30∘

60

20 km50 km10m

]

]

]

]

]

]

(22)


5 10 15 20 25 301100120013001400150016001700180019002000

Impa

ct v

eloc

ity (m

s)

Initial velocity 3200 ms

Initial height 1366 kmInitial height 1466 kmInitial height 1266 km

Range to target (km)

(a) Distribution for initial velocity 3200ms

5 10 15 20 25 301400

1500

1600

1700

1800

1900

2000

2100

Impa

ct v

eloc

ity (m

s)




(b) Distribution for initial velocity 3400ms

5 10 15 20 25 301400

1500

1600

1700

1800

1900

2000

2100

2200

Impa

ct v

eloc

ity (m

s)




(c) Distribution for initial velocity 3600ms

5 10 15 20 25 301400150016001700180019002000210022002300

Impa

ct v

eloc

ity (m

s)




(d) Distribution for initial velocity 3800ms

Figure 2 Distribution of the impact velocity and range to target computed by Monte Carlo method


[

[

[

[

1205790

1198810

119910119891

120579119891

]

]

]

]

=

[

[

[

[

minus15∘

119881initial0

minus90∘

]

]

]

]

(23)

Suppose that the mass of guided projectile is 450 kg theoptimal parameters for maximum final velocity in differentinitial velocities are shown in Table 1 With the increase ofinitial velocity the optimal initial height becomes higher theoptimal range to target becomes farther and the maximumimpact velocity becomes faster Comparing with the resultsas shown in Figure 2 the optimal heights and ranges to targetfor maximum impact velocities in different initial velocitiesare almost similar to the results computed by Monte Carlomethod

For instance when the initial velocity and path angleare respectively 3400ms and minus15∘ the optimal height andrange to target are respectively 138 km and 1148 km (shown

Table 1 The optimal initial position in different velocities formaximum impact velocity

1205790(∘) 119881

0(ms) 119910

0(km) 119909

0(km) 119881

119891(ms)

minus15 3200 1266 1058 1945minus15 3400 1380 1148 2045minus15 3600 1528 1319 2134minus15 3800 1676 1469 2205

in Table 1) and the maximum terminal velocity is 2045msCompared to the results shown in Figure 2(b) the resultsfind that while the initial height becomes higher or lowerthan 138 km the maximum final velocity becomes slowerAs shown in Figure 2(b) when the initial height is 138 kmthe range to target for the maximum final velocity is close to115 km and the impact velocity is close to 2050ms

Using the initial parameters as shown in Table 1 theequation of motion (1) with guidance logic (3) is integratedto get the flight characteristics of projectile in different initial


Initial velocity 3200 msInitial velocity 3400 ms


18

16

14

12

10

8

6

4

2

0

Hei

ght (

km)

0 25 5 75 10 125 15Range to target (km)

(a) Comparison of height and range to target



0 1 2 3 4 5 6 7 8 9minus90

minus60

minus30

0

Time (s)

120579(d

eg)

4000

3000

2000

1000

Velo

city

(ms

)

0 1 2 3 4 5 6 7 8 9Time (s)

(b) Comparison of velocity and path angle

0 1 2 3 4 5 6 7 8 9minus25

minus20

minus15

minus10

minus5

0

5

10

Time (s)

AOA

(deg

)



(c) Comparison of AOA

0 1 2 3 4 5 6 7 8 9minus70minus60minus50minus40minus30minus20minus10

0102030

Time (s)

ny



(d) Comparison of overloading

Figure 3 Comparison of flight characteristics of projectile in different initial condition

velocities Analyzing the results shown in Figure 3(a) toFigure 3(d) the result finds that the trajectories of projectilefor maximum impact velocity are close to the parabolasand the final impact direction AOA and overloading satisfythe constraints So the feasibility of new initial parameteroptimization method used in this work is validated bycomparison to the results shown in Figures 2 and 3 The newmethod can obtain the optimal initial parameters of guidedprojectile for different cost functions with the multipleconstraints

As shown in Figure 3(d) the maximum overloadingof projectile is saturated for long time during flight indifferent initial conditions it goes against the adaptability ofprojectile to the error and disturbance In fact the maximumoverloading should be smaller than the limit With theconstraint 119899

119910lt 60 the optimal initial parameters for

maximum impact velocity are solved by the new method asshown in Table 2

Table 2 Optimal initial position with more severe constraint

1205790(∘) 119881

0(ms) 119910

0(km) 119909

0(km) 119881

119891(ms)

minus15 3400 15781 14098 2021minus15 3600 16562 14149 2114minus15 3800 18232 16094 2175

Contrasting the results shown in Table 1 in the sameinitial velocities the initial heights are higher and the rangesto target are farther but the maximum final velocities withconstraint 119899

119910lt 60 are slower than the results with constraint

119899119910le 60The comparison of flight characteristics in different initial

conditions set out in Table 2 is shown in Figure 4 The max-imum overloading in different initial conditions as shownin Figure 4(d) is smaller than 60 it is propitious to theadaptability of projectile to the error and disturbance


0 2 4 6 8 10 12 14 16 1802468

101214161820

Hei

ght (

km)

Initial velocity 3400 msInitial velocity 3600 msInitial velocity 3800 ms



0 1 2 3 4 5 6 7 8 9 1020002500300035004000

Velo

city

(ms

)

0 1 2 3 4 5 6 7 8 9 10minus90

minus60

minus30

0Time (s)

Time (s)

120579(d

eg)



0 1 2 3 4 5 6 7 8 9 10minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

Time (s)

AOA

(deg

)



0 1 2 3 4 5 6 7 8 9 10minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

Time (s)

ny



Figure 4 Comparison of flight characteristics with severe overloading constraint

4 Conclusions

A new algorithm is developed for initial parameters opti-mization of guided projectiles by discretizing equations ofmotionwith guidance logic based on pseudospectralmethodThe method can solve the initial parameters optimizationproblem with multiple constraints using SQP The initialparameters solved by the new method for maximum impactvelocity are similar to the results solved by Monte Carlomethod and the flight characteristics solved by integratingthe equations of motion in the optimal initial conditionssatisfy the all constraints

References

[1] P Lu D B Doman and J D Schierman ldquoAdaptive terminalguidance for hypersonic impact in specified directionrdquo OMB0704-0188

[2] V C Lam ldquoCircular guidance laws with and without terminalvelocity direction constraintsrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit HonoluluHawaii USA August 2008

[3] Y Zhang W Zhang J Chen and L Shen ldquoAir-to-groundweapon delivery trajectory planning for UCAVs using gausspseudospectral methodrdquo Acta Aeronautica et AstronauticaSinica vol 32 no 7 pp 1240ndash1251 2011

[4] V L Siewert J C Sussingham and J A Farm ldquo6-DOFenhancement of precision guided munitions testingrdquo in Pro-ceedings of the 36th AIAA Paper on Aerospace Sciences Meetingand Exhibit Reno Nev USA 1998

[5] H Guo-Qiang N Ying C Fang et al ldquoOptimal releasecondition of no-power gliding bombrdquo Flight Dynamics vol 27no 1 pp 93ndash96 2009

[6] Z Han-YuanReentry Dynamics andGuidance Changsha Pressof National University of Defense Technology 1997


[7] M-S Hou ldquoOptimal gliding guidance in vertical plane forstandoff released guided bomb unitrdquo Acta Armamentarii vol28 no 5 pp 572ndash575 2007

[8] P Williams ldquoJacobi pseudospectral method for solving optimalcontrol problemsrdquo Journal of Guidance Control and Dynamicsvol 27 no 2 pp 293ndash297 2004

[9] T R Jorris C S Schulz F R Friedl andAV Rao ldquoConstrainedtrajectory optimization using pseudospectral methodsrdquo in Pro-ceedings of the AIAA Atmospheric Flight Mechanics Conferenceand Exhibit Honolulu Hawaii USA August 2008

[10] C L Darby and A V Rao ldquoAn initial examination of usingpseudospectral methods for timescale and differential geomet-ric analysis of nonlinear optimal control problemsrdquo in Proceed-ings of the AIAAAAS Astrodynamics Specialist Conference andExhibit Ontario Canada August 2008

[11] E R Hartman and P J Johnston ldquoMach 6 experimental andtheoretical stability and performance of a cruciform missile atangle of attack up to 65∘rdquo NASA Technical Paper 2733 1987

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Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation



DistributedSensor Networks



X

Y

V0

120582D

120574D

M

x0

y0

Otarget

Figure 1 Coordinate system and geometry

2 Theoretical Development

21 Guidance Logic with Impact Direction Constraint inLongitudinal Plane For development purposes a simplifiedflat earth coordinate system is defined to be shown in Figure 1The target is at the origin of the coordinate system We willfocus on a fixed target in the following discussion The 119909-axis is pointed to the projectile and the 119910-axis is vertical tothe horizontal plane The line-of-sight (LOS) from the originto the projectile is defined by the elevation angle 120582

119863 where

0 le 120582119863le 1205872 is measured from positive 119909-axis in a clockwise

direction The flight path angle 120579 is the angle between therelative velocity vector and the local horizontal plane andminus1205872 le 120579 le 0 is measured from local horizon in a clockwisedirection as defined in Figure 1 120574

119863(minus1205872 le 120574

119863le 0) is the

azimuth angle of relative velocity vector in vertical plane asdefined in Figure 1

By the definition of the coordinate system shown inFigure 1 the standard 2D equations of motion of the projec-tile in the longitudinal plane can be described as follows

=

minus1205881198812119878119862119909 (120572)

2119872

+ 119892119909ℎ

120579 =

1

119881

(

1205881198812119878119862119910 (120572)

2119872

+ 119892119910ℎ)

= minus119881 cos 120579

119910 = 119881 sin 120579

(1)

As described in (1) the position coordinates are 119909 and 119910The earth-relative velocity is 119881 the mass of projectile is119872the atmospheric density model is 120588 = 120588

0sdot 119890(minus119910119867) the angle of

attack is 120572 and 119892119909ℎ

and 119892119910ℎ

can be defined as follows

119892119909ℎ= minus119892ℎsin 120579 119892

119910ℎ= minus119892ℎcos 120579 (2)

The additional parameters are defined as follows

119892ℎ= 981ms2 gravity acceleration

1205880= 1226 kgm3 standard sea-level density

119867 = 725424m atmospheric scale height

119878 = 0102m2 aerodynamic reference area

In this vertical plane the guidance logic of the projectilewith impact direction constraint may be represented by [6]

120574119863=

119870119866119863120582119863+ 119870119871119863(120582119863+ 120574119863119865)

119879119892

(3)

Here 120574119863119865

is defined as the desired impact direction

22 Differential Equations Expressed by State VariablesBefore the parameter optimization the equations of motionwith control variable are transformed to the differentialequations only including state variables By the discretizationof new differential equations the equation of motion canbe described by some algebraic equations So the initialparameter optimization can be transformed to the nonlinearprogramming problem

As shown in Figure 1 the 120582119863is also easily obtained

120582119863= arctan(

119910

119909

) (4)

So the derivative of 120582119863can be described as

120582119863=


119863

radic1199092+ 1199102

119879119892=

radic1199092+ 1199102

119881 cos 120579 cos 120582119863+ 119881 sin 120579 sin 120582

119863

(5)

Using (5) the guidance logic equation (3) can be repre-sented as follows

120574119863= 119870119866119863


119863

radic1199092+ 1199102

+ 119870119871119863(120582119863+ 120574119863119865)

119881 cos 120579 cos 120582119863+ 119881 sin 120579 sin 120582

119863

radic1199092+ 1199102

(6)

While the equation 120579 = 120574119863satisfies in the vertical plane

the lift coefficient 119862119910(120572) yields from (1)

119862119910 (120572) =

2 ( 120574119863119881 minus 119892

119910ℎ) sdot 119872

1205880sdot 119890(minus119910119867)

1198812119878

(7)

The aerodynamic coefficients 119862120572119910 1198620

119909 and 119862120572

119909come from

[11] and the lift coefficient can be defined as 119862119910(120572) = 119862

120572

119910120572 so

the 120572 is obtained

120572 =

119862119910 (120572)

119862120572

119910

=

1

119862120572

119910

sdot

2 ( 120574119863119881 minus 119892

119910ℎ) sdot 119872

1205880sdot 119890(minus119910119867)

1198812119878

(8)



0

119909+

119862120572


119863 =

1205880sdot 119890(minus119910119867)

1198812119878

2

1198620

119909+ 119862120572

119909[

119862119910 (120572)

119862120572

119910

]

2

(9)


119889119881

119889119905

= minus

1205880sdot 119890(minus119910119867)

1198812119878

2119872

1198620

119909+ 119862120572

119909[

2 ( 120574119863119881 minus 119892

119910ℎ) sdot 119872

119862120572

1199101205880sdot 119890(minus119910119867)

1198812119878

]

2

+ 119892119909ℎ

119889120579

119889119905

= 119870119866119863


119863

radic1199092+ 1199102

+119870119871119863(120582119863+ 120574119863119865)

119881 cos 120579 cos 120582119863+ 119881 sin 120579 sin 120582

119863

radic1199092+ 1199102

119889119909

119889119905

= minus119881 cos 120579

119889119910

119889119905

= 119881 sin 120579(10)


119891is


Cost function

min 120593 (x0 x119891) (11)

Subject to


x = f (x) (12)


|120572| =

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119862120572

119910

sdot

2 ( 120574119863119881 minus 119892

119910ℎ) sdot 119872

1205880sdot 119890(minus119910119867)

1198812119878

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120572LIMIT

10038161003816100381610038161003816119899119910

10038161003816100381610038161003816=

10038161003816100381610038161003816120574119863119881 minus 119892

119910ℎ

10038161003816100381610038161003816

119892ℎ

le 119899119910 LIMIT

10038161003816100381610038161003816119909119891

10038161003816100381610038161003816le CEP 119909

0gt 0 119910

0gt 0 119881

119891gt 0

(13)


1205790= 120579initial 119881

0= 119881initial 119910

119891= 119910terminal

120579119891= 120574119863119865

(14)


Cost functional

min 119869 asymp Φ (X(1)1 1199050X(119870)119873119870+1

119905119870) (15)

Equation constraint

equation of motion

119873119896+1

sum

119895=1

119863(119896)

119894119895X(119896)119895minus

119905119896minus 119905119896minus1

2

f (X(119896)119894 120591(119896)

119894 119905119896minus1 119905119896) = 0

(119894 = 1 119873119896)

(16)


120601 (X(1)1 1199050X(119870)119873119870+1

119905119870) = 0 (17)



C(119896) (X(119896)119894 120591119894 119905119896minus1 119905119896) le 0 (119894 = 1 119873

119896) (18)


120601 (X(1)1 1199050X(119870)119873119870+1

119905119870) le 0 (19)




Cost function

min minus119881119891 (20)

Subject to


x = f (x) (21)


[

[

[

[

[

[

minus30∘

minus60

0

0

minus10m

]

]

]

]

]

]

le

[

[

[

[

[

[

120572

119899119910

1199100

1199090

119909119891

]

]

]

]

]

]

le

[

[

[

[

[

[

30∘

60

20 km50 km10m

]

]

]

]

]

]

(22)


5 10 15 20 25 301100120013001400150016001700180019002000

Impa

ct v

eloc

ity (m

s)





5 10 15 20 25 301400

1500

1600

1700

1800

1900

2000

2100

Impa

ct v

eloc

ity (m

s)





5 10 15 20 25 301400

1500

1600

1700

1800

1900

2000

2100

2200

Impa

ct v

eloc

ity (m

s)





5 10 15 20 25 301400150016001700180019002000210022002300

Impa

ct v

eloc

ity (m

s)







[

[

[

[

1205790

1198810

119910119891

120579119891

]

]

]

]

=

[

[

[

[

minus15∘

119881initial0

minus90∘

]

]

]

]

(23)




1205790(∘) 119881

0(ms) 119910

0(km) 119909

0(km) 119881

119891(ms)







18

16

14

12

10

8

6

4

2

0

Hei

ght (

km)





0 1 2 3 4 5 6 7 8 9minus90

minus60

minus30

0

Time (s)

120579(d

eg)

4000

3000

2000

1000

Velo

city

(ms

)

0 1 2 3 4 5 6 7 8 9Time (s)


0 1 2 3 4 5 6 7 8 9minus25

minus20

minus15

minus10

minus5

0

5

10

Time (s)

AOA

(deg

)





0102030

Time (s)

ny










1205790(∘) 119881

0(ms) 119910

0(km) 119909

0(km) 119881

119891(ms)







0 2 4 6 8 10 12 14 16 1802468

101214161820

Hei

ght (

km)




0 1 2 3 4 5 6 7 8 9 1020002500300035004000

Velo

city

(ms

)

0 1 2 3 4 5 6 7 8 9 10minus90

minus60

minus30

0Time (s)

Time (s)

120579(d

eg)



0 1 2 3 4 5 6 7 8 9 10minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

Time (s)

AOA

(deg

)



0 1 2 3 4 5 6 7 8 9 10minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

Time (s)

ny




4 Conclusions


References















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0

119909+

119862120572


119863 =

1205880sdot 119890(minus119910119867)

1198812119878

2

1198620

119909+ 119862120572

119909[

119862119910 (120572)

119862120572

119910

]

2

(9)


119889119881

119889119905

= minus

1205880sdot 119890(minus119910119867)

1198812119878

2119872

1198620

119909+ 119862120572

119909[

2 ( 120574119863119881 minus 119892

119910ℎ) sdot 119872

119862120572

1199101205880sdot 119890(minus119910119867)

1198812119878

]

2

+ 119892119909ℎ

119889120579

119889119905

= 119870119866119863


119863

radic1199092+ 1199102

+119870119871119863(120582119863+ 120574119863119865)

119881 cos 120579 cos 120582119863+ 119881 sin 120579 sin 120582

119863

radic1199092+ 1199102

119889119909

119889119905

= minus119881 cos 120579

119889119910

119889119905

= 119881 sin 120579(10)


119891is


Cost function

min 120593 (x0 x119891) (11)

Subject to


x = f (x) (12)


|120572| =

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119862120572

119910

sdot

2 ( 120574119863119881 minus 119892

119910ℎ) sdot 119872

1205880sdot 119890(minus119910119867)

1198812119878

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120572LIMIT

10038161003816100381610038161003816119899119910

10038161003816100381610038161003816=

10038161003816100381610038161003816120574119863119881 minus 119892

119910ℎ

10038161003816100381610038161003816

119892ℎ

le 119899119910 LIMIT

10038161003816100381610038161003816119909119891

10038161003816100381610038161003816le CEP 119909

0gt 0 119910

0gt 0 119881

119891gt 0

(13)


1205790= 120579initial 119881

0= 119881initial 119910

119891= 119910terminal

120579119891= 120574119863119865

(14)


Cost functional

min 119869 asymp Φ (X(1)1 1199050X(119870)119873119870+1

119905119870) (15)

Equation constraint

equation of motion

119873119896+1

sum

119895=1

119863(119896)

119894119895X(119896)119895minus

119905119896minus 119905119896minus1

2

f (X(119896)119894 120591(119896)

119894 119905119896minus1 119905119896) = 0

(119894 = 1 119873119896)

(16)


120601 (X(1)1 1199050X(119870)119873119870+1

119905119870) = 0 (17)



C(119896) (X(119896)119894 120591119894 119905119896minus1 119905119896) le 0 (119894 = 1 119873

119896) (18)


120601 (X(1)1 1199050X(119870)119873119870+1

119905119870) le 0 (19)




Cost function

min minus119881119891 (20)

Subject to


x = f (x) (21)


[

[

[

[

[

[

minus30∘

minus60

0

0

minus10m

]

]

]

]

]

]

le

[

[

[

[

[

[

120572

119899119910

1199100

1199090

119909119891

]

]

]

]

]

]

le

[

[

[

[

[

[

30∘

60

20 km50 km10m

]

]

]

]

]

]

(22)


5 10 15 20 25 301100120013001400150016001700180019002000

Impa

ct v

eloc

ity (m

s)





5 10 15 20 25 301400

1500

1600

1700

1800

1900

2000

2100

Impa

ct v

eloc

ity (m

s)





5 10 15 20 25 301400

1500

1600

1700

1800

1900

2000

2100

2200

Impa

ct v

eloc

ity (m

s)





5 10 15 20 25 301400150016001700180019002000210022002300

Impa

ct v

eloc

ity (m

s)







[

[

[

[

1205790

1198810

119910119891

120579119891

]

]

]

]

=

[

[

[

[

minus15∘

119881initial0

minus90∘

]

]

]

]

(23)




1205790(∘) 119881

0(ms) 119910

0(km) 119909

0(km) 119881

119891(ms)







18

16

14

12

10

8

6

4

2

0

Hei

ght (

km)





0 1 2 3 4 5 6 7 8 9minus90

minus60

minus30

0

Time (s)

120579(d

eg)

4000

3000

2000

1000

Velo

city

(ms

)

0 1 2 3 4 5 6 7 8 9Time (s)


0 1 2 3 4 5 6 7 8 9minus25

minus20

minus15

minus10

minus5

0

5

10

Time (s)

AOA

(deg

)





0102030

Time (s)

ny










1205790(∘) 119881

0(ms) 119910

0(km) 119909

0(km) 119881

119891(ms)







0 2 4 6 8 10 12 14 16 1802468

101214161820

Hei

ght (

km)




0 1 2 3 4 5 6 7 8 9 1020002500300035004000

Velo

city

(ms

)

0 1 2 3 4 5 6 7 8 9 10minus90

minus60

minus30

0Time (s)

Time (s)

120579(d

eg)



0 1 2 3 4 5 6 7 8 9 10minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

Time (s)

AOA

(deg

)



0 1 2 3 4 5 6 7 8 9 10minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

Time (s)

ny




4 Conclusions


References















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










5 10 15 20 25 301100120013001400150016001700180019002000

Impa

ct v

eloc

ity (m

s)





5 10 15 20 25 301400

1500

1600

1700

1800

1900

2000

2100

Impa

ct v

eloc

ity (m

s)





5 10 15 20 25 301400

1500

1600

1700

1800

1900

2000

2100

2200

Impa

ct v

eloc

ity (m

s)





5 10 15 20 25 301400150016001700180019002000210022002300

Impa

ct v

eloc

ity (m

s)







[

[

[

[

1205790

1198810

119910119891

120579119891

]

]

]

]

=

[

[

[

[

minus15∘

119881initial0

minus90∘

]

]

]

]

(23)




1205790(∘) 119881

0(ms) 119910

0(km) 119909

0(km) 119881

119891(ms)







18

16

14

12

10

8

6

4

2

0

Hei

ght (

km)





0 1 2 3 4 5 6 7 8 9minus90

minus60

minus30

0

Time (s)

120579(d

eg)

4000

3000

2000

1000

Velo

city

(ms

)

0 1 2 3 4 5 6 7 8 9Time (s)


0 1 2 3 4 5 6 7 8 9minus25

minus20

minus15

minus10

minus5

0

5

10

Time (s)

AOA

(deg

)





0102030

Time (s)

ny










1205790(∘) 119881

0(ms) 119910

0(km) 119909

0(km) 119881

119891(ms)







0 2 4 6 8 10 12 14 16 1802468

101214161820

Hei

ght (

km)




0 1 2 3 4 5 6 7 8 9 1020002500300035004000

Velo

city

(ms

)

0 1 2 3 4 5 6 7 8 9 10minus90

minus60

minus30

0Time (s)

Time (s)

120579(d

eg)



0 1 2 3 4 5 6 7 8 9 10minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

Time (s)

AOA

(deg

)



0 1 2 3 4 5 6 7 8 9 10minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

Time (s)

ny




4 Conclusions


References















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












18

16

14

12

10

8

6

4

2

0

Hei

ght (

km)





0 1 2 3 4 5 6 7 8 9minus90

minus60

minus30

0

Time (s)

120579(d

eg)

4000

3000

2000

1000

Velo

city

(ms

)

0 1 2 3 4 5 6 7 8 9Time (s)


0 1 2 3 4 5 6 7 8 9minus25

minus20

minus15

minus10

minus5

0

5

10

Time (s)

AOA

(deg

)





0102030

Time (s)

ny










1205790(∘) 119881

0(ms) 119910

0(km) 119909

0(km) 119881

119891(ms)







0 2 4 6 8 10 12 14 16 1802468

101214161820

Hei

ght (

km)




0 1 2 3 4 5 6 7 8 9 1020002500300035004000

Velo

city

(ms

)

0 1 2 3 4 5 6 7 8 9 10minus90

minus60

minus30

0Time (s)

Time (s)

120579(d

eg)



0 1 2 3 4 5 6 7 8 9 10minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

Time (s)

AOA

(deg

)



0 1 2 3 4 5 6 7 8 9 10minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

Time (s)

ny




4 Conclusions


References















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 2 4 6 8 10 12 14 16 1802468

101214161820

Hei

ght (

km)




0 1 2 3 4 5 6 7 8 9 1020002500300035004000

Velo

city

(ms

)

0 1 2 3 4 5 6 7 8 9 10minus90

minus60

minus30

0Time (s)

Time (s)

120579(d

eg)



0 1 2 3 4 5 6 7 8 9 10minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

Time (s)

AOA

(deg

)



0 1 2 3 4 5 6 7 8 9 10minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

Time (s)

ny




4 Conclusions


References















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article A New Method for Initial Parameters...

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Transcript of Research Article A New Method for Initial Parameters...