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tan� (8)

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0BB@ g sinY � TmV + D0

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%OvwW|tA = 1

V 2 g cos

B = L� (D0 � T +mg sin )

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C = mV 3 (2� + T=L�)B(T=L� + 1)2 L�

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�sec�dR (12)

|R=Uxv}tm O}k w |}x]kv sQH ZQi =@ xr}Uwl} |=Q@ |rm |x]@=Ql} 12 |x]@=Q|R=Uxv}tm O}k =@ xv}y@ pQDvm |xr�Ut pL 'wQ V}B Vr=J uwvm = "CU= xv} Ry `@=Du=mt |xR=@ w 8 w 7 \@=wQ R= xO=iDU= =@ "CU= |Owta |xLiY QO CaQU|yOpmW ?} Q[ u=wD|t 'uwDr}ty |x]@=Q p}mWD w 12 |x]@=Q QO QiY =D Rp=ta= u}@t `k=w QO G |yOpmW ?} Q[ "OQm |iQat 13 |x]@=Q =@ =Q G2 Q}UtCU=B=m C}=Oy uwv=k QO xar=]t OQwt |xr}Uw |UOvy w |m}t=v}OwQ}; C=YNWt

u=}@ Q} R CQwYx@ u=wD|t =Q xOW|]N |=B=m C}=Oy uwv=k |rm Cr=L QO[11]%OQm

am = K1tg0

(v1 � v)� K2tg0

[r1 � (r � vtg0)] (1)

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[11]"OvDUy Q=PoQ}F-=D C}=Oy uwv=k \UwD xOW|L=Q] Q}Ut

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[13]%CU= xOW u=}@ 2 |x]@=Q CQwYx@ |]NQ}ea = K1

RV 2 sin � cos� � K2

RV 2 sin� (2)

xORu}tND u=tR tg '|}x_Lr CaQU V xm CU= RQ=sy V ts cos� |x]@=Q =@ R"CU= OYkt w |}x_Lr u=mt pY=w Q=OQ@ x@ C@Uv CtU � w 'OYkt =D xOW

%OwW|t u=}@ k =@ xv}y@ Q}Ut |=vLv=

k = 1Vd dt

= 1V 2 a (3)

xv}y@ Q}Ut |=vLv= '3 w 2 |=yx]@=Q R= xO=iDU= =@ "CU= R=wQB Q}Ut |x} w=R xm%O@=}|t Q}}eD Q} R CQwYx@

k = K1RV 2 sin � cos� � K2

RV 2 sin� (4)

hOy =@ w xv}y@ pQDvm |xr�Ut l} CQwYx@ xr�Ut '=B=m C}=Oy |R=U|]N QO%CU= xOW |Ov@pwtQi Q} R OQmrta Q=}at |R=Uxv}tm

J = �V (If ) (5)

%OwW|t ZQi 6 |x]@=Q CQwYx@ R}v |RQt \}=QW�(tf ) = 0 �(tf ) = 0 (6)

Q@ 1 pmW j@] Q@ xv}y@ pQDvm |x} Q_v '|r}rLD CQwYx@ 5 |x]@=Q pL Qw_vtx@w R \@=wQ "CU= O}O \N |x} w=R � w 'R=wQB |x} w=R '� CtU |x} w=R ?UL

%OwW|t u}}aD Q} R CQwYx@ �R = �V cos� � = V sin�

R

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�x}v=F/QDt� CaQU �QDt� u=mt[5 0 � 2] [0 0 1000] x}rw= \}=QW

[5 5 0] [2000 2000 0] |}=yv \}=QW"xvwtv p}wi=Q=B C=YNWt "2 pwOH

Q=Okt QDt=Q=B Q=Okt QDt=Q=B0 25 CD0 0 9 C�

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p} wi=Q=Bl} xm |R=wQB |xr}Uw |UOvy w |m}t=v}OwQ}; C=YNWt 2 pwOH QO"CU= xOW xOQw; 'CU= xvwtv

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"OyO|t V}=tv xyz |xir-wt xU |=Q@ =Q Q}Ut \=kv |xJN} Q=DQ_vOQwt |}=wy |xv=t=U x@ xOWOQ=w |=y?=DW |v=tR |xJN} Q=D 5 pmW QOpt=W xm Q}Ut |}=yDv= CtUk RHx@ xm OwW|t xOy=Wt "CU= xOW xO=O V}=tv|O=tv Ov=wD|t xm CU= |D=Q}}eD |=Q=O ?=DW 'Q}Ut |x}k@ QO 'CU= OwQi C=_LrxOWp=ta= ?=DW C=Q}}eD |xawtHt x@ |rQDvm VqD u}= xm OW=@ |rQDvm VqD R=C=t=Rr= xOvQB |xr}Uw \UwD u; |R=UxOQw;Q@ w OwW|t jq]= C}=Oy uwv=k \UwD6 pmW QO R}v CaQU |=yxir-wt |v=tR |xJN} Q=D "CW=O Oy=wN =Q OwN X=N

"CU= xOW xO=O u=Wv|R=Uxv}tm|=Q@ xDi=}s}taD |Q@ w=v?}=Q[R= xO=iDU= =@ =B=mC}=Oy x@\w@ QtG}=Dvu}vJty "CU= xOW x�=Q= 11 =D 7 |=ypmW QO |Owta |xLiY QO |}=yv CaQU

(V ) CaQU 'OwW|t |v=UQ RwQx@ x_Lr Qy QO xm u; QO Q}eDt |xir-wt =yvD xmQ[=L VywSB QO xOWx�=Q= Q=my=Q |rY= R}=tD xHw 'xYqN Qw]x@ [25w13�11]"CU=QO Cw=iD R= |W=v xm CU= G |yOpmW ?} Q[ Q=DN=U QO [25]Qo}O `H=Qt =@

%CU= |R=wQB C} Qwt-=t C=t=Rr= w xv} Ry `@=D h} QaD

G2 = (D0 + 2mg sin )L�2m2V 4� +A2 (13)

%OwW|t xO=iDU= Q} R |xv}y@ pQDvm |xrO=at QO xmR dkdR

+G2R sin� � k + c = 0 (14)

Vwt=N QwDwt CQwYx@ |DWoR=@ |xrwtLt 'OYkt =D QDt Q=Ry R= QDW}@ |xrY=i QO[25]"CU= xOW u=}@ 15 |x]@=Q CQwYx@ xm xOWZQi RwQm \}=QW QO w

G2 = D0L�2m2V 4� (15)

A2 Q=Okt QO pw= Cw=iD "CU= xrtH wO QO 15 w 13 \@=wQ u}@ OwHwt Cw=iD2mg sin Q=Okt 13 |x]@=Q QO xm CU=UB |wQ}v |xrtH QO swO Cw=iD "CU=

"CU= xOW xi=[=[25]%OvwW|t u=}@ Q} R CQwYx@ |Q@ w=v ?}=Q[ '14 |xrO=at pL =@

K1 =2G2R2 +GR

�eGR � e�GR�

eGR (GR� 2)� e�GR (GR+ 2) + 4 (16)

K2 =G2R2 �eGR + e�GR � 2�

eGR (GR� 2)� e�GR (GR+ 2) + 4 (17)

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|OQwt xar=]t w G}=Dv "4u}vJty w |SQv= hqD= w ?=DW C=Q}}eD MQv u} QDtm =@ Q}Utl} |L=Q] xm OW xQ=W=|xr�Ut |OOa =} |r}rLD |UQQ@ R= xm CU= |D}=Oy |Q=my=Q OvtR=}v q=@ CkO|rN=O w |HQ=N Ow}k x=Qtyx@ |RQt \}=QW O}=@ w OW=@ xOW pY=L Q_v OQwt|xv=t=U OwQi u=mt= 'xr=kt u}= QO xOWx�=Q= C}=Oy Q=my=Q "OvwW =[Q= R}v xr�UtC}=Oy x@ \w@ Qt G}=Dv "CU= xOQm sy=Qi OQwNQ@ CaQU u} QDtm =@ =Q xvwtv p=aiQ}eQO =@ |Q@ w=v |xv}y@ ?}=Q[ �? '|Q@ w=v C@=F ?}=Q[ �hr= %Cr=L wO QO =B=mCQ=@ax@ "CU= xOW x�=Q= |Owta |xLiY QO |}=yv CaQU |xv}tm uDiQo Q_vK2 w K1 |Q@ w=v ?}=Q[ =@� =B=m C}=Oy G}=Dv =@ xDiQos=Hv= s}taD G}=Dv 'Qo}O"CU= xOt; xv=t=U |R=wQB Q}Ut |RQt \}=QW 1 pwOH QO "CU= xOW xU}=kt �C@=Fx@ xHwD =@ xv=t=U |OQmrta Ow}k "CU= pwRv CaQU uOv=UQ pk=OL x@ 'hOyCQ=@a xm CU= xOW p=ta= ptLD p@=k ?=DW |wQ Q@ 'xOWs=Hv= |=y|R=Ux}@W

"amax = [3 3 0 5]% R= CU=

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"xDi=}s}taD |Q@w=v ?}=Q[ =@ =B=m C}=Oy uwv=k |Oa@ xU Q}Ut "7 pmW

|Q@w=v ?}=Q[ =@ =B=m C}=Oy uwv=k R= xO=iDU= =@ xv=t=U u=mt Q=Owtv "8 pmW"xDi=}s}taD

xOWOQ=w ?=DW |v=tR |xJN} Q=D Q=Owtv OwW|t xOy=Wt 9 pmW QO xmv=vJw C@=F ,=@} QkD CQwYx@ Q}Ut |x}k@ QO '|}=yDv= w |}=OD@= |xR=@ QO RHx@ xv=t=U x@

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"CU= xOW xO=O u=Wv xDi=}s}taD |Q@ w=v ?}=Q[

w K1 = �2 C@=F |Q@ w=v ?}=Q[ =@ =B=m C}=Oy uwv=k |Oa@ xU Q}Ut "3 pmW"K2 = 6

C@=F |Q@ w=v ?}=Q[ =@ =B=m C}=Oy uwv=k R= xO=iDU= =@ xv=t=U u=mt Q=Owtv "4 pmW"K2 = 6 w K1 = �2

?}=Q[ =@ =B=m C}=Oy uwv=k \UwD xOWx@U=Lt R=}v OQwt ?=DW Q=Owtv "5 pmW"K2 = 6 w K1 = �2 C@=F |Q@w=v

|=yQwLt QO u}at pLt x@ |DWoR=@ |xv=t=U |=Q@ ptLD p@=k ?=DW |xv}W}@ O}kxO=iDU= =@ xOWO}rwD Q}Ut '7 pmW QO "CU= xOW p=ta= C}=Oy uwv=k Q@ 'hrDNt|=vLv= xm OwW|t xOy=Wt "CU= xOW xO=O u=Wv xDi=}s}taD |=B=m C}=Oy uwv=k R=CmQL CtU 'Q}Ut |=OD@= u=ty QO xv=t=U w xDi=} Vy=m |UwULt OL QO Q}Ut'8 pmW Q=Owtv "OW=@v QDW}@ C=Q}}eD x@ R=}v xm CU= xOQm u}}aD |}xvwo x@ =Q OwN

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&(Sec) |}x_Lr u=tR %t&(Sec) |}=yv u=tR %t1

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=yCWwv=B1. neighboring optimized control2. singular perturbations3. shaping trajectory4. virtual sliding target

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=B=mC}=

OyQ=my

=Qs}taD21. Prabhakra, N., Chandra Sekhar, K. and Vaidianathan,

V. \Target hit interceptor mid-course guidance schemefor ballistic missile interception", International Jour-nal of Applied Research in Mechanical Engineering(IJARME), 3(2), pp. 7-11 (2013).

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23. Tahk, M.-J. \A tutorial on linear quadratic optimal guid-

ance for missile applications", J. KSIAM, 19(3), pp. 217-234 (2015).

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82