Character weight list, fall speed list and random things

MrSilver

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A little while ago I had a big math project for school. Since I’d just gotten Brawl and very much preferred playing it over doing homework I decided to combine the two and make my math project about Brawl. So I set out to figure out how the knock back system in Brawl works. To achieve this I looked at the “Fasted Launch Speed” at the end of match statistics. This shows how fast a character was sent flying and as such represents the knock back of a move. If you only connect with one move during a match this number will tell you the knock back of that move. So using the handicap function to set the percentage I tested many different moves on many different characters and found several patterns. First up, I found out that knock back is calculated after damage. So the percentage after the attack is what determines its strength. Next I figured out that like Melee each move can be divided in two parts, the base knock back that determines how much an attack would hit for at 0% (Which is theoretically impossible since knock back is calculated after damage) and an increment which determines how much the knock back increases for each attack. Both these parts are each affected by different attributes of the character that is being hit.


Weight:

This one was pretty simple to figure out since it’s the same as it was in Melee. The base knock back of a move is not affected by weight. So this only leaves the incremental knock back with damage. By calculating this incremental knock back against different characters and comparing it to the value I found for Mario I made the following weight table:

Bowser 1.111
Donkey Kong 1.091
Snake 1.076
Dedede 1.071
Charizard 1.061
Ganondorf 1.056
Samus 1.051
Yoshi 1.045
Wario 1.045
R.O.B. 1.040
Ike 1.035
Captain Falcon 1.030
Link 1.030
Wolf 1.020
Lucario 1.010
Ivysaur 1.010
Mario 1
Luigi 0.995
Sonic 0.985
Ness 0.980
Pit 0.980
Lucas 0.980
Diddy Kong 0.975
Toon Link 0.970
Ice Climber 0.970
Peach 0.960
Marth 0.944
Sheik 0.935
Zelda 0.935
Falco 0.919
Olimar 0.919
Zero Suit Samus 0.914
Fox 0.910
Pikachu 0.904
Meta Knight 0.904
Kirby 0.899
Squirtle 0.884
Mr. Game & Watch 0.884
Jigglypuff 0.849

The numbers in this table show the weight of a character compared to Mario, so Bowser is 1.111 times as heavy as Mario, Jigglypuff is 0.849 times as heavy as Mario. This table has proved to be 100% consistent with all moves I’ve tested.


Fall speed:

As I already said, base knock back is not affected by the weight of a character. However, different characters still have different base knock back for the same move. Upon further testing I found that this seemed to be related to the fall speed of characters. With characters like Jiggly and Samus having a low base knock back and characters like Fox and Wolf having a high base knock back. My theory is that this was added to account for the increased susceptibility that fast falling characters have to combos. I also found that the amount by which the base knock back for characters differs is not the same for all moves. For weaker moves it’s the same, but for stronger moves it seems to differ a lot. To account for this I took Ike’s fully charged forward Smash as a bench mark and assigned it a value of 100 for the deviation of its base knock back per character. This number is purely arbitrary but it does allow me to make a list to compare fall speed. This gave me the following list. Again I took Mario as a benchmark, hence he has a value of “0”.

Jigglypuff -0.78
Samus -0.54
Olimar -0.47
Kirby -0.44
Lucario -0.40
Peach -0.37
Luigi -0.32
R.O.B. -0.32
Zelda -0.26
Marth -0.25
Toon Link -0.16
Ivysaur -0.16
Ness -0.12
Yoshi -0.05
Pit 0.00
Mario 0
Mr. Game & Watch 0.01
Ice Climber 0.07
Snake 0.16
Donkey Kong 0.20
Ike 0.28
Dedede 0.29
Wario 0.30
Charizard 0.33
Sonic 0.33
Pikachu 0.40
Zero Suit Samus 0.44
Link 0.46
Lucas 0.49
Diddy Kong 0.55
Meta Knight 0.68
Bowser 0.91
Ganondorf 0.91
Captain Falcon 0.92
Squirtle 1.05
Falco 1.23
Wolf 1.84
Sheik 2.21
Fox 3.41

It should be noted that these numbers shouldn’t be takes as absolute values. They show how much a character deviates from Mario. So you can’t say “Fox falls 3 times as fast as Squirtle” but you can say “The difference between Fox and Mario is 3 times larger then the difference between Squirtle and Mario”.


How much of an impact do these two values have on knock back?

As a general rule you can say that weight has far more of an impact on knock back then fall speed. For example, Jiggly falls significantly slower then Bowser and as such takes less base knock back. But in most situations she’s still fly faster because of the weight difference. The only time fall speed makes a noticeable difference is with fairly weak moves or at low percentages. As such the added knock back fast fallers get from attacks should not have much of an impact on their survivability.


The mathematical formula

For the purpose of posting this here I’ve cut out most of the math. However, I would like to include the formula for calculating knock back:

v = d * a / m + b + f * c

v = is the velocity a character has when it flies away.
d = the damage the character has after the attack
a = is the incremental knock back of a move on Mario
m = the weight of the character being hit from the list I posted
b = the base knock back of a move against Mario
f = is the value in the fall speed list above
c = is a value that determines how much the base knock back differs based on fall speed. For attacks where b is smaller then 2200 (the vast majority of moves in the game) this value is 100. For attack above 2200 it seems to vary between 70 and 170.


Other interesting facts I found while testing

First up, the percentage given in the game is not very accurate. The percentage a character has is internally tracked as a decimal number, but only displayed as a number without decimals. So a character with 90.5% will fly further then one with 90.0% even though the game will display both as having 90% damage. The same is true for moves, many moves don’t do a whole number of damage. Ike’s Forward Smash for example does roughly 32.3% damage when fully charged. Also, the game does not stop counting at 999%. So taking damage above 999% will still increase the knock back you take, even if the added damage is not shown in the game. To account for this I worked out what real damage that moves did by using it over and over on a single character in a box shaped level I made (obviously with a suicide after each attack to reset the diminishing returns) and then for all my other tests I set the starting damage with the handicap mode to ensure it was a number without decimals. This way I could calculate the exact damage after a given attack with decimals.

There is a maximum knock back a character can take, namely 80529mph. However, this takes an enormous amount of damage to reach. For example, Marth Final Smash, which is the strongest move I tested will do roughly 80000mph against a Jiggly at 300%. Many other moves won’t even reach this maximum at 999% damage. Obviously such a knock back results in instant death.

The knock back displayed is given in “Miles per hour”. However, this is quite obviously complete bull**** since the numbers don’t match up with what’s happening on the screen. A knock back of 1000mph will hardly move a character, which is clearly nonsense. However, this does not matter for the weight list I’ve derived from it since it’s based on the relative differences in knock back between different situations, not on what the absolute numbers are. My guess is that it’s simply an internal number that they’ve added a unit to too show what it represents.

Wiimotes don’t give a perfectly consistent signal. I used Wiimotes for testing because they’re wireless and it would happen fairly frequently that an attack would go of early while charging it. Which probably happens because of a break in the signal which the game interprets as the player releasing the button. However, these breaks seemed to be very short, probably only a few frames, so for anything other then charging attacks it shouldn’t be very noticeable.
 

WhiteWingDemon

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Nice Work. I wish I did something like this for my math project.
For the formula, is it v = (d*a)/(m+b+f+c) or v = (d*(a/m))+b+f+c ?
Did you round with significant digits?
 

Tyr_03

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Wow yeah this actually answers a few questions for me. Explains why squirtle gets destroyed by Lucas's PK Thunder edgeguard (low weight but higher fall speed) and why Fox gets trapped in the tail of PK Thunder worse than anyone because of his high fall speed. Thanks a ton.
 

KURiSUTARU

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Wow, that's just awesome that you would base a math project on Brawl. Creative, interesting, and a heck lot more fun than most other possibilities, I'm guessing.

Great job, this is a cool list, and also should help some people understand a few things. I'd say more than this but I'm not too good when it comes to math...
 

usea

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Wow thanks for posting this information matt. Really appreciated.
 

MrSilver

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Nice Work. I wish I did something like this for my math project.
For the formula, is it v = (d*a)/(m+b+f+c) or v = (d*(a/m))+b+f+c ?
Did you round with significant digits?
It follows normal mathematical precedence, so that would make it v = (d*a/m) + b +(f*c). And yes I did round with significant digits.

Good job, except M2K released his weight list yesterday.
Yeah, I saw it. But I don't wish to rain on his parade but his way of gathering data is not as accurate as mine. As I already said weight does not influence base knock back. Since his data is based on the total knock back this includes the base knock back. Since different moves have different base knock back that means he'd find a slightly different weight list for every move he tried. Also, some of his tests where influenced by other character attributes like fall speed.
 

Magus420

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Can you use your formula to explain this sample of KO percents for Snake's U-Throw in M2K's list?

140 194 99 101 .... Lucas
140 200 100 100 ... Marth
140 195 99 100 .... Toon Link
131 185 93 94 ..... Zelda
131 185 92 93 ..... Sheik

Lucas 194
T Link 195
Marth 200
Zelda 185
Sheik 185

I believe the throw does 10%, and I'm guessing he listed it before the damage (I don't have the game to be able to test anything).

Lucas .... V1 = (204 * A / 0.980) + B + (+0.49 * C)
T Link .... V2 = (205 * A / 0.970) + B + (-0.16 * C)
Marth .... V3 = (210 * A / 0.944) + B + (-0.25 * C)
Zelda .....V4 = (195 * A / 0.935) + B + (-0.26 * C)
Sheik .....V5 = (195 * A / 0.935) + B + (+2.21 * C)

For starters, if Marth and Zelda have nearly identical weight and fall speed, how could there be such a large difference in KO percents between them?


EDIT: Also...

144 201 102 103 ... Mario
143 203 102 103 ... Pit

Weight
Mario 1.000
Pit ... 0.980

Fall Speed
Pit ... 0.00
Mario 0.00
 

Magus420

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Also, with the decimal damages, do you know if the display ever rounds up for the damage it shows, or does it only show full %s of damage? After reading a thread Simna made about damage done in training mode sometimes showing 1% less than in versus mode, I hypothesized that training mode does not keep track of damage as a decimal, and only tracks whole damage points. This also explained why KO damages appeared as though they were 1-2% lower than in training mode. If the game NEVER rounds up for the damage displayed and/or training mode does keep track of decimal damages then well... I'm not quite sure what to make of the damage difference between the two.

http://allisbrawl.com/forum/topic.aspx?id=5731
 

MrSilver

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It doesn't round of, so 10.9% will be displayed as 10%. As for training mode, I didn't use it because it lacks end of game statistics, so I don't know how it works with percentages. However, it should be pretty easy to test if it tracks whole percentages. Just take Fox and start blastering a character. If after a while each shot only does 1% without every doing 2% then decimal numbers are not tracked since fully diminished from repeated use Fox' blaster does somewhere around 1.3 or 1.4 damage from my experience.

About those kill percentages you listed. When you start looking at how far characters fly a lot of other things have to be taken into account. For example, in my experience it seems some character decelerate faster then others, so they have to be sent flying faster then others to get the same distance. Fall speed also influences how how far a character flies with a certain initial speed with upwards trajectories. There might be more things that come into play since the physics engine after the initial knock back is a lot more complicated then the part I researched. This is also exactly why I based my lists of the in game statistics for initial speed, since it greatly reduces the number of factors that influence my results, making them more accurate and consistent.
 

Magus420

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It doesn't round of, so 10.9% will be displayed as 10%. As for training mode, I didn't use it because it lacks end of game statistics, so I don't know how it works with percentages. However, it should be pretty easy to test if it tracks whole percentages. Just take Fox and start blastering a character. If after a while each shot only does 1% without every doing 2% then decimal numbers are not tracked since fully diminished from repeated use Fox' blaster does somewhere around 1.3 or 1.4 damage from my experience.
There is actually no stale moves in training mode at all though, which is pretty strange because not only is there no stale moves, but the attacks aren't even at full power and just do ever so slightly less damage than normal everytime regardless of how much you use it. Also, I don't have the game myself which is why I asked you about them since few people are even aware that the smash games use decimal damage totals.

About those kill percentages you listed. When you start looking at how far characters fly a lot of other things have to be taken into account. For example, in my experience it seems some character decelerate faster then others, so they have to be sent flying faster then others to get the same distance. Fall speed also influences how how far a character flies with a certain initial speed with upwards trajectories.
Yes, I was thinking the same thing about it at first and figured it was due to fall speed acceleration or something else that resists the movement while they are moving. However, with each of those attacks on that particular character they are travelling almost the same exact distance and with the same vertical trajectory. Staying within the scope of a single character's physics, wouldn't the minimum launch velocity required from an attack for that particular character to travel the same distance while following the same trajectory in more than one instance be the same?

With the case of Mario and Pit: Pit and Mario have the exact same 'fall speed' for the knockback calculation in your formula and Mario is also heavier than Pit. When calculated with the same ending damage using the formula, Mario will always have a lower launch velocity than Pit because he is heavier and more weight can only decrease the velocity.

144 201 102 103 ... Mario
143 203 102 103 ... Pit

In all 8 cases they are travelling essentially the same distance, and for each 4 per character that char needs a minimum launch velocity to be able to cover that distance in total when factoring in the effect of that character's physics slowing them down over that distance. If Pit will always have a higher launch velocity than Mario at a given %, then logically if Pit is able to survive for ~2% longer than Mario from an attack, Pit must be decelerating faster over the course of that distance than Mario. However, if Pit is able to decelerate faster enough over that distance from the ground to blastzone to make up for the faster launch velocity and more by surviving ~2% longer, then why would he not decelerate nearly as quickly when covering that same distance when being hit by the first attack and actually die sooner than Mario?
 

mugwhump

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Yeah, knockback is slightly reduced in training mode. It is teh mysterious.
 

OlimarFan

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Wow nice work this is going to be very useful. I must admit, I was pleasantly surprised that Yoshi was quite heavy: I would imagine him just above Mario or something.
 

Doval

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Mr. Silver, I don't mean to undermine your work, but I measured each character's fall speed, and I got vastly different results from what you got. I did my tests by letting the characters freefall a very long distance, and measuring the time (at 1/4 speed in training, for greater precision. Obviously the time was divided by 4 afterwards.) I repeated this thrice, and averaged the times. I had the characters jump before going through the platform on top of the stage I made, so they'd already have accelerated when I started the stopwatch and thus acceleration wouldn't be a factor. These are the results I got:

Fighter Ratio Speed (b/s)
King Dedede 1.521 5.749
Captain Falcon 1.433 5.417
Fox 1.421 5.371
Wolf 1.402 5.3
Snake 1.351 5.107
Falco 1.329 5.024
Ike 1.288 4.869
Ganon 1.28 4.838
Link 1.248 4.717
Sheik 1.233 4.661
Donkey Kong 1.232 4.657
Pikachu 1.172 4.43
Marth 1.168 4.415
Sonic 1.138 4.302
Zero Suit Samus 1.123 4.245
Diddy Kong 1.121 4.237
Pit 1.106 4.181
Wario 1.088 4.113
Meta Knight 1.084 4.098
Bowser 1.083 4.094
Lucas 1.068 4.037
Ivysaur 1.051 3.973
Charizard 1.049 3.965
Ness 1.024 3.871
Olimar 1.02 3.856
Squirtle 1.02 3.856
Yoshi 1.007 3.806
Toon Link 1.005 3.799
Mario 1 3.78
Mr. Game & Watch 0.969 3.663
Luigi 0.954 3.606
Ice Climbers 0.941 3.557
Lucario 0.941 3.557
R.O.B. 0.939 3.549
Kirby 0.939 3.549
Zelda 0.884 3.342
Samus 0.833 3.149
Peach 0.825 3.119
Jigglypuff 0.768 2.903

(Please excuse the poor formatting, I pulled these numbers straight out of Excel.) The first number is the ratio between that character's speed and Mario's, and the third number is the actual speed value in terms of Stage Editor blocks/second. The fastest-falling character in the game is DEFINITELY King Dedede, yet your list shows him as being nowhere near the fastest fallers. Could it be that what you measured wasn't actually falling speed, but falling acceleration, or some other internal statistic the game doesn't let us know about?
 

TKD+ITA+Mar=

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I think it could be very possible that there is a fall acceleration and a fall top speed. In Melee there was at least, I read something by M2K that said C. Falcon and Falco fall faster than Fox, but Fox accelerates to his top falling speed faster.
 

Gindler

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I can't help but wonder if people took into account that yoshi has 2 different falling speeds, one after his first jump and a slower falling speed after he uses his flutter (2nd) jump.
 

Doval

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Yoshi doesn't fall faster after his aerial jump. I checked by having two Yoshi's fall a very long distance, and having one of them use its aerial jump. It took longer than most characters to accelerate to top speed, but once it did, the distance between the Yoshis remained the same.
 

Sosuke

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Wow. So this is when math pays off.
 

fissionprime

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this is brilliant work and im sure you must have spent tons of time on it. Hope your teacher plays brawl so they can see how great your work is.
 

ph00tbag

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Mr. Silver, I don't mean to undermine your work, but I measured each character's fall speed, and I got vastly different results from what you got.
Judging by Mr. Silver's own analysis of his work, I'd say you two found two completely different values. You've found the free fall state falling speed, and he found the fast fall compensation factor. Falling speed itself has nothing to do with initial launch speed calculations, but rather with deceleration. On the other hand, launch speed does seem to be influenced by the values Silver found.
 

Amarkov

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I don't think that base knockback depends on fall speed. I've been testing things, and it seems to be constant for a given move.
 

brg

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I had the impression Dedede was one of the fastest falling characters, yet he is close to mario in your values...why is that?
 

Ilex

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One thing I wish you did do was post actual values instead of just how they relate to Mario. Or at least, I think Jigglypuff would've been a better model for weight since she is the very end of the spectrum. And falling speed doesn't really show how fast they fall, just how much faster they are than a randomly chosen character. Maybe I'll make a similar list that times how many blocks they fall in one second or something.
 

MrSilver

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Judging by Mr. Silver's own analysis of his work, I'd say you two found two completely different values. You've found the free fall state falling speed, and he found the fast fall compensation factor. Falling speed itself has nothing to do with initial launch speed calculations, but rather with deceleration. On the other hand, launch speed does seem to be influenced by the values Silver found.
You're probably right on that. I'd even go as far to say that there are even more factors involved, such as the rate of acceleration when falling. The values I found may be a direct result of one of these factors or they may be independent values that where added to compensate for the effects of the other factors of falling. As such they might not be an exact analog for fall speed but you'll still see fast falling characters at the top and slow falling characters at the bottom.

I don't think that base knockback depends on fall speed. I've been testing things, and it seems to be constant for a given move.
The impact on base knockback is very small in most cases. It also doesn't scale at all with percentages so at higher percentages it's almost negligible.

One thing I wish you did do was post actual values instead of just how they relate to Mario. Or at least, I think Jigglypuff would've been a better model for weight since she is the very end of the spectrum. And falling speed doesn't really show how fast they fall, just how much faster they are than a randomly chosen character. Maybe I'll make a similar list that times how many blocks they fall in one second or something.
The problem is that the game doesn't give you anything to reference weight by. A number only has value if you have something to relate it to. What you relate it to is mostly arbitrary. Like you said, I could have picked Jiggly or Bowser as my benchmark weight because they're at the extreme ends of the spectrum. However, this wouldn't have made any difference in how the numbers relate to each other. It would have just produced a higher or lower value for each character depending on which character I picked. I just went with Mario since I did my initial testing with him so I had a lot more data on him compared to the other characters, making him more accurate as a benchmark character.
 

Sonic The Hedgedawg

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you SURE link only has that fall speed? he falls like a brick when he fast falls
 
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