research article velocity control for coning motion

Research ArticleVelocity Control for Coning Motion Missile System Using DirectDiscretization Method

Rui-sheng Sun and Chao Ming

Nanjing University of Science and Technology Nanjing 210094 China

Correspondence should be addressed to Rui-sheng Sun srscom163com

Received 3 September 2015 Accepted 19 November 2015

Academic Editor Jing Na

Copyright copy 2015 R-s Sun and C Ming This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper presents a new coning motion control methodology which takes into account the terminal speed constraint to designthe velocity control system for a missile By using a direct discretization method to transform the optimal control problem intoa nonlinear dynamic programming problem the optimal trajectory and velocity profile are obtained to satisfy the design indexrequirement In order to perform the velocity control a virtualmoving target is proposed for themissile to chase along the optimizedtrajectory Consequently after building velocity control model a velocity control law and control parameters of the coning motionare completed through the dynamic inversion theory The simulation results suggest that the proposed control law has a goodperformance and could be applied to the guidance for the missile with terminal speed control constraint

1 Introduction

Modern warfare demands increasingly high performancerequirements for missile for example reentry warheadcluster bombs and rocket-assisted torpedo It needs not onlyhigh guidance precision but also desired velocity in terminalphase The reason for this is threefold The first one is theconceptual requirement in terminal phase For instance arocket-assisted torpedo should fly not too fast to guaranteethat the matrix and the subtorpedo can separate securelyThesecond is the requirement of the large launch window andlarge envelope For a missile system it should not only get aslarge range as possible but also hit the enough short-rangetargets which requires the missile to consume superfluouskinetic energy in a timely manner The third is to ensurethe performance of control system For example the radarsignals can fail in transfer due to the high-speed motion ofradar-guided missile which makes reentry warhead surfacebe surrounded by a serious aerodynamic heating of plasmaConsequently it is necessary to control the deceleration ofthe missile flight speed in a proper time which has become afocus for researchers

Generally speaking there are three kinds of approaches tocontrol the velocity of missile when flying in the air Firstlywe can control the velocity by changing the thrust forcemagnitude Enomoto et al [1] introduced a velocity controlsystem for a leader-following UAV through changing thethrust forcemagnitude and using the dynamic inversion withthe two-time scale approach However this control approachcan only be suitable for the missile with propulsion systemstill on

Secondly according to the flight status and the terminalvelocity constraint the missile velocity can be governedby using the trajectory optimization method By using thisapproach the flight velocity is controlled by changing theflight height profile of the missile which can change its dragforce due to its changing dynamic pressure In [2] a definedproblem of optimal control was transformed into a two-point boundary problem through applying the Pontryaginminimum principle In this work the optimal control andthe optimal trajectory corresponding to this control at a skipthree-dimensional entry of space vehicle into the planetaryatmosphere were determined related to the obtaining of

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 716547 10 pageshttpdxdoiorg1011552015716547

2 Discrete Dynamics in Nature and Society

the maximum terminal velocity Saraf et al [3] presentedan entry control algorithm for future space transportationvehicles through tracking the reference drag acceleration andheading angle profiles so as to satisfy all entry constraintsincluding flight velocity control Bruyere et al [4 5] designeda sideslip velocity autopilot for a model of tactical missile inorder to meet the requirements of sideslip velocity over fullflight envelope

Finally the missile velocity can be controlled by means ofthe coning motion method It is a type of control techniquethat the missile axis movement is tapered at a fixed angleof rotation around the velocity vector which will produce avelocity-centered conical surface a coning motion [6] thatcan make the missile increase induced drag in order to adjustthe flight velocity through producing induced angle of attackObviously the coningmotion control is an effective approachto control the flight velocity of missile [7ndash9] Song [10]presented a new velocity control method of coning motionthrough simulating the reentry to improve flight controlaccuracy and control robustness of the terminal speed ofthe reentry vehicle when antidesigning the speed controlmethod for Pershing II Reference [11] put forward a velocitymagnitude control program which separates decelerationmotion into two different forms in the capability range ofwarhead guidance and control systems in order to meetthe requirement of fall velocity of a reentry maneuveringwarhead and to avoid designing a complicated ideal velocitycurve In the field of noncontrolled rocket Mao et aldid some work for the coning motion on the producingmechanism [12] motion characteristics and stability analysismethods [8 13] optimal control [14] and the approach ofreducing and avoiding coning motion [15]

Consequently in order to design the velocity controlsystem for a missile with the constraint of the terminalvelocity a deep understanding of the interaction among flightmechanics optimizations and control is necessary The aimof this research is to propose a design methodology for thevelocity control system by utilizing the coning motion basedon a direct discretization and dynamic inversion methodThe remaining of this paper is organized as follows Section 2describes the problem formulation and design scheme forvelocity control In Section 3 a direct discretization methodis deduced to develop the standard trajectory optimizationand the velocity control profile and the velocity controlmethodology is presented in detail through the coningmotion technique Moreover the structure and parameterdesign of velocity controller are explored for a coningmotionmissile In Section 4 a simulation case is demonstrated togovern the terminal velocity of the missile through using theconing motion algorithm In order to illustrate the velocitycontrol performance a traditional control method is alsocarried out for comparative study Finally some conclusionsare presented in Section 5

2 Problem Formulation

21 Motion Equations For the optimal flight velocity controlproblem under consideration a mathematical model based

on point mass dynamics can be used for determining thetrajectory and velocity yielding

119898119872

= minus119863 minus 119898119892 sin 120579

119898119881119872

120579 = 119871 cos 120574

119881minus 119885 sin 120574

119881minus 119898119892 cos 120579

minus119898119881119872cos 120579

119881= 119884 sin 120574

119881+ 119885 cos 120574

119881

119872

= 119881119872cos 120579 cos120595

119881

119910119872

= 119881119872sin 120579

119872

= minus119881119872sin 120579 sin120595

119881

(1)

where 119881119872 120579 Ψ119881 and 120574

119881denote velocity trajectory inclina-

tion angle path angle and bank angle of missile respectively119909119872 119910119872 and 119911

119872represent distance along 119909- 119910- 119911-axis and

119898 and 119892 are mass of missile and acceleration of gravityseparately

Here the earth is assumed to be flat and themodel for thedrag (119863) the lift (119871) and the side force (119885) can be formulatedas

119863 = 119902119878119862119863

119871 = 119902119878119862119871

119885 = 119902119878119862119911

(2)

where 119902 is dynamic pressure and 119878 is reference area Herethe coefficients of aerodynamics 119862

119863 119862119871 and 119862

119885are usually

obtained by CFD or wind tunnel test

22 Cost Function The performance index used for mini-mization of missile miss-distance and control is

119869 = 119909 (119905119891)

2

+ int

119905119891

1199050

120572 (119905)2 d119905 (3)

where 119905 denotes flight time and 120572 is angle of attack (AOA)Here subscript 0 and 119891 represent initial and terminal timerespectively

23 Flight Path Constraints The constraints correspondingto the flight path optimization for a given missile are asfollows

The constraints on states and parameters control are

119910 isin [119910lower 119910upper]

120579 isin [120579lower 120579upper] (4)

where the subscript lower and upper represent lower andupper limit respectively

These numbers are completely based on the flight enve-lope characteristics representing the possible values for themaximum and minimum state variables

The terminal constraints of the missile are

119881(119905119891) le 119881119891

120579 (119905119891) ge 120579119891

(5)

Discrete Dynamics in Nature and Society 3

Optimizetrajectory andvelocity profile

Proposea virtual moving

target

Model velocitycontrol and conduct

control law

Integration design forvelocity and

trajectory control

Figure 1 The process of the velocity control system design

Similarly the nonlinear state inequality constraints are

120572lower le 120572 (119905) le 120572upper

119886119910lower le 119886

119910 (119905) le 119886

119910upper(6)

For the purpose of the velocity control design twosubproblems are imperative to be solved One is to obtain theoptimal trajectory in the ideal condition In consideration ofthis subproblem the flight performance should be optimizedto produce the trajectory and velocity profile satisfying theabove requirements The other one is to control the missileflying path along the optimal trajectory in the same timeto ensure the coherence of the missile speed and the idealvelocity profile that is the integration of the trajectory andvelocity control should be considered

3 Velocity Control System Design

Theprocess of the velocity control systemdesign is conductedin Figure 1 Primarily the optimal trajectory and velocityprofile are carried out as a criterion for meeting the indexrequirementsThen in order to perform this velocity controla virtual moving target moving along the critical trajectoryis applied to guide the missile In terms of this the velocitycontrol schematic and model can be built and the velocitycontrol law and control parameters will be implementedincluding the integration design for velocity control andtrajectory control for a good performance of the coningmotion missile system

31 Trajectory and Velocity Profile Optimization With theadvent of computers and evolution of modern theories ofoptimal control the numerical computation techniques foroptimal atmospheric trajectories have been an active researcharea since the early 1970s The approaches on trajectory opti-mization are of two distinct categories such as direct methodbased on the mathematical programming and parameteriza-tion of state and control histories and the indirect methodgrounded on the solution of two-point boundary valueproblem (TPBVP) using optimal control principle [16ndash18]Because the guess of the initial values of costate variablesis random and there is a lack of physical implication theTPBVP is difficult to solve especially when the optimalsystem is accompanied with high nonlinearity and multipleconstraints (eg nonlinear trajectory optimization problem)The direct method has better convergence properties andthus can perform well with a poor initial guess [17] Fromthe mathematical point of view the KKT conditions of NLPtransformed by a discretization are equivalent to first-ordernecessary conditions of the original optimal control problemwhich means the solution of the NLP is equivalent to that ofthe original optimal control problem

Therefore the direct method is derived for the trajec-tory and the velocity profile optimization of the missileFurthermore the hp-adaptive Pseudospectral Method asa kind of efficient direct method is combining LegendrePseudospectral Method [19 20] and hp-adaptive method[21] which discretizes the state variables and control vari-ables on a series of Legendre-Gauss-Lobatto (LGL) pointsThrough constructing a Lagrange interpolation polynomialto approach the state variables and the control variables bytaking these discrete points as nodes the optimal controlproblem is transformed into a nonlinear dynamic program-ming problemThen the constraints of differential equationscan be altered into the form of algebra equations and sodo the integration item and the terminal status of the costfunction namely they can be calculated by using Gauss-Lobatto integration and integrating the right function frominitial status respectively When the calculation precisionduring some intervals fails to meet the requirement the col-location numbers and order number of global interpolationpolynomial should be adjusted adaptively in accordance withthe hp-adaptive method

Generally the discretization for the trajectory is con-ducted as follows For transforming flight time [119905

0 119905119891] into the

formof Legendre PseudospectralMethod [minus1 1] the variable119905 is converted to

120591 =

2119905

(119905119891minus 1199050)

minus

(119905119891+ 1199050)

(119905119891minus 1199050)

(7)

If 119870 is for the collocation numbers the state vector andcontrol vector will be approximately described by119870 Lagrangepolynomials 119871

119894(120591) considered as basis functions that is

x (120591) asymp X (120591) =

119870

sum

119894=1

L119894 (120591)X (120591

119894)

u (120591) asymp U (120591) =

119870

sum

119894=1

L119894 (120591)U (120591

119894)

(8)

where state vector x = [119881119872

120579 120595119881119909119872

119910119872

119911119872] and control

vector u = [120572]Derive (7) and the approximate derivation of state vector

is

x (120591119896) asymp X (120591

119896) =

119870

sum

119894=1

L119894(120591119896)X (120591119894) (9)

The constraint of dynamics equations thus is converted tothe form of algebra constraint that is

119870

sum

119894=1

D119896119894X (120591119894) minus

119905119891minus 1199050

2

119891 (X (120591119896)) = 0 (10)


Solve NLP

No

Yes

No

Calculate costfunction J

Updatecontrolvariable

Calculate ratio of maximumcurvature to average curvature r

YesNo

Generatenew mesh

Solutionmeets errortolerance

Update numberof collocation

points in mesh

Transcribe trajectoryoptimization into NLP

Yes

Quit

Start

Update orderof interpolating

polynomial

Compute LGR pointsweights

and differentiation matrices

UntranscribeNLP

solution todiscreteoptimalcontrol

problem

Is Jminimum

Updateparameters

r lt rmax

Figure 2 Schematic of hp-adaptive pseudospectral algorithm to solve optimal control problem

whereD119896119894= L119894(120591119896)means differentiation matrix of Legendre

Pseudospectral MethodIn terms of the above process of discretization the

interior-point method is applied to deal with the boundedconstraints of the inequality such as (4) (5) and (6) thatis a barrier term 120583 is brought in to the objective functionfor completely avoiding this problem of bound constraintsFortunately C++ software pack (Ipopt) [22] of the interior-point method can be utilized to solve this discrete nonlinearprogramming (NLP) A design philosophy is described toenable the optimal generic design by using the hp-adaptivePseudospectral Method outlined briefly in Figure 2 Thenthe schematic of algorithm to solve optimal control problemcan be described as follows

(1) Initialize a new mesh grid

(2) Discretize the continuous optimization control prob-lem through the Legendre Pseudospectral Method

and transform it into NLP problem through com-puting LGR points (X(120591

119894)U(120591119894)) weights L

119894(120591) and

differentiation matricesD119896

(3) Solve the NLP problem by using the interior-pointmethod

(4) Update control variable and go back to Step (3) if theindex 119869 of cost function is not minimized

(5) Quit the optimization process otherwise updateparameters and go back to Step (2) if the solution ofstate and path constraints meets error tolerance

32 Trajectory Control by Tracking Virtual Moving TargetAfter obtaining the standard optimization trajectory andvelocity control profile the task shifts to pursuit of a designto make the missile fly along the ideal trajectory as well asthe velocity profileThe conventional guidancemethod of theshaped trajectory can ensure the performance of the precisionto the fixed target which does not consider the requirement


(xT0 yT0)

VT0

T

M

A

O

VT

VM

xT y

xVTflowast

Figure 3 Kinematical relationship of the missile and the virtualtarget

of the velocity control Therefore a virtual target proposedhere is to afford a possibility of velocity control for themissileConcretely a virtual target is designed tomove exactly like theideal movement of the missile that is to move strictly alongthe standard optimized trajectory 1006704AOwith the same speed tothe velocity profile If a control design is developed to makethe missile precisely chase the virtual target the requirementof the velocity will be naturally satisfied for the missile flyingalong the ideal trajectory Thus in terms of the longitudinalplane of the kinematics described in Figure 3 all parametersof the position (119909

119879 119910119879) and velocity 119881

119879including initial

position (1199091198790 1199101198790) ideal initial velocity 119881

1198790 and anticipant

terminal velocity 119881119879119891lowast can be obtained from the information

of the optimized trajectory

33 Velocity Control Methodology Figure 4 depicts the totalcontrol flow of the coning motion control It is known thatthe desired initial velocity119881

1198720lowast (start from point A shown in

Figure 3) can be conducted by using the above hp-adaptivepseudospectral trajectory optimization method for decidingwhether the coning motion is need Besides in considerationof the influence of the control dynamics to the velocity theconing motion will be made to descent of the flight velocitywhen real velocity 119881

1198720is more than 11 119881

1198720lowast

Obviously the relative velocity and distance between themissile and the virtual target can be written as Δ119881 = 119881

119872minus119881119879

and Δ119877 = 119877119879minus 119877119872 respectively Here let Δ119877 be larger

than zero when the target moves in front of the missile Ifthe missile motion satisfies the condition of Δ119877 gt 0 or119881119872

lt 119881119879 the terminal velocity will surely meet the index

constraint Thus the velocity control system can be built upin accordance with the relationship of the relative distanceand speed between the missile and the virtual target (seeFigure 5) Then the velocity control law can be formulatedas

119886119909119888

= 119891 (Δ119881 Δ119877) = 119870119877(Δ119877 minus Δ119877

119905) + 119870119881(119881119879minus 119881119872) (11)

where Δ119877119905is the desired distance between the missile and

the virtual target and 119870119877and 119870

119881denote the velocity control

gainsFor the sake of convenience of analysis it is good to

suppose the anticipant distance between the missile and

NoSuspend coning motion

Restart coning motionGuidance withconing motion

Yes

Proportionalguidance

No

Start

No Coning motionend

Need decelerationMissilemotion

Yes

Yes

Δayc Δazc

ayc azc


VM0 gt 11 middot VM0lowast

VM gt VTlowast

Figure 4 Control flow chart of the coning motion control

1

s

1

s

+

+

+

minus

minus

VTKV

KR

ΔR ΔRt

VM

axc

Figure 5 Velocity control structure of the missile

the virtual target to be 0 that isΔ119877119905= 0Therefore the veloc-

ity control law will be simplified as

119891 (Δ119881 Δ119877) = 119870119877Δ119877 + 119870

119881Δ119881 (12)

If the missile overlaps at the position of virtual targetfor example 119891(Δ119881 Δ119877) = 0 Δ119881 = 0 Δ119877 = 0 theindex of terminal velocity is satisfied naturally Moreover if119891(Δ119881 Δ119877) lt 0 there are two circumstances (1) the missilemoves in back of the virtual target and (2) the missile is infront of the virtual target but the speedmagnitude is less thanthat of virtual target Since the virtual target moves along theideal flight trajectory all over the circumstances of (1) and(2) will both satisfy the terminal velocity index Additionallythe kinetic energy of the missile is unnecessarily large when119891(Δ119881 Δ119877) gt 0 which will result in the missile decelerationby coning motion

Consequently the cost function 119891(Δ119881 Δ119877) determineswhether the velocity control is required For example if119891(Δ119881 Δ119877) is less than zero the missile coningmotion shouldbe paused and the proportional navigation guidance to


the target should be switched on as well Otherwise if119891(Δ119881 Δ119877) is greater than zero the coning motion will berestarted

According to the control structure of themissile dynamicinversion theory [23] can be used to design the controlparameters 119870

119877and 119870

119881 The equation related to the velocity

shown in Figure 3 is expressed as

119898119872

= minus119902119878119862119909minus 119898119892 sin 120579 (13)

Replacing 119872with 119886

119909119888gives

119886119909119888

=

minus119902119878119862119909

119898

minus 119892 sin 120579 (14)

From the velocity control structure outlined in Figure 4transfer function 119881

119879to 119881119872becomes

119882(119904) =

119881119872 (119904)

119881119879 (

119904)

=

119870119881119904 + 119870119877

1199042+ 119870119881119904 + 119870119877

(15)

where the 2nd-order characteristic equation is

1199042+ 119870119881119904 + 119870119877= 0 (16)

Control gains 119870119881and 119870

119877are determined such that 119881

119872

may become its desiring velocity 119881119879 Apparently from the

linear system theory the stability of the system is ascertainedby the pole points and the system is stable only when all polepoints are located at the left of 119904 plane

(1) If 1198702119881

ge 4119870119877 since 119870

119881and 119870

119877are both over zero

the two characteristic roots of the system will be bothminus real numbers which can be written as

11990412

=

minus119870119881plusmn radic119870

2

119881minus 4119870119877

2

(17)

(2) If 1198702119881lt 4119870119877 11990412

changes to

11990412

=

minus119870119881plusmn 119894radic4119870

119877minus 1198702

119881

2

(18)

Apparently the nearer pole to the imaginary axis pri-marily determines the speed of output response of the abovesystem Therefore if the poles 119904

12have the same minus real

part that is when 1198702

119881le 4119870

119877 it is possible to obtain the

fastest output response Synchronously the imaginary partof the characteristic roots determines the magnitude of theovershoot In general the relative damping value of system isoptimally set toradic22 and then the equation

radic4119870119877minus 1198702

119881

119870119881

=

radic2

2

(19)

gives

119870119877=

3

8

1198702

119881 (20)

Let 119881119879(119904) = 1119904 substituting it into transfer function (15)

yields

119881119872 (119904) =

1

119904

sdot

119870119881119904 + 119870119877

1199042+ 119870119881119904 + 119870119877

=

1

119904

minus

1

119904 + 05119870119881

+

05119870119881

(119904 + 05119870119881)2

(21)

which can be described in the time-domain form by applyingLaplace inverse transform that is

119881119872 (119905) = 1 + 119890

minus05119870119881sdot119905(05119870

119881sdot 119905 minus 1) (22)

Besides by differentiating (22) it becomes

dd119905119881119872 (119905) =

1

2

119870119881119890minus(12)119870119881sdot119905

(2 minus

1

2

119870119881119905) (23)

Let (dd119905)119881119872(119905) = 0 and the settling time of the system can

be given as

119879119901=

4

119870119881

(24)

In order to ensure the performance of the system thesettling time 119879

119901is generally set to the triple of the time

constant of the missile 119879119889 namely

119879119901= 3119879119889 (25)

By combining (20) (24) and (25) the solution to thevelocity control law is

119870119881=

4

3119879119889

119870119877=

3

8

1198702

119881

(26)

When the missile does pure coning motion the acceler-ation commands in longitudinal and lateral plane 119886

120575119910119888and

119886120575119911119888

are

119886120575119910119888

= 119886120575sin120596119905

119886120575119911119888

= 119886120575cos120596119905

(27)

where 120596 represents a circular frequency and 119886120575is the accel-

eration command corresponding to command of AOA 120575 incondition of the pure coning motion

In accordance with the velocity control law shown in (12)drag force command119883

119888can be acquired as

minus119883119888= 119898119886119909119888+ 119898119892 sin 120579 (28)

which gives the lift force command

119884119888= 119901119883119888= 119901 sdot (119898119886

119909119888+ 119898119892 sin 120579) (29)

where 119901means lift-drag ratio So

119886120575=

119884119888

119898

=

119901119883119888

119898

=

119901 sdot (119898119886119909119888+ 119898119892 sin 120579)119898

asymp 119901 (119870119881Δ119881 + 119870

119877Δ119877)

(30)


34 Integrated Design for Velocity Control and TrajectoryControl On the other hand in order to make the missile hitthe target precisely it is necessary to design an appropriatecontrol law under which the normal acceleration 119886

119898can be

formulated with the relative motion between the missile andthe target as follows

119886119898= 119891 (119903 120585) (31)

where 119903 denotes directive distance betweenmissile and targetand 120585 is angle of line-of-sight (LOS)

Traditional control laws are determined to approach theconvergence of the distance and the LOS rate between themissile and the target For instance in terms of classicproportional navigation (PN) method it is concluded thatmissile acceleration commands 119886

119898are in proportion to the

rate of LOS 120585

From (27) the integrated guidance commands can bewritten as follows

119886119910119888

= Δ119886119910119888+ 119886120575sin120596119905

119886119911119888= Δ119886119911119888+ 119886120575cos120596119905

(32)

where Δ119886119910119888

and Δ119886119911119888

denote the normal PN commands inlongitudinal and lateral plane respectively

As given in (32) if the velocity meets the anticipantrequirement that is 119886

120575= 0 the missile will stop coning

motion andwill fly through the proportion guidance law so asto hit the target accurately Otherwise the missile will do theconingmotion whichmakes a continuous alternate changingbetween the angle of attack 120575 sin120596119905 and sideslip angle120575 cos120596119905 respectively In an alternating cycle the equivalentlift force produced by the alternate AOA is equal to zero andthe same to the side force That is to say only the drag forceincreases during the period of coning motion

Additionally consistent with the normal control com-mands Δ119886

119910119888and Δ119886

119911119888 the total AOA of the missile can be

given as

Δ = radic1205722+ 1205732

= radic(Δ120572 + 120575 sin120596119905)2 + (Δ120573 + 120575 cos120596119905)2

= radicΔ1205722+ Δ1205732+ ℎ (120575 Δ120572 Δ120573) + 120575

2

(33)

where 120573 represents sideslip angle and ℎ(120575 Δ120572 Δ120573) =

2120575(Δ120572 sin120596119905 + Δ120573 cos120596119905)From (33) if the missile flies without coning motion that

is 120575 = 0 the total AOA will be simplified as

Δ = radicΔ1205722+ Δ1205732 (34)

Since ℎ(120575 Δ120572 Δ120573) changes periodically with time asshown in (33) it is possible that the missile with coningmotion produces less total AOA than that without coningmotion when the item ℎ(120575 Δ120572 Δ120573) + 120575

2lt 0 That is to

say coning motion control results in the missilersquos dragdecrease especially when the weight of the missile gravity

in the direction of the velocity is over the drag force that isminus119902119878119862119909minus 119898119892 sin 120579 gt 0 and it is concluded that the speed of

the missile will increase and not decrease Consequently toprevent this from happening the inequation can be supposedas

Δ120572 sin120596119905 + Δ120573 cos120596119905 gt 0 (35)

which gives

radicΔ1205722+ Δ1205732 sin (120596119905 + 120596

0) gt 0 (36)

where 1205960is equal to acos(Δ120572radicΔ120572

2+ Δ1205732)

In order to meet inequation (36) all along circularfrequency 120596 should satisfy

2120587119899 le 120596119905 + 1205960le 2120587119899 + 120587 119899 isin 119864 (37)

So2120587119899 minus 120596

0

119905

le 120596 le

2120587119899 + 120587 minus 1205960

119905

(38)

In view of the dynamic characteristics of the missile andthe performance of the actuator the bandwidth of the coiningmotion command must be less than that of the actuatorOtherwise it is difficult for the missile guidance system totrack the guidance commandThus the angular ratio 120596mustbe subject to 120596 lt 120596

119879 that is

2120587119899 + 120587 minus 1205960

119905

lt 120596119879

119899 = Round [120596119879119905 minus 120587 + 120596

0

2120587

] (39)

where 120596119879represents the bandwidth of the system

4 Simulation Example

In this section a simulation example of the velocity controlsystem by using coning motion technique is demonstratedfor a missile whose indexes include terminal speed terminalpath angle and miss distance In order to exhibit the velocitycontrol performance the conventional PN mode withoutthe coning motion and particle swarm optimization (PSO)approach are involved for comparing studyThe initial condi-tions that the missile makes coning motion are set as followsthe initial values of 119910 120579 119881 and Ψ

119881are set to be 119909(0) =

minus19 km 119910(0) = 10 km 120579(0) = minus20 deg 119881(0) = 500msand Ψ

119881(0) = 0 deg The initial position of the virtual target is

selected as (minus7292 9329 0)m besides the total flight time 119905119879

638 s The flight constraints are set as 120579low = minus30∘ 120579top = 30

∘119910low = 0 km 119910top = 10 km119881

119891= 250ms 120579

119891= minus25

∘ 120572down =minus20 deg 120572up = 20 deg 119886

119910down = minus50ms2 and 119886119910up = 50m

s2 Additionally the design process of PSO trajectory isreferred to [24] and the parameters of the adopted PSOmethod are 119898 = 50 119899 = 2 and 119888

1= 1198882= 18 Inertial weight

119908 is set to be reduced linearly from 10 to 04 and the searchwill be terminated if the fitness value is less than 04 or thenumber of iteration reaches 150

According to the above integrated design method forvelocity control and trajectory control the simulation results


Table 1 Simulation results produced by different guidance laws

Guidance withvelocity control

Guidance withoutvelocity control

Optimizationtrajectory profile PSO trajectory profile

Final velocity (ms) 25409 28136 2550 2556Final flying path angle (deg) minus1565 minus2018 minus2146 minus2093Miss distance (m) 22 16 06 08

minus20minus15

minus10minus50

minus1minus05

005

1

0

2

4

6

8

10

x (km)

y(k

m)

z (km)

Optimal trajectoryPSO trajectory

PN modeConing motion mode

Figure 6 3D curve of the flight trajectories with different modes

020

4060

80

minus10minus5

05

1015

minus10

minus5

0

5

10

15

B

A



t (s)

120572(d

eg)

120573 (deg)

Figure 7 History curve of the coning motion

are illustrated in Figures 6ndash13 Comparing the results of theoptimalmethod proposed in this paper with those of the PSOapproach it is concluded that the optimization solutions oftwo approaches are almost the same which implies the pre-sented direct discretizationmethod has sufficient precision ofoptimization stratifying the requirements of the constraintsas exhibited in Table 1

Additionally as the result shown in Figure 6 the velocitycontrol system has got the same normal control performance

minus15 minus10 minus5 0 5 10 15

minus10

minus5

0

5

10

A

B

120572(d

eg)

120573 (deg)

StartEnd

Figure 8 Variation of the nutation angle

minus20 minus15 minus10 minus5 0250

300

350

400

450

500

550

600

(m

s)

x (km)



Figure 9 Response of the velocity in different condition

to the conventional proportional navigation scheme FromFigures 7-8 where points A and B denote the start andend of the coning motion respectively the control systemimproves the magnitude amount of induced AOA whichmakes the missilersquos drag increase Therefore the speed of themissile decreases apparently in this period as illustrated in


minus20 minus15 minus10 minus5 0minus10

minus5

0

5

10

15

120572(d

eg)

x (km)



Figure 10 Variation of the angle of attack


minus5

0

5

10

15

120573(d

eg)

x (km)



Figure 11 Variation of the sideslip angle

Figure 9 Meanwhile Figures 10-11 indicate that the coningmotion control suspends at the range of 13 km and switchesinto the pure proportion guidance mode which suggeststhat the missile velocity is gradually moving to the indexof requirement because the magnitudes of the attack andsideslip angles are both getting smaller with the missileapproaching to the virtual target According to the path anglecurves outlined in Figures 12-13 it is helpful to suspendthe coning motion control before hitting the target becausethe coning motion reduces the tracking precision to theideal trajectory Finally as shown in Table 1 both guidancemodes meet the requirements of high guidance precision andterminal path angle but the coning motion control systemproduces lower amount of terminal velocity which is in linewith the desired velocity index


minus35

minus30

minus25

minus20

minus15

minus10

x (km)



120579(d

eg)

Figure 12 Curve of the trajectory inclination angle


minus2

minus1

0

1

2

3

4

x (km)



ΨV

(deg

)

Figure 13 Curve of the path angle versus range

5 Conclusions

In this paper a novel coning motion based guidance methodis proposed for the velocity control of missile system withterminal velocity constraint An ideal velocity profile is pre-sented by using nonlinear programming method in which avirtual moving target is put forward for missile to chase soas to perform the velocity control Moreover the dynamicinversion theory is applied to design the velocity controlsystem parameters To show the effectiveness of the controllaw comparative simulations of a missile in which pureproportional navigation mode has been taken into consid-eration are provided Simulation results demonstrate thatthe proposed velocity control law has a good performanceproviding reference to the design for velocity control system


of the missileThe extensions of the presented coningmotioncontrol laws to a spinning missile and the inclusion of thedynamics in the velocity control system are possible areas offurther research

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Natural Science Founda-tion of China (NSFC) under Grant no 11176012 and theResearch Innovation Project for Graduate Student of Jiangsu-Provincial Ordinary University under Grant no KLYX15-0394

References

[1] K Enomoto T Yamasakiy H Taknoz and Y Baba ldquoA study ona velocity control system design using the dynamic inversionmethodrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference AIAA 2010-8204 Toronto Canada August2010

[2] V Istratie ldquoThree-dimensional optimal skip entrywith terminalmaximum velocityrdquo in Proceedings of the 22nd AtmosphericFlight Mechanics Conference AIAA 97-3483 New Orleans LaUSA August 1997

[3] A Saraf J A Leavitt D T Chen and K D Mease ldquoDesignand evaluation of an acceleration guidance algorithm for entryrdquoJournal of Spacecraft and Rockets vol 41 no 6 pp 986ndash9962004

[4] L Bruyere ldquoDynamic inversion for missile lateral velocity con-trol via polynomial eigenstructure assignmentrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit Austin Tex USA August 2003

[5] D M Acosta and S S Joshi ldquoAdaptive nonlinear dynamicinversion control of an autonomous airship for the explorationof titanrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit Hilton Head Island SC USAAugust 2007

[6] Y-X Jiang and J Cui ldquoEffectiveness of weaving maneuverstrategy of a missilerdquo Journal of Beijing University of Aeronauticsand Astronautics vol 28 no 2 pp 133ndash136 2002

[7] W H Curry and J F Reed ldquoMeasurement of Magnus effectson a sounding rocket model in a supersonic wind tunnelrdquo inProceedings of the Aerodynamic Testing Conference AmericanInstitute of Aeronautics and Astronautics Los Angeles CalifUSA September 1966

[8] X R Mao S X Yang and Y Xu ldquoConing motion stability ofwrap around fin rocketsrdquo Science in China Series E Technologi-cal Sciences vol 50 no 3 pp 343ndash350 2007

[9] J Morote and G Liano ldquoStability analysis and flight trials ofa clipped wrap around fin configurationrdquo in Proceedings ofthe Aiaa Atmospheric Flight Mechanics Conference and ExhibitAAAI 2004-5055 American Institute of Aeronautics andAstro-nautics Providence RI USA August 2004

[10] J Song ldquoA velocity control method of the reentry vehicle basedon flight simulationrdquoMissiles and Space Vehicles no 2 pp 5ndash72012

[11] FWang X Liu J Liang and X Fang ldquoVelocity control methodbased on terminal restraintrdquo Flight Dynamics vol 29 no 6 pp52ndash55 2011

[12] X R Mao S X Yang and Y Xu ldquoResearch on the coningmotion of wrap-around fin projectilesrdquo Canadian Aeronauticsand Space Journal vol 52 no 3 pp 119ndash125 2006

[13] K Y Li L Y Zhao andW Zhou ldquoStability of coning motion ofspinning rocket projectiles at high altituderdquo Journal ofDynamicsand Control vol 10 no 3 pp 239ndash243 2012

[14] X R Mao and S X Yang ldquoOptimal control of coning motionof spinning missilesrdquo Journal of Aerospace Engineering vol 28no 2 Article ID 04014068 pp 1ndash10 2015

[15] H-B Wang and J-S Wu ldquoConing motion of rockets itsnumerical simulation and restraintrdquo Transaction of BeijingInstitute of Technology vol 27 no 3 pp 196ndash199 2007

[16] P K A Menon V H L Cheng C A Lin and M MBriggs ldquoHigh-performance missile synthesis with trajectoryand propulsion system optimizationrdquo Journal of Spacecraft andRockets vol 24 no 6 pp 552ndash557 1987

[17] X-P Bai Y-H Zhao and Y-N Liu ldquoA novel approach tostudy real-time dynamic optimization analysis and simulationcomplex mine logistics transportation hybrid system with beltand surge linksrdquo Discrete Dynamics in Nature and Society vol2015 Article ID 601578 8 pages 2015

[18] J T Betts Practical Methods for Optimal Control Using Nonlin-ear Programming SIAM 1st edition 2001

[19] F Fahroo and I M Ross ldquoCostate estimation by a Legendrepseudospectral methodrdquo Journal of Guidance Control andDynamics vol 24 no 2 pp 270ndash277 2001

[20] M M Khader N H Sweilam and W Y Kota ldquoCardinalfunctions for Legendre pseudo-spectral method for solving theintegro-differential equationsrdquo Journal of the Egyptian Mathe-matical Society vol 22 no 3 pp 511ndash516 2014

[21] C L Darby W W Hager and A V Rao ldquoDirect trajectoryoptimization using a variable low-order adaptive pseudospec-tral methodrdquo Journal of Spacecraft and Rockets vol 48 no 3pp 433ndash445 2011

[22] A Wachter and L T Biegler ldquoOn the implementation ofan interior-point filter line-search algorithm for large-scalenonlinear programmingrdquoMathematical Programming vol 106no 1 pp 25ndash57 2006

[23] L Bruyere B A Whitey and A Tsourdos ldquoDynamic inversionformissile lateral velocity control via polynomial eigenstructureassignmentrdquo in Proceedings of the AIAA Guidance Navigationand Control Conference and Exhibit AIAA 2003-5792 AustinTex USA August 2003

[24] R S Sun C Ming and C J Sun ldquoTrajectory optimizationdesign for morphing wing missilerdquo Journal of Harbin Instituteof Technology vol 22 no 5 pp 48ndash53 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the maximum terminal velocity Saraf et al [3] presentedan entry control algorithm for future space transportationvehicles through tracking the reference drag acceleration andheading angle profiles so as to satisfy all entry constraintsincluding flight velocity control Bruyere et al [4 5] designeda sideslip velocity autopilot for a model of tactical missile inorder to meet the requirements of sideslip velocity over fullflight envelope

Finally the missile velocity can be controlled by means ofthe coning motion method It is a type of control techniquethat the missile axis movement is tapered at a fixed angleof rotation around the velocity vector which will produce avelocity-centered conical surface a coning motion [6] thatcan make the missile increase induced drag in order to adjustthe flight velocity through producing induced angle of attackObviously the coningmotion control is an effective approachto control the flight velocity of missile [7ndash9] Song [10]presented a new velocity control method of coning motionthrough simulating the reentry to improve flight controlaccuracy and control robustness of the terminal speed ofthe reentry vehicle when antidesigning the speed controlmethod for Pershing II Reference [11] put forward a velocitymagnitude control program which separates decelerationmotion into two different forms in the capability range ofwarhead guidance and control systems in order to meetthe requirement of fall velocity of a reentry maneuveringwarhead and to avoid designing a complicated ideal velocitycurve In the field of noncontrolled rocket Mao et aldid some work for the coning motion on the producingmechanism [12] motion characteristics and stability analysismethods [8 13] optimal control [14] and the approach ofreducing and avoiding coning motion [15]

Consequently in order to design the velocity controlsystem for a missile with the constraint of the terminalvelocity a deep understanding of the interaction among flightmechanics optimizations and control is necessary The aimof this research is to propose a design methodology for thevelocity control system by utilizing the coning motion basedon a direct discretization and dynamic inversion methodThe remaining of this paper is organized as follows Section 2describes the problem formulation and design scheme forvelocity control In Section 3 a direct discretization methodis deduced to develop the standard trajectory optimizationand the velocity control profile and the velocity controlmethodology is presented in detail through the coningmotion technique Moreover the structure and parameterdesign of velocity controller are explored for a coningmotionmissile In Section 4 a simulation case is demonstrated togovern the terminal velocity of the missile through using theconing motion algorithm In order to illustrate the velocitycontrol performance a traditional control method is alsocarried out for comparative study Finally some conclusionsare presented in Section 5

2 Problem Formulation

21 Motion Equations For the optimal flight velocity controlproblem under consideration a mathematical model based

on point mass dynamics can be used for determining thetrajectory and velocity yielding

119898119872

= minus119863 minus 119898119892 sin 120579

119898119881119872

120579 = 119871 cos 120574

119881minus 119885 sin 120574

119881minus 119898119892 cos 120579

minus119898119881119872cos 120579

119881= 119884 sin 120574

119881+ 119885 cos 120574

119881

119872

= 119881119872cos 120579 cos120595

119881

119910119872

= 119881119872sin 120579

119872

= minus119881119872sin 120579 sin120595

119881

(1)

where 119881119872 120579 Ψ119881 and 120574

119881denote velocity trajectory inclina-

tion angle path angle and bank angle of missile respectively119909119872 119910119872 and 119911

119872represent distance along 119909- 119910- 119911-axis and

119898 and 119892 are mass of missile and acceleration of gravityseparately

Here the earth is assumed to be flat and themodel for thedrag (119863) the lift (119871) and the side force (119885) can be formulatedas

119863 = 119902119878119862119863

119871 = 119902119878119862119871

119885 = 119902119878119862119911

(2)

where 119902 is dynamic pressure and 119878 is reference area Herethe coefficients of aerodynamics 119862

119863 119862119871 and 119862

119885are usually

obtained by CFD or wind tunnel test

22 Cost Function The performance index used for mini-mization of missile miss-distance and control is

119869 = 119909 (119905119891)

2

+ int

119905119891

1199050

120572 (119905)2 d119905 (3)

where 119905 denotes flight time and 120572 is angle of attack (AOA)Here subscript 0 and 119891 represent initial and terminal timerespectively

23 Flight Path Constraints The constraints correspondingto the flight path optimization for a given missile are asfollows

The constraints on states and parameters control are

119910 isin [119910lower 119910upper]

120579 isin [120579lower 120579upper] (4)

where the subscript lower and upper represent lower andupper limit respectively

These numbers are completely based on the flight enve-lope characteristics representing the possible values for themaximum and minimum state variables

The terminal constraints of the missile are

119881(119905119891) le 119881119891

120579 (119905119891) ge 120579119891

(5)




target


control law


trajectory control



120572lower le 120572 (119905) le 120572upper

119886119910lower le 119886

119910 (119905) le 119886

119910upper(6)







0 119905119891] into the


120591 =

2119905

(119905119891minus 1199050)

minus

(119905119891+ 1199050)

(119905119891minus 1199050)

(7)



x (120591) asymp X (120591) =

119870

sum

119894=1

L119894 (120591)X (120591

119894)

u (120591) asymp U (120591) =

119870

sum

119894=1

L119894 (120591)U (120591

119894)

(8)


120579 120595119881119909119872

119910119872



is

x (120591119896) asymp X (120591

119896) =

119870

sum

119894=1

L119894(120591119896)X (120591119894) (9)


119870

sum

119894=1

D119896119894X (120591119894) minus

119905119891minus 1199050

2

119891 (X (120591119896)) = 0 (10)


Solve NLP

No

Yes

No




YesNo

Generatenew mesh



points in mesh


Yes

Quit

Start


polynomial



UntranscribeNLP


problem

Is Jminimum

Updateparameters

r lt rmax








119894)U(120591119894)) weights L

119894(120591) and







(xT0 yT0)

VT0

T

M

A

O

VT

VM

xT y

xVTflowast



119879 119910119879) and velocity 119881









1198720is more than 11 119881

1198720lowast


119872minus119881119879





119886119909119888

= 119891 (Δ119881 Δ119877) = 119870119877(Δ119877 minus Δ119877

119905) + 119870119881(119881119879minus 119881119872) (11)








Yes


No

Start

No Coning motionend


Yes

Yes

Δayc Δazc

ayc azc



VM gt VTlowast


1

s

1

s

+

+

+

minus

minus

VTKV

KR

ΔR ΔRt

VM

axc




119891 (Δ119881 Δ119877) = 119870119877Δ119877 + 119870

119881Δ119881 (12)






119877and 119870



119898119872

= minus119902119878119862119909minus 119898119892 sin 120579 (13)


119909119888gives

119886119909119888

=

minus119902119878119862119909

119898

minus 119892 sin 120579 (14)


119879to 119881119872becomes

119882(119904) =

119881119872 (119904)

119881119879 (

119904)

=

119870119881119904 + 119870119877

1199042+ 119870119881119904 + 119870119877

(15)


1199042+ 119870119881119904 + 119870119877= 0 (16)



119872



(1) If 1198702119881

ge 4119870119877 since 119870

119881and 119870



11990412

=


2

119881minus 4119870119877

2

(17)

(2) If 1198702119881lt 4119870119877 11990412

changes to

11990412

=


119877minus 1198702

119881

2

(18)




119881le 4119870



radic4119870119877minus 1198702

119881

119870119881

=

radic2

2

(19)

gives

119870119877=

3

8

1198702

119881 (20)


yields

119881119872 (119904) =

1

119904

sdot

119870119881119904 + 119870119877

1199042+ 119870119881119904 + 119870119877

=

1

119904

minus

1

119904 + 05119870119881

+

05119870119881

(119904 + 05119870119881)2

(21)


119881119872 (119905) = 1 + 119890

minus05119870119881sdot119905(05119870

119881sdot 119905 minus 1) (22)


dd119905119881119872 (119905) =

1

2

119870119881119890minus(12)119870119881sdot119905

(2 minus

1

2

119870119881119905) (23)


be given as

119879119901=

4

119870119881

(24)




119879119901= 3119879119889 (25)


119870119881=

4

3119879119889

119870119877=

3

8

1198702

119881

(26)


120575119910119888and

119886120575119911119888

are

119886120575119910119888

= 119886120575sin120596119905

119886120575119911119888

= 119886120575cos120596119905

(27)





minus119883119888= 119898119886119909119888+ 119898119892 sin 120579 (28)


119884119888= 119901119883119888= 119901 sdot (119898119886

119909119888+ 119898119892 sin 120579) (29)


119886120575=

119884119888

119898

=

119901119883119888

119898

=

119901 sdot (119898119886119909119888+ 119898119892 sin 120579)119898

asymp 119901 (119870119881Δ119881 + 119870

119877Δ119877)

(30)



119898can be


119886119898= 119891 (119903 120585) (31)




rate of LOS 120585


119886119910119888

= Δ119886119910119888+ 119886120575sin120596119905

119886119911119888= Δ119886119911119888+ 119886120575cos120596119905

(32)

where Δ119886119910119888

and Δ119886119911119888






119910119888and Δ119886


given as

Δ = radic1205722+ 1205732

= radic(Δ120572 + 120575 sin120596119905)2 + (Δ120573 + 120575 cos120596119905)2

= radicΔ1205722+ Δ1205732+ ℎ (120575 Δ120572 Δ120573) + 120575

2

(33)




Δ = radicΔ1205722+ Δ1205732 (34)


2lt 0 That is to




Δ120572 sin120596119905 + Δ120573 cos120596119905 gt 0 (35)

which gives

radicΔ1205722+ Δ1205732 sin (120596119905 + 120596

0) gt 0 (36)


2+ Δ1205732)


2120587119899 le 120596119905 + 1205960le 2120587119899 + 120587 119899 isin 119864 (37)

So2120587119899 minus 120596

0

119905

le 120596 le

2120587119899 + 120587 minus 1205960

119905

(38)


119879 that is

2120587119899 + 120587 minus 1205960

119905

lt 120596119879

119899 = Round [120596119879119905 minus 120587 + 120596

0

2120587

] (39)




119881are set to be 119909(0) =





∘119910low = 0 km 119910top = 10 km119881

119891= 250ms 120579

119891= minus25













minus20minus15

minus10minus50

minus1minus05

005

1

0

2

4

6

8

10

x (km)

y(k

m)

z (km)




020

4060

80

minus10minus5

05

1015

minus10

minus5

0

5

10

15

B

A



t (s)

120572(d

eg)

120573 (deg)





minus10

minus5

0

5

10

A

B

120572(d

eg)

120573 (deg)

StartEnd



300

350

400

450

500

550

600

(m

s)

x (km)







minus5

0

5

10

15

120572(d

eg)

x (km)





minus5

0

5

10

15

120573(d

eg)

x (km)






minus35

minus30

minus25

minus20

minus15

minus10

x (km)



120579(d

eg)



minus2

minus1

0

1

2

3

4

x (km)



ΨV

(deg

)


5 Conclusions






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












target


control law


trajectory control



120572lower le 120572 (119905) le 120572upper

119886119910lower le 119886

119910 (119905) le 119886

119910upper(6)







0 119905119891] into the


120591 =

2119905

(119905119891minus 1199050)

minus

(119905119891+ 1199050)

(119905119891minus 1199050)

(7)



x (120591) asymp X (120591) =

119870

sum

119894=1

L119894 (120591)X (120591

119894)

u (120591) asymp U (120591) =

119870

sum

119894=1

L119894 (120591)U (120591

119894)

(8)


120579 120595119881119909119872

119910119872



is

x (120591119896) asymp X (120591

119896) =

119870

sum

119894=1

L119894(120591119896)X (120591119894) (9)


119870

sum

119894=1

D119896119894X (120591119894) minus

119905119891minus 1199050

2

119891 (X (120591119896)) = 0 (10)


Solve NLP

No

Yes

No




YesNo

Generatenew mesh



points in mesh


Yes

Quit

Start


polynomial



UntranscribeNLP


problem

Is Jminimum

Updateparameters

r lt rmax








119894)U(120591119894)) weights L

119894(120591) and







(xT0 yT0)

VT0

T

M

A

O

VT

VM

xT y

xVTflowast



119879 119910119879) and velocity 119881









1198720is more than 11 119881

1198720lowast


119872minus119881119879





119886119909119888

= 119891 (Δ119881 Δ119877) = 119870119877(Δ119877 minus Δ119877

119905) + 119870119881(119881119879minus 119881119872) (11)








Yes


No

Start

No Coning motionend


Yes

Yes

Δayc Δazc

ayc azc



VM gt VTlowast


1

s

1

s

+

+

+

minus

minus

VTKV

KR

ΔR ΔRt

VM

axc




119891 (Δ119881 Δ119877) = 119870119877Δ119877 + 119870

119881Δ119881 (12)






119877and 119870



119898119872

= minus119902119878119862119909minus 119898119892 sin 120579 (13)


119909119888gives

119886119909119888

=

minus119902119878119862119909

119898

minus 119892 sin 120579 (14)


119879to 119881119872becomes

119882(119904) =

119881119872 (119904)

119881119879 (

119904)

=

119870119881119904 + 119870119877

1199042+ 119870119881119904 + 119870119877

(15)


1199042+ 119870119881119904 + 119870119877= 0 (16)



119872



(1) If 1198702119881

ge 4119870119877 since 119870

119881and 119870



11990412

=


2

119881minus 4119870119877

2

(17)

(2) If 1198702119881lt 4119870119877 11990412

changes to

11990412

=


119877minus 1198702

119881

2

(18)




119881le 4119870



radic4119870119877minus 1198702

119881

119870119881

=

radic2

2

(19)

gives

119870119877=

3

8

1198702

119881 (20)


yields

119881119872 (119904) =

1

119904

sdot

119870119881119904 + 119870119877

1199042+ 119870119881119904 + 119870119877

=

1

119904

minus

1

119904 + 05119870119881

+

05119870119881

(119904 + 05119870119881)2

(21)


119881119872 (119905) = 1 + 119890

minus05119870119881sdot119905(05119870

119881sdot 119905 minus 1) (22)


dd119905119881119872 (119905) =

1

2

119870119881119890minus(12)119870119881sdot119905

(2 minus

1

2

119870119881119905) (23)


be given as

119879119901=

4

119870119881

(24)




119879119901= 3119879119889 (25)


119870119881=

4

3119879119889

119870119877=

3

8

1198702

119881

(26)


120575119910119888and

119886120575119911119888

are

119886120575119910119888

= 119886120575sin120596119905

119886120575119911119888

= 119886120575cos120596119905

(27)





minus119883119888= 119898119886119909119888+ 119898119892 sin 120579 (28)


119884119888= 119901119883119888= 119901 sdot (119898119886

119909119888+ 119898119892 sin 120579) (29)


119886120575=

119884119888

119898

=

119901119883119888

119898

=

119901 sdot (119898119886119909119888+ 119898119892 sin 120579)119898

asymp 119901 (119870119881Δ119881 + 119870

119877Δ119877)

(30)



119898can be


119886119898= 119891 (119903 120585) (31)




rate of LOS 120585


119886119910119888

= Δ119886119910119888+ 119886120575sin120596119905

119886119911119888= Δ119886119911119888+ 119886120575cos120596119905

(32)

where Δ119886119910119888

and Δ119886119911119888






119910119888and Δ119886


given as

Δ = radic1205722+ 1205732

= radic(Δ120572 + 120575 sin120596119905)2 + (Δ120573 + 120575 cos120596119905)2

= radicΔ1205722+ Δ1205732+ ℎ (120575 Δ120572 Δ120573) + 120575

2

(33)




Δ = radicΔ1205722+ Δ1205732 (34)


2lt 0 That is to




Δ120572 sin120596119905 + Δ120573 cos120596119905 gt 0 (35)

which gives

radicΔ1205722+ Δ1205732 sin (120596119905 + 120596

0) gt 0 (36)


2+ Δ1205732)


2120587119899 le 120596119905 + 1205960le 2120587119899 + 120587 119899 isin 119864 (37)

So2120587119899 minus 120596

0

119905

le 120596 le

2120587119899 + 120587 minus 1205960

119905

(38)


119879 that is

2120587119899 + 120587 minus 1205960

119905

lt 120596119879

119899 = Round [120596119879119905 minus 120587 + 120596

0

2120587

] (39)




119881are set to be 119909(0) =





∘119910low = 0 km 119910top = 10 km119881

119891= 250ms 120579

119891= minus25













minus20minus15

minus10minus50

minus1minus05

005

1

0

2

4

6

8

10

x (km)

y(k

m)

z (km)




020

4060

80

minus10minus5

05

1015

minus10

minus5

0

5

10

15

B

A



t (s)

120572(d

eg)

120573 (deg)





minus10

minus5

0

5

10

A

B

120572(d

eg)

120573 (deg)

StartEnd



300

350

400

450

500

550

600

(m

s)

x (km)







minus5

0

5

10

15

120572(d

eg)

x (km)





minus5

0

5

10

15

120573(d

eg)

x (km)






minus35

minus30

minus25

minus20

minus15

minus10

x (km)



120579(d

eg)



minus2

minus1

0

1

2

3

4

x (km)



ΨV

(deg

)


5 Conclusions






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Solve NLP

No

Yes

No




YesNo

Generatenew mesh



points in mesh


Yes

Quit

Start


polynomial



UntranscribeNLP


problem

Is Jminimum

Updateparameters

r lt rmax








119894)U(120591119894)) weights L

119894(120591) and







(xT0 yT0)

VT0

T

M

A

O

VT

VM

xT y

xVTflowast



119879 119910119879) and velocity 119881









1198720is more than 11 119881

1198720lowast


119872minus119881119879





119886119909119888

= 119891 (Δ119881 Δ119877) = 119870119877(Δ119877 minus Δ119877

119905) + 119870119881(119881119879minus 119881119872) (11)








Yes


No

Start

No Coning motionend


Yes

Yes

Δayc Δazc

ayc azc



VM gt VTlowast


1

s

1

s

+

+

+

minus

minus

VTKV

KR

ΔR ΔRt

VM

axc




119891 (Δ119881 Δ119877) = 119870119877Δ119877 + 119870

119881Δ119881 (12)






119877and 119870



119898119872

= minus119902119878119862119909minus 119898119892 sin 120579 (13)


119909119888gives

119886119909119888

=

minus119902119878119862119909

119898

minus 119892 sin 120579 (14)


119879to 119881119872becomes

119882(119904) =

119881119872 (119904)

119881119879 (

119904)

=

119870119881119904 + 119870119877

1199042+ 119870119881119904 + 119870119877

(15)


1199042+ 119870119881119904 + 119870119877= 0 (16)



119872



(1) If 1198702119881

ge 4119870119877 since 119870

119881and 119870



11990412

=


2

119881minus 4119870119877

2

(17)

(2) If 1198702119881lt 4119870119877 11990412

changes to

11990412

=


119877minus 1198702

119881

2

(18)




119881le 4119870



radic4119870119877minus 1198702

119881

119870119881

=

radic2

2

(19)

gives

119870119877=

3

8

1198702

119881 (20)


yields

119881119872 (119904) =

1

119904

sdot

119870119881119904 + 119870119877

1199042+ 119870119881119904 + 119870119877

=

1

119904

minus

1

119904 + 05119870119881

+

05119870119881

(119904 + 05119870119881)2

(21)


119881119872 (119905) = 1 + 119890

minus05119870119881sdot119905(05119870

119881sdot 119905 minus 1) (22)


dd119905119881119872 (119905) =

1

2

119870119881119890minus(12)119870119881sdot119905

(2 minus

1

2

119870119881119905) (23)


be given as

119879119901=

4

119870119881

(24)




119879119901= 3119879119889 (25)


119870119881=

4

3119879119889

119870119877=

3

8

1198702

119881

(26)


120575119910119888and

119886120575119911119888

are

119886120575119910119888

= 119886120575sin120596119905

119886120575119911119888

= 119886120575cos120596119905

(27)





minus119883119888= 119898119886119909119888+ 119898119892 sin 120579 (28)


119884119888= 119901119883119888= 119901 sdot (119898119886

119909119888+ 119898119892 sin 120579) (29)


119886120575=

119884119888

119898

=

119901119883119888

119898

=

119901 sdot (119898119886119909119888+ 119898119892 sin 120579)119898

asymp 119901 (119870119881Δ119881 + 119870

119877Δ119877)

(30)



119898can be


119886119898= 119891 (119903 120585) (31)




rate of LOS 120585


119886119910119888

= Δ119886119910119888+ 119886120575sin120596119905

119886119911119888= Δ119886119911119888+ 119886120575cos120596119905

(32)

where Δ119886119910119888

and Δ119886119911119888






119910119888and Δ119886


given as

Δ = radic1205722+ 1205732

= radic(Δ120572 + 120575 sin120596119905)2 + (Δ120573 + 120575 cos120596119905)2

= radicΔ1205722+ Δ1205732+ ℎ (120575 Δ120572 Δ120573) + 120575

2

(33)




Δ = radicΔ1205722+ Δ1205732 (34)


2lt 0 That is to




Δ120572 sin120596119905 + Δ120573 cos120596119905 gt 0 (35)

which gives

radicΔ1205722+ Δ1205732 sin (120596119905 + 120596

0) gt 0 (36)


2+ Δ1205732)


2120587119899 le 120596119905 + 1205960le 2120587119899 + 120587 119899 isin 119864 (37)

So2120587119899 minus 120596

0

119905

le 120596 le

2120587119899 + 120587 minus 1205960

119905

(38)


119879 that is

2120587119899 + 120587 minus 1205960

119905

lt 120596119879

119899 = Round [120596119879119905 minus 120587 + 120596

0

2120587

] (39)




119881are set to be 119909(0) =





∘119910low = 0 km 119910top = 10 km119881

119891= 250ms 120579

119891= minus25













minus20minus15

minus10minus50

minus1minus05

005

1

0

2

4

6

8

10

x (km)

y(k

m)

z (km)




020

4060

80

minus10minus5

05

1015

minus10

minus5

0

5

10

15

B

A



t (s)

120572(d

eg)

120573 (deg)





minus10

minus5

0

5

10

A

B

120572(d

eg)

120573 (deg)

StartEnd



300

350

400

450

500

550

600

(m

s)

x (km)







minus5

0

5

10

15

120572(d

eg)

x (km)





minus5

0

5

10

15

120573(d

eg)

x (km)






minus35

minus30

minus25

minus20

minus15

minus10

x (km)



120579(d

eg)



minus2

minus1

0

1

2

3

4

x (km)



ΨV

(deg

)


5 Conclusions






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(xT0 yT0)

VT0

T

M

A

O

VT

VM

xT y

xVTflowast



119879 119910119879) and velocity 119881









1198720is more than 11 119881

1198720lowast


119872minus119881119879





119886119909119888

= 119891 (Δ119881 Δ119877) = 119870119877(Δ119877 minus Δ119877

119905) + 119870119881(119881119879minus 119881119872) (11)








Yes


No

Start

No Coning motionend


Yes

Yes

Δayc Δazc

ayc azc



VM gt VTlowast


1

s

1

s

+

+

+

minus

minus

VTKV

KR

ΔR ΔRt

VM

axc




119891 (Δ119881 Δ119877) = 119870119877Δ119877 + 119870

119881Δ119881 (12)






119877and 119870



119898119872

= minus119902119878119862119909minus 119898119892 sin 120579 (13)


119909119888gives

119886119909119888

=

minus119902119878119862119909

119898

minus 119892 sin 120579 (14)


119879to 119881119872becomes

119882(119904) =

119881119872 (119904)

119881119879 (

119904)

=

119870119881119904 + 119870119877

1199042+ 119870119881119904 + 119870119877

(15)


1199042+ 119870119881119904 + 119870119877= 0 (16)



119872



(1) If 1198702119881

ge 4119870119877 since 119870

119881and 119870



11990412

=


2

119881minus 4119870119877

2

(17)

(2) If 1198702119881lt 4119870119877 11990412

changes to

11990412

=


119877minus 1198702

119881

2

(18)




119881le 4119870



radic4119870119877minus 1198702

119881

119870119881

=

radic2

2

(19)

gives

119870119877=

3

8

1198702

119881 (20)


yields

119881119872 (119904) =

1

119904

sdot

119870119881119904 + 119870119877

1199042+ 119870119881119904 + 119870119877

=

1

119904

minus

1

119904 + 05119870119881

+

05119870119881

(119904 + 05119870119881)2

(21)


119881119872 (119905) = 1 + 119890

minus05119870119881sdot119905(05119870

119881sdot 119905 minus 1) (22)


dd119905119881119872 (119905) =

1

2

119870119881119890minus(12)119870119881sdot119905

(2 minus

1

2

119870119881119905) (23)


be given as

119879119901=

4

119870119881

(24)




119879119901= 3119879119889 (25)


119870119881=

4

3119879119889

119870119877=

3

8

1198702

119881

(26)


120575119910119888and

119886120575119911119888

are

119886120575119910119888

= 119886120575sin120596119905

119886120575119911119888

= 119886120575cos120596119905

(27)





minus119883119888= 119898119886119909119888+ 119898119892 sin 120579 (28)


119884119888= 119901119883119888= 119901 sdot (119898119886

119909119888+ 119898119892 sin 120579) (29)


119886120575=

119884119888

119898

=

119901119883119888

119898

=

119901 sdot (119898119886119909119888+ 119898119892 sin 120579)119898

asymp 119901 (119870119881Δ119881 + 119870

119877Δ119877)

(30)



119898can be


119886119898= 119891 (119903 120585) (31)




rate of LOS 120585


119886119910119888

= Δ119886119910119888+ 119886120575sin120596119905

119886119911119888= Δ119886119911119888+ 119886120575cos120596119905

(32)

where Δ119886119910119888

and Δ119886119911119888






119910119888and Δ119886


given as

Δ = radic1205722+ 1205732

= radic(Δ120572 + 120575 sin120596119905)2 + (Δ120573 + 120575 cos120596119905)2

= radicΔ1205722+ Δ1205732+ ℎ (120575 Δ120572 Δ120573) + 120575

2

(33)




Δ = radicΔ1205722+ Δ1205732 (34)


2lt 0 That is to




Δ120572 sin120596119905 + Δ120573 cos120596119905 gt 0 (35)

which gives

radicΔ1205722+ Δ1205732 sin (120596119905 + 120596

0) gt 0 (36)


2+ Δ1205732)


2120587119899 le 120596119905 + 1205960le 2120587119899 + 120587 119899 isin 119864 (37)

So2120587119899 minus 120596

0

119905

le 120596 le

2120587119899 + 120587 minus 1205960

119905

(38)


119879 that is

2120587119899 + 120587 minus 1205960

119905

lt 120596119879

119899 = Round [120596119879119905 minus 120587 + 120596

0

2120587

] (39)




119881are set to be 119909(0) =





∘119910low = 0 km 119910top = 10 km119881

119891= 250ms 120579

119891= minus25













minus20minus15

minus10minus50

minus1minus05

005

1

0

2

4

6

8

10

x (km)

y(k

m)

z (km)




020

4060

80

minus10minus5

05

1015

minus10

minus5

0

5

10

15

B

A



t (s)

120572(d

eg)

120573 (deg)





minus10

minus5

0

5

10

A

B

120572(d

eg)

120573 (deg)

StartEnd



300

350

400

450

500

550

600

(m

s)

x (km)







minus5

0

5

10

15

120572(d

eg)

x (km)





minus5

0

5

10

15

120573(d

eg)

x (km)






minus35

minus30

minus25

minus20

minus15

minus10

x (km)



120579(d

eg)



minus2

minus1

0

1

2

3

4

x (km)



ΨV

(deg

)


5 Conclusions






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119877and 119870



119898119872

= minus119902119878119862119909minus 119898119892 sin 120579 (13)


119909119888gives

119886119909119888

=

minus119902119878119862119909

119898

minus 119892 sin 120579 (14)


119879to 119881119872becomes

119882(119904) =

119881119872 (119904)

119881119879 (

119904)

=

119870119881119904 + 119870119877

1199042+ 119870119881119904 + 119870119877

(15)


1199042+ 119870119881119904 + 119870119877= 0 (16)



119872



(1) If 1198702119881

ge 4119870119877 since 119870

119881and 119870



11990412

=


2

119881minus 4119870119877

2

(17)

(2) If 1198702119881lt 4119870119877 11990412

changes to

11990412

=


119877minus 1198702

119881

2

(18)




119881le 4119870



radic4119870119877minus 1198702

119881

119870119881

=

radic2

2

(19)

gives

119870119877=

3

8

1198702

119881 (20)


yields

119881119872 (119904) =

1

119904

sdot

119870119881119904 + 119870119877

1199042+ 119870119881119904 + 119870119877

=

1

119904

minus

1

119904 + 05119870119881

+

05119870119881

(119904 + 05119870119881)2

(21)


119881119872 (119905) = 1 + 119890

minus05119870119881sdot119905(05119870

119881sdot 119905 minus 1) (22)


dd119905119881119872 (119905) =

1

2

119870119881119890minus(12)119870119881sdot119905

(2 minus

1

2

119870119881119905) (23)


be given as

119879119901=

4

119870119881

(24)




119879119901= 3119879119889 (25)


119870119881=

4

3119879119889

119870119877=

3

8

1198702

119881

(26)


120575119910119888and

119886120575119911119888

are

119886120575119910119888

= 119886120575sin120596119905

119886120575119911119888

= 119886120575cos120596119905

(27)





minus119883119888= 119898119886119909119888+ 119898119892 sin 120579 (28)


119884119888= 119901119883119888= 119901 sdot (119898119886

119909119888+ 119898119892 sin 120579) (29)


119886120575=

119884119888

119898

=

119901119883119888

119898

=

119901 sdot (119898119886119909119888+ 119898119892 sin 120579)119898

asymp 119901 (119870119881Δ119881 + 119870

119877Δ119877)

(30)



119898can be


119886119898= 119891 (119903 120585) (31)




rate of LOS 120585


119886119910119888

= Δ119886119910119888+ 119886120575sin120596119905

119886119911119888= Δ119886119911119888+ 119886120575cos120596119905

(32)

where Δ119886119910119888

and Δ119886119911119888






119910119888and Δ119886


given as

Δ = radic1205722+ 1205732

= radic(Δ120572 + 120575 sin120596119905)2 + (Δ120573 + 120575 cos120596119905)2

= radicΔ1205722+ Δ1205732+ ℎ (120575 Δ120572 Δ120573) + 120575

2

(33)




Δ = radicΔ1205722+ Δ1205732 (34)


2lt 0 That is to




Δ120572 sin120596119905 + Δ120573 cos120596119905 gt 0 (35)

which gives

radicΔ1205722+ Δ1205732 sin (120596119905 + 120596

0) gt 0 (36)


2+ Δ1205732)


2120587119899 le 120596119905 + 1205960le 2120587119899 + 120587 119899 isin 119864 (37)

So2120587119899 minus 120596

0

119905

le 120596 le

2120587119899 + 120587 minus 1205960

119905

(38)


119879 that is

2120587119899 + 120587 minus 1205960

119905

lt 120596119879

119899 = Round [120596119879119905 minus 120587 + 120596

0

2120587

] (39)




119881are set to be 119909(0) =





∘119910low = 0 km 119910top = 10 km119881

119891= 250ms 120579

119891= minus25













minus20minus15

minus10minus50

minus1minus05

005

1

0

2

4

6

8

10

x (km)

y(k

m)

z (km)




020

4060

80

minus10minus5

05

1015

minus10

minus5

0

5

10

15

B

A



t (s)

120572(d

eg)

120573 (deg)





minus10

minus5

0

5

10

A

B

120572(d

eg)

120573 (deg)

StartEnd



300

350

400

450

500

550

600

(m

s)

x (km)







minus5

0

5

10

15

120572(d

eg)

x (km)





minus5

0

5

10

15

120573(d

eg)

x (km)






minus35

minus30

minus25

minus20

minus15

minus10

x (km)



120579(d

eg)



minus2

minus1

0

1

2

3

4

x (km)



ΨV

(deg

)


5 Conclusions






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119898can be


119886119898= 119891 (119903 120585) (31)




rate of LOS 120585


119886119910119888

= Δ119886119910119888+ 119886120575sin120596119905

119886119911119888= Δ119886119911119888+ 119886120575cos120596119905

(32)

where Δ119886119910119888

and Δ119886119911119888






119910119888and Δ119886


given as

Δ = radic1205722+ 1205732

= radic(Δ120572 + 120575 sin120596119905)2 + (Δ120573 + 120575 cos120596119905)2

= radicΔ1205722+ Δ1205732+ ℎ (120575 Δ120572 Δ120573) + 120575

2

(33)




Δ = radicΔ1205722+ Δ1205732 (34)


2lt 0 That is to




Δ120572 sin120596119905 + Δ120573 cos120596119905 gt 0 (35)

which gives

radicΔ1205722+ Δ1205732 sin (120596119905 + 120596

0) gt 0 (36)


2+ Δ1205732)


2120587119899 le 120596119905 + 1205960le 2120587119899 + 120587 119899 isin 119864 (37)

So2120587119899 minus 120596

0

119905

le 120596 le

2120587119899 + 120587 minus 1205960

119905

(38)


119879 that is

2120587119899 + 120587 minus 1205960

119905

lt 120596119879

119899 = Round [120596119879119905 minus 120587 + 120596

0

2120587

] (39)




119881are set to be 119909(0) =





∘119910low = 0 km 119910top = 10 km119881

119891= 250ms 120579

119891= minus25













minus20minus15

minus10minus50

minus1minus05

005

1

0

2

4

6

8

10

x (km)

y(k

m)

z (km)




020

4060

80

minus10minus5

05

1015

minus10

minus5

0

5

10

15

B

A



t (s)

120572(d

eg)

120573 (deg)





minus10

minus5

0

5

10

A

B

120572(d

eg)

120573 (deg)

StartEnd



300

350

400

450

500

550

600

(m

s)

x (km)







minus5

0

5

10

15

120572(d

eg)

x (km)





minus5

0

5

10

15

120573(d

eg)

x (km)






minus35

minus30

minus25

minus20

minus15

minus10

x (km)



120579(d

eg)



minus2

minus1

0

1

2

3

4

x (km)



ΨV

(deg

)


5 Conclusions






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















minus20minus15

minus10minus50

minus1minus05

005

1

0

2

4

6

8

10

x (km)

y(k

m)

z (km)




020

4060

80

minus10minus5

05

1015

minus10

minus5

0

5

10

15

B

A



t (s)

120572(d

eg)

120573 (deg)





minus10

minus5

0

5

10

A

B

120572(d

eg)

120573 (deg)

StartEnd



300

350

400

450

500

550

600

(m

s)

x (km)







minus5

0

5

10

15

120572(d

eg)

x (km)





minus5

0

5

10

15

120573(d

eg)

x (km)






minus35

minus30

minus25

minus20

minus15

minus10

x (km)



120579(d

eg)



minus2

minus1

0

1

2

3

4

x (km)



ΨV

(deg

)


5 Conclusions






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus5

0

5

10

15

120572(d

eg)

x (km)





minus5

0

5

10

15

120573(d

eg)

x (km)






minus35

minus30

minus25

minus20

minus15

minus10

x (km)



120579(d

eg)



minus2

minus1

0

1

2

3

4

x (km)



ΨV

(deg

)


5 Conclusions






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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