10 Fingers of Death: Algorithms for Combat Killing Roger Smith and Don Stoner Titan Corporation

Transcription

1 10 Fingers of Deat: Algoritms for Combat Killing Roger Smit and Don Stoner Titan Cororation Good sooting games need good killing algoritms. Tis gem rovides a series of combat algoritms tat can be used to imrove te realism of combat decisions and do so wit faster algoritms. Most of te algoritms discussed ere were develoed for te United States military and ave been validated for use in one or more real combat simulations. First-erson sooter killing algoritms are fine, but some situations can be andled more accurately and efficiently by including geometry, statistics, robability, and aggregation. Massively Multi-layer and Real-time Strategy games articularly include a lot of action tat does not ave to be modeled wit individual line-of-sigt (LOS) and targeting for a eadsot. Tese and oter games can also benefit from te inclusion of multile kill tyes tat are based on real live-fire exeriments. First-erson sooters may equi AI s wit some of tese algoritms wile leaving te more detailed LOS algoritms for te avatar controlled by te uman layer. Hitting a Ribbon Te first finger of deat resents a simle metod for determining weter a sooter will it a ribbon target like a road, river, long convoy of veicles, or a serentine creature (Figure 1). If te target is so long tat it is imossible to oversoot or undersoot its lengt, te robability of itting te target is deendent only uon te widt of te target and te standard deviation of te sot attern of te weaon. Tis simle algoritm is also a good way to introduce te logic and matematics beind several of te attrition algoritms tat follow [arry95]. Deviations in te imact oint of te munitions being fired are due to factors suc as te quality of te weaon, steadiness and skill of te uman oerator, variations in te construction of te roectile, and wind conditions. W W Figure 1. robability of Hitting a Ribbon Target 1

2 Te equation for calculating te robability of it ( ) for a ribbon target is: = W πσ were, x W is te widt of te target, and is te standard deviation of te bullet disersion in te x dimension. Tis assumes tat te attern is normally distributed wit te same standard deviation in bot te x and y dimensions. [Note: Te code for all of tese algoritms can be found on te CD-ROM.] Hitting te Bullseye Te second finger of deat describes te mat and robability of itting a round target. Like te revious algoritm, tis one is based on te fact tat all sooters, uman and macine alike, ave built-in variation in every sot fired. Te algoritm is driven by two very simle variables te radius of te target and te standard deviation of te rounds. Tis deviation is based on a normal distribution in wic te mean value is zero because te sooter is aimed directly at te center of te target [arry95]. Te algoritm determines weter eac sot will it te target, but does not calculate te actual imact oint of te round. Tis simlification eliminates calculations tat would ave to be done to distribute te round normally in bot te x and y dimension. r Figure. robability of Hitting te Bullseye Te equation for calculating te robability of it ( ) for a round target is: ( r ) σ x = 1 e

3 were, r is te radius of te target, and is te standard deviation of te bullet disersion in te x dimension. Hitting a Rectangle Most targets are not saed like bullseyes, so we need a more flexible algoritm to soot rectangular targets like te torso of a uman or a veicle. Tis algoritm includes measures for te lengt and widt of a rectangular target [arry95]. σ y L W Figure 3. robability of Hitting a Rectangle Te equation for calculating te robability of it ( ) for a rectangular target is: = A* B ( L π A = 1 e ) ( W πσ y ) B = 1 e were, L is te lengt of te target in te x dimension, W is te widt of te target in te y dimension, is te standard deviation of te bullet disersion in te x dimension, and σ y is te standard deviation of te bullet disersion in te y dimension. Weaons often ave different standard deviations in te x and y dimensions. For examle, wen a football quarterback trows a ass, te variation from te aim oint along te axis of fligt is usually greater tan te variation left or rigt of te aim oint. Te same is true for missiles being fired at a combat veicle or rocks being trown at a dinosaur. Sotgunning a Small Target 3

4 Some weaons unleas a barrage of rockets, bomblets, or exlosive munitions all at once in an attemt to totally overwelm te target and blow it to smitereens. Wen tis aens, tere are muc faster ways of determining te killing effect of te entire barrage tan calculating te imact oints and letality of eac rocket individually and ten accumulating tem. Tis algoritm calculates te robability tat one of te munitions letal areas will overla wit te oint target. Te size of te target is not considered in tese calculations because it is assumed tat te letal blast area can encomass an entire target [arry95]. a T n Simultaneous Rounds Sooter Figure 4. robability of Killing a Target wit a Simultaneous Barrage of Munitions were, ( na πσ x 1 ) k = e n is te number of rounds in te barrage, a is te letal area of a single round against tis target, and is te standard deviation of te bullet disersion in te x dimension. Deat by Walking Artillery Artillery and catault rounds are often adusted by a sotting team tat radios corrections back to te firing battery and allows tem to lace te next round closer to te target. Wen tis occurs, te letality of te barrage is iger tan te revious sotgunning metod. Te letality of tis walking artillery is calculated troug a summation series in te exonent [arry95]. 4

5 Sotter a Simulated Correction Communication T n Sequential Corrected Rounds Sooter Figure 5. Letality of Walking Artillery wit a Sotter were, n ( ) a σ x ( i 1) i i = k = 1 e n is te number of rounds fired in te barrage, a is te letal area of a single round against tis target, and is te standard deviation of te bullet disersion in te x dimension. Kills Come in Four Flavors To ararase a famous ig, all kills are equal, but some are more equal tan oters. Military simulations usually model four different tyes of kills tat are most often found in real-world combat. Te first flavor is a mobility kill in wic te target is no longer able to move, but remains alive enoug to fire its weaon or communicate wit oter veicles. Te second is a fireower kill in wic te weaon is damaged, but te veicle or erson is still able to move. Te tird is a mobility and fireower kill in wic te veicle or erson is still alive, but cannot move or use its weaon. Tis target may still be able to observe enemy oerations, communicate, consume sulies and, in some simulations, trigger a rescue oeration. Te final kill tye is te catastroic kill or K- kill, often ictured as an aircraft exloding into a million ieces, a flaming tank turret sinning troug te air, or a erson being turned into fres cunks of meat. Tese four kill tyes can be ictured as a Venn diagram (Figure 6). Toug tis form clearly communicates te relationsis between te kill tyes, in order to be alied, it as to be searated so tat a secific kill tye can be determined quickly for eac engagement. Tis searated data is usually reresented as a kill termometer (Figure 7). Normalizing te kill tyes in a single sace as reresented in te termometer allows a 5

6 rogram to determine te kill tye of an engagement by drawing a single random number. Mobility Kill M&F Kill Fireower Kill K Kill Figure 6. Standard Kill Tyes No Kill 1.0 Catastroic Kill n + m + f + mf Mobility and Fireower Kill n + m + f n + m Fireower Kill Mobility Kill n 0.0 No Kill Figure 7. Kill Termometer Tere are live-fire roects tat determine te robability of eac kill tye under different conditions by firing real weaons at real targets and measuring te result. Most simulations and games do not ave access to suc a ric source of information. Terefore, we ave to identify common trends in exerimental data and create equations tat mimic tose wile remaining flexible enoug to be alied to new weaon/target airs. One simulation roect noticed a distinct relationsi between te mobility, fireower, and catastroic kill data tey ad received from live-fire exeriments. Tis relationsi allowed tem to create simle equations tat use te root of a single robability of mobility or fireower kill ( is te union of all of te saded areas above) to calculate all of te oter robabilities. Te trend tey noticed was tat a mobility kill occurred 90% of te time tat damage was done ( M = 0.9* ); a fireower kill occurred 90% of te time ( F = 0.9* ); and a catastroic kill occurred 50% of te time tat damage was done ( K = 0.5* ). 6

7 However, tis information cannot be alied directly to te kill termometer in Figure 7. M does not say tat 90% of all engagements result in a mobility kill. It says tat 90% of mobility-or-fireower kills include a mobility kill. Terefore, it as to be searated to make it ossible to draw a single random number and determine wic kill to aly to te target. Tese indeendent kill robabilities can be extracted as sown below. = 1.0 n m f k mf = = = 0.5 = F M m = 0.1 = 0.1 f = 0.3 k were, te small subscrit indicates te robability tat only one tye of kill occurs. For examle, m is te robability of only getting a mobility kill, but not getting any oter form of kill. n is te robability of no kill occurring. Tese indeendent kill robabilities determine were te breakoints fall in a kill termometer and can be easily rogrammed as sown in te code on te CD-ROM. Cemicals, Fireballs, and Area Magic Tere ave been many models of te disersion of cemicals and oter agents. Te following simle algoritm calculates te robability of a kill based on te volume of cemical released and te distance tat te release occurs from te target. For games, tis algoritm could be used for exanding fireballs, area magic, or any oter exotic and evil weaon. k ( ) ( k d nw ) r nw e = π r were, n is te number of rounds imacting at a secific oint, w r is te weigt of te cemical inside of eac round (in kilograms), d is te distance tat te rounds fall from te target location (in meters), and k is a constant reresenting te disersion caracteristic of te cemical. For tese exeriments we recommend beginning wit a value of Tis equation allows you to deal wit eac round individually or to aggregate multile rounds into a single attack centered at te same imact oint. Te equation also 7

8 incororates te constant k tat reresents te density and viscosity of a cemical comound. You can adust tis value to create te effect desired. Te Sranel Wedge Wen an aircraft is sot down wit a missile it is seldom accomlised by te missile flying directly into te aircraft. More often, te missile reaces a oint of closest aroac and exlodes near te aircraft. Te sranel from te missile ten sreads out in a donut or serical attern from te oint of exlosion and oefully, te aircraft is caugt in tat sranel attern and destroyed [Ball85]. Tis algoritm can be used wit exloding roectiles, fireballs, and magic targeted at aircraft, dragons, and sacebugs. r φ φ 1 Figure 8. robability of Killing a Target in a Sranel Wedge were, x = na k v = 1 e ( πr ( cosφ cosφ )) 1 x n is te number of fragments or roectiles in te missile waread, A v is te vulnerable area of te target resented to te missile (in sq meters), r is te range from te detonation oint to te target (in meters), φ 1 is te angle from te traectory of te missile to te near edge of te vulnerability area of te target, and φ is te angle from te traectory of te missile to te far edge of te vulnerability area of te target. Beating te Buses Some engagements involve teams of unters searcing te terrain or buses for idden rey [Subik83]. Wen a large grou of unters is looking for a large grou of rey, it is ossible to model te cature or kill of te rey in an aggregate form, rater tan reresenting te individual movement and line-of-sigt of every unter and every rey. 8

9 As before, tis aroac is very valuable wen te unting and killing is being conducted by AI controlled unters and esecially wen it is aening off te layer s screen. Te algoritm is structured to calculate te cange in te oulation of te rey based on te number and efficiency of te unters. It also accounts for different tyes of rey and unter animals, e.g. small rodents, medium-sized wolves, and large eleants. To use te algoritm, we must define a robability of detection for eac tye of unter against eac tye of rey under te given conditions (oen terrain, forest, city, etc.). We must also select a ardness factor tat differentiates te ability of te rey to elude, escae, or survive te actions of te unter. Tese numbers are usually determined euristically troug exerimentation and observation. rey are Ligt Hunters are Dark Figure 9. Multile Tyes of Hunters Searcing for Multile Tyes of rey were, x = A = ( k ) n D * (1 e i, i= 1 x ) * i A is te number of kills of animal tye, is te number of rey of tye, k is a ardness measure of te rey in te range [0,1], is te total number of rey of all tyes, n is te number of rey tyes D i, is te robability tat a unter of tye i can detect a rey of tye, and i is te number of unters of tye i. Beating te Buses wit rey Sacing Te final finger of deat is a modification of te revious one. Matematicians and analysts noticed tat te revious algoritm did not account for differences in te density of rey iding in te buses. It is clearly muc easier to find and kill rey wen tere are 9

10 a undred of tem in te searc area tan if tere are ust two or tree. Terefore, tey created a variation known as te Luleian model [Subik83] in wic te sacing between te rey is an imortant factor. Te visual icture for tis algoritm is te same as tat above, but te matematics differ to account for te sacing of rey. Te definition of k also varies sligtly in tat Lileian defines k as te average destruction of te unters on rey tye. were, A n x = k i= 1 = * i * (1 e ( s * ) x ) A is te number of kills of rey tye, is te number of rey of tye, s is te average sacing between te rey in te searc area (in meters), is te total number of rey of all tyes, n is te number of rey tyes k is te average destruction of te unters on rey tye, in te range [0,1], i is te number of unters of tye i. Conclusion Te ten fingers of deat described in tis cater are ust a few of te combat killing algoritms tat can be alied to comuter games. Te concets of geometry, robability, statistics, and ysics used in te ten fingers of deat are good examles of aroaces to many roblems. Game develoers sould do wat military modelers do to imrove tese aly exerience, matematics, creativity, and oter sciences to find equations tat work well for your game. Don t be afraid to exeriment! References [Ball85] Ball, Robert E. Te Fundamentals of Aircraft Combat Survivability Analysis and Design. AIAA ress, [Estein85] Estein, Josua M. Te Calculus of Conventional War: Dynamic Analysis witout Lancester Teory. Brookings Institution, [arry95] arry, Samuel, Editor. Military OR Analyst s Handbook: Conventional Weaons Effects. Military Oerations Researc Society, [May0] May, Janet O. OneSAF Killer/Victim Scoreboard Caability for C Exerimentation, roceedings of te 00 Conference on Beavioral Reresentation in Modeling and Simulation, 00. [Subik83] Subik, Martin, Editor. Matematics of Conflict. Elsevier Science ublisers, [Army90] U.S. Army. Field Artillery Handbook. U.S. Army,