Brawl Mechanics: Damage & Knockback Formulae

[hotgarbage]

Wow, this thread is chock full of good stuff. I mean seriously.... it's fascinating to see exactly how the game works out the things we take for granted. Weight2 and "c" in particular are very interesting.

And it's about time someone completely cracked the stale moves system . Geeze it's harsh.....


Anyways great job on this ColinJF and Amazing Ampharos; keep it up!

 

[zacharia zako]

they should make awhole magazine dedicated to smash and u guys should be creators.

I never new brawl is so in depth like this with mathematical equations. Im gonna half to show this one to my dad who thinks brawl is a dumb game, and my algebra teacher while im at it.

now i know why some moves are excellent "KOing" moves: some moves have higher base knockback than others so it doesnt matter what percent they're at.

you guys should reasearch stuff like this more often like the multiplyer of GAW's down special, oil panick or something that has to do with grabs and throws.

all this could lead to tier list changes

 

[kupo15]

I have one question about your data. Are you sure your not missing a variable to be extra clear?

I think there is a multiplier for both the stale move and damage and that they are separate. I think Brawl basically added another stale variable (stale knockback) in addition to the stale damage variable. You use the same name in both sections so I'm wondering if you meant for that to be the same variable or two separate ones.

 

[ColinJF]

The stale move multiplier we figured out is the one for damage, which is why it appears in the damage section. As you can see in "Further Research", stale moves for knockback remains to be figured out (although the stale moves for damage affect knockback as well since knockback is based on damage). We're planning on looking at it when we get around to it. I've already made a few observations on it though.

For example, if a move launches a character with speed v at x% after the hit in training mode then it will launch the character with the same speed v at (x/1.05)% in versus mode if the move is fully fresh. (That's the only observation we've made on stale moves for knockback so far though. Expect more on it later.)

It's possible it's the same multiplier, but it remains to be seen.

[Edit: damage before hit -> damage after hit]

 

[kupo15]

The stale move multiplier we figured out is the one for damage, which is why it appears in the damage section. As you can see in "Further Research", stale moves for knockback remains to be figured out (although the stale moves for damage affect knockback as well since knockback is based on damage). We're planning on looking at it when we get around to it. I've already made a few observations on it though.

For example, if a move launches a character with speed v at x% before the hit in training mode then it will launch the character with the same speed v at (x/1.05)% in versus mode if the move is fully fresh. (That's the only observation we've made on stale moves for knockback so far though. Expect more on it later.)

It's possible it's the same multiplier, but it remains to be seen.

ok just checking. Maybe you should differential by using "stale damage multiplier"

From our brawl+ discussions, I feel that they are separate. As you said, you get decreased kb with stale damage and I believe brawl just added another one to that. But a hacker is looking into it to give us no kb decay but only dam decay so we will find out soon enough I suspect.

 

[Thinkaman]

Hey guys, what you are listing as "w2" is vertical acceleration.

If you want to check, run comparisons between Ness, Lucas, and Pit.

The more you know.

 

[ColinJF]

We speculated as much, but it's difficult to directly measure vertical acceleration.

 

[Thinkaman]

Do you guys have a Gecko? I just did it frame by frame on a tall custom stage with wavebirds on the same channel. This was ages ago though...

 

[leafgreen386]

Hey guys, what you are listing as "w2" is vertical acceleration.

If you want to check, run comparisons between Ness, Lucas, and Pit.

The more you know.

This doesn't make a whole lot of sense to me. Although my research isn't 100% concrete, I had thought I had least figured out all the basics from testing. From what I've seen, there is no such thing as "acceleration" in brawl. This is especially true considering we're talking about launch speed here. I'm quite sure that launch speed is just the force with which your character is launched. It's nothing fancy (besides the fact that it's also used for calculating hitstun). After the initial force is applied to your character, another force is constantly applied in the opposite direction of that force until they cancel out. This opposite force is applied starting straight from frame 1, so although I have called launch speed "the number of units you travel on the first frame of knockback" in the past, this is not quite accurate.

All forces in brawl are either applied initially or continuously. If you're rising due to a jump or something similar, you rise less each frame because the jump force was applied once, and gravity is applied each consecutive frame. When the jump force is extinguished, you fall at a constant rate equal to gravity, which is a separate calculation from the jump calculation that was going on the previous frames. This force is applied once and there is no other force that is applied to counter it. It's the same thing with fastfalls - you immediately start falling at the new rate. Additionally, knockback and fallspeed are two completely separate calculations and neither has an effect upon the other. I haven't investigated horizontal motion yet, so I can't comment on that. I can say with confidence, however, that it is also a completely separate calculation.

Do you guys have a Gecko? I just did it frame by frame on a tall custom stage with wavebirds on the same channel. This was ages ago though...

Hm, very interesting. How did you perform this test? Like, did you just have everyone start on a tower made of stage builder blocks and then run off the side? Everything I've seen so far seems to have supported that characters fall at a constant rate in the air due to gravity. If there is any acceleration, it has to be incredibly minuscule, considering it took the same number of frames for a character to travel each of 4 stage builder blocks in my tests. Even if it didn't last for long, the first should've taken more time to travel than the others if there was acceleration.

 

[SuSa]

What about moves that cannot stale, or have a set knockback?


(Peach's Jab, Sonic's first hit of Uair, Snake's first hit of Ftilt, Bowser's uthrow, etc.)

 

[ColinJF]

Those are covered in the original post, but they aren't the same class of moves. Having fixed knockback is neither a necessary nor a sufficient condition for not being affected by stale move decay.

 

[Villi]

What's the difficulty in finding out how knockback decay works? I would look into it myself, but I'm not very confident in my math prowess. The only problem I could think of was that the game only records the highest launch speed, but couldn't you just hit the pillars of Luigi's Mansion to decay attacks without it registering as a launch?

 

[ColinJF]

The "difficulty" is that it's not obvious how it works. It isn't in collecting the data.

At this point, it's clear there is only one stale moves multiplier, and I have an hypothesis how stale moves work for knockback, but it still need some more fleshing out.

 

[Adapt]

The "difficulty" is that it's not obvious how it works. It isn't in collecting the data.

At this point, it's clear there is only one stale moves multiplier, and I have an hypothesis how stale moves work for knockback, but it still need some more fleshing out.

I can already tell you that... It was worked out mainly by FadedImage, but I played a small part.

I'll check with him, and if he doesn't mind I will send you the info in a PM

 

[Thinkaman]

From what I've seen, there is no such thing as "acceleration" in brawl.

...wait... what? Characters certainly do not instantly obtain their max fall speed. Each character accelerates from a free-fall state to their max fall speed at a different rate. Instead of trying to saying something like "there's no such thing as acceleration", why don't you take 3 minutes to test it? I suggested Ness, Lucas, and Pit, due to their identical weights.

Spoilers: Pit has the fastest fall speed, but Lucas will fall faster initially due to higher acceleration, before Pit builds up his faster speed and slowly overtakes them.

 

[ColinJF]

Fall speed acceleration definitely seems to exist, and Thinkaman seems to be right that w2 corresponds to it. I've verified it in an indirect way that also gives us a great new result.

First, I defined a = (w[2] + 1.78)/1.78 so that a is the ratio of a character's fall speed acceleration to Mario's fall speed acceleration.

Then the average fall speed over a distance s is given by

average fall speed = s/(s/f - (a * (f/2a)^2)/(2 * f) + f/2a)

Now, the unit system for distance is a natural consequence of defining a and f to be ratio to Mario's fall speed. It turns out that the distance between the ground of final destination and the top blast zone is around 12.

Taking that into consideration, I plotted this graph of average fall speed over 12 units versus the launch speed required to kill a character straight vertically from the ground of final destination and this is the result:

Pretty linear, eh? In fact, r^2 = 0.9875.

I take this as confirmation that it is average fall speed that governs vertical survivability. With this we can already predict vertical kill percents. More on that later though.

(Edit: Fixed some computation errors from earlier...)

 

[leafgreen386]

...wait... what? Characters certainly do not instantly obtain their max fall speed. Each character accelerates from a free-fall state to their max fall speed at a different rate. Instead of trying to saying something like "there's no such thing as acceleration", why don't you take 3 minutes to test it? I suggested Ness, Lucas, and Pit, due to their identical weights.

Spoilers: Pit has the fastest fall speed, but Lucas will fall faster initially due to higher acceleration, before Pit builds up his faster speed and slowly overtakes them.

Well then it appears I was wrong. But when I was observing a character (mario) falling off the side of the stage frame by frame, there did not appear to be any change in velocity as he fell. Certainly the acceleration cannot be very significant, can it?

 

[Thinkaman]

What? When Mario first walks off the stage, his fall speed is zero. It increases over time to his max fall speed. Do you know realize how awkward the game would play if what you were saying was true, and characters jumped at a constant speed until reaching the apex, then immediately fell back down at that speed? The game would have absolutely zero smooth movements, as all moves would be completely jerky.

 

[CR4SH]

So, maybe I'm not getting it completely. Does that suggest that fall speed is calculated during the knockback of a move? In essence that a character is falling at their fall speed, and rising at their launch speed, simultaneously? That would make sense, but does it hold with your data? Plz correct me.

Also, I love to see that you guys are researchers. Well deserved, this is definitely the most valuable piece of smash research I've seen so far.

Lol, I want to look into the research room, I don't need to comment, I just want to sit back and love it.

 

[leafgreen386]

What? When Mario first walks off the stage, his fall speed is zero. It increases over time to his max fall speed. Do you know realize how awkward the game would play if what you were saying was true, and characters jumped at a constant speed until reaching the apex, then immediately fell back down at that speed? The game would have absolutely zero smooth movements, as all moves would be completely jerky.

Oh, I wasn't saying that characters jumped at a constant speed at all. My theory was that they jump using a one time force, and then gravity was a constant force applied each frame until the jump force was depleted, at which point they fell at a constant speed downward. The first part of this should still be correct, although I now see that characters continue to accelerate downward afterward until they reach a max fall speed, which admittedly makes more sense than my hypothesis did.

Really, my errors came from insufficient data and my inability to collect more.

 

[ph00tbag]

If you haven't found it yet, I can say for sure that stale move decay only affects knockback growth (fixed and set knockback do not decay), and the amount of decay is dependent on the amount of damage the move does. This is to say, G&W's full bucket will decay to a smaller percent of its fresh knockback growth constant than, say, Wario's uair. The following list (compiled by Faded) shows the percent to which a move's knockback growth constant decays fully in relation to its damage.

60 - 45.53%
38 - 46.56%
24 - 47.94%
22 - 48.27%
13 - 51.01%
10 - 52.79%
6  - 57.35%
6  - 57.34%
5  - 59.27%

The following chart shows what appears to be a logarithmic relationship between the two.

FadedImage and I haven't yet determined a precise equation for effect of damage on the stale move decay for each place in the queue, though.

 

[Adapt]

If you haven't found it yet, I can say for sure that stale move decay only affects knockback growth (fixed and set knockback do not decay), and the amount of decay is dependent on the amount of damage the move does. This is to say, G&W's full bucket will decay to a smaller percent of its fresh knockback growth constant than, say, Wario's uair. The following list (compiled by Faded) shows the percent to which a move's knockback growth constant decays fully in relation to its damage.

60 - 45.53%
38 - 46.56%
24 - 47.94%
22 - 48.27%
13 - 51.01%
10 - 52.79%
6  - 57.35%
6  - 57.34%
5  - 59.27%

FadedImage and I haven't yet determined a precise equation for effect of damage on the stale move decay for each place in the queue, though.

I took a look into determining the formula there, It didn't fit an exponential (a+b^-x) or logarithmic (a+logbaseb(x) )

So in the end I tried fitting polynomial curves to it and ended up with:

c = 0.0000179*x.^4-0.002509*x.^3+0.123844174*x.^2-2.675723221*x+69.6575725

fitting to an R^2 of ~.95-.99 (can't remember)

 

[ph00tbag]

I took a look into determining the formula there, It didn't fit an exponential (a+b^-x) or logarithmic (a+logbaseb(x) )

So in the end I tried fitting polynomial curves to it and ended up with:

c = 0.0000179*x.^4-0.002509*x.^3+0.123844174*x.^2-2.675723221*x+69.6575725

fitting to an R^2 of ~.95-.99 (can't remember)

Word to the wise: Excel is horrible at finding trendlines that aren't polynomial. ;-)

Actually, looking at it more closely, it could be exponential. It's so hard to tell. I'd want data for attacks that do less than five percent, but most of those have set or fixed knockback, so are pretty much useless.

 

[Adapt]

Word to the wise: Excel is horrible at finding trendlines that aren't polynomial. ;-)

Actually, looking at it more closely, it could be exponential. It's so hard to tell. I'd want data for attacks that do less than five percent, but most of those have set or fixed knockback, so are pretty much useless.

Didn't use excel... used a graphing calculator

And I think you, me, and Faded should make a thread in the ZSS boards for compiling all this data we have for random testing we have done. Maybe make it under TheZeroSuit so we all have access to the OP

 

[leafgreen386]

Hey uh... guys... it's sorta listed in the first post what the effect of stale moves is lol

And it specifies exactly what each slot does to the launch speed.

 

[ColinJF]

Well, we only figured it out precisely for damage.

It's known that launch speed uses the same multiplier though, but not exactly how, except that it only affects the launch speed growth.

The models that Adapt and friends are looking at are certainly just approximations though, as I'm not convinced the game actually uses any equation with a slew of random constants. The in game launch speed decay model is probably elegant. By the way, are those damages you are looking at supposed to be the base damage?

 

[ph00tbag]

The models that Adapt and friends are looking at are certainly just approximations though, as I'm not convinced the game actually uses any equation with a slew of random constants. The in game launch speed decay model is probably elegant. By the way, are those damages you are looking at supposed to be the base damage?

I think this is the point I'm trying to make. I'm positive that the decay on knockback is modeled very simply, and that it involves damage, I just don't know how it works precisely.

And yes, those are the base damages, which might account for the curve of the graph that I've put up. It might become less steep if the damages after decay are used. If the amount of damage a decayed move does is used in the equation at each step of the way, it might just all work out.

 

[Adapt]

Well, we only figured it out precisely for damage.

It's known that launch speed uses the same multiplier though, but not exactly how, except that it only affects the launch speed growth.

The models that Adapt and friends are looking at are certainly just approximations though, as I'm not convinced the game actually uses any equation with a slew of random constants. The in game launch speed decay model is probably elegant. By the way, are those damages you are looking at supposed to be the base damage?

I realize that my equation is most likely an approximation... but it was the only curve of best fit that I tried that showed any promise.

I think this is the point I'm trying to make. I'm positive that the decay on knockback is modeled very simply, and that it involves damage, I just don't know how it works precisely.

And yes, those are the base damages, which might account for the curve of the graph that I've put up. It might become less steep if the damages after decay are used. If the amount of damage a decayed move does is used in the equation at each step of the way, it might just all work out.

That's a very interesting point you both brought up... that would be awesome if by taking into account the decayed damage it turned into a very simple equation. We must take a closer look at that...

 

[Cherry64]

I'm just going to pop a hello, and that this is really interesting, but mainly so I subscribe to this thread and can read it when i'm not falling asleep. See ya!

 

[Blue Ninjakoopa]

Can someone tell me this in a way I can understand? Are the negative numbers corresponding to how floaty the character is?

 

[AlphaZealot]

On the MLG get better fast website I've also gone and done stale moves for about 15 characters through 10 moves deep-the data can be used to help confirm all of this stuff.

 

[Blue Ninjakoopa]

I just got finished with some tests. Fox's U-smash KOs Bowser at 103% damage, while it KOs Dedede at 111% damage. Snake and Donkey Kong also survive from the attack longer than Bowser does. Your list says Bowser falls faster (?) than D3, Snake, and DK. I'm confused.

 

[vbdood1337]

Is a perfectly fresh move (has not been used at all) any different from a move that has been used but has been refreshed completely by 9+ moves? So, if I use a usmash at 100% and kill off the top, will a usmash that has been used but been refreshed also kill at 100%?

I think they are the same, but some of the TL boards say otherwise.

 

[ColinJF]

On the MLG get better fast website I've also gone and done stale moves for about 15 characters through 10 moves deep-the data can be used to help confirm all of this stuff.

After we published this document, brawl+ hackers were able to crack the actual stale moves code in the game and our model was verified to be correct, so it's not in doubt.

We did actually refer to your work before we published this though as a quick check (since you didn't find all the decimals, it wasn't useful for actually figuring out the model in the first place though). So thanks for publishing those.

I just got finished with some tests. Fox's U-smash KOs Bowser at 103% damage, while it KOs Dedede at 111% damage. Snake and Donkey Kong also survive from the attack longer than Bowser does. Your list says Bowser falls faster (?) than D3, Snake, and DK. I'm confused.

This document doesn't say a thing about top fall speeds, so I don't know what you're talking about.

There are, however, a lot of factors involved in vertical kills: launch resistance, fall acceleration, and top fall speed. I have written a whole other document on the dynamics of knockback, so you should read it:

http://www.smashboards.com/showthread.php?t=216700

This should clear up your confusion.

Is a perfectly fresh move (has not been used at all) any different from a move that has been used but has been refreshed completely by 9+ moves? So, if I use a usmash at 100% and kill off the top, will a usmash that has been used but been refreshed also kill at 100%?

I think they are the same, but some of the TL boards say otherwise.

There are no magical stale move properties not covered in this document. If a move has been pushed out of the stale move queue then it is fully fresh again and the stale move multiplier will be 1.05 again.

 

[vbdood1337]

There are no magical stale move properties not covered in this document. If a move has been pushed out of the stale move queue then it is fully fresh again and the stale move multiplier will be 1.05 again.

Thanks for clearing that up.

 

[EverAlert]

The following moves are not affected by stale move decay: Luigi's down taunt; DK's cargo throws; tether attacks; Zero Suit Samus's neutral air.

I'm just looking for a little confirmation here. By tether attacks, are you talking every attack that can tether, or simply ZAirs? Originally when I read this (excellent, by the way) article, I assumed you were talking the latter, but after testing the Ice Climbers' up special, this has been thrown into question. Looking at the following damage list, the move is obviously not decaying just by glancing over the figures. I suppose it's obvious that you meant the former, but again, I'm looking for a little clarification here just to make sure.

Also, another thing I have been curious about for a while: do "glancing blows" actually do any damage/knockback or are they purely for show?

016
033
050
067
084
100
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386
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[Zarez]

I'll have to read more of this later (skimmed through), but I wanted to know if this will ever lead to a list of average knockback and damage for each character (like the weight list in the first post). I have always assumed that there are too many factors for there to be an accurate list, or a list that's worth the time that would be spent putting it together, but I'm still curious about what such a list would actually look like.

 

[fornaxxx]

is there a list of moves that are affected by decay.. really dont feel like testing them all with my chars

or are mosts move affected by decay and there some exception?

..thinking about it, does grabs are affected by decay? i think yes. They are registered in the decay list right?

 

[Corpsecreate]

Grab throws are registered in the decay list.

Did the stale effect on Knockback ever end up being calculated?