LINE-OF-SIGHT HEIGHT AND THE CANT ANGLE EFFECT IN AIR-RIFLE SHOOTING

Copyright Jeroen Hogema, 5 September 1999

Minor revision 16 February 2001

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SUMMARY

This document describes an experiment that was carried out to study the effects of line-of-sight height and cant angle variation on the point of impact in air rifle shooting. To enable accurate shooting with large cant angle variations (-90° to +90°), a laser was used to replace the conventional diopter and ring sights. For a low (1.6 cm) and a high (4.7 cm) line of sight height condition, the effect of cant angle on the point-of-impact was investigated. The results were in line with the theory: cant angle variations cause a point of impact displacement following a circular pattern, and the line of sight height does not influence this effect. The experimental results were compared with the ballistic calculations obtained from the computer program PCB.

 

SAMENVATTING

Dit document beschrijft een experiment waarin is onderzocht hoe bij sportschieten het trefpunt verschuift onder invloed van de vizierlijn-hoogte en van kanteling van het geweer om de visuele as.

Om nauwkeurig richten mogelijk te maken bij een grote variatie aan kantel-hoeken (-90° tot +90°), werd een laser gebruikt in plaats van de normale diopter en ringkorrel. Bij een lage (1.6 cm) en een hoge (4.7 cm) vizierlijn werd het effect van kantelen op het trefpunt onderzocht. De resultaten klopten met de theorie: kantel-variatie veroorzaakt een cirkelvormig patroon van trefpuntverplaatsing, en de vizierlijn-hoogte is hierop niet van invloed. De experimentele resultaten werden vergeleken met de ballistische berekeningen van het computerprogramma PCB.

 

1 INTRODUCTION

One of the many elements a spot shooter must pay attention to is that he or she holds the rifle under a constant cant angle. Variation in the cant angle (i.e. rotating around the line of sight) will cause variation in the point of impact, even with perfectly centred sights and with a perfectly motionless weapon.

In another document (Hogema, 1999a), it has been argued that when a rifle is zeroed under 0° cant angle and subsequently fired with cant unequal to 0°, the point of impact (POI) will move according to a circular pattern. The amount of displacement along this circle is proportional to the cant angle. The top of this circle is in the centre of the target, and the centre of the circle is located at a distance drop below the centre of the target. Here, drop is a ballistic variable, indicating the height the bullet loses between the moment it leaves the rifle and the moment it reaches the target. In this theory, only the drop determines the POI displacement due to a given cant angle error. Therefore, the LOS height is not expected to be of influence on the POI displacement.

In an earlier experiment, cant angle was varied from –90o to +90o, with 45o increments (Hogema, 1998b). The results confirmed the circular displacement of the POI. However, only one LOS height was used in that study. This document describes an experiment that was carried out to investigate empirically if LOS height does have an effect on POI displacement when canting the rifle.

In the earlier study (Hogema, 1998b), the normal sights (diopter and ring sights) were used. Aiming at all cant angles included in the study lead to rather awkward (bench rest) shooting positions. Therefore, in the current study, a laser sight was used instead of the conventional sights.

Again, the experimental results are compared with ballistic calculations carried out with the computer program PCB (Skevik, 1996).

 

2 METHOD

The experiment was carried out on June 26, 1999, at the indoor shooting range of S.V. Prins Hendrik, Amersfoort, The Netherlands. Standard ISSF 10 m air rifle targets were used. The height of the target was such, that the line of sight was about horizontal.

A Feinwerkbau (FWB) air rifle model 603 was used. This model is equipped with an absorber, that eliminates the recoil (small as it is for a match air rifle even without an absorber). Selected RWS R10 diabolos were used. The weight of this bullet is 0.53 grams (specified on the package), and the Ballistic Coefficient (BC) is 0.011 (Eichstädt, 1995)

To obtain large POI displacements due to cant variation (compared with the random error), the following measures were taken.

The diopter and ring sight were replaced by a laser sight. This was a laser diode (Laser Devices, Inc. S/N 93217, class IIIa laser product, wavelength: 632 to 670 NM, power: 5 mW max).

The laser was applied in two conditions:

Figure 1: Laser mounted on the rifle. A: low line of sight (1.6 cm). B: high line of sight (4.7 cm). The upper arrow indicates the laser aperture and the lower arrow the bore axis.

For comparison, the standard front sight of the FWB 603 has a LOS height of 2.7 cm. ISSF regulations allow a maximum LOS height of 4.0 cm (Special Technical Rules for Rifle Shooting, see 7.4.3.6 and 7.4.3.7). This shows that the range of LOS heights applied in this experiment range from much smaller than typically found on a rifle to higher than allowed.

Procedure
First, the rifle was zeroed in the cant = 0° condition, with the laser mounted in the low LOS condition. Zeroing was done by fixing the rifle in a clamp (i.e. upright), firing 5 pellets on a single target and then aligning the laser dot on the resulting shot group on the target.

Then the rifle was fired five times: under (approximately) +90° and +45° cant angle (i.e. rotated clockwise), under 0 degrees cant angle, and under -45° and –90° (i.e. rotated counter-clockwise). For each shot, a new target was used. All shots were fired with the rifle fully supported by kneeling rolls, that were arranged such that the laser dot was aimed at the centre of the target. A spotting scope was used in aligning the centre of the laser dot with the centre of the target. No special equipment was used to measure or control the cant angle. This entire procedure was repeated three times.

Then the laser was mounted for the high LOS condition. The zeroing and the shooting procedure were repeated in this condition. Thus, 30 shots were fired in total: 2 (LOS height) x 3 (series) x 5 (cant angles).

Data analysis
The position of the centre of each shot with respect to the centre of the target was determined using a scoring machine (Disag International, model DISAG RM III, Ser.Nr. 10037.). This yielded, for each shot:

The shooting results were statistically analysed with an Analysis of Variance (ANOVA). LOS condition and cant angle were used as the independent variables, and the x and y co-ordinates of the shot locations as dependent variables.

Next, for each series of 5 shots, the best-fitting circle was determined. The radius as well as the co-ordinates of the centre were estimated by means of a hill-climbing routine. The advantage of this approach is that it is insensitive to errors in the realisation of the cant angles prescribed by the experimental design. A Sum of Squared Errors (SSE) criterion was used, where the error was defined for each shot as the distance between the centre of the point of impact and circle. Hence, the cost function to be minimised was:

J = SUM (from i=1 to i=5) [r – {(xi – xc)2 + (yi – yc)2}0.5 ]2

with

Parameters that were varied in the procedure were xc, yc, and r. These resulting parameters were analysed in t-tests to test for possible differences between high and low LOS.

 

3 RESULTS

A graphical representation of the shot groups is given in Figure 2. The expected semi-circle is clearly visible from these results for both LOS conditions.

Figure 2 Shot groups for low and high LOS condition.

An ANOVA was carried out on the x co-ordinate of the shot location, with LOS height and cant angle as the independent variables. There was a significant main effect of LOS [F(1,20)=816, p<0.001], showing that the x co-ordinate increases when cant angle is varied from –90º to 90º. There was also a significant interaction between LOS and cant angle [F(4,20)=2.9, p<0.05]. A Tukey test showed that:

Secondly, an ANOVA was carried out on the y co-ordinate of the shot location, with LOS height and cant angle as the independent variables. There was a significant effect of cant angle [F(4,20)=113, p<0.001]. No other effect or interaction was found.

Looking back at Figure 2, the difference found between the x co-ordinates between high and low LOS conditions can be explained as differences in the cant angle. Probably, the cant angles applied in the high LOS condition were a bit smaller than in the low LOS condition, causing a sift along the circle towards the top of the circle.

Next, the POI circle parameters were estimated using the hill climbing procedure, as described in Chapter 2. Using different initial search values, the procedure converged to the same optimum which shows that the search procedure was not disturbed by local minima.

The results are shown in Figure 3. The resulting means and standard deviations are given in Table I, together with the results from the t-tests carried out.

Figure 3 Best-fitting circles through shots for low and high line of sight.

Table I Means and standard deviations (mm) of the circle parameters, and results of t-tests

 

Low LOS

High LOS

     

Variable

Mean

s.d.

Mean

s.d.

t-value

df

p

xc

0.41

1.38

0.12

0.65

0.32

4

0.764

yc

-32.70

2.29

-32.14

3.38

-0.24

4

0.823

r

29.98

1.39

28.28

2.16

1.15

4

0.314

As Table I shows, no statistically significant difference was found between the low and high LOS condition for any of the circle parameters.

Comparison with PCB results
As Hogema (1999a,b) has shown, the effect of cant angle predicted by PCB is a circular pattern of POI displacement. This pattern was found in the current study, showing a qualitative agreement. The remaining question is whether there is also a quantitative agreement. To investigate this, the drop of the bullet is derived from the current experimental data. Next, this drop value will be compared against the PCB drop calculation for the situation at hand.

As Figure 3 shows, it appears that the rifle was zeroed slightly too low for both LOS conditions: the overall mean of the y co-ordinate is –3.2 mm when cant angle is 0°. This can be seen in Table I as well: the mean r values are somewhat smaller than the mean yc values.

The value found for yc will be used as an estimate for the bullet’s drop rather than the circle’s radius. When zeroed too high, the radius of the POI circle increases, but it’s centre remains at the same location, i.e. at drop below the centre of the target. Likewise, when zeroed too low, the radius of the POI circle decreases, but it’s centre still remains at the same location, i.e. at drop below the centre of the target (see also Appendix B). Ultimately, when the rifle is zeroed such, that the point of impact is at drop below the centre of the target, the barrel is aimed at the centre of the target. Now, canting the rifle has no effect on the POI, i.e. the radius of the POI circle is 0.

Hence, the mean drop of the bullet is estimated at 3.24 cm from Table I. The 95% confidence interval is 3.52 to 2.97 cm.

At the range of 14 m applied here, PCB predicts the following relationship between drop and V0 (see Appendix A for a further explanation).

Figure 4: Drop as a function of muzzle velocity V0 (range = 14 m, B.C. = 0.011).

According to PCB, a drop of 3.24 cm is achieved with a VO of 182 m/s. The V0 of the rifle used in this experiment (with other ammunition though) was measured later on to be 173.5 m/s (mean based on 5 shots; standard deviation 0.2 m/s). This is in the same order of magnitude; a more precise comparison can only be made when V0 is measured for the same rifle-ammunition combination that was used in the experiment.

 

4 CONCLUSIONS

The results of this experiment confirmed that varying cant angle causes a movement of the point of impact, following a circular pattern. This effect is not influenced by the line of sight height: the circles found in both conditions are the same. So increasing the line of sight height (by applying riser blocks) does not increase the shooting error caused by variations in cant angle.

Comparison of the experimental results with those obtained from the computer program PCB showed that there is a qualitative agreement. The quantitative results seem to be reasonably in line; however, a better measurement of V0 is required before a more solid comparison can be made.

 

ACKNOWLEDGEMENTS

Many thanks to Joost Doedens for letting me use his rifle for this experiment, to Arno Brinkman (S.V. Op de Korrel, Bemmel) for processing the targets with the DISAG, and to Harm Krale for measuring V0.

 

REFERENCES

Eichstädt, U. (1995). Visier, 10, 42-48.

Hogema, J.H. (1999a). Sport shooting: the effect of rifle cant on the point of impact. <http://home.wanadoo.nl/jhogema/skeetn/ballist/kantel01/kantel1e.htm> [2006, 7 March].

Hogema, J.H. (1999b). Effect of cant angle variation on point-of-impact in air-rifle shooting. <http://home.wanadoo.nl/jhogema/skeetn/ballist/cantexp1/cantexp1.htmt > [2006, 7 March].

Skevik, O.H. (1996). PCB v1.8, Exterior Ballistics Software for the IBM PC. Freeware found at: <http://www.stud.unit.no/~oddske/ballistics.html> [now here; 2006, 7 March]

 

APPENDIX A RELATION BETWEEN MUZZLE VELOCITY AND DROP

To determine the relationship between the bullet’s muzzle velocity V0 and the its drop at a range of 14.0 m, PCB version 1.8 was used. The following Trajectory chart shows the input parameters used for this purpose. Data for the RWS R10 diabolo were obtained from Eichstädt (1995).

PCB 1.8 Trajectory chart
Bullet name..................... RWS R10 diabolo
Bullet weight................... 0.53 gram
Bullet diameter................. 4.50 mm
Muzzle velocity................. 170 m/s
Ballistic coefficient........... 0.011
Zero............................ 0.0 cm at 14.0 meters
Crosswind....................... 0.00 m/s from 90 degs
Line of sight above bore axis... 2.7 cm
Temperature..................... 15.0 grad C
Altitude........................ 0 meters
Sights zeroed at 0 degs
Firing angle.................... 0 degs
Cant angle...................... 0 degs

Range

Velocity

Energy

Flight time

Drop

Max height

Path

Drift

Click
up

Click side

[m]

[m/s]

[J]

[s]

[cm]

[cm]

[cm]

[cm]

   

0

170

8

0.000

0.0

-2.7

-2.7

0.0

0.0

0.0

2

166

7

0.012

0.1

-1.1

-1.9

0.0

92.6

0.0

4

162

7

0.024

0.3

-1.0

-1.1

0.0

28.7

0.0

6

158

7

0.037

0.6

-0.9

-0.6

0.0

9.9

0.0

8

155

6

0.049

1.2

-0.8

-0.2

0.0

2.4

0.0

10

151

6

0.062

1.8

-0.6

0.0

0.0

-0.5

0.0

12

147

6

0.076

2.7

-0.4

0.1

0.0

-1.0

0.0

14

144

5

0.090

3.7

-0.1

0.0

0.0

0.0

0.0

16

140

5

0.104

4.9

0.2

-0.3

0.0

2.0

0.0

18

137

5

0.118

6.4

0.6

-0.8

0.0

4.6

0.0

20

133

5

0.133

8.0

1.1

-1.6

0.0

7.8

0.0

 

As this example shows, a V0 of 170 m/s corresponds to a drop of 3.7 cm. In the same manner, the data from Table A1 were obtained. See Figure 4 for a graphical representation.

Table A1 Drop at range = 14 m as a function of VO.

VO (m/s) Drop (cm)
140 5.5
150 4.8
160 4.2
170 3.7
180 3.3
190 3.0
200 2.7

 

 

APPENDIX B EFFECT OF ZEROING BELOW THE CENTRE OF THE TARGET

PCB has the possibility to study the effect of zeroing the rifle too low or too high (with respect to the centre of the target) on the trajectory.

For example, take a situation for air rifle that has 2.0 cm drop:

PCB 1.8 Trajectory chart
Bullet name..................... RWS R10 diabolo
Bullet weight................... 0.53 gram
Bullet diameter................. 4.50 mm
Muzzle velocity................. 163 m/s
Ballistic coefficient........... 0.011
Zero............................ 0.0 cm at 10.0 meters
Crosswind....................... 0.00 m/s from 90 degs
Line of sight above bore axis... 2.7 cm
Temperature..................... 15.0 grad C
Altitude........................ 0 meters
Sights zeroed at 0 degs
Firing angle.................... 0 degs
Cant angle...................... 0 degs

 

Range

Velocity

Energy

Flight time

Drop

max height

Path

Drift

Click up

Click side

[m]

[m/s]

[J]

[s]

[cm]

[cm]

[cm]

[cm]

   

0

163

7

0.000

0.0

-2.7

-2.7

0.0

0.0

0.0

10

144

6

0.065

2.0

-0.6

0.0

0.0

0.0

0.0

When, in this situation, a 90° cant angle is applied, the POI is expected to move along a quarter circle, i.e. down by distance = drop = 2.0 cm, and sideways by the same distance. Indeed, this is what is found:

PCB 1.8 Trajectory chart
Bullet name..................... RWS R10 diabolo
Bullet weight................... 0.53 gram
Bullet diameter................. 4.50 mm
Muzzle velocity................. 163 m/s
Ballistic coefficient........... 0.011
Zero............................ 0.0 cm at 10.0 meters
Crosswind....................... 0.00 m/s from 90 degs
Line of sight above bore axis... 2.7 cm
Temperature..................... 15.0 grad C
Altitude........................ 0 meters
Sights zeroed at 0 degs
Firing angle.................... 0 degs
Cant angle...................... 90 degs

 

Range

Velocity

Energy

Flight time

Drop

max height

Path

Drift

Click up

Click side

[m]

[m/s]

[J]

[s]

[cm]

[cm]

[cm]

[cm]

   

0

163

7

0.000

0.0

-2.7

-2.7

0.0

0.0

0.0

10

144

6

0.065

2.0

-0.6

0.0

0.0

0.0

0.0

Next, the cant angle is re-set to 0°, and the rifle is re-zeroing the sights such that the POI is 1 cm too low:

PCB 1.8 Trajectory chart
Bullet name..................... RWS R10 diabolo
Bullet weight................... 0.53 gram
Bullet diameter................. 4.50 mm
Muzzle velocity................. 163 m/s
Ballistic coefficient........... 0.011
Zero............................ -1.0 cm at 10.0 meters
Crosswind....................... 0.00 m/s from 90 degs
Line of sight above bore axis... 2.7 cm
Temperature..................... 15.0 grad C
Altitude........................ 0 meters
Sights zeroed at 0 degs
Firing angle.................... 0 degs
Cant angle...................... 0 degs

Range

Velocity

Energy

Flight time

Drop

Max height

Path

Drift

Click up

Click side

[m]

[m/s]

[J]

[s]

[cm]

[cm]

[cm]

[cm]

   

0

163

7

0.000

0.0

-2.7

-2.7

0.0

0.0

0.0

10

144

6

0.065

2.0

-0.6

-1.0

0.0

10.0

0.0

Conform the input parameters, the Path is now –1.0 cm at the range of 10 m. From this situation, changing to cant angle 90° yields the following result:

PCB 1.8 Trajectory chart
Bullet name..................... RWS R10 diabolo
Bullet weight................... 0.53 gram
Bullet diameter................. 4.50 mm
Muzzle velocity................. 163 m/s
Ballistic coefficient........... 0.011
Zero............................ -1.0 cm at 10.0 meters
Crosswind....................... 0.00 m/s from 90 degs
Line of sight above bore axis... 2.7 cm
Temperature..................... 15.0 grad C
Altitude........................ 0 meters
Sights zeroed at 0 degs
Firing angle.................... 0 degs
Cant angle...................... 90 degs

Range

Velocity

Energy

Flight time

Drop

Max height

Path

Drift

Click up

Click side

[m]

[m/s]

[J]

[s]

[cm]

[cm]

[cm]

[cm]

   

0

163

7

0.000

0.0

-2.7

-2.7

0.0

0.0

0.0

10

144

6

0.065

2.0

-0.6

-2.0

1.0

19.9

9.9

Due to the 90° cant, the POI moves another 1.0 cm down, and 1.0 cm sideways.

The POI still moves along a circle. The centre of this circle is at drop distance below the centre of the target. The radius of the circle equals the drop minus the distance the rifle was zeroed below then target centre.

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