Interception of Ballistic Targets 

by Andrew A. Thompson 

ARL-MR-740 April 2010 

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Army Research Laboratory 

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ARL-MR-740 April 2010 

Interception of Ballistic Targets 

Andrew A. Thompson 

Weapons and Materials Research Directorate, ARL 

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Interception of Ballistic Targets 

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Andrew A. Thompson 

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14. ABSTRACT 

Intercepting an incoming artillery or mortar round is receiving interest as an area-protection mechanism. A problem has been 

that estimation, based on 6-DOF equations for both the incoming round and the interceptor round or missile, is too time 

consuming from a computational perspective. This report presents a basic simulation in MATLAB that is used to find the 

maneuver control and the aimpoint or lead angle needed to intercept an incoming round. The method uses reduced dynamics 

flight models for the interceptor and the incoming round. Also, questions relating to the probability of a hit given a specific 

standard deviation and bias are addressed. 

15. SUBJECT TERMS

ballistic trajectory, offset probability, interceptor 

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Andrew A. Thompson 

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iii

Contents 

List of Figures iv

1. Introduction 1

2. Simulation 1

3. Probability of Hit 4

4. Extension of Problem 6

5. Conclusions 9 

Distribution List 10 
iv

List of Figures 

Figure 1. Target – interceptor diagram. ..........................................................................................2

Figure 2. Graphic realization of the simulation. .............................................................................3

Figure 3. Rayleigh inverse cumulative distribution command. ......................................................4

Figure 4. Rayleigh CDF command. ................................................................................................5

Figure 5. Rectangular target probability calculations. ....................................................................6

Figure 6. Commands to setup calculation of a Riemann sum. ........................................................8

Figure 7. MATLAB calculation of upper and lower probabilities. ................................................8
1

1. Introduction 

Intercepting an incoming artillery or mortar round is receiving interest as an area-protection 

mechanism. A problem has been that estimation, based on 6-DOF equations for both the 

incoming round and the interceptor round or missile, is too time consuming from a 

computational perspective. This report presents a basic simulation in MATLAB that is used to 

find the maneuver control and the aimpoint or lead angle needed to intercept an incoming round. 

The method uses reduced dynamics flight models for the interceptor and the incoming round. 

Also, questions relating to the probability of a hit given a specific standard deviation and bias are 

addressed. 

A trajectory can be adequately modeled by a differential equation. Using the initial conditions, 

an expected projectile path can be generated. Range and velocity measurements can be used to 

improve the estimate of the projectile’s path. The differential equation and the measurement 

process are combined to form a Kalman filter. The differential equation models the physics of 

the trajectory while the measurement updates the parameters of the equation based on least 

squares estimation. The tradeoffs and dynamics are discussed by Thompson.1

2. Simulation 

A simulation was devised to indicate the amount of maneuver authority needed to hit an 

incoming target. The incoming round was modeled using the point mass equations and the 

interceptor round was modeled with a single dimension differential equation; see Thompson1

 for 

a discussion. Using these equations, it was possible to compute the lead angle needed for the 

gun interceptor round. This calculation is based on estimated time of arrival of the interceptor at 

predicted points of the target trajectory. Figure 1 attempts to illustrate the ideas associated with 

aimpoint selection. The location of the interceptor is at (5,0). At a given time, the blue line 

represents the estimated target trajectory moving from the left towards the right. The red points 

signify points along the trajectory that have the time of arrival of the target also calculated. The 

green lines represent the interceptor motion to a specific trajectory point. The expected time of 

the interceptor’s arrival at each point can be calculated. At each point, the predicted time until 

arrival of the target and the interceptor are differenced. An aimpoint/lead angle can be selected 

for any positive difference by adjusting the firing time of the interceptor so that the time 

difference is zero. It is also possible to search the interval that goes from negative to positive for 

 1

Thompson, A. A. Ballistic Filtering; ARL-TR-4735; U.S. Army Research Laboratory: Aberdeen Proving Ground, MD, 

March 2009. 
2

Figure 1. Target-interceptor diagram. 

the zero delay aimpoint. The terminal geometry could also be considered in selecting an 

aimpoint. Post launch adjustments are made by selecting a different predicted target trajectory 

point for the interceptor to hit. After the interceptor is fired, it could only adjust its course by 

one degree at a time; thus, corrections could only take place if there was over one degree of error 

in the predicted impact point. 

The accuracy of the target track would have to meet certain criteria before an interceptor could 

be fired. Meeting this accuracy requirement would ensure that the target would be within the 

interceptor’s maneuver space. Also, in-flight corrections would only be made if the correction 

statistically exceeds a noise threshold. This latter threshold would keep the interceptor from 

cycling or correcting and recorrecting the intercept trajectory. 

In order to introduce uncertainty in the model, the interceptor speed was changed part way 

through the flight. In these cases, the initial speed perturbation was followed by several thruster 

firings. After these firings, the interceptor stayed on an unaltered course until it got near the 

target. Close to the target several “fine tuning” thruster firings were made. Figure 2 shows a 

realization of the simulation. In this simulation, the speed of the interceptor was reduced by 

30%. The red curve is the target trajectory, the figure shows the entire trajectory as if the target 

was not intercepted; the blue line is the path of the interceptor; and the green points are the 

-3 -2 -1 0 1 2 3 4 5 6

0

5

10

15

20

25

30

35

40

45

Diagram for target intercept estimation
3

Figure 2. Graphic realization of the simulation. 

locations of thruster firings. If one visualizes the predicted target trajectory, increases in the 

interceptor speed cause the interception point to move back along the trajectory while decreases 

move the desired aimpoint further forward along the target trajectory. It is obvious that the 

accuracy of the estimate of the target’s trajectory is of basic importance for a system to be 

successful. Likewise the estimate of the interceptor’s trajectory is of equal concern. Thompson1

discusses these issues. As time increases, the accuracy of both the estimate of the target 

trajectory and the interceptor trajectory will increase. Many times, polynomial filters are 

discussed as candidates for trajectory estimation; these filters do not attempt to capture the 

dynamics of the situation and for longer prediction times are inadequate. Although filters based 

on 6-DOF models are conceivable, they impose a computational load that cannot be met at this 

time. Extended Kalman filters using the modified point mass equations present a technique that 

can be used for this problem. Particle filters also offer a potential method to increase trajectory 

estimation accuracy over that obtained through recursive least squares polynomial estimation. 

4

6

0

200

400

600

800

1000
4

3. Probability of Hit 

In this section, the statistical issues associated with the probability of hitting a target are 

discussed using MATLAB as the computational language. There are two families of functions 

of interest. First, cumulative distribution functions (CDF) give the probability of an event 

occurring up to a specific input value; since probability can not be negative this function can 

never decrease as the input value increases. It is often the case that the desired information is to 

find the value of a distribution associated with a specified cumulative probability. In this latter 

case, the inverse cumulative distribution function is used. 

The root mean square of the target and interceptor estimator errors will result in the overall 

system error. First, consider a plane perpendicular to the interceptor flight centered on the target, 

in this plane the position the interceptor strikes is of interest. Consider the target being a circle 

centered at (0, 0); the probability that the interceptor strikes within this circle is of interest. The 

first situation to consider is with equal normal errors in both dimensions (X, Y). This question is 

approached using either a Chi-Square distribution with two degrees of freedom or a Rayleigh 

distribution. The unit of measure will be a standard deviation unit (sd). 

The inverse cumulative distribution function is used to find the values of a distribution associated 

with specific probabilities. The following MATLAB command is used to find the target radius 

in standard deviation units for hit probabilities of 0.1 to 0.9 in increments of 0.1 for a Rayleigh 

distribution. Figure 3 shows the MATLAB syntax and the command output. 

Figure 3. Rayleigh inverse cumulative distribution command. 
5

The following comments show how to interpret the above result. A target with a radius of 

0.4590 standard deviation units will have a 0.1 chance of being hit by one round. A target with a 

radius of 1.1774 standard deviation units will have a 0.5 chance of being hit; this is the value 

associated with the circular error probable (CEP) for equal errors. Another example would be 

for standard deviation of 0.1 the target size would have to be 0.21460 to have a probability of 0.9 

of being hit. 

For a volley of rounds, the probability of missing the target is used to find the probability of at 

least one round hitting the target. For example, if the standard deviation is 0.2 m and the target 

radius is 0.2 (0.459) = 0.0918 m, there is a probability of 0.9 of missing the target. A volley of 

six rounds will have a probability of 0.96 or 0.53 of missing the target six times; thus, there is a 

probability of 0.47 of hitting the target. 

The cumulative distribution function returns the probability of an event up to the value given. 

The next MATLAB command used calculates the cumulative probability of the Rayleigh 

distribution for targets of radius of 0.2–3 standard deviation units in increments of 0.2 standard 

deviation units. Figure 4 shows the command and the output generated by the command. The 

first entry column 1 gives the probability of hitting a target of 0.2 standard deviation units. 

Figure 4. Rayleigh CDF command. 

The final entry, column fifteen could indicate that if the standard deviation was 0.1 m there is a 

probability of 0.9889 of hitting a 0.3-m target. 

These ideas can be used to get an idea of the accuracy required to hit a target of a given size. 

These numbers can be used to generate requirements of outputs of filtering routines and then as 

requirements for radar data. The issues here are the fidelity of the filter and its prediction ability 

and then the quality of the data put into the filter. Filters that do a good job of capturing the 
6

dynamics of the situation typically yield lower estimation and prediction errors. Other filters may 

do well at smoothing but may suffer in prediction. The baseline candidate for prediction and 

estimation is an extended Kalman filter based on the dynamics of a point mass model. Other 

models should be compared to these estimators. 

This problem can be extended to discuss rectangular targets and targets that are offset from the 

center (bias); also, the case of different horizontal and vertical errors can be discussed. 

4. Extension of Problem 

Next, the problem of offset targets will be explored. In this situation, the target is not at the 

aimpoint this situation can be created by timing errors. Assume the aimpoint is at (0, 0) and the 

target is at an offset along the x axis. First, rectangular targets will be discussed and then 

approximations will be made for circular targets. We will assume the standard deviation of the 

aim error is 1. If this is not the case, just divide all distances by the standard deviation. 

Consider a 2  2 rectangle or square target centered on (1, 0). What is the probability of hitting 

this target when aiming at (0, 0)? The x-axis goes from 0 to 2 and the y-axis goes from –1 to 1. 

By integrating the normal probability distribution between these limits, a solution is obtained via 

MATLAB commands. Figure 5 shows the execution of the appropriate commands. 

Figure 5. Rectangular target probability calculations.
7

So, the probability of hitting a square 2 sd square target at an offset of 1 sd unit is 0.3258. If a 

circular target is desired instead, this result can be multiplied to get an approximation. If the 

multiplication factor is the ratio of the areas of the inscribed circle to the square, the factor is 

pi/4. A comparison of the formulas for a rectangle and an ellipse results in the same correction 

factor. 

 0.3258*pi/4 = 0.2559. (1) 

So, there is about a 26% chance of hitting a circular target. Consulting DARCOM P 706-101 

chapter 14 table 14-2,2

 a precise estimate is found that is 0.2671. The discussion in the 

DARCOM pamphlet is based on references 3 and 4. The given estimate is precise enough for 

many applications. The absolute error is 0.01 while the relative error is low at about 4%. 

 (0.2671–0.2559)/0.2671 = 0.0420. (2) 

If this value is not of sufficient precision, the problem can be solved by using many small 

rectangles that approximate the target. The Riemann sum can be used to increase the accuracy of 

the estimate. Using the Riemann sum, a lower and upper probability will be calculated so that 

the true probability is contained in the resulting closed interval. The Riemann sum is discussed 

in most elementary calculus books. 

Figure 6 shows the preliminary commands used to create the variables needed to calculate the 

upper and lower probabilities. The first command defines the x values (0 to 2 in steps of 0.01), 

the second creates the vector of y values that go with the x values (assuming a circle). The offset 

is represented by x-1. The third line displays the area of interest; this is just the half circle above 

the x-axis. Next the values of the cumulative distribution are found for each of the x values; and 

then likewise those for the y values. The y values used are between the x-axis and the calculated 

y value; thus the calculation will be for a half circle and the result multiplied by two. The final 

command above finds the probability between consecutive x-values, or the probability associated 

with the different x intervals. The final steps of the procedure follow. For this problem, we can 

get an upper and lower Riemann sum as shown in figure 7. 

Thus the true probability is between 0.2654 and 0.2686. The average of these is 0.267. As 

previously mentioned the true value was 0.2671; error is 0.001. This leads to relative errors of 

 (0.2671-0.2654)/0.2671 = 0.0063. (3) 

The relative error is an order of magnitude less than that of the previous method, similarly 

 (0.2686–0.2671)/0.2671 = 0.0056. (4) 

 2

DARCOM P 706-101. Chapter 14, table 14-2; U.S. Army Materiel Development and Readiness Command: Alexandria, 

VA, November 1977. 
8

Figure 6. Commands to setup calculation of a Riemann sum. 

Figure 7. Matlab calculation of upper and lower probabilities. 

If the step size is decreased to 0.001, then the corresponding calculations yield a lower sum of 

0.2670 and an upper sum of 0.2672 and the error interval is only 0.002. Thus, the procedure 

given can achieve high accuracy based on the divisions of the x intervals used, or DARCOM P 

706-101 can be used. If the divisions of x become extremely small, it is possible that numeric 

problems could arise; thus, one should gradually shrink the size of the x intervals. 
9

5. Conclusions 

Two issues associated with defeating incoming projectiles have been discussed. The aimpoint 

algorithm is based on timing information associated with the estimators of the target and the 

interceptor trajectories. Errors in the timing will lead to errors in impact. The simulations 

indicated two areas of maneuver: first, when errors associated with the trajectories are observed 

there were firings of 1–4 thrusters; second, as the interceptor closed in on the target there were 

one to four “fine tuning” thrusters fired. Given there are errors in impact position, methods have 

been discussed that can be used to quantify performance based on the bias and standard deviation 

of the impact errors. The performance of the two projectile estimation techniques determines the 

system error. The methods presented can be used to analyze the ability of an interceptor to 

defeat an incoming ballistic object. 
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