Ballistic ranging methods and systems for inclined shooting   

Abstract: A portable system for facilitating inclined shooting of projectile weapons comprises a ranging system, an inclinometer and a processor. The ranging system measures a line-of sight range distance from a vantage point to a target that is elevated or depressed relative to the vantage point, and the inclinometer measures an inclination angle of a line of sight between the vantage point and the target. Based on information from the rangefinder and inclinometer, the processor determines a predicted altitude-compensated inclined shooting (ACIS) trajectory at the line-of sight range distance for a preselected projectile. The ACIS trajectory is based on a bullet path height correction between a bullet path height at a first altitude and a bullet path height at a second altitude, a range distance of the target from the vantage point, and selected meteorological atmospheric information.

The subject matter disclosed herein relates to methods and systems for compensating for ballistic drop and to portable devices (such as various equipments embodying various target locating and designators) implementing such methods. More particularly, the subject matter disclosed herein relates to method and system for compensating for ballistic drop for inclined shooting and to rangefinders and other portable devices implementing such methods.

BRIEF DESCRIPTION OF THE DRAWINGS

The subject matter disclosed herein is illustrated by way of example and not by limitation in the accompanying figures in which like reference numerals indicate similar elements and in which:

FIG. 1 depicts a schematic diagram of level-fire and inclined-fire trajectories for a projectile;

FIG. 2 depicts a schematic diagram illustrating measurements and factors in calculating an Equivalent Horizontal Range;

FIG. 3 depicts a flow chart for one exemplary embodiment of a method for determining the Equivalent Horizontal Range for accurately aiming a projectile weapon at an elevated or depressed target located at a inclined line of sight;

FIG. 4 depicts a summary of one exemplary method for calculating a trajectory parameter of bullet path and Equivalent Horizontal Range for bullets;

FIG. 5 depicts a summary of one exemplary method for calculating a trajectory parameter of an arrow path and equivalent horizontal range for arrows

FIG. 6 depicts a flow diagram for one exemplary embodiment of operations and a computational process performed by a master processor for generating reference trajectory information for computing Altitude-Compensated Inclined Shooting (ACIS) trajectory information for a selected cartridge according to the subject matter disclosed herein, the ACIS method being an alternative to the equivalent horizontal range method for generating reference trajectory information;

FIG. 7 depicts a flow diagram for one exemplary embodiment of operations and computational process performed by a device processor for generating Altitude-Compensated Inclined Shooting (ACIS) trajectory information for a selected cartridge according to the subject matter disclosed herein;

FIG. 8 depicts an exemplary embodiment of a portable handheld rangefinder that generates Altitude-Compensated Inclined Shooting (ACIS) trajectory information for a selected cartridge;

FIG. 9 depicts an enlarged view of an exemplary embodiment of an electronic display as viewed through an eyepiece of the exemplary portable handheld rangefinder depicted in FIG. 8;

FIG. 10 depicts an exemplary block diagram for an exemplary embodiment of rangefinder device according to the subject matter disclosed herein; and

FIG. 11 depicts an exemplary embodiment of a telescopic sighting device for use with the subject matter disclosed herein.

DETAILED DESCRIPTION

The word “exemplary,” as used herein, means “serving as an example, instance, or illustration.” Any embodiment described herein as “exemplary” is not to be construed as necessarily preferred or advantageous over other embodiments. Also, six technical terms used repeatedly herein require explanation. These terms are “ballistic path,” “bullet path,” “arrow path,” “ballistic path height,” “bullet path height,” and “arrow path height.” A projectile flying through the air without propulsion follows a ballistic trajectory, which may also be called a “ballistic path.” Two types of projectiles are addressed herein, bullets from firearms and arrows from bows. Thus, the “ballistic path” of a bullet is a “bullet path,” and similarly, the “ballistic path” of an arrow is an “arrow path”. When the word “height” is added to any of these terms (e.g., “bullet path height”), it refers specifically to the perpendicular distance between the instantaneous position of the projectile (e.g., the bullet) in flight and the extended line of sight of a shooter through the sighting device on the weapon which launched the projectile. The path height is considered positive when the projectile is above the extended line of sight, and negative when the projectile is below the extended line of sight.

FIG. 1 depicts a schematic diagram illustrating the effect on the trajectory of a projectile of the inclination of the line along which projectile is fired, cast, or otherwise launched (the “line of initial trajectory” or, in the case of guns, the “bore line”). For purposes of illustration, the trajectory curves and angles between various lines in FIG. 1 are greatly exaggerated and are not to scale.

With reference to FIG. 1, a “level fire” trajectory is the path along which a projectile moves when shot at a target T at range R0 and at substantially the same geographic elevation as a vantage point VP of the shooter. The weapon launching the projectile has a line of initial trajectory (“level-fire bore line”) that is not actually level, but rather is inclined relative to the level-fire line of sight (level-fire LOS) by an elevation angle α. The angle α is quite small, typically about one minute of angle (MoA) for a firearm, and larger (several MoA) for a bow. The level-fire line of sight, which is therefore approximately horizontal, begins at a height h above the beginning of the bore line. The height h and elevation angle α represent the typical mounting arrangement of a sighting device (i.e., riflescope, open sights, etc.) on a firearm or an archery sight on a bow. The level-fire trajectory intersects the level-fire line of sight at range R0 and is known as the “sighted-in range” or “zero range” or “zeroed-in range” (also referred to herein as zero-range distance RZ) of the weapon and sight combination. The sighted-in range R0 is typically established by shooting the weapon at a target at a known horizontal reference distance R0, such as 100 yards, and adjusting the elevation angle α of the riflescope or other sighting device until projectiles shot by the weapon impact the target at a point that coincides with the cross hairs or other aiming mark of the riflescope or other sighting device.

An “inclined-fire trajectory” is also depicted in FIG. 1. The inclined-fire trajectory represents the path along which the same projectile travels when aimed at a target that is elevated relative to vantage point VP. The height h and elevation angle α of the inclined-fire line of sight relative to the bore line are the same as in the level-fire scenario, because there can be no adjustment to the sighting device on the firearm or bow to anticipate the target elevation in the field. The inclined-fire line of sight will be inclined by an angle of inclination θ. As illustrated in FIG. 1, the inclined-fire trajectory crosses the inclined-fire line of sight at a distance substantially greater than the sighted-in range R0. This overshoot is due to the effect of gravity, which always acts in the vertically downward direction, regardless of the angle of inclination θ. The overshoot phenomena and prior methods of correcting for it are discussed in detail by W. T. McDonald in his paper titled “Inclined Fire” (June 2003), available from sierrabullets.com. The effects of inclination are typically even more pronounced in archery than for bullets and are caused by differences in the initial speed and aerodynamic characteristics of the projectiles used. The line-of-sight range distance and the inclination angle of a target relative to the shooter may be measured or estimated in the field where the target is encountered.

In accordance with exemplary embodiments described herein, many hunters (including bow hunters) and other shooters, such as military and law enforcement snipers, are versed in holdover techniques for compensating for ballistic drop in horizontal fire scenarios. A holdover adjustment involves aiming high by a measured or estimated amount. For example, a countersniper shooting a rifle with a riflescope sighted in at 200 yards may know that a killing shot for his target (in the heart-lung area) at a level-fire range of approximately 375 yards involves aiming the cross hairs of the riflescope at the top of the target\'s head. Holdover adjustments are much faster in practice than elevation adjustments, which involve manually adjusting an elevation setting of the riflescope or other aiming device to change the elevation angle α of the aiming device relative to the weapon. Holdover adjustments are also the primary mode of aiming adjustment for most archers. Holdover and holdunder techniques also avoid the need to re-zero the aiming device after making a temporary elevation adjustment.

Many varieties of ballistic reticles are employed in riflescopes to facilitate holdover and holdunder. For archery, a common ballistic aiming sight, known as a pin sight, is often employed for holdover aiming adjustment. Ballistic reticles and other ballistic aiming sights generally include multiple aiming marks spaced apart along a vertical axis. Exemplary ballistic reticles include mil-dot reticles and variations, such as the LEUPOLD TACTICAL MILLING RETICLE™ (TMR™) available from Leupold & Stevens, Inc., Leupold® DUPLEX™ reticles; the LEUPOLD SPECIAL PURPOSE RETICLE™ (SPR™); and LEUPOLD BALLISTIC AIMING SYSTEM™ (BAS™) reticles, such as the LEUPOLD BOONE & CROCKETT BIG GAME RETICLE™ and the LEUPOLD VARMINT HUNTER\'S RETICLE™ BAS reticles and methods of using them are described in U.S. Pat. No. 7,603,804 B2 to Zaderey et al., entitled “Ballistic Reticle for Projectile Weapon Aiming Systems and Method of Aiming” (“the \'804 patent”), the disclosure of which is incorporated herein by reference. As described in the \'804 patent, BAS reticles include secondary aiming marks that are spaced at progressively increasing distances below a primary aiming mark and positioned to compensate for ballistic drop at preselected regular incremental ranges for a group of ammunition having similar ballistic characteristics.

In accordance with one exemplary embodiment depicted in FIGS. 2 and 3, a method 300 of inclined shooting involves calculation of an equivalent horizontal range (EHR) that may be used by a shooter to make a holdover or elevation adjustment for accurately aiming a projectile weapon at an elevated or depressed target located at a inclined line-of-sight (LOS) range that is different from the EHR. With reference to FIG. 2, a shooter at vantage point VP determines a line-of-sight range to a target. As in FIG. 1, a zero range R0 represents the horizontal-fire distance at which the trajectory of the projectile launched from the projectile weapon and the line of sight from the aiming device of the weapon intersect. Line-of-sight ranges R1 and R2 to two different targets are depicted in FIG. 2, illustrating the usefulness of the method with respect to both positive and negative ballistic path heights BP1 and BP2 relative to the inclined-fire LOS. For purposes of illustration, the steps of method 300 (FIG. 3) will be described with reference to a generic LOS range R to a target T, shown in FIG. 2 at range R2. It should be appreciated that the methods described herein are equally applicable to “near” LOS ranges R1 at which the ballistic path height BP1 is positive, as well as to “far” LOS ranges R2 at which the ballistic path height BP2 is negative. The LOS range R may be determined by a relatively accurate ranging technique, such as use of a lidar (laser ranging) or radar, or by a method of range estimation, such as optical range estimating methods in which a distant target of known size is bracketed in a scale of an optical device, as described in the \'804 patent.

Methods 300 in accordance with the present disclosure also involve determining an inclination θ of the inclined LOS between vantage point VP and the target T. The angle of inclination θ may be determined by an electronic inclinometer, calibrated tilt sensor circuit, or other similar device. For accuracy, ease of use, and speed, an electronic inclinometer for determining the angle of inclination θ may be mounted in a common housing with a handheld laser rangefinder 800 of the kind described below with reference to FIGS. 8-10.

FIG. 3 is a flow diagram depicting steps of inclined shooting method 300, including the initial steps of determining the LOS range R (step 312) and determining the inclination θ of the inclined LOS (step 314). With reference to FIG. 3, after LOS range R and inclination θ have been determined (steps 312 and 314), method 300 may involve a check (step 316) to determine whether the absolute value of inclination θ is less than a predetermined limit under which the effects of inclination can be disregarded and the LOS range R can be regarded as the Equivalent Horizontal Range (EHR) (step 318).

Archery ballistics exhibit a more significant difference between positive and negative lines of initial trajectory (uphill and downhill shots) because the initial velocity is relatively low, giving the effects of gravity more time to affect the trajectory over a given distance than with bullets, which reach their targets much faster than arrows. Especially at long ranges, uphill shots experience more drop than downhill shots; therefore, when applying method 300 for archery, check 316 may involve comparing a positive inclination θ against a positive limit and a negative inclination θ against a negative limit that is different from the positive limit. Mathematically, such a check would be expressed as: {lower_limit}>θ<{upper limit}?

If the result of check 316 is negative, then a predicted trajectory parameter TP is calculated or otherwise determined at the LOS range for a preselected projectile P shot from vantage point VP toward the target T (step 320). Trajectory parameter TP may comprise any of a variety of trajectory characteristics or other characteristics of a projectile that are calculable using ballistics software. For example, trajectory parameter TP at LOS range R may comprise one or more of ballistic path height (e.g., arrow path or bullet path), ballistic drop relative to line of initial trajectory (e.g., the bore line in FIG. 1), observed ballistic drop perpendicular to LOS (i.e., (vertical ballistic drop) cos(θ+α)), velocity, energy, and momentum. In accordance with the exemplary embodiment described below with reference to FIGS. 2 and 4, for R=R2, trajectory parameter TP may comprise ballistic path height BP2 (e.g., bullet path height). In another embodiment, described below with reference to FIG. 5, the trajectory parameter of ballistic path height comprises arrow path height (AP).

Nothing in the figures or written description should, however, be construed as limiting the scope of possible trajectory parameters to only ballistic path height. Of the many possible choices of trajectory parameters, ballistic path height (bullet path height or arrow path height) usually is best for the shooter, who needs to know where the projectile will be relative to the line of sight when the projectile reaches the line-of-sight range distance to the target so that the shooter can make appropriate aiming adjustments.

After the trajectory parameter TP has been calculated, the method may then output the trajectory parameter TP (step 321) or calculate EHR based on the trajectory parameter TP or parameters (step 322). At step 321, the trajectory parameter TP output may comprise ballistic path height BP expressed as a linear distance in inches or millimeters (mm) of apparent drop, or as a corresponding angle subtended by the ballistic path height (e.g., BP2 in FIG. 2) in minutes of angle (MOA) or milliradians (mils). The TP output (step 321) may comprise a display of numerical ballistic path data in an electronic display device, such as a display 900 (FIG. 9) of rangefinder 800 (FIG. 8) or a reticle in a riflescope. Alternatively or additionally, the TP output (step 321) may comprise a graphical display of a holdover aiming recommendation in a rangefinder display, a riflescope reticle, an archery sight, or another aiming sight, based on the trajectory parameter of ballistic path height BP.

In one exemplary method of calculating EHR, a reference ballistics equation for a level-fire scenario (θ=0 ) comprising a polynomial series is reverted (i.e., through series reversion) to solve for EHR based on a previously calculated ballistic path height BP (e.g., BP2). As depicted in FIG. 2, BP2 corresponds to EHR2 under level-fire conditions. Thus, EHR is calculated as the range at which trajectory parameter TP would occur if shooting projectile P in a level-fire condition from the vantage point VP toward a theoretical target Tth in a common horizontal plane with vantage point VP, such that the horizontal plane coincides with the level-fire LOS. The reference ballistics equation may be established to deviate slightly from horizontal without appreciable error. Consequently, the terms “horizontal,” “level-fire LOS,” and other similar terms are construed to allow for equations to deviate from perfect horizontal unless the context indicates otherwise. For example, when solving for EHR, the degree of levelness of the reference equations should facilitate calculation of EHR with sufficient accuracy to allow aiming adjustments for inclined shooting resulting in better than ±6 inches of error at 500 yards throughout the range of between −60 and +60 degrees inclination. Ballistic trajectories are generally flatter at steeper shooting angles and trajectories of different projectiles are therefore more similar. Consequently, the deviation tends to be less significant at very steep inclines.

The calculation of trajectory parameter TP, the calculation of equivalent horizontal range EHR, or both, may also be based on a ballistic coefficient of the projectile P and one or more shooting conditions. The ballistic coefficient and shooting conditions may be specified by a user or automatically determined at step 324. Automatically-determined shooting conditions may include meteorological conditions, such as temperature, relative humidity, and/or barometric pressure, which may be measured by micro-sensors in communication with a computer processor for operating method 300. Meteorological conditions may also be determined by receiving local weather data via radio transmission signal, received by an antenna and receiver in association with the computer processor. Similarly, geospatial shooting conditions, such as the compass heading of the LOS to the target and the geographic location of the vantage point VP (including latitude, longitude, altitude, or all three), may be determined automatically by a GPS receiver and an electronic compass sensor in communication with the computer processor, to ballistically compensate for the Coriolis effect (caused by the rotation of the Earth). Alternatively, such meteorological and geospatial shooting conditions may be specified by a user and input into a memory associated with the computer processor, based on observations made by the user. It may be noted that for bows, geospatial conditions are unnecessary because maximum range distances are short, while for high-powered rifles, Coriolis corrections to a trajectory are necessary only if range distances exceed about 1000 yards or meters.

User selection of shooting conditions and ballistic coefficient may also involve preselecting or otherwise inputting non-meteorological and non-geospatial conditions for storage in a memory associated with a computer processor on which method 300 is executed. The ballistic coefficient and certain shooting conditions, such as the initial velocity of projectile P (e.g., muzzle velocity, in the case of bullets), may be set by a user simply by selecting from two or more weapon types (such as guns and bows), and from two or more ballistic groupings and possibly three, four, five, six, seven or more groups, such that each group has a nominal ballistic characteristic representative of different sets of projectiles having similar ballistic properties. The sets (groups) may be mutually-exclusive or overlapping (intersecting). A sighted-in range of a weapon aiming device and a height of the weapon aiming device above a bore line of a weapon may also be entered in this manner. In a rangefinder device 800 for operating the method, described below with reference to FIGS. 8 and 9, the weapon type and ballistic group may be selected from a menu of possible choices during a menu mode or setup mode of rangefinder device 800.

After a trajectory parameter TP has been calculated at step 320 or EHR has been calculated at step 322, method 300 then involves outputting TP or EHR in some form (step 321 or 326). For example, TP or EHR may be displayed via a display device, such as an LCD display, in the form of a numeric value specified in a convenient unit of measure. For example, TP output may be expressed as ballistic path height BP in inches or mm of apparent drop or as an angle (in MOA or mils) subtended by the ballistic path height BP. EHR may be expressed in yards or meters, for example. In other embodiments, BP or EHR may be effectively output via a graphical representation of the data through the identification of a reticle aiming mark corresponding to the holdover or holdunder (holdover or holdunder is always the negative of the BP), or the EHR, for example, as described below.

Once EHR is output at 326, EHR can then be employed to aim the projectile weapon at target T (step 328) along the inclined LOS at R2. In one embodiment, a shooter merely makes a holdover or holdunder adjustment based on the calculated EHR, as if he or she were shooting under level-fire conditions—it being noted that wind effects, firearm inaccuracy, and wiggle of the shooter are still in effect over the entire LOS range R2. In another embodiment, the shooter adjusts an elevation adjustment mechanism of a riflescope or other aiming device based on the displayed EHR. Similar elevation adjustments may be made based on the display of the calculated trajectory parameter TP (step 321).

FIG. 4 summarizes details of one possible sequence of steps 400 for calculating a trajectory parameter of bullet path height (BP ) and equivalent horizontal range (EHR) for bullets. The calculation sequence 400 begins with selection of a ballistic group (A, B, or C) in which the bullet and cartridge are listed (step 401). Ballistic grouping may effectively normalize groups of bullets having similar characteristics, based on their ballistic coefficients, muzzle velocities and masses. Listings of cartridges in the various groupings may be provided to the user by a printed table or software-generated information display, facilitating selection of the appropriate ballistic group. Reference trajectories for ballistic groups A, B, and C are set forth in Table 1. The other inputs to the calculations include the LOS range R and the inclination angle θ, which may be determined automatically by a handheld laser rangefinder with inclinometer (step 402). The calculation method involves solving the following polynomial equation for bullet path height BP :

BP=α0+α1R+α2R2+α3R3+α4R4   (1)

(step 406), in which the coefficients α0, α1, α2, α3 and α4 are calculated from the inclination angle θ based on a series of polynomial equations 404 in which the coefficients thereof (identified in FIG. 4 as A00, A01, A02, etc.) are different stored parameters for each ballistic group A, B, and C. A single equation 406 (Equation (1)) is suitable for both positive and negative angles of inclination, expressed as absolute angular values. After bullet path height BP has been determined, the BP is then used as an input to one of two different reversions of the bullet path equation for θ=0 to solve for EHR. If bullet path height BP is positive (test 408), then a “short-range EHR” polynomial equation is used (step 410), such that B0, B1, . . . , B6 are parameters corresponding to the selected ballistic group. If BP is negative (test 408), then a “long-range EHR” polynomial equation is used (step 412), such that C0, C1, . . . , C6 are parameters corresponding to the selected ballistic group. Each ballistic group also has an associated coefficient named BPLIM, which is an upper limit for BP in the computations shown in FIG. 4. Parameters A00 to A43, B0 to B6, and C0 to C6 are constants that are stored for each of the ballistic groups and recalled based on the selected ballistic group for purposes completing the calculations 400.

FIG. 5 illustrates a similar sequence of calculations 500 for archery. In FIG. 5, reference numerals 501, 502, 506, etc., indicate steps that respectively correspond to steps 401, 402, 406, etc., of FIG. 4. Unlike the calculations 400 (FIG. 4) for bullets, the calculation of ballistic path for arrows 500 (hereinafter arrow path (AP) must take into account whether the inclination angle is positive or negative (branch 503), due to the increased flight time of arrows and attendant increased effects of gravity on their trajectory. For this reason, the calculations involve one of two different sets of coefficients Aij and Dij, (for i=1, 2, 3, 4, 5 and j=1, 2, 3, 4, 5) depending on whether the inclination is positive (step 504a′) or negative (step 504b′). Parameters A00 to A43, B0 to B6, C0 to C6, D00 to D43, APLIM, and EHRLIM are constants that are stored in memory for each of the ballistic groups and recalled based on the selected ballistic group for purposes completing the calculations 500.

Table 1 lists one example of criteria for ballistic grouping of arrows and bullets:

TABLE 1 BALLISTIC CHARACTERISTIC GROUP BALLISTIC DROP (WITHOUT INCLINE) Arrow group A Arrow drop of 20-30 in from the 20 yd sight pin at 40 yd Arrow group B Arrow drop of 30-40 in from the 20 yd sight pin at 40 yd Arrow group C Arrow drop of 10-20 in from the 20-yd sight pin at 40 yd Bullet group A Rifles sighted in at 200 yards with 30-40 in drop at 500 yd Bullet group B Rifles sighted in at 200 yards with 40-50 in drop at 500 yd Bullet group C Rifles sighted in at 300 yards with 20-30 in drop at 500 yd

Arrow groupings may be more dependent on the launch velocity achieved than the actual arrow used, whereas bullet groupings may be primarily based on the type of cartridge and load used. Table 2 lists exemplary reference trajectories from which the calculation coefficients of FIG. 4 may be determined for ballistic groups A, B, and C.

TABLE 2 BALLISTIC GROUP REFERENCE TRAJECTORY A Winchester Short Magnum with Winchester 180 grain Ballistic Silvertip bullet at 3010 fps, having a level fire bullet path height of −25.21 in at 500 yds. B 7 mm Remington Magnum with Federal 150 grain SBT GameKing bullet at 3110 fps, having a level fire Bullet Path height of −34.82 in at 500 yds. C 7 mm-08 Remington with Remington Pointed Soft Point Core-Lokt bullet at 2890 fps, having a level fire Bullet Path height of −45.22 in at 500 yds.

Alternatives to solving a series of polynomial equations also exist, although many of them will not provide the same accuracy as solving a polynomial series. For example, a single simplified equation for ballistic drop or ballistic path height may be used to calculate a predicted trajectory parameter, and then a second simplified equation used to calculate EHR from the predicted trajectory parameter. Another alternative method of calculating EHR involves the “Sierra Approach” described in W. T. McDonald, “Inclined Fire” (June 2003), incorporated herein by reference. Still another alternative technique for calculating ballistic drop or ballistic path involves a table lookup of a predicted trajectory parameter and/or interpolation of table lookup results, followed by calculation of EHR using the formula identified in FIG. 4. Yet another alternative involves determining both the predicted trajectory parameter and EHR by table lookup and interpolation, using stored sets of inclined-shooting data at various angles.

Table 3 illustrates an example of an EHR calculation using the sequence of steps for calculating a trajectory parameter of bullet path height (BP) and equivalent horizontal range (EHR) for bullets described above in connection with FIG. 4. The EHR calculations in Table 3 are also compared with the results of aiming using EHR to aiming with no compensation for incline, and aiming by utilizing the horizontal distance to the target (rifleman\'s rule).

TABLE 3 .300 WSM, 165 GRAIN NOSLER PARTITION, 3050 FPS MUZZLE LOAD VELOCITY Angle of inclination 50° Inclined line-of-sight range 500 yds Equivalent Horizontal Range (EHR) 389 yds Ballistic table holdover for 389 yds 18 in level fire

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