Chapte 8 The Line of Sight Guidance Law odule 7: Lectue 19 Intoduction; LOS Guidance Keywods. Beam ide Guidance; CLOS Guidance; LOS Guidance As mentioned in the pevious chapte, the Line-of-sight (LOS) guidance law functions on the philosophy that if the missile emains on the line joining the launch platfom and the taget then it eventually must hit the taget. By its vey natue LOS guidance is a thee-point guidance. Basically, it may be implemented in two ways: Beam ide (B): Hee the missile senses its own deviation fom the LOS (defined by an electo-optical beam focussed continuously on the taget) and uses this infomation to geneate guidance commands. This can be thought of as simila to semi-active homing but not pecisely so since the missile does not use the eflected enegy fom the taget, but athe it uses the beam fom the launch platfom to the taget itself. Command-to-Line-of-Sight (CLOS): Hee the ada at the launch platfom tacks both the missile and the taget. The deviation is sensed by the tacking ada and the guidance command is computed and sent to the missile though an uplink. This is 123
124 Guidance of issiles/nptel/2012/d.ghose V T T V L ef Figue 8.1: Engagement geomety fo LOS guidance simila to command guidance. Essentially B and CLOS ae the two diffeent mechanizations of the basic LOS guidance philosophy. In this chapte we will fist study the basic LOS guidance law and then discuss its two mechanizations. 8.1 LOS Guidance In Figue 8.1 we show the engagement geomety of the LOS guidance law. As in ealie cases we estict ou attention to a non-maneuveing taget. We assume that V T, T, and V ae constants. The distance fom the launch platfom to the missile is L= and the distance fom the missile to the taget is T =. In the ideal case, the missile should always be on the LOS (given by LT) between the launch platfom and the taget. Note that in LOS guidance the LOS efes to the LOS between points L and T, unlike in pusuit guidance analysis whee the LOS was between the missile and the taget. We have the following equations of motion, V =ṙ = V cos( ) (8.1) V = Ṙ = V T cos( T ) V cos( ) (8.2) V = = V sin( ) (8.3) V = = V T sin( T ) V sin( ) (8.4)
Guidance of issiles/nptel/2012/d.ghose 125 Note that in this case V is the ate of change of the LOS sepaation between L and, and V is the ate of change of the LOS sepaation between and T. Similaly, V is the angula ate of the LOS between L and, and V is the angula ate of the LOS between and T. Now, fom (8.3) and (8.4) we can wite, = V sin( ) = V T sin( T ) V sin( ) = V T sin( T ) (8.5) ()V sin( ) =V T sin( T ) (8.6) Diffeentiating (8.6) with espect to time, using the fact that = a /V, and substituting (8.1)-(8.4) appopiately, we obtain the following sequence of equations, (Ṙ +ṙ)v sin( )+()V cos( )( ) =ṙv T sin( T )+V T cos( T )( ) V T cos( T )V sin( )+()a cos( ) V T sin( T )V cos( ) = V T sin( T )V cos( ) V T cos( T )V sin( ) ()a cos( ) =2V T V sin( T ) a = 2V T V sin( T ) () cos( ) (8.7) This gives an expession fo the missile acceleation a equied in an ideal implementation of the LOS guidance law in ode to keep the missile on the LOS between the launch platfom L and the taget T. Howeve, the expession fo a given in (8.7) is a function of seveal time-vaying quantities:,,, and. Of these, it is possible to expess and () as functions of time as follows: Conside Figue 8.1. Fom the geomety, since the taget moves in a staight line, we can see that, tan = ( 0 + 0 ) sin 0 + V T t sin T ( 0 + 0 ) cos 0 + V T t cos T = tan 1 ( 0 + 0 ) sin 0 + V T t sin T (8.8) ( 0 + 0 ) cos 0 + V T t cos T = ( 0 + 0 ) 2 +(V T t) 2 +2( 0 + 0 )V T t cos( 0 T ) (8.9) Whee, the subscipt 0 denotes the initial values which ae known. This gives an expession fo the LOS angle and as functions of time.
126 Guidance of issiles/nptel/2012/d.ghose V T V T a a V T L ef Figue 8.2: Anothe engagement geomety fo LOS guidance Now, fom (8.6), we have, ()V sin( ) =V T sin( T ) sin( ) = V T sin( T ) ()V So, can be finally expessed as a function of t and the distance of the missile fom the launch platfom. These expessions can be used to geneate the missile latax fo simulating an LOS guided missile taget engagement fo a non-maneuveing taget. An altenative way to obtain the missile acceleation, that is usually given in most books is also given below. Conside the Figue 8.2. In this figue we conside the missile to be a body moving along a otating line (actually the LOS between the launch platfom L and the taget T that otates at ate ) with speed V. Then the acceleation expeienced by nomal to the otating line is denoted by a and is given by, a =2V + = d dt ( )+ṙ (8.10) Note that the expession fo a is obtained by adding the Coiolis acceleation ṙ to the ate of change of the angula velocity obtained fom d/dt( ). Now, the actual missile velocity and acceleations ae as as shown in Figue 8.2.
Guidance of issiles/nptel/2012/d.ghose 127 They ae elated to V and a as, V = V cos( ) a = a cos( ) (8.11) Substituting in (8.10), we get a cos( ) = 2V cos( ) + a = 2V + cos( ) (8.12) Fom Figue 8.2, = V T sin( T ) (8.13) which on eaangement and diffeentiation yields, d dt ()+() = V T cos( T ) (8.14) Substituting and (8.13), we get d dt () =V T cos( T ) (8.15) = 2V T 2 sin( T ) cos( T ) () 2 (8.16) Substituting these expessions fo and in (8.12), we get a = 2V V T sin( T ) Substituting fom 2V T 2 sin( T ) cos( T ) cos( )() 2 (8.17) = V sin( ) = V T sin( T ) (8.18) we get a = 2V V T sin( T ) 2V sin( )V T cos( T ) cos( )() 2V V T () cos( ) [sin( T ) cos( ) sin( ) cos( T )] 2V V T sin( T ) (8.19) () cos( ) which is the same as (8.7) deived ealie.