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SIMULATIONS FOR BASIC PROPORTIONAL NAVIGATION Introduction 1. The proportional guidance law derived earlier is tested for its properties by using Matlab and SIMULINK. This will help in understanding the effectiveness of proportional navigation. The SIMULINK blocks used for the simulation help in directly applying the Runge-Kutta fourth order numerical integration technique for solving the differential equations derived earlier. A fixed step size of 0.001 is selected. The inputs to the simulation are target and missile speeds, effective navigation ratio, initial location of the target and missile, initial missile acceleration, flight path angle, target manoeuvre parameter and the heading error. By varying the navigation ratio, target manoeuvre parameter, flight path angle and the heading error the properties of proportional navigation in terms of missile-target engagement (down-range vs cross-range/altitude) and missile acceleration were studied. Simulation Data Listing 2.
Matlab Program Listing for initial conditions and input data.
%%------------------------------------------------------------------------%SIMULATION OF PROPORTIONAL NAVIGATION GIVEN IN BOOK BY P.ZARCHAN %-------------------------------------------------------------------------global RT1 RT2 VT NT Beta VT1 VT2 RM1 RM2 RTM1 RTM2 Lambda L VM1 VM2 VTM1 VTM2 Vc Nc aM1 aM2 K HE %-----INITIAL CONDITIONS------Beta0=0;%for head-on-fire target %Beta0=pi%for target moving away or tail fire target RT10=4000;RT20=5000;RM10=2;RM20=1;aM10=0;aM20=0; HE=-20*pi/180;%Heading Error VT=250;VM=500; NT=0;%For non-maneuvring target(3*9.6 for target with 3g maneuvre) K=3;%navigation ratio RTM10=RT10-RM10;RTM20=RT20-RM20; Lambda0=atan(RTM20/RTM10); L0=asin(VT*sin(Beta0+Lambda0)/VM); VM10=VM*cos(L0+HE+Lambda0);VM20=VM*sin(L0+HE+Lambda0); %------Simulation parameters----tstart=0;tstop=9.9;tstep=0.001; sim('PNJan10');figure; set(gcf,'DefaultLineLineWidth',2); set(gca,'FontSize',16,'FontWeight','bold'); plot(RT1,RT2,'r',RM1,RM2,'b') title('TARGET MOVING HEAD ON WITH 0g MANEUVRE K=3 HE=-20deg');%figure xlabel('RT1,RM1(m)','FontSize',10,'FontWeight','bold');ylabel('RT2,RM2 (m)','FontSize',10,'FontWeight','bold'); legend('TARGET','MISSILE');figure; set(gcf,'DefaultLineLineWidth',2); set(gca,'FontSize',16,'FontWeight','bold'); plot(t,aM1) title('MISSILE ACCELERATION FOR TARGET MOVING HEAD ON 0g MANEUVRE K=3 HE=20deg'); xlabel('time(sec)','FontSize',10,'FontWeight','bold');ylabel('MISSILE ACCELERATION n_c (g)','FontSize',10,'FontWeight','bold');
3.
Matlab Function Program Listing
%MATLAB FUNCTION FOR PROPORTIONAL NAVIGATION CALCULATIONS function output = PN_con1(u); global RT1 RT2 VT NT Beta VT1 VT2 RM1 RM2 RTM1 RTM2 Lambda L VM1 VM2 VTM1 VTM2 Vc Nc aM1 aM2 K Beta=u(1);RT1=u(2);RT2=u(3);RTM1=u(4);RTM2=u(5);Lambda=u(6); VM1=u(7);VM2=u(8);RM1=u(9);RM2=u(10);%VTM1=u(11);VTM2=u(12); Betadot=NT/VT; VT1=-VT*cos(Beta);VT2=VT*sin(Beta); RT1dot=VT1; RT2dot=VT2; % L=asin(VT*sin(Beta+Lambda)/VM); % VM1=VM*cos(L+HE+Lambda);VM2=VM*sin(L+HE+Lambda); VTM1=VT1-VM1;VTM2=VT2-VM2; RTM1dot=VTM1; RTM2dot=VTM2; RTM=sqrt(RTM1^2+RTM2^2); Vc=-(RTM1*VTM1+RTM2*VTM2)/RTM; Lambdadot=(RTM1*VTM2-RTM2*VTM1)/(RTM^2); Nc=K*Vc*Lambdadot; Nc1=Nc/9.8; aM1=-Nc*sin(Lambda);aM2=Nc*cos(Lambda); VM1dot=aM1; VM2dot=aM2; RM1dot=VM1; RM2dot=VM2; output=[Betadot RT1dot RT2dot RTM1dot RTM2dot Lambdadot VM1dot VM2dot RM1dot RM2dot Nc1 aM2];
4. Engagement Geometry for a non-manoeuvring target in Head-on-Fire, Effective Navigation Ratio=3 with zero heading error. TARGET MOVING HEAD ON WITH 0g MANEUVRE K=3 6000 TARGET MISSILE 5000
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5. Missile Acceleration Vs Time Plot for a non-manoeuvring target in Head-on-Fire, Effective Navigation Ratio=3 with zero heading error.
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6. Engagement Geometry for a non-manoeuvring target in Head-on-Fire, Effective Navigation Ratio=3 with heading error of -20 degrees.
TARGET MOVING HEAD ON WITH 0g MANEUVRE K=3 HE=-20deg 6000 TARGET MISSILE 5000
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7. Missile Acceleration Vs Time Plot for a non-manoeuvring target in Head-on-Fire, Effective Navigation Ratio=3 with heading error -20 degrees. It can be noted that when there is a heading error, the missile acceleration is maximum initially and reduces gradually to become zero at the time of intercept.
MISSILE ACCELERATION FOR TARGET MOVING HEAD ON 0g MANEUVRE K=3 HE=-20deg 6 5
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8. Engagement Geometry for a non-manoeuvring target in Head-on-Fire, Effective Navigation Ratio=3 with heading error of +20 degrees.
TARGET MOVING HEAD ON WITH 0g MANEUVRE K=3 HE=20deg 6000 TARGET MISSILE 5000
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9. Missile Acceleration Vs Time Plot for a non-manoeuvring target in Head-on-Fire, Effective Navigation Ratio=3 with heading error +20 degrees. It can be noted that when there is a heading error, the missile acceleration is maximum initially and reduces gradually to become zero at the time of intercept.
MISSILE ACCELERATION FOR TARGET MOVING HEAD ON 0g MANEUVRE K=3 HE=20deg 0.5 0
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10. Engagement Geometry for a target manoeuvring with 1-g in Tail-Fire, Effective Navigation Ratio=4 with zero heading error. TARGET MOVING AWAY WITH 1g MANEUVRE 5000 TARGET MISSILE
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10. Engagement Geometry for a target manoeuvring with 1-g in Tail-Fire, Effective Navigation Ratio=3 with zero heading error. TARGET MOVING AWAY WITH 1g MANEUVRE K=3 5000 TARGET MISSILE
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11. Missile Acceleration Vs Time Plots for a target manoeuvring with 1-g in Tail-Fire, Effective Navigation Ratio=3 and 4 with zero heading error. It can be seen from the two plots that the maximum acceleration attained during the flight time decreases when the effective navigation ratio=4 as compared to that when effective navigation ratio=3. MISSILE ACCELERATION FOR TARGET MOVING AWAY WITH 1g MANEUVRE K=3 0.1 0 -0.1
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MISSILE ACCELERATION FOR TARGET MOVING AWAY WITH 1g MANEUVRE K=4 0.1 0
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12. Engagement Geometry for a target manoeuvring with 3-g in Head-on-Fire, Effective Navigation Ratio=3 with zero heading error.
TARGET MOVING HEAD ON WITH 3g MANEUVRE K=3 8000 TARGET MISSILE
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13. Missile Acceleration Vs Time Plots for a target manoeuvring with 3-g in Head-on-Fire, Effective Navigation Ratio=3 with zero heading error. It can be noted that the maximum missile acceleration during the entire flight time is around 7-g which is greater than the target acceleration of 3-g.
MISSILE ACCELERATION FOR TARGET MOVING HEAD ON WITH 3g MANEUVRE K=3 2 1 0
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14. Engagement Geometry for a target manoeuvring with 3-g in Tail-Fire, Effective Navigation Ratio=3 with zero heading error. TARGET MOVING AWAY WITH 3g MANEUVRE K=3 5000 TARGET MISSILE
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15. Missile Acceleration Vs Time Plots for a target manoeuvring with 3-g in Tail-Fire, Effective Navigation Ratio=3 with zero heading error. It can be noted that the maximum missile acceleration during the entire flight time is around 7-g which is greater than the target acceleration of 3-g.
MISSILE ACCELERATION FOR TARGET MOVING AWAY WITH 3g MANEUVRE K=3 7 6
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Conclusion
Simulations show that proportional navigation is highly effective even for targets manoeuvring with accelerations as high as 3-g. Increase in effective navigation ratio is seen to reduce the maximum missile acceleration attained during the entire flight time.