novel fractional order calculus extended pn for...

Research ArticleNovel Fractional Order Calculus Extended PN forManeuvering Targets

Jikun Ye Humin Lei and Jiong Li

Air and Missile Defense College Air Force Engineering University Shaanxi Xirsquoan 710051 China

Correspondence should be addressed to Jikun Ye jikunbosinacom

Received 22 July 2016 Revised 30 November 2016 Accepted 18 December 2016 Published 12 January 2017

Academic Editor Christopher J Damaren

Copyright copy 2017 Jikun Ye et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Based on the theory of fractional order calculus (FOC) a novel extended proportional guidance (EPN) law for intercepting themaneuvering target is proposed In the first part considering the memory function and filter characteristic of FOC the novelextended PN guidance algorithm is developed based on the conventional PN after introducing the properties and operation rulesof FOC Further with the help of FOC theory the average load and ballistics characteristics of proposed guidance law are analyzedThen using the small offset kinematic model the robustness of the new guidance law against autopilot parameters is studiedtheoretically by analyzing the sensitivity of the closed loop guidance system At last representative numerical results show thatthe designed guidance law obtains a better performance than the traditional PN for maneuvering target

1 Introduction

The proportional navigation (PN) has been widely usedbecause of its simplicity ease of implementation and effec-tiveness in practical systems [1ndash5] Due to the fact that endof the line of sight (LOS) angular rate rapidly rotating easilyleads to overload saturation when PN against maneuveringtarget many scholars modified traditional PN guidancelaw to improve the performance of guidance law [6ndash9] ALyapunov method is used by Dhananjay to analyze the effectof LOS rate delay on the performance of PN guidance law ina head-on or tail-chase missile-target engagement scenario[10] the author proposed the way for selecting the navigationgain to overcome the problem of LOS rate delay Ghosh etal derived a modified PN guidance law against higher speednonmaneuvering targets to control terminal impact anglewhich uses standard pure PN and retro-PN guidance laws ina 3D engagement scenario based on the initial engagementgeometry and terminal engagement requirements And thecapture region of the new guidance law was researched [11]Jeon and Lee modified PN for an ideal pursuer with astationary target by obtaining an optimal feedback solutionminimizing a performance index of the range-weightedcontrol energy the authors presented the exact analyses onoptimality of PN based on a nonlinear formulation [12] The

core idea of the above guidance law design is to add thecompensation term against LOS rate delay target maneuveracceleration of gravity and so onThe above studies centeredon the integer order differential LOS rate while the fractionalcalculus LOS rate has not been deeply researched and widelyused

With the development of fractional order calculus (FOC)theory the FOC has been more and more widely used inengineering [13ndash15] Considering the different applicationscenario there are three common FOC definition RiemannLiouville (RL) definition Grumwald Letnkov (GL) defini-tion and Caputo definition [16 17] Zhang et al discussed theorder range of FOC and point out the differences of fractionalorder derivatives and integer order derivatives the relationsbetween RL definition GL definition and Caputo definitionwere studied too [18] Unlike discontinuous change of integerorder calculus FOC expands calculus from the integer tofraction the order of FOC can be changed continuously andreflects the nature of continuity better The physical modelestablished based on the FOC theory can describe physicalphenomena with memory and time-dependent

The major advantage of FOC is that it has fewer param-eters and more simple forms thus it is easy to be applied inrealityThe specificmemory functions and stability character-istics of fractional system have been more and more applied

HindawiInternational Journal of Aerospace EngineeringVolume 2017 Article ID 5931967 9 pageshttpsdoiorg10115520175931967

2 International Journal of Aerospace Engineering

in the area of guidance and control [19] Considering that theatmospheric viscosity and interactions between air vehiclesmakes aircraft have aerodynamic properties of FOC thus theFOC theory can describe the characteristics of the aircraftkinematic comprehensive and truly

Sun and Zhu developed a new fractional order tensioncontrol law for the fast and stable deployment controlproblem of a space tether system and proved its stability [20]Almeida and Torres presented a method to solve fractionaloptimal control problems where the dynamic control systemdepended on integer order and Caputo fractional derivatives[21] Vishal et al used the theory of fractional order derivativeto discuss the local stability of the Mathieundashvan der Polhyperchaotic system and studied the analysis of control timewith respect to different fractional order derivatives [22]

Reference [23] studied the application of FOC controllerfor guided projectiles attitude control the author concludedthat fractional order control was better than classical PIDcontrol in robustness and control performance Li et alstudied the stability of the fractional order unified chaoticsystem by the equivalent passivity method With the FOCtheory the Lyapunov function was constructed by which itwas proved that the controlled fractional order system wasstable [24] The FOC theory was introduced to the guidancearea in reference [25] the author employed the PD120582 guidancelaw to carry forward the advantages of traditional PN andavoid its drawbacks based on FOC theory and concludedthat the PD120582 guidance had the higher guidance precision andshorter interception time Zhu et al proposed a modifiedFOC pure PN guidance law by means of Lyapunov-likemethod which solved the control problem for the capture of atarget with random maneuvers effectively [26] However theconclusion of [25 26] was obtained in the two-dimensionalplane under some certain conditions so there were somestrict restrictions on the designed guidance law and thisgreatly limits its practical engineering applications

This paper presents a novel 3D extended PN guidancealgorithm for maneuvering target based on FOC theoryand it overcomes the disadvantage of PN First the basicsknowledge and relative nature of FOC are outlined Secondthe proposed guidance law is given which is composed oftraditional LOS rate term and FOC LOS rate term And thetrajectory features the load character and the robustnessof the new guidance law are analyzed theoretically Atlast simulations show that the proposed guidance law hassuperior performance especially for a maneuvering target

2 Preliminary Knowledge

The fractional calculus differential operator can be expressedas follows [13 25]

120572119863119901119905 =

119889119901119889119905119901 119877 (119901) gt 01 119877 (119901) = 0int119905

0(119889119905)minus119901 119877 (119901) lt 0

(1)

where 120572 119905 denote the upper limit and lower limit respectively119901 is the calculus operator its value can be any plurality and119877(119901) presents the real part of 119901 If 119901 gt 0 then 120572119863119901119905 denotesfractional derivative

Definition 1 If the function 119891(119905) can be taken 119899 orderderivation here 119898 minus 1 le 119901 lt 119898 and 119899 isin N then we havethe definition of Caputo derivative as follows

119863119901119905 = 1Γ (119898 minus 119901) int

119905

0

119891119898 (120591)(119905 minus 120591)119901minus119898+1 119889120591 (2)

where Γ(119909) denotes Gamma function which is generalizationof the factorial function 119899 and we can take derivationof Gamma function with noninteger or even plurality Theexpression of Gamma function can be denoted as

Γ (119909) = intinfin

0119905119909minus119905119890minus119905119889119905 (3)

TheLaplace transformof Caputo fractional derivative canbe shown as

119871 [0119863119901119905 119891 (119905)] = intinfin

0119890minus1199041199050119863119901119905 119891 (119905) 119889119905

= 119904119901119865 (119904) minus119899minus1

sum119896=0

119904120572minus119896minus1119891119896 (0) (4)

In order to understand the links and differences betweenthe integer order and fractional order calculus we supposethat 119891(119905) is flat functions in the range [119886 119905] Taking thederivation of 119891(119905) with 119899 order we have

119891119899 (119905) = limℎrarr0

1ℎ119899119899

sum119895=0

(minus1)119895 119899119895 (119899 minus 119895)119891 (119905 minus 119895ℎ) (5)

When the noninteger 119901 replaces integer 119899 and Γ func-tion replaces binomial coefficients we get the definition ofGrunwald-Letnikov derivative as follows

119863119901119905 119891 (119905) = limℎrarr0

1ℎ119901119899

sum119895=0

(minus1)119895 Γ (119901 + 1)119895Γ (119901 minus 119895 + 1)119891 (119905 minus 119895ℎ) (6)

As can be seen from the above derivation the mostsignificant difference between integer order derivative andfractional derivative is that the former contains the recentinformation of several points while the latter contains thepast information of all the points We can see that fromequation (6) the fractional derivative has the ability ofmemory function and the closer the point the stronger theimpact factor [14]

In fact the integer order calculus is a special case offractional calculus when the orders of fractional calculus119901 isin 119873 the above two are equivalent Another point tonote is that FOC uniforms the forms of the differential andintegral expressions by fractional operator When 119901 gt 0 itdenotes fractional differential operator while when 119901 lt 0 itrepresents fractional integral operator so the study of integraland differential operation can be considered as an operator

International Journal of Aerospace Engineering 3

Y

XL120593t

120579t

VtZT

YT

YL

R

o

120593L

120593m

120579m

XM

X

Z

ZM

ZL

YM

Vm

120579L

XT

LOS

Figure 1 Pursuit geometry in 3-dimensional space

Compared with the integer order linear system the frac-tional linear system described by the FOC has the advantageof better memory function better stability and a greaterchoice of controller parameters

3 Design of EPN Guidance Algorithm

We assume that the missile is an aerodynamic-controlledinterceptor in a surface-to-air tactical missile defense sce-nario with a flat no rotating Earth Consider a three-dimensional homing scenario shown in Figure 1 where 119872119879 and119872119879 denote missile target and the missile-target LOSrespectively The inertial frame is defined as 119874119883119884119885 and it isinertially fixed and centered at the launch site at the instantof launching 119874119883119871119884119871119885119871 denotes the LOS coordinate 119903 is thedistance between missile and target the relative velocity ofmissile and target is represented as 119881119903 and the elevation andazimuth of LOS are denoted by 120579119871 and 120593119871 respectively 119881119898 isthe velocity of missile and 120579119898 and 120593119898 are the elevation andazimuth of 119881119898 respectively And 120579119905 and 120593119905 are the elevationand azimuth of 119881119905 respectively

According to the TPN law the guidance command isdefined as

119880119886 = 1198701119881119903119871119880119887 = 1198701119881119903119871 cos 120579119871

(7)

where the guidance coefficient 1198701 gt 2 ensures convergenttrajectory 119871 and 119871 denote the LOS rate which can beobtained by detection system

The conventional TPN consists of the relative velocity119881119903the LOS rate (119871 and 119871) and the proportional coefficient1198701 119881119903 can be detected by the seeker and 1198701 can be set inadvance before launching the missile the derivative of LOSangle (120579119871 and 120593119871) contains the information of instantaneousrelativemotion betweenmissile and targetThe instantaneous

LOS rate is used to calculate the load of TPN however thehistorical information of LOS rate is neglected which maybe useful to adjust the trajectory of missile Considering thatthe derivative of FOC has unique historical memory effecthere we add the term of fractional order LOS rate to theguidance loop in order to carry forward the guidance effectof conventional TPN In this way the guidance commandcontains the more comprehensive information of relativemotion to control the trajectory of missile

Based on the theory of fractional order here we constructa new form of EPN guidance law as follows

119880119901119886 = 1198701119881119903119871 + 1198701119881119903119901119863119901119905 120579119871 (8)

119880119901119887= 1198701119881119903119871 cos 120579119871 + 1198701119881119903119863119901119905 (120593119871) cos 120579119871 (9)

From (8) and (9) we find that the EPN guidance lawis composed of two terms one term is the traditional PNcontrol and the other term is fractional order differentialof the LOS rotation rate The fractional order term used ascompensation for the PN contains the past information ofLOS rotation rate Note that when 119901 = 0 the proposedEPN guidance law can be transformed to conventional PNguidance law

4 Ballistic Characteristics Analysis ofEPN Guidance Law

41 Characteristics Analysis of Average Overload In order tosimplify the expression of EPN (7) can be rewritten as follows

119880119901119886 = 1198701198861119871 + 1198701198862119863119901119905 (120579119871) 119880119901119887= 1198701198871119871 + 1198701198872119863119901119905 (120593119871)

(10)

where 1198701198861 = 1198701119881119903 1198701198871 = 1198701119881119903 cos 120579119871 1198701198862 = 1198701119881119903119901 and1198701198872 = 1198701119881119903119863119901119905 (120593119871) cos 120579119871


According to the EPN expression the average overloadguidance law can be expressed as

119880 = 1119905119891 int119905119891

0(119880119886 + 119880119887) 119889119905 = 1

119905119891 int119905119891

0(1198701198861119871

+ 1198701198862119863119901119905 (120579119871) + 1198701198871119871 + 1198701198872119863119901119905 (120593119871)) 119889119905 = 1119905119891

sdot int119905119891

01198701198861119871119889119905 + 1

119905119891 int119905119891

01198701198871119871119889119905 + 1

119905119891sdot int119905119891

01198701198872119863119901119905 (120593119871) 119889119905 + 1

119905119891 int119905119891

01198701198862119863119901119905 (120579119871) 119889119905

(11)

Equation (11) can be simplified by the following term

1198801198861 = 1119905119891 int119905119891

01198701198861119871119889119905

1198801198862 = 1119905119891 int119905119891

01198701198862119863119901119905 (120579119871) 119889119905

1198801198871 = 1119905119891 int119905119891

01198701198871119871119889119905

1198801198872 = 1119905119891 int119905119891

01198701198872119863119901119905 (120593119871) 119889119905

(12)

The integration operation of 1198801198861 1198801198862 1198801198871 and 1198801198872 withthe interval [0 119905119891] (119905119891 represents the final time) yields

1198801198861 = 1119905119891 int119905119891

01198701198861119871119889119905 =

1198701198861 (120579119871119891 minus 1205791198710)119905119891

1198801198871 = 1119905119891 int119905119891

01198701198871119871119889119905 =

1198701198871 (120593119871119891 minus 1205931198710)119905119891

1198801198862 = 1119905119891 int119905119891

01198701198862119863119901119905 (120579119871) 119889119905

= limℎrarr0

1198701198862119905119891 int119905119891

0ℎminus119901[119905ℎ]

sum119895=0

120596119901119895 119871 (119905 minus 119895ℎ) 119889119905

= limℎrarr0

1198701198862119905119891 ℎminus119901

[119905ℎ]

sum119895=0

120596119901119895 int119905119891

0119871 (119905 minus 119895ℎ) 119889119905

asymp 1198701198862119905119891 ℎminus119901

[119905ℎ]

sum119895=0

120596119901119895 (120579119871119891 minus 1205791198710)

(13)

The following equation can be obtained according tooperation of 1198801198862

1198801198862 = 1198862(120579119871119891 minus 1205791198710)

119905119891 (14)

where 1198862 = (1198701198862119905119891)ℎminus119901sum[119905ℎ]119895=0 120596119901119895 and the subscripts 0 and119891 denote the initial state and final state respectively

The process of solving 1198801198872 is similar to 1198801198862 so we have1198801198872 = (1198872120593119871119891 minus 1205931198710)119905119891 where 1198872 = (1198701198872119905119891)ℎminus119901sum[119905ℎ]119895=0 120596119901119895

Let 119880119886 119880119887 denote the average pitch and lateral loadrespectively they can be expressed as

119880119886 = 1198801198861 + 1198801198862 = (1198701198861 + 1198862)(120579119871119891 minus 1205791198710)

119905119891

119880119887 = 1198801198871 + 1198801198872 = (1198701198871 + 1198872)(120579119871119891 minus 1205791198710)

119905119891 (15)

From the above operation we get the average overload ofEPN in the vertical plane as follows

119880 = 119880119886 + 119880119887

= (1198701198861 + 1198701198862)(120579119871119891 minus 1205791198710)

119905119891

+ (1198701198871 + 1198701198872)(120593119871119891 minus 1205931198710)

119905119891

(16)

Consider that the average overload of traditional TPN canbe expressed as

119880TPN = 119870TPN1(120579119871119891 minus 1205791198710)

119905119891 + 119870TPN1(120593119871119891 minus 1205931198710)

119905119891 (17)

where 119870TPN1 denotes the proportional coefficient of con-ventional PN According to (16) and (17) we find that theaverage overload is closely related to change of LOS angleTheabsolute value of average overload will increase as the LOSrotation rate becomes bigger And calculations demonstratethat the average overload of EPN is closer to PN it meansthat the ballistic controllerrsquos quantity of EPN is closer to PN

42 Analysis of Required Overload If we want to keep thetrajectory of designed guidance law placidly and reduce theoverload requirements of interceptor the following condi-tions must be met

1003816100381610038161003816120579119871 minus 12057911987101003816100381610038161003816 997888rarr min10038161003816100381610038161003816120579119871119891 minus 120579119871010038161003816100381610038161003816 997888rarr min10038161003816100381610038161003816120579119871119891 minus 12057911987110038161003816100381610038161003816 997888rarr min1003816100381610038161003816120593119871 minus 12059311987101003816100381610038161003816 997888rarr min10038161003816100381610038161003816120593119871119891 minus 120593119871010038161003816100381610038161003816 997888rarr min10038161003816100381610038161003816120593119871119891 minus 12059311987110038161003816100381610038161003816 997888rarr min

(18)

From (8) (9) and (18) the absolute load of EPN meetsthe following condition

1003816100381610038161003816119880119901119886 1003816100381610038161003816 le 1198701198861 1003816100381610038161003816100381611987110038161003816100381610038161003816 + 1198701198862 10038161003816100381610038161003816119863119901119905 (120579119871)

10038161003816100381610038161003816 10038161003816100381610038161003816119880119901119887

10038161003816100381610038161003816 le 1198701198871 10038161003816100381610038161198711003816100381610038161003816 + 1198701198872 10038161003816100381610038161003816119863119901119905 (120593119871)10038161003816100381610038161003816

(19)


From (19) we conclude that the maximum load of EPNcan be denoted as follows

max 119880119901119886 le max 1198701198861119871 + 1198701198862119863119901119905 (120579119871) max 119880119901

119887 le max 1198701198871119871 + 1198701198872119863119901119905 (120593119871)

(20)

Note that as the variation range of LOS angle is small thetrajectory of missile is closer to the straight line When theLOS angle approaches zero the acceleration of EPN 119880119901119886 rarr 0and 119880119901

119887rarr 0 The extreme requirement loads in vertical and

horizontal direction can be denoted as 119873119898119910 = max|119880119886|119892and 119873119898119911 = max|119880119887|119892 When 119880119901119886 rarr 0 and 119880119901

119887rarr 0 we

have 119873119898119910 rarr 0 and 119873119898119911 rarr 0 Thus the proposed guidanceEPN guidance law has the features of minimizing energycontrol and keeping trajectory straightly

43 Characteristics Analysis of Ballistics Considering the upsand downs of trajectory are closely related to the incrementof LOS angle if the LOS angles are controlled in a smallrange the more flat trajectory can be obtained Using thisidea compare the trajectory characteristic of EPN and PN byLOS angle changes during the process of flyingThe lead angleof missile in pitch and lateral direction can be represented asfollows

120578119898 = 120579119871 minus 120579119898 (21)

120593119898 = 120593119871 minus 120593119898 (22)

Considering the analysis process of trajectory in verticalplane is similar to the horizontal plane here we analyze thechanges of lead angle in vertical plane only

Taking the derivation of (21) we have

119898 = 119871 minus 119898 (23)

The ballistic inclination rate of missile can be denoted as

119898 = 1198701119881119903119871 + 1198701119881119903119901119863119901119905 (120579119871)119881119898

= 1198701205791119871 + 1198701205792119863119901119905 (120579119871) (24)

The integration of (23) for 119905 isin [0 119905119891] yields120578119898119891 = 1205781198980 + (1 minus 1198701205791) (120579119871119891 minus 1205791198710)

minus 1198701205792 int119905119891

0119863119901119905 (119871) 119889119905

= 1205781198980 + (1 minus 1198701205791) (120579119871119891 minus 1205791198710)

minus 1198701205792ℎminus119901[119905ℎ]

sum119895=0

120596119901119895 (120579119871119891 minus 1205791198710)

(25)

Let 1205792 = 1198701205792ℎminus119901sum[119905ℎ]119895=0 120596119901119895 Substituting 1205792 to (25)results in

120578119898119891 = 1205781198980 + (1 minus 1198701205791) (120579119871119891 minus 1205791198710)minus 1205792 (120579119871119891 minus 1205791198710)

(26)

YL

Vrtgo

at

U120579L

Vm

Figure 2 Illustration of small offset motion

When 1205781198980 = 0 the following equations can be obtainedbased on (26)

120579119871119891 minus 1205791198710 =120578119898119891

1198701205791 + 1205792 minus 1 (27)

For the traditional PN guidance law the lead anglechanges in vertical plane can be denoted as follows

120579119871119891 minus 1205791198710 =120578119898119891

1198701205791 minus 1 (28)

According to (27) and (28) we find that the introducedfractional order term of EPN further increases the trajectorysmoothness by decreasing the variation range of ballisticelevation So the trajectory of EPN can be controlled easilyand the requirements of actuator design reduced greatlyConsidering the analysis in horizontal is similar to thevertical therefore we no longer study it

44 Robustness Analysis of EPN against Autopilot Figure 2shows the kinematic relation of small offset

By using the approximation sin 120579119871 asymp 120579119871 (this approxima-tion is rational because 119884119871 is small offset) we have

120579119871 = 119884119871119881119903119905go (29)

where 119905go denotes time-to-go the solution of the differentialequation (29) can be obtained as

119871 = 119884119871 +119871119905go1198811199031199052go (30)

The transform function of autopilot can be expressed asfollows

119866 (119904) = 119887119898119904119898 + 119887119898minus1119904119898minus1 + sdot sdot sdot + 1198871119904 + 1119886119899119904119899 + 119886119899minus1119904119899minus1 + sdot sdot sdot + 1198861119904 + 1 (119899 gt 119898) (31)

Taking the terminal homing guidance as an examplethe guidance loop of small offset model can be denoted asFigure 3 Note that comparing with traditional PN guidanceloop the EPN increases the fractional order term which isparallel with the PN term in the homing guidance loop


Let 119880119910(119904)119884(119904) = 119866119910(119904) according to the properties offractional order we have

119871 [119863119901119905 119891 (119905)] = 119904119901119871 [119891 (119905)] minus119899minus1

sum119896=1

119904119896 [119863119901minus119896minus1119905 119891 (119905)]119905=0

≃ 119904120582 1 + 119904119905go1198811198881199052go minus 1199040ℎminus119901

[119905ℎ]

sum119895=0

120596minus1119895 119891 (0)

minus 1199041ℎminus119901[119905ℎ]

sum119895=0

120596119901minus2119895 119891 (0)

(32)

The transfer function 119866119910(119904) can be obtained as

119866119910 (119904) = 11987011988611 + 119904119905go1198811199031199052go + 119904119901 1 + 119904119905go

1198811199031199052go

+ ℎminus119901[119905ℎ]

sum119895=0

120596119901minus11198952 + 119904119905go1198811199031199053go

+ ℎminus119901[119905ℎ]

sum119895=0

120596119901minus2119895119904 + 1199042119905go1198811199031199053go

(33)

The guidance loop transfer coefficient of EPN can beexpressed as

119870119910FOCPN = lim119904rarr0

1199042 11199042119866119910 (119904) 119866 (119904)

= 11198811199031199053go (1198701198861119905go + 21198862)

(34)

where 1198862 = 1198701198862ℎminus119901sum[119905ℎ]119895=0 120596119901119895 The guidance loop transfer coefficient of PN and PD

(proportional differential) guidance law can be denotedrespectively as follows

119870119910PN = 1198701119905go1198811199031199053go

119870119910PD = (1198701198861119905go + 21198701198862)1198811199031199053go

(35)

Comparing the transfer coefficients of EPN with PN andPD we have

119870119910FOCPN gt 119870119910PD gt 119870119910PN (36)

The conclusion of transfer coefficients in horizontal planeis similar to the vertical plane the relationship can beexpressed as

119870119909FOCPN gt 119870119909PD gt 119870119909PN (37)

According to (36) and (37) we can see the guidanceability against maneuvering target strengthened as a resultof increasing the gain coefficients of transfer function bythe introduction of fractional order term So the guidancesystematic errors will be reduced and the guidance accuracywill be improved in some sense

at +

+minus

L[Dpt

120579L(t)] ka2

UyG(s)

am1 + tgos

Vrt2go

1

s2YL ka1

Figure 3 Illustration of the closed loop EPN guidance system

Definition 2 The sensitivity of systems 119879 against coefficient119870 is defined as follows

119878119879119870 = 119889 (ln (119879))119889 (ln (119870)) (38)

The sensitivity of stability guidance system against thecoefficients 1198701198861 and1198701198862 can be calculated as follows

1198781198901198701198861 = minus 1198701198861119905go1198701198861119905go + 21198701198861

1198781198901198701198862 = minus 11987011988621198701198861119905go + 21198701198861

(39)

When using traditional PN guidance the sensitivity1198781198901198701 = minus1 In contrast with formula (39) we have |1198781198901198701 | gt|1198781198901198701198861 | |119878119890119870| gt |1198781198901198701198862 | this means that the sensitivity of EPNguidance against each coefficient (1198701198861 1198701198862) is smaller thanPN guidance Thus the proposed EPN guidance law has astronger response against parameter changes

According to the homing guidance system loop inFigure 3 the differential functions of system transmissionerror can be obtained

119889119866119890 (119904)119889119866 (119904) = minus 11990421198661 (119904)

[1199042 + 1198661 (119904) 119866 (119904)]2

119889119866 (119904)119889119887119894 = 119904119894

119875 (119904) 119894 = 1 2 119898

119889119866 (119904)119889119886119895 = minus119866 (119904) 119904119895

119875 (119904) 119895 = 1 2 119899

(40)

where 119875(119904) is the intermediate computing term thereforeaccording to (40) we obtain

119878119890119887119894 = 119864 (119904) 119887119894119904119894119875 (119904) 119894 = 1 2 119898

119878119890119886119895 = minus119864 (119904) 119866 (119904) 119886119895119904119895119875 (119904) 119895 = 1 2 119899

(41)

where 119864(119904) = minus119866(119904)(1199042 + 1198661(119904)119866(119904)) as 119904 = 0 we have119878119890119887119894119904119894

1003816100381610038161003816100381610038161003816100381610038161003816119904=0= minus 119887119894

1198701198861119905go + 21198701 119894 = 1 2 119898

119878119890119886119895119904119895

1003816100381610038161003816100381610038161003816100381610038161003816119904=0= minus 119886119895

1198701198861119905go + 21198701 119895 = 1 2 119899


Table 1 Engagement specifications

Parameters of missile and target(1198831198980 1198841198980 1198851198980) (0 0 0) km(1198831199050 1198841199050 1198851199050 ) (10 10 10) km120579119905 0120593119905 180120579119898 36120593119898 45119881119898 800119881119905 400

119878119890119887119894119904119894

1003816100381610038161003816100381610038161003816100381610038161003816PN

119904=0

= minus 1198871198941198701198861119905go 119894 = 1 2 119898

119878119890119886119895119904119895

1003816100381610038161003816100381610038161003816100381610038161003816

PN

119904=0

= minus 1198861198951198701198861119905go 119895 = 1 2 119899

(42)

It can be seen from (42) that the absolute sensitivity ofEPN against autopilot parameter decreases because of theincrease in denominator by introducing the fractional termTherefore the robustness of EPN against autopilot parameterchanges is more robust than PN guidance

5 Simulation Results and Analysis

In this section some results of simulation are presented for3D engagement to verify the performance of the designedEPN simulation initial step is 001 s and the proportionalcoefficient is taken as 35 Some other initial parametersof missile and target are shown in Table 1 The airframedynamics including autopilot has been modeled as a second-order system with time lag as described by [27] The totalautopilot lag is modeled as

119879 = 1205962119899(120591119904 + 1) (1199042 + 2120577120596119899119904 + 1205962119899)

(43)

where the autopilot parameters are 120591 = 05 s 120585 = 06and 120596119899 = 18 rads Due to the order of FOC in therange [0 1] according to traverse optimization methodsthe optimum value of differential order 119901 is taken as 06Consider that the LOS rate is contaminated with uncertaintyduring the terminal guidance phase and we assume that themeasurement error of the onboard seeker is 001 degs

To investigate the performance of the proposed guidancelaw efficiently three cases of simulations are carried out (1)the target with no maneuver (2) the target maneuver in theinclined plane with the acceleration 40ms2 (3) the targetmaneuver in a snake-like mode with the acceleration 20ms2in both the vertical and horizontal planes

The comparison of the interception performancesbetween the EPN guidance law and PN guidance law ispresented in Table 2 where MD is the mean miss distancedefined as the closest distance between the missile and

Table 2 Comparison of the interception performance

Guidance law MDm ETs

Case 1 PN 12129 3555EPN 06272 3505

Case 2 PN 39193 3334EPN 1978 3289

Case 3 PN 23398 4055EPN 1178 4032

0 05 1 15 20

5000

10000

15000

EPNTargetPN

Target

X (m)

Y (m)

Z (m

)

times104

times104

0

05

1

15

2

25

Figure 4 Trajectories of missile and target

the target before divergence and where ET denotes theengagement time All performance results are based on 200run Monte-Carlo simulations

Small miss distance is the most important in the homingguidance According to the simulation results shown inTable 2 both of the conventional PN and the proposed EPNguidance law are able to intercept targets effectively Howeverthe guidance precision of EPN is nearly 50 better thanthe PN guidance which indicates that the EPN has greatlyimproved the guidance accuracy It is also interesting to notethat the engagement time of EPN is always shorter than PNwhich means that the EPN will shot down the target earlierthan PN

In order to describe the guidance character of accelerationand ballistic trajectory we give the simulation figures of case3 only which are shown as Figures 4ndash8

As can be seen from Figure 4 the trajectory of the EPNis smoother than the PN The curvature of the missilersquostrajectory guided by EPN is smaller than PN especially in theend of engagement which is in favor of ballistic control Asillustrated in Figures 5 and 6 the main difference betweenEPN and PN is the adjustment rate of elevation and azimuthIn the initial phase of engagement the proposed EPN adjustsits attitude quicker than PN and this makes the trajectory ofEPN smoother in the later stage of interception Figures 7 and8 denote that the acceleration of the PN guidance law tends


5 10 15 20 25 30 35 40 450T (s)

20

25

30

35

40

45

50

55

EPNPN

120579 m(∘)

Figure 5 Time history of trajectory elevation

5 10 15 20 25 30 35 40 450T (s)

25

30

35

40

45

50

55

60

65

EPNPN

120595m

(∘)

Figure 6 Time history of trajectory azimuth

to achieve its saturation which compensates for the targetrsquosmaneuver at the end whereas the loads of EPN undergoesa relatively stable tendency during the engagement whichpositively adjusts the missile attitude at the expense of loadsin the initial phase of interception without dramatic increaseat the end So the load of EPN undergoes a relatively stabletendency during the flight envelope while the load of PNdiverges at the point of impact caused by the divergence ofLOS rate

With respect to the PN guidance law the proposedEPN guidance law maintains the tracking performance andexhibit more favorable guidance performance in interceptingmaneuvering target In fact EPN solves the problem of LOSrate estimation by the fractional order differential treatmentof LOS rate In other words the fractional order differential

0

5 10 15 20 25 30 35 40 450T (s)

EPNPN

minus100

minus80

minus60

minus40

minus20

20

40

60

80

100

a my(m

s2)

Figure 7 Time history of normal acceleration

5 10 15 20 25 30 35 40 450T (s)

0

EPNPN

minus50

minus40

minus30

minus20

minus10

10

20

30a m

z(m

s2)

Figure 8 Time history of lateral acceleration

controller plays the role of filter which improves the controlprecision and stability of the guidance system In a word theEPN extends the guidance performance ofmissile by carryingforward the advantages of PN guidance law and avoids itsshortcomings in some sense

6 Conclusions

To improve the performance of conventional PN guidancethis paper proposed a novel EPN guidance law based onthe fractional order calculus theory The guidance charac-teristics of EPN such as average load trajectory stabilityand robustness are analyzed theoretical The inducementof fractional order term extended the performance of PNgreatly especially in the guidance precision and interception


time and overcome the load saturation caused by LOS ratedivergence in the end of engagement And the EPN guidancelaw has more flexible structure and better robustness Exten-sive simulations of various engagements demonstrate thatthe proposed guidance law provides satisfactory performanceagainst maneuvering targets

The proposed guidance law is designed with the consid-eration of maneuvering target in the atmosphere howeverthe guidance law needs further analysis especially to achievedirect collision interception in the near space against hyper-sonic target

Competing Interests

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

Acknowledgments

The first author would like to thank the Air and MissileDefenseCollege ofAir Force EngineeringUniversity of Chinafor giving permission for conducting guidance research atthe laboratory This work was supported by National NaturalScience Foundation of China (nos 61573374 61503408 and61603410) and Aviation Science Innovation Foundation ofChina (no 20150196006)

References

[1] L Chen and B Zhang ldquoNovel TPN control algorithm forexoatmospheric interceptrdquo Journal of Systems Engineering andElectronics vol 20 no 6 pp 1290ndash1295 2009

[2] Q Z Zhang Z B Wang and F Tao ldquoOptimal guidance lawdesign for impact with terminal angle of attack constraintrdquoOptik vol 125 no 1 pp 243ndash251 2014

[3] K Li L Chen and X Bai ldquoDifferential geometric modeling ofguidance problem for interceptorsrdquo Science China TechnologicalSciences vol 54 no 9 pp 2283ndash2295 2011

[4] J Ye H Lei D Xue J Li and L Shao ldquoNonlinear differentialgeometric guidance for maneuvering targetrdquo Journal of SystemsEngineering and Electronics vol 23 no 5 pp 752ndash760 2012

[5] F Tyan ldquoUnified approach to missile guidance laws a 3Dextensionrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 41 no 4 pp 1178ndash1199 2005

[6] K Li T Zhang and L Chen ldquoIdeal proportional navigation forexoatmospheric interceptionrdquo Chinese Journal of Aeronauticsvol 26 no 4 pp 976ndash985 2013

[7] S-C Han H Bang and C-S Yoo ldquoProportional navigation-based collision avoidance for UAVsrdquo International Journal ofControl Automation and Systems vol 7 no 4 pp 553ndash5652009

[8] C-Y Li and W-X Jing ldquoGeometric approach to captureanalysis of PN guidance lawrdquoAerospace Science and Technologyvol 12 no 2 pp 177ndash183 2008

[9] Y Xia and M Fu ldquoMissile guidance laws based on SMC andFTC techniquesrdquo in Compound Control Methodology for FlightVehicles vol 438 of Lecture Notes in Control and InformationSciences pp 211ndash224 Springer Berlin Heidelberg 2013

[10] N Hananjay K-Y Lum and J-X Xu ldquoProportional navigationwith delayed line-of-sight raterdquo IEEE Transactions on ControlSystems Technology vol 21 no 1 pp 247ndash253 2013

[11] S Ghosh D Ghose and S Raha ldquoThree dimensional PN basedimpact angle control for higher speed nonmaneuvering targetsrdquoin Proceedings of the AmericanControl Conference (ACC rsquo13) pp31ndash36 Washington DC USA June 2013

[12] I-S Jeon and J-I Lee ldquoOptimality of proportional naviga-tion based on nonlinear formulationrdquo IEEE Transactions onAerospace and Electronic Systems vol 46 no 4 pp 2051ndash20552010

[13] R Caponetto G Dongola and L Fortuna Fractional OrderSystems Modeling and Control Applications World ScientificSingapore 2010

[14] Q Zeng G Cao and X Zhu ldquoResearch on controllability fora class of fractional-order linear control systemsrdquo Journal ofSystems Engineering and Electronics vol 16 no 2 pp 376ndash3812005

[15] I P Cristina I Clara D K Robain and E H Dulf ldquoRobustnessevaluation of fractional order control for varying time delayprocessesrdquo Signal Image and Video Processing vol 6 no 3 pp453ndash461 2012

[16] K SMiller and B RossAn Introduction to the Fractional Calcu-lus and Fractional Differential Equations A Wiley-IntersciencePublication John Wiley amp Sons New York NY USA 1993

[17] S G Smako A A Kilbas and O I Marichev FractionalIntegrals and Derivitives Theory and Applications Gorden andBreach Science Publishers New York NY USA 1987

[18] X-X Zhang T-S Qiu andH Sheng ldquoA physical interpretationof fractional calculus and fractional calculus with constantextent of integralrdquoActa Electronica Sinica vol 41 no 3 pp 508ndash512 2013

[19] A A Kilbas H M Srivastava and J J Trujillo Theory andapplications of fractional differential equations vol 204 ofNorth-HollandMathematics Studies Elsevier Science BV AmsterdamNetherlands 2006

[20] G Sun and Z H Zhu ldquoFractional-order tension control lawfor deployment of space tether systemrdquo Journal of GuidanceControl and Dynamics vol 37 no 6 pp 2057ndash2061 2014

[21] R Almeida and D F M Torres ldquoA discrete method to solvefractional optimal control problemsrdquo Nonlinear Dynamics vol80 no 4 pp 1811ndash1816 2015

[22] K Vishal S K Agrawal and S Das ldquoHyperchaos controland adaptive synchronization with uncertain parameter forfractional-order Mathieu-van der Pol systemsrdquo Pramana vol86 no 1 pp 59ndash75 2016

[23] J-G Shi Z-YWang S-J Chang and B-S Zhang ldquoFractional-order control systems for guided projectilesrdquo Journal of NanjingUniversity of Science and Technology (Natural Science) vol 35no 1 pp 52ndash56 2011

[24] T-Z Li Y Wang and M-K Luo ldquoControl of fractionalchaotic and hyperchaotic systems based on a fractional ordercontrollerrdquo Chinese Physics B vol 23 no 8 Article ID 0805012014

[25] F Wang and H M Lei ldquoPD120582 guidance law based on fractionalcalculusrdquo Control Theory amp Applications vol 27 no 1 pp 126ndash130 2010

[26] Z-T Zhu Z Liao C Peng and Y Wang ldquoA fractional-order modified proportional navigation lawrdquo Control Theory ampApplications vol 29 no 7 pp 945ndash949 2012

[27] S Rituraj N Prabhakar A K Sarkar et al ldquoThree dimensionalnonlinear inverse dynamics guidance law for parallel naviga-tionrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit pp 1ndash8 Providence RI USA2004

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



in the area of guidance and control [19] Considering that theatmospheric viscosity and interactions between air vehiclesmakes aircraft have aerodynamic properties of FOC thus theFOC theory can describe the characteristics of the aircraftkinematic comprehensive and truly

Sun and Zhu developed a new fractional order tensioncontrol law for the fast and stable deployment controlproblem of a space tether system and proved its stability [20]Almeida and Torres presented a method to solve fractionaloptimal control problems where the dynamic control systemdepended on integer order and Caputo fractional derivatives[21] Vishal et al used the theory of fractional order derivativeto discuss the local stability of the Mathieundashvan der Polhyperchaotic system and studied the analysis of control timewith respect to different fractional order derivatives [22]

Reference [23] studied the application of FOC controllerfor guided projectiles attitude control the author concludedthat fractional order control was better than classical PIDcontrol in robustness and control performance Li et alstudied the stability of the fractional order unified chaoticsystem by the equivalent passivity method With the FOCtheory the Lyapunov function was constructed by which itwas proved that the controlled fractional order system wasstable [24] The FOC theory was introduced to the guidancearea in reference [25] the author employed the PD120582 guidancelaw to carry forward the advantages of traditional PN andavoid its drawbacks based on FOC theory and concludedthat the PD120582 guidance had the higher guidance precision andshorter interception time Zhu et al proposed a modifiedFOC pure PN guidance law by means of Lyapunov-likemethod which solved the control problem for the capture of atarget with random maneuvers effectively [26] However theconclusion of [25 26] was obtained in the two-dimensionalplane under some certain conditions so there were somestrict restrictions on the designed guidance law and thisgreatly limits its practical engineering applications

This paper presents a novel 3D extended PN guidancealgorithm for maneuvering target based on FOC theoryand it overcomes the disadvantage of PN First the basicsknowledge and relative nature of FOC are outlined Secondthe proposed guidance law is given which is composed oftraditional LOS rate term and FOC LOS rate term And thetrajectory features the load character and the robustnessof the new guidance law are analyzed theoretically Atlast simulations show that the proposed guidance law hassuperior performance especially for a maneuvering target

2 Preliminary Knowledge

The fractional calculus differential operator can be expressedas follows [13 25]

120572119863119901119905 =

119889119901119889119905119901 119877 (119901) gt 01 119877 (119901) = 0int119905

0(119889119905)minus119901 119877 (119901) lt 0

(1)

where 120572 119905 denote the upper limit and lower limit respectively119901 is the calculus operator its value can be any plurality and119877(119901) presents the real part of 119901 If 119901 gt 0 then 120572119863119901119905 denotesfractional derivative

Definition 1 If the function 119891(119905) can be taken 119899 orderderivation here 119898 minus 1 le 119901 lt 119898 and 119899 isin N then we havethe definition of Caputo derivative as follows

119863119901119905 = 1Γ (119898 minus 119901) int

119905

0

119891119898 (120591)(119905 minus 120591)119901minus119898+1 119889120591 (2)

where Γ(119909) denotes Gamma function which is generalizationof the factorial function 119899 and we can take derivationof Gamma function with noninteger or even plurality Theexpression of Gamma function can be denoted as

Γ (119909) = intinfin

0119905119909minus119905119890minus119905119889119905 (3)

TheLaplace transformof Caputo fractional derivative canbe shown as

119871 [0119863119901119905 119891 (119905)] = intinfin

0119890minus1199041199050119863119901119905 119891 (119905) 119889119905

= 119904119901119865 (119904) minus119899minus1

sum119896=0

119904120572minus119896minus1119891119896 (0) (4)

In order to understand the links and differences betweenthe integer order and fractional order calculus we supposethat 119891(119905) is flat functions in the range [119886 119905] Taking thederivation of 119891(119905) with 119899 order we have

119891119899 (119905) = limℎrarr0

1ℎ119899119899

sum119895=0

(minus1)119895 119899119895 (119899 minus 119895)119891 (119905 minus 119895ℎ) (5)

When the noninteger 119901 replaces integer 119899 and Γ func-tion replaces binomial coefficients we get the definition ofGrunwald-Letnikov derivative as follows

119863119901119905 119891 (119905) = limℎrarr0

1ℎ119901119899

sum119895=0

(minus1)119895 Γ (119901 + 1)119895Γ (119901 minus 119895 + 1)119891 (119905 minus 119895ℎ) (6)

As can be seen from the above derivation the mostsignificant difference between integer order derivative andfractional derivative is that the former contains the recentinformation of several points while the latter contains thepast information of all the points We can see that fromequation (6) the fractional derivative has the ability ofmemory function and the closer the point the stronger theimpact factor [14]

In fact the integer order calculus is a special case offractional calculus when the orders of fractional calculus119901 isin 119873 the above two are equivalent Another point tonote is that FOC uniforms the forms of the differential andintegral expressions by fractional operator When 119901 gt 0 itdenotes fractional differential operator while when 119901 lt 0 itrepresents fractional integral operator so the study of integraland differential operation can be considered as an operator


Y

XL120593t

120579t

VtZT

YT

YL

R

o

120593L

120593m

120579m

XM

X

Z

ZM

ZL

YM

Vm

120579L

XT

LOS






119880119886 = 1198701119881119903119871119880119887 = 1198701119881119903119871 cos 120579119871

(7)





119880119901119886 = 1198701119881119903119871 + 1198701119881119903119901119863119901119905 120579119871 (8)

119880119901119887= 1198701119881119903119871 cos 120579119871 + 1198701119881119903119863119901119905 (120593119871) cos 120579119871 (9)




119880119901119886 = 1198701198861119871 + 1198701198862119863119901119905 (120579119871) 119880119901119887= 1198701198871119871 + 1198701198872119863119901119905 (120593119871)

(10)

where 1198701198861 = 1198701119881119903 1198701198871 = 1198701119881119903 cos 120579119871 1198701198862 = 1198701119881119903119901 and1198701198872 = 1198701119881119903119863119901119905 (120593119871) cos 120579119871



119880 = 1119905119891 int119905119891

0(119880119886 + 119880119887) 119889119905 = 1

119905119891 int119905119891

0(1198701198861119871

+ 1198701198862119863119901119905 (120579119871) + 1198701198871119871 + 1198701198872119863119901119905 (120593119871)) 119889119905 = 1119905119891

sdot int119905119891

01198701198861119871119889119905 + 1

119905119891 int119905119891

01198701198871119871119889119905 + 1

119905119891sdot int119905119891

01198701198872119863119901119905 (120593119871) 119889119905 + 1

119905119891 int119905119891

01198701198862119863119901119905 (120579119871) 119889119905

(11)


1198801198861 = 1119905119891 int119905119891

01198701198861119871119889119905

1198801198862 = 1119905119891 int119905119891

01198701198862119863119901119905 (120579119871) 119889119905

1198801198871 = 1119905119891 int119905119891

01198701198871119871119889119905

1198801198872 = 1119905119891 int119905119891

01198701198872119863119901119905 (120593119871) 119889119905

(12)


1198801198861 = 1119905119891 int119905119891

01198701198861119871119889119905 =

1198701198861 (120579119871119891 minus 1205791198710)119905119891

1198801198871 = 1119905119891 int119905119891

01198701198871119871119889119905 =

1198701198871 (120593119871119891 minus 1205931198710)119905119891

1198801198862 = 1119905119891 int119905119891

01198701198862119863119901119905 (120579119871) 119889119905

= limℎrarr0

1198701198862119905119891 int119905119891

0ℎminus119901[119905ℎ]

sum119895=0

120596119901119895 119871 (119905 minus 119895ℎ) 119889119905

= limℎrarr0

1198701198862119905119891 ℎminus119901

[119905ℎ]

sum119895=0

120596119901119895 int119905119891

0119871 (119905 minus 119895ℎ) 119889119905

asymp 1198701198862119905119891 ℎminus119901

[119905ℎ]

sum119895=0

120596119901119895 (120579119871119891 minus 1205791198710)

(13)


1198801198862 = 1198862(120579119871119891 minus 1205791198710)

119905119891 (14)




119880119886 = 1198801198861 + 1198801198862 = (1198701198861 + 1198862)(120579119871119891 minus 1205791198710)

119905119891

119880119887 = 1198801198871 + 1198801198872 = (1198701198871 + 1198872)(120579119871119891 minus 1205791198710)

119905119891 (15)


119880 = 119880119886 + 119880119887

= (1198701198861 + 1198701198862)(120579119871119891 minus 1205791198710)

119905119891

+ (1198701198871 + 1198701198872)(120593119871119891 minus 1205931198710)

119905119891

(16)


119880TPN = 119870TPN1(120579119871119891 minus 1205791198710)

119905119891 + 119870TPN1(120593119871119891 minus 1205931198710)

119905119891 (17)




(18)


1003816100381610038161003816119880119901119886 1003816100381610038161003816 le 1198701198861 1003816100381610038161003816100381611987110038161003816100381610038161003816 + 1198701198862 10038161003816100381610038161003816119863119901119905 (120579119871)

10038161003816100381610038161003816 10038161003816100381610038161003816119880119901119887

10038161003816100381610038161003816 le 1198701198871 10038161003816100381610038161198711003816100381610038161003816 + 1198701198872 10038161003816100381610038161003816119863119901119905 (120593119871)10038161003816100381610038161003816

(19)



max 119880119901119886 le max 1198701198861119871 + 1198701198862119863119901119905 (120579119871) max 119880119901

119887 le max 1198701198871119871 + 1198701198872119863119901119905 (120593119871)

(20)




119887rarr 0 we



120578119898 = 120579119871 minus 120579119898 (21)

120593119898 = 120593119871 minus 120593119898 (22)



119898 = 119871 minus 119898 (23)


119898 = 1198701119881119903119871 + 1198701119881119903119901119863119901119905 (120579119871)119881119898

= 1198701205791119871 + 1198701205792119863119901119905 (120579119871) (24)


minus 1198701205792 int119905119891

0119863119901119905 (119871) 119889119905

= 1205781198980 + (1 minus 1198701205791) (120579119871119891 minus 1205791198710)

minus 1198701205792ℎminus119901[119905ℎ]

sum119895=0

120596119901119895 (120579119871119891 minus 1205791198710)

(25)



(26)

YL

Vrtgo

at

U120579L

Vm



120579119871119891 minus 1205791198710 =120578119898119891

1198701205791 + 1205792 minus 1 (27)


120579119871119891 minus 1205791198710 =120578119898119891

1198701205791 minus 1 (28)




120579119871 = 119884119871119881119903119905go (29)


119871 = 119884119871 +119871119905go1198811199031199052go (30)






119871 [119863119901119905 119891 (119905)] = 119904119901119871 [119891 (119905)] minus119899minus1

sum119896=1

119904119896 [119863119901minus119896minus1119905 119891 (119905)]119905=0


[119905ℎ]

sum119895=0

120596minus1119895 119891 (0)

minus 1199041ℎminus119901[119905ℎ]

sum119895=0

120596119901minus2119895 119891 (0)

(32)


119866119910 (119904) = 11987011988611 + 119904119905go1198811199031199052go + 119904119901 1 + 119904119905go

1198811199031199052go

+ ℎminus119901[119905ℎ]

sum119895=0

120596119901minus11198952 + 119904119905go1198811199031199053go

+ ℎminus119901[119905ℎ]

sum119895=0

120596119901minus2119895119904 + 1199042119905go1198811199031199053go

(33)


119870119910FOCPN = lim119904rarr0

1199042 11199042119866119910 (119904) 119866 (119904)

= 11198811199031199053go (1198701198861119905go + 21198862)

(34)



119870119910PN = 1198701119905go1198811199031199053go

119870119910PD = (1198701198861119905go + 21198701198862)1198811199031199053go

(35)


119870119910FOCPN gt 119870119910PD gt 119870119910PN (36)


119870119909FOCPN gt 119870119909PD gt 119870119909PN (37)


at +

+minus

L[Dpt

120579L(t)] ka2

UyG(s)

am1 + tgos

Vrt2go

1

s2YL ka1



119878119879119870 = 119889 (ln (119879))119889 (ln (119870)) (38)


1198781198901198701198861 = minus 1198701198861119905go1198701198861119905go + 21198701198861

1198781198901198701198862 = minus 11987011988621198701198861119905go + 21198701198861

(39)



119889119866119890 (119904)119889119866 (119904) = minus 11990421198661 (119904)

[1199042 + 1198661 (119904) 119866 (119904)]2

119889119866 (119904)119889119887119894 = 119904119894

119875 (119904) 119894 = 1 2 119898

119889119866 (119904)119889119886119895 = minus119866 (119904) 119904119895

119875 (119904) 119895 = 1 2 119899

(40)


119878119890119887119894 = 119864 (119904) 119887119894119904119894119875 (119904) 119894 = 1 2 119898

119878119890119886119895 = minus119864 (119904) 119866 (119904) 119886119895119904119895119875 (119904) 119895 = 1 2 119899

(41)


1003816100381610038161003816100381610038161003816100381610038161003816119904=0= minus 119887119894

1198701198861119905go + 21198701 119894 = 1 2 119898

119878119890119886119895119904119895

1003816100381610038161003816100381610038161003816100381610038161003816119904=0= minus 119886119895

1198701198861119905go + 21198701 119895 = 1 2 119899




119878119890119887119894119904119894

1003816100381610038161003816100381610038161003816100381610038161003816PN

119904=0

= minus 1198871198941198701198861119905go 119894 = 1 2 119898

119878119890119886119895119904119895

1003816100381610038161003816100381610038161003816100381610038161003816

PN

119904=0

= minus 1198861198951198701198861119905go 119895 = 1 2 119899

(42)




119879 = 1205962119899(120591119904 + 1) (1199042 + 2120577120596119899119904 + 1205962119899)

(43)






Case 1 PN 12129 3555EPN 06272 3505

Case 2 PN 39193 3334EPN 1978 3289

Case 3 PN 23398 4055EPN 1178 4032

0 05 1 15 20

5000

10000

15000

EPNTargetPN

Target

X (m)

Y (m)

Z (m

)

times104

times104

0

05

1

15

2

25







5 10 15 20 25 30 35 40 450T (s)

20

25

30

35

40

45

50

55

EPNPN

120579 m(∘)


5 10 15 20 25 30 35 40 450T (s)

25

30

35

40

45

50

55

60

65

EPNPN

120595m

(∘)




0

5 10 15 20 25 30 35 40 450T (s)

EPNPN

minus100

minus80

minus60

minus40

minus20

20

40

60

80

100

a my(m

s2)


5 10 15 20 25 30 35 40 450T (s)

0

EPNPN

minus50

minus40

minus30

minus20

minus10

10

20

30a m

z(m

s2)



6 Conclusions





Competing Interests


Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Y

XL120593t

120579t

VtZT

YT

YL

R

o

120593L

120593m

120579m

XM

X

Z

ZM

ZL

YM

Vm

120579L

XT

LOS






119880119886 = 1198701119881119903119871119880119887 = 1198701119881119903119871 cos 120579119871

(7)





119880119901119886 = 1198701119881119903119871 + 1198701119881119903119901119863119901119905 120579119871 (8)

119880119901119887= 1198701119881119903119871 cos 120579119871 + 1198701119881119903119863119901119905 (120593119871) cos 120579119871 (9)




119880119901119886 = 1198701198861119871 + 1198701198862119863119901119905 (120579119871) 119880119901119887= 1198701198871119871 + 1198701198872119863119901119905 (120593119871)

(10)

where 1198701198861 = 1198701119881119903 1198701198871 = 1198701119881119903 cos 120579119871 1198701198862 = 1198701119881119903119901 and1198701198872 = 1198701119881119903119863119901119905 (120593119871) cos 120579119871



119880 = 1119905119891 int119905119891

0(119880119886 + 119880119887) 119889119905 = 1

119905119891 int119905119891

0(1198701198861119871

+ 1198701198862119863119901119905 (120579119871) + 1198701198871119871 + 1198701198872119863119901119905 (120593119871)) 119889119905 = 1119905119891

sdot int119905119891

01198701198861119871119889119905 + 1

119905119891 int119905119891

01198701198871119871119889119905 + 1

119905119891sdot int119905119891

01198701198872119863119901119905 (120593119871) 119889119905 + 1

119905119891 int119905119891

01198701198862119863119901119905 (120579119871) 119889119905

(11)


1198801198861 = 1119905119891 int119905119891

01198701198861119871119889119905

1198801198862 = 1119905119891 int119905119891

01198701198862119863119901119905 (120579119871) 119889119905

1198801198871 = 1119905119891 int119905119891

01198701198871119871119889119905

1198801198872 = 1119905119891 int119905119891

01198701198872119863119901119905 (120593119871) 119889119905

(12)


1198801198861 = 1119905119891 int119905119891

01198701198861119871119889119905 =

1198701198861 (120579119871119891 minus 1205791198710)119905119891

1198801198871 = 1119905119891 int119905119891

01198701198871119871119889119905 =

1198701198871 (120593119871119891 minus 1205931198710)119905119891

1198801198862 = 1119905119891 int119905119891

01198701198862119863119901119905 (120579119871) 119889119905

= limℎrarr0

1198701198862119905119891 int119905119891

0ℎminus119901[119905ℎ]

sum119895=0

120596119901119895 119871 (119905 minus 119895ℎ) 119889119905

= limℎrarr0

1198701198862119905119891 ℎminus119901

[119905ℎ]

sum119895=0

120596119901119895 int119905119891

0119871 (119905 minus 119895ℎ) 119889119905

asymp 1198701198862119905119891 ℎminus119901

[119905ℎ]

sum119895=0

120596119901119895 (120579119871119891 minus 1205791198710)

(13)


1198801198862 = 1198862(120579119871119891 minus 1205791198710)

119905119891 (14)




119880119886 = 1198801198861 + 1198801198862 = (1198701198861 + 1198862)(120579119871119891 minus 1205791198710)

119905119891

119880119887 = 1198801198871 + 1198801198872 = (1198701198871 + 1198872)(120579119871119891 minus 1205791198710)

119905119891 (15)


119880 = 119880119886 + 119880119887

= (1198701198861 + 1198701198862)(120579119871119891 minus 1205791198710)

119905119891

+ (1198701198871 + 1198701198872)(120593119871119891 minus 1205931198710)

119905119891

(16)


119880TPN = 119870TPN1(120579119871119891 minus 1205791198710)

119905119891 + 119870TPN1(120593119871119891 minus 1205931198710)

119905119891 (17)




(18)


1003816100381610038161003816119880119901119886 1003816100381610038161003816 le 1198701198861 1003816100381610038161003816100381611987110038161003816100381610038161003816 + 1198701198862 10038161003816100381610038161003816119863119901119905 (120579119871)

10038161003816100381610038161003816 10038161003816100381610038161003816119880119901119887

10038161003816100381610038161003816 le 1198701198871 10038161003816100381610038161198711003816100381610038161003816 + 1198701198872 10038161003816100381610038161003816119863119901119905 (120593119871)10038161003816100381610038161003816

(19)



max 119880119901119886 le max 1198701198861119871 + 1198701198862119863119901119905 (120579119871) max 119880119901

119887 le max 1198701198871119871 + 1198701198872119863119901119905 (120593119871)

(20)




119887rarr 0 we



120578119898 = 120579119871 minus 120579119898 (21)

120593119898 = 120593119871 minus 120593119898 (22)



119898 = 119871 minus 119898 (23)


119898 = 1198701119881119903119871 + 1198701119881119903119901119863119901119905 (120579119871)119881119898

= 1198701205791119871 + 1198701205792119863119901119905 (120579119871) (24)


minus 1198701205792 int119905119891

0119863119901119905 (119871) 119889119905

= 1205781198980 + (1 minus 1198701205791) (120579119871119891 minus 1205791198710)

minus 1198701205792ℎminus119901[119905ℎ]

sum119895=0

120596119901119895 (120579119871119891 minus 1205791198710)

(25)



(26)

YL

Vrtgo

at

U120579L

Vm



120579119871119891 minus 1205791198710 =120578119898119891

1198701205791 + 1205792 minus 1 (27)


120579119871119891 minus 1205791198710 =120578119898119891

1198701205791 minus 1 (28)




120579119871 = 119884119871119881119903119905go (29)


119871 = 119884119871 +119871119905go1198811199031199052go (30)






119871 [119863119901119905 119891 (119905)] = 119904119901119871 [119891 (119905)] minus119899minus1

sum119896=1

119904119896 [119863119901minus119896minus1119905 119891 (119905)]119905=0


[119905ℎ]

sum119895=0

120596minus1119895 119891 (0)

minus 1199041ℎminus119901[119905ℎ]

sum119895=0

120596119901minus2119895 119891 (0)

(32)


119866119910 (119904) = 11987011988611 + 119904119905go1198811199031199052go + 119904119901 1 + 119904119905go

1198811199031199052go

+ ℎminus119901[119905ℎ]

sum119895=0

120596119901minus11198952 + 119904119905go1198811199031199053go

+ ℎminus119901[119905ℎ]

sum119895=0

120596119901minus2119895119904 + 1199042119905go1198811199031199053go

(33)


119870119910FOCPN = lim119904rarr0

1199042 11199042119866119910 (119904) 119866 (119904)

= 11198811199031199053go (1198701198861119905go + 21198862)

(34)



119870119910PN = 1198701119905go1198811199031199053go

119870119910PD = (1198701198861119905go + 21198701198862)1198811199031199053go

(35)


119870119910FOCPN gt 119870119910PD gt 119870119910PN (36)


119870119909FOCPN gt 119870119909PD gt 119870119909PN (37)


at +

+minus

L[Dpt

120579L(t)] ka2

UyG(s)

am1 + tgos

Vrt2go

1

s2YL ka1



119878119879119870 = 119889 (ln (119879))119889 (ln (119870)) (38)


1198781198901198701198861 = minus 1198701198861119905go1198701198861119905go + 21198701198861

1198781198901198701198862 = minus 11987011988621198701198861119905go + 21198701198861

(39)



119889119866119890 (119904)119889119866 (119904) = minus 11990421198661 (119904)

[1199042 + 1198661 (119904) 119866 (119904)]2

119889119866 (119904)119889119887119894 = 119904119894

119875 (119904) 119894 = 1 2 119898

119889119866 (119904)119889119886119895 = minus119866 (119904) 119904119895

119875 (119904) 119895 = 1 2 119899

(40)


119878119890119887119894 = 119864 (119904) 119887119894119904119894119875 (119904) 119894 = 1 2 119898

119878119890119886119895 = minus119864 (119904) 119866 (119904) 119886119895119904119895119875 (119904) 119895 = 1 2 119899

(41)


1003816100381610038161003816100381610038161003816100381610038161003816119904=0= minus 119887119894

1198701198861119905go + 21198701 119894 = 1 2 119898

119878119890119886119895119904119895

1003816100381610038161003816100381610038161003816100381610038161003816119904=0= minus 119886119895

1198701198861119905go + 21198701 119895 = 1 2 119899




119878119890119887119894119904119894

1003816100381610038161003816100381610038161003816100381610038161003816PN

119904=0

= minus 1198871198941198701198861119905go 119894 = 1 2 119898

119878119890119886119895119904119895

1003816100381610038161003816100381610038161003816100381610038161003816

PN

119904=0

= minus 1198861198951198701198861119905go 119895 = 1 2 119899

(42)




119879 = 1205962119899(120591119904 + 1) (1199042 + 2120577120596119899119904 + 1205962119899)

(43)






Case 1 PN 12129 3555EPN 06272 3505

Case 2 PN 39193 3334EPN 1978 3289

Case 3 PN 23398 4055EPN 1178 4032

0 05 1 15 20

5000

10000

15000

EPNTargetPN

Target

X (m)

Y (m)

Z (m

)

times104

times104

0

05

1

15

2

25







5 10 15 20 25 30 35 40 450T (s)

20

25

30

35

40

45

50

55

EPNPN

120579 m(∘)


5 10 15 20 25 30 35 40 450T (s)

25

30

35

40

45

50

55

60

65

EPNPN

120595m

(∘)




0

5 10 15 20 25 30 35 40 450T (s)

EPNPN

minus100

minus80

minus60

minus40

minus20

20

40

60

80

100

a my(m

s2)


5 10 15 20 25 30 35 40 450T (s)

0

EPNPN

minus50

minus40

minus30

minus20

minus10

10

20

30a m

z(m

s2)



6 Conclusions





Competing Interests


Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119880 = 1119905119891 int119905119891

0(119880119886 + 119880119887) 119889119905 = 1

119905119891 int119905119891

0(1198701198861119871

+ 1198701198862119863119901119905 (120579119871) + 1198701198871119871 + 1198701198872119863119901119905 (120593119871)) 119889119905 = 1119905119891

sdot int119905119891

01198701198861119871119889119905 + 1

119905119891 int119905119891

01198701198871119871119889119905 + 1

119905119891sdot int119905119891

01198701198872119863119901119905 (120593119871) 119889119905 + 1

119905119891 int119905119891

01198701198862119863119901119905 (120579119871) 119889119905

(11)


1198801198861 = 1119905119891 int119905119891

01198701198861119871119889119905

1198801198862 = 1119905119891 int119905119891

01198701198862119863119901119905 (120579119871) 119889119905

1198801198871 = 1119905119891 int119905119891

01198701198871119871119889119905

1198801198872 = 1119905119891 int119905119891

01198701198872119863119901119905 (120593119871) 119889119905

(12)


1198801198861 = 1119905119891 int119905119891

01198701198861119871119889119905 =

1198701198861 (120579119871119891 minus 1205791198710)119905119891

1198801198871 = 1119905119891 int119905119891

01198701198871119871119889119905 =

1198701198871 (120593119871119891 minus 1205931198710)119905119891

1198801198862 = 1119905119891 int119905119891

01198701198862119863119901119905 (120579119871) 119889119905

= limℎrarr0

1198701198862119905119891 int119905119891

0ℎminus119901[119905ℎ]

sum119895=0

120596119901119895 119871 (119905 minus 119895ℎ) 119889119905

= limℎrarr0

1198701198862119905119891 ℎminus119901

[119905ℎ]

sum119895=0

120596119901119895 int119905119891

0119871 (119905 minus 119895ℎ) 119889119905

asymp 1198701198862119905119891 ℎminus119901

[119905ℎ]

sum119895=0

120596119901119895 (120579119871119891 minus 1205791198710)

(13)


1198801198862 = 1198862(120579119871119891 minus 1205791198710)

119905119891 (14)




119880119886 = 1198801198861 + 1198801198862 = (1198701198861 + 1198862)(120579119871119891 minus 1205791198710)

119905119891

119880119887 = 1198801198871 + 1198801198872 = (1198701198871 + 1198872)(120579119871119891 minus 1205791198710)

119905119891 (15)


119880 = 119880119886 + 119880119887

= (1198701198861 + 1198701198862)(120579119871119891 minus 1205791198710)

119905119891

+ (1198701198871 + 1198701198872)(120593119871119891 minus 1205931198710)

119905119891

(16)


119880TPN = 119870TPN1(120579119871119891 minus 1205791198710)

119905119891 + 119870TPN1(120593119871119891 minus 1205931198710)

119905119891 (17)




(18)


1003816100381610038161003816119880119901119886 1003816100381610038161003816 le 1198701198861 1003816100381610038161003816100381611987110038161003816100381610038161003816 + 1198701198862 10038161003816100381610038161003816119863119901119905 (120579119871)

10038161003816100381610038161003816 10038161003816100381610038161003816119880119901119887

10038161003816100381610038161003816 le 1198701198871 10038161003816100381610038161198711003816100381610038161003816 + 1198701198872 10038161003816100381610038161003816119863119901119905 (120593119871)10038161003816100381610038161003816

(19)



max 119880119901119886 le max 1198701198861119871 + 1198701198862119863119901119905 (120579119871) max 119880119901

119887 le max 1198701198871119871 + 1198701198872119863119901119905 (120593119871)

(20)




119887rarr 0 we



120578119898 = 120579119871 minus 120579119898 (21)

120593119898 = 120593119871 minus 120593119898 (22)



119898 = 119871 minus 119898 (23)


119898 = 1198701119881119903119871 + 1198701119881119903119901119863119901119905 (120579119871)119881119898

= 1198701205791119871 + 1198701205792119863119901119905 (120579119871) (24)


minus 1198701205792 int119905119891

0119863119901119905 (119871) 119889119905

= 1205781198980 + (1 minus 1198701205791) (120579119871119891 minus 1205791198710)

minus 1198701205792ℎminus119901[119905ℎ]

sum119895=0

120596119901119895 (120579119871119891 minus 1205791198710)

(25)



(26)

YL

Vrtgo

at

U120579L

Vm



120579119871119891 minus 1205791198710 =120578119898119891

1198701205791 + 1205792 minus 1 (27)


120579119871119891 minus 1205791198710 =120578119898119891

1198701205791 minus 1 (28)




120579119871 = 119884119871119881119903119905go (29)


119871 = 119884119871 +119871119905go1198811199031199052go (30)






119871 [119863119901119905 119891 (119905)] = 119904119901119871 [119891 (119905)] minus119899minus1

sum119896=1

119904119896 [119863119901minus119896minus1119905 119891 (119905)]119905=0


[119905ℎ]

sum119895=0

120596minus1119895 119891 (0)

minus 1199041ℎminus119901[119905ℎ]

sum119895=0

120596119901minus2119895 119891 (0)

(32)


119866119910 (119904) = 11987011988611 + 119904119905go1198811199031199052go + 119904119901 1 + 119904119905go

1198811199031199052go

+ ℎminus119901[119905ℎ]

sum119895=0

120596119901minus11198952 + 119904119905go1198811199031199053go

+ ℎminus119901[119905ℎ]

sum119895=0

120596119901minus2119895119904 + 1199042119905go1198811199031199053go

(33)


119870119910FOCPN = lim119904rarr0

1199042 11199042119866119910 (119904) 119866 (119904)

= 11198811199031199053go (1198701198861119905go + 21198862)

(34)



119870119910PN = 1198701119905go1198811199031199053go

119870119910PD = (1198701198861119905go + 21198701198862)1198811199031199053go

(35)


119870119910FOCPN gt 119870119910PD gt 119870119910PN (36)


119870119909FOCPN gt 119870119909PD gt 119870119909PN (37)


at +

+minus

L[Dpt

120579L(t)] ka2

UyG(s)

am1 + tgos

Vrt2go

1

s2YL ka1



119878119879119870 = 119889 (ln (119879))119889 (ln (119870)) (38)


1198781198901198701198861 = minus 1198701198861119905go1198701198861119905go + 21198701198861

1198781198901198701198862 = minus 11987011988621198701198861119905go + 21198701198861

(39)



119889119866119890 (119904)119889119866 (119904) = minus 11990421198661 (119904)

[1199042 + 1198661 (119904) 119866 (119904)]2

119889119866 (119904)119889119887119894 = 119904119894

119875 (119904) 119894 = 1 2 119898

119889119866 (119904)119889119886119895 = minus119866 (119904) 119904119895

119875 (119904) 119895 = 1 2 119899

(40)


119878119890119887119894 = 119864 (119904) 119887119894119904119894119875 (119904) 119894 = 1 2 119898

119878119890119886119895 = minus119864 (119904) 119866 (119904) 119886119895119904119895119875 (119904) 119895 = 1 2 119899

(41)


1003816100381610038161003816100381610038161003816100381610038161003816119904=0= minus 119887119894

1198701198861119905go + 21198701 119894 = 1 2 119898

119878119890119886119895119904119895

1003816100381610038161003816100381610038161003816100381610038161003816119904=0= minus 119886119895

1198701198861119905go + 21198701 119895 = 1 2 119899




119878119890119887119894119904119894

1003816100381610038161003816100381610038161003816100381610038161003816PN

119904=0

= minus 1198871198941198701198861119905go 119894 = 1 2 119898

119878119890119886119895119904119895

1003816100381610038161003816100381610038161003816100381610038161003816

PN

119904=0

= minus 1198861198951198701198861119905go 119895 = 1 2 119899

(42)




119879 = 1205962119899(120591119904 + 1) (1199042 + 2120577120596119899119904 + 1205962119899)

(43)






Case 1 PN 12129 3555EPN 06272 3505

Case 2 PN 39193 3334EPN 1978 3289

Case 3 PN 23398 4055EPN 1178 4032

0 05 1 15 20

5000

10000

15000

EPNTargetPN

Target

X (m)

Y (m)

Z (m

)

times104

times104

0

05

1

15

2

25







5 10 15 20 25 30 35 40 450T (s)

20

25

30

35

40

45

50

55

EPNPN

120579 m(∘)


5 10 15 20 25 30 35 40 450T (s)

25

30

35

40

45

50

55

60

65

EPNPN

120595m

(∘)




0

5 10 15 20 25 30 35 40 450T (s)

EPNPN

minus100

minus80

minus60

minus40

minus20

20

40

60

80

100

a my(m

s2)


5 10 15 20 25 30 35 40 450T (s)

0

EPNPN

minus50

minus40

minus30

minus20

minus10

10

20

30a m

z(m

s2)



6 Conclusions





Competing Interests


Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











max 119880119901119886 le max 1198701198861119871 + 1198701198862119863119901119905 (120579119871) max 119880119901

119887 le max 1198701198871119871 + 1198701198872119863119901119905 (120593119871)

(20)




119887rarr 0 we



120578119898 = 120579119871 minus 120579119898 (21)

120593119898 = 120593119871 minus 120593119898 (22)



119898 = 119871 minus 119898 (23)


119898 = 1198701119881119903119871 + 1198701119881119903119901119863119901119905 (120579119871)119881119898

= 1198701205791119871 + 1198701205792119863119901119905 (120579119871) (24)


minus 1198701205792 int119905119891

0119863119901119905 (119871) 119889119905

= 1205781198980 + (1 minus 1198701205791) (120579119871119891 minus 1205791198710)

minus 1198701205792ℎminus119901[119905ℎ]

sum119895=0

120596119901119895 (120579119871119891 minus 1205791198710)

(25)



(26)

YL

Vrtgo

at

U120579L

Vm



120579119871119891 minus 1205791198710 =120578119898119891

1198701205791 + 1205792 minus 1 (27)


120579119871119891 minus 1205791198710 =120578119898119891

1198701205791 minus 1 (28)




120579119871 = 119884119871119881119903119905go (29)


119871 = 119884119871 +119871119905go1198811199031199052go (30)






119871 [119863119901119905 119891 (119905)] = 119904119901119871 [119891 (119905)] minus119899minus1

sum119896=1

119904119896 [119863119901minus119896minus1119905 119891 (119905)]119905=0


[119905ℎ]

sum119895=0

120596minus1119895 119891 (0)

minus 1199041ℎminus119901[119905ℎ]

sum119895=0

120596119901minus2119895 119891 (0)

(32)


119866119910 (119904) = 11987011988611 + 119904119905go1198811199031199052go + 119904119901 1 + 119904119905go

1198811199031199052go

+ ℎminus119901[119905ℎ]

sum119895=0

120596119901minus11198952 + 119904119905go1198811199031199053go

+ ℎminus119901[119905ℎ]

sum119895=0

120596119901minus2119895119904 + 1199042119905go1198811199031199053go

(33)


119870119910FOCPN = lim119904rarr0

1199042 11199042119866119910 (119904) 119866 (119904)

= 11198811199031199053go (1198701198861119905go + 21198862)

(34)



119870119910PN = 1198701119905go1198811199031199053go

119870119910PD = (1198701198861119905go + 21198701198862)1198811199031199053go

(35)


119870119910FOCPN gt 119870119910PD gt 119870119910PN (36)


119870119909FOCPN gt 119870119909PD gt 119870119909PN (37)


at +

+minus

L[Dpt

120579L(t)] ka2

UyG(s)

am1 + tgos

Vrt2go

1

s2YL ka1



119878119879119870 = 119889 (ln (119879))119889 (ln (119870)) (38)


1198781198901198701198861 = minus 1198701198861119905go1198701198861119905go + 21198701198861

1198781198901198701198862 = minus 11987011988621198701198861119905go + 21198701198861

(39)



119889119866119890 (119904)119889119866 (119904) = minus 11990421198661 (119904)

[1199042 + 1198661 (119904) 119866 (119904)]2

119889119866 (119904)119889119887119894 = 119904119894

119875 (119904) 119894 = 1 2 119898

119889119866 (119904)119889119886119895 = minus119866 (119904) 119904119895

119875 (119904) 119895 = 1 2 119899

(40)


119878119890119887119894 = 119864 (119904) 119887119894119904119894119875 (119904) 119894 = 1 2 119898

119878119890119886119895 = minus119864 (119904) 119866 (119904) 119886119895119904119895119875 (119904) 119895 = 1 2 119899

(41)


1003816100381610038161003816100381610038161003816100381610038161003816119904=0= minus 119887119894

1198701198861119905go + 21198701 119894 = 1 2 119898

119878119890119886119895119904119895

1003816100381610038161003816100381610038161003816100381610038161003816119904=0= minus 119886119895

1198701198861119905go + 21198701 119895 = 1 2 119899




119878119890119887119894119904119894

1003816100381610038161003816100381610038161003816100381610038161003816PN

119904=0

= minus 1198871198941198701198861119905go 119894 = 1 2 119898

119878119890119886119895119904119895

1003816100381610038161003816100381610038161003816100381610038161003816

PN

119904=0

= minus 1198861198951198701198861119905go 119895 = 1 2 119899

(42)




119879 = 1205962119899(120591119904 + 1) (1199042 + 2120577120596119899119904 + 1205962119899)

(43)






Case 1 PN 12129 3555EPN 06272 3505

Case 2 PN 39193 3334EPN 1978 3289

Case 3 PN 23398 4055EPN 1178 4032

0 05 1 15 20

5000

10000

15000

EPNTargetPN

Target

X (m)

Y (m)

Z (m

)

times104

times104

0

05

1

15

2

25







5 10 15 20 25 30 35 40 450T (s)

20

25

30

35

40

45

50

55

EPNPN

120579 m(∘)


5 10 15 20 25 30 35 40 450T (s)

25

30

35

40

45

50

55

60

65

EPNPN

120595m

(∘)




0

5 10 15 20 25 30 35 40 450T (s)

EPNPN

minus100

minus80

minus60

minus40

minus20

20

40

60

80

100

a my(m

s2)


5 10 15 20 25 30 35 40 450T (s)

0

EPNPN

minus50

minus40

minus30

minus20

minus10

10

20

30a m

z(m

s2)



6 Conclusions





Competing Interests


Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119871 [119863119901119905 119891 (119905)] = 119904119901119871 [119891 (119905)] minus119899minus1

sum119896=1

119904119896 [119863119901minus119896minus1119905 119891 (119905)]119905=0


[119905ℎ]

sum119895=0

120596minus1119895 119891 (0)

minus 1199041ℎminus119901[119905ℎ]

sum119895=0

120596119901minus2119895 119891 (0)

(32)


119866119910 (119904) = 11987011988611 + 119904119905go1198811199031199052go + 119904119901 1 + 119904119905go

1198811199031199052go

+ ℎminus119901[119905ℎ]

sum119895=0

120596119901minus11198952 + 119904119905go1198811199031199053go

+ ℎminus119901[119905ℎ]

sum119895=0

120596119901minus2119895119904 + 1199042119905go1198811199031199053go

(33)


119870119910FOCPN = lim119904rarr0

1199042 11199042119866119910 (119904) 119866 (119904)

= 11198811199031199053go (1198701198861119905go + 21198862)

(34)



119870119910PN = 1198701119905go1198811199031199053go

119870119910PD = (1198701198861119905go + 21198701198862)1198811199031199053go

(35)


119870119910FOCPN gt 119870119910PD gt 119870119910PN (36)


119870119909FOCPN gt 119870119909PD gt 119870119909PN (37)


at +

+minus

L[Dpt

120579L(t)] ka2

UyG(s)

am1 + tgos

Vrt2go

1

s2YL ka1



119878119879119870 = 119889 (ln (119879))119889 (ln (119870)) (38)


1198781198901198701198861 = minus 1198701198861119905go1198701198861119905go + 21198701198861

1198781198901198701198862 = minus 11987011988621198701198861119905go + 21198701198861

(39)



119889119866119890 (119904)119889119866 (119904) = minus 11990421198661 (119904)

[1199042 + 1198661 (119904) 119866 (119904)]2

119889119866 (119904)119889119887119894 = 119904119894

119875 (119904) 119894 = 1 2 119898

119889119866 (119904)119889119886119895 = minus119866 (119904) 119904119895

119875 (119904) 119895 = 1 2 119899

(40)


119878119890119887119894 = 119864 (119904) 119887119894119904119894119875 (119904) 119894 = 1 2 119898

119878119890119886119895 = minus119864 (119904) 119866 (119904) 119886119895119904119895119875 (119904) 119895 = 1 2 119899

(41)


1003816100381610038161003816100381610038161003816100381610038161003816119904=0= minus 119887119894

1198701198861119905go + 21198701 119894 = 1 2 119898

119878119890119886119895119904119895

1003816100381610038161003816100381610038161003816100381610038161003816119904=0= minus 119886119895

1198701198861119905go + 21198701 119895 = 1 2 119899




119878119890119887119894119904119894

1003816100381610038161003816100381610038161003816100381610038161003816PN

119904=0

= minus 1198871198941198701198861119905go 119894 = 1 2 119898

119878119890119886119895119904119895

1003816100381610038161003816100381610038161003816100381610038161003816

PN

119904=0

= minus 1198861198951198701198861119905go 119895 = 1 2 119899

(42)




119879 = 1205962119899(120591119904 + 1) (1199042 + 2120577120596119899119904 + 1205962119899)

(43)






Case 1 PN 12129 3555EPN 06272 3505

Case 2 PN 39193 3334EPN 1978 3289

Case 3 PN 23398 4055EPN 1178 4032

0 05 1 15 20

5000

10000

15000

EPNTargetPN

Target

X (m)

Y (m)

Z (m

)

times104

times104

0

05

1

15

2

25







5 10 15 20 25 30 35 40 450T (s)

20

25

30

35

40

45

50

55

EPNPN

120579 m(∘)


5 10 15 20 25 30 35 40 450T (s)

25

30

35

40

45

50

55

60

65

EPNPN

120595m

(∘)




0

5 10 15 20 25 30 35 40 450T (s)

EPNPN

minus100

minus80

minus60

minus40

minus20

20

40

60

80

100

a my(m

s2)


5 10 15 20 25 30 35 40 450T (s)

0

EPNPN

minus50

minus40

minus30

minus20

minus10

10

20

30a m

z(m

s2)



6 Conclusions





Competing Interests


Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119878119890119887119894119904119894

1003816100381610038161003816100381610038161003816100381610038161003816PN

119904=0

= minus 1198871198941198701198861119905go 119894 = 1 2 119898

119878119890119886119895119904119895

1003816100381610038161003816100381610038161003816100381610038161003816

PN

119904=0

= minus 1198861198951198701198861119905go 119895 = 1 2 119899

(42)




119879 = 1205962119899(120591119904 + 1) (1199042 + 2120577120596119899119904 + 1205962119899)

(43)






Case 1 PN 12129 3555EPN 06272 3505

Case 2 PN 39193 3334EPN 1978 3289

Case 3 PN 23398 4055EPN 1178 4032

0 05 1 15 20

5000

10000

15000

EPNTargetPN

Target

X (m)

Y (m)

Z (m

)

times104

times104

0

05

1

15

2

25







5 10 15 20 25 30 35 40 450T (s)

20

25

30

35

40

45

50

55

EPNPN

120579 m(∘)


5 10 15 20 25 30 35 40 450T (s)

25

30

35

40

45

50

55

60

65

EPNPN

120595m

(∘)




0

5 10 15 20 25 30 35 40 450T (s)

EPNPN

minus100

minus80

minus60

minus40

minus20

20

40

60

80

100

a my(m

s2)


5 10 15 20 25 30 35 40 450T (s)

0

EPNPN

minus50

minus40

minus30

minus20

minus10

10

20

30a m

z(m

s2)



6 Conclusions





Competing Interests


Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










5 10 15 20 25 30 35 40 450T (s)

20

25

30

35

40

45

50

55

EPNPN

120579 m(∘)


5 10 15 20 25 30 35 40 450T (s)

25

30

35

40

45

50

55

60

65

EPNPN

120595m

(∘)




0

5 10 15 20 25 30 35 40 450T (s)

EPNPN

minus100

minus80

minus60

minus40

minus20

20

40

60

80

100

a my(m

s2)


5 10 15 20 25 30 35 40 450T (s)

0

EPNPN

minus50

minus40

minus30

minus20

minus10

10

20

30a m

z(m

s2)



6 Conclusions





Competing Interests


Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Competing Interests


Acknowledgments


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









novel fractional order calculus extended pn for...

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