Ball Trajectory Planning
 

Ugh, just got off the phone from a friend of mine with a masters in physics. I'm sure I'm not the only one starting down the painful math road of calculating accurate trajectory planning for small basketballs. I'm posting this in the hope it will help others or that others here who know more than me can contribute back.

According to my friend the problem is both very simple and very hard and he was unsure which terms would have a large enough impact to be relevant. In fact, that is ultimately the crux of the problem. If we account for everything in the equation then we are going to be far more accurate either way so I'm working from that assumption though I'm not sure if it's feasible.

To that end, we need to know quite a few things starting with the ball diameter, density, and mass (does someone have the weight of the ball measured accurately?). He was uncertain but he did feel that with the light weight of the ball the wind resistance might actually matter. Unless someone has access to a wind tunnel and wants to share results we are going to be guessing this one in part because determining the coefficient of friction for a pebbled surface seems exceptionally difficult. At best, we can use the friction coefficient for something like rubber to air and calculate it for a 7" diameter sphere at the temperature/pressure of the playing field.

This paper says drag through the air on a full size basketball is negligible:
http://www.phys.ubbcluj.ro/~evintele...na/Baschet.pdf

But then it gets shot down a bit here:
http://www.wired.com/wiredscience/20...n-basketballs/ You can see in the graph at the end here that air drag has a huge impact on a baseball:
http://wps.aw.com/wps/media/objects/...cs/topic01.pdf

Another potential nasty problem is the Magnus Effect. This is what happens when something is spinning and creating lift. This is one my friend felt we could likely ignore due to the low spin RPM though it's also one of the easier ones to do:
http://en.wikipedia.org/wiki/Magnus_effect

So what is everyone else currently looking at? How deep down this rabbit hole are you going? Thanks!

-Mike

Re: Ball Trajectory Planning
 

Keep in mind that this very topic is being researched in universities. It goes VERY FAR.


For the context of FIRST, I personally think only basic kinematics is required... Unless you are making an arm that will catch the ball mid air.

Re: Ball Trajectory Planning
 

If you're unsure, go with the simplest, and then physically test it. Math is great, but unless you test it IRL, there can be many variables you're forgetting about.

Re: Ball Trajectory Planning
 

I think this is the case for shorter shots. I'm not so sure for longer shots due to the light weight of the ball.
I appreciate the input but I actually disagree agree with this suggestion. I know that drag, magnus effect, etc. all have a real world impact. I am unsure if it's significant enough to cause a noticeable impact in practice. I think it would be better to start with the math being as accurate as it can be first rather than over-simplify and be left guessing where the inconsistency came from if it does matter. It's quite easy to compare the complicated math solution against the simple math solution for a sanity check. It's extremely hard to dial in accuracy from an inaccurate starting point while it's quite easy to take something super accurate and make it less so.

Now with that said, this is likely beyond my math/physics skill so it could very easily all become a moot point in the end :)

-Mike

Re: Ball Trajectory Planning
 

Well, being only a sophomore in High School, I must first applaud you for attempting the math involved in this. With my current skill level, I would never be able to calculate stuff like this. Being on a robotics team, I will secondly tell you that you can skip a lot of long, confusing math by making a prototype. While the math still exists, it's sometimes easier to mock up something and just change it little by little until it's perfect.

Hope this helps, and whatever path you choose for this game, I wish you the best! :D

Re: Ball Trajectory Planning
 

Once you have a basic shooter that is very similar to the final design, you can begin some experimentation. Obviously, all experimental data will have to be retaken with the final design. But what you should do is set your shooter to shoot at 0-degrees relative to the ground, or as horizontally as you can.

Using the same rpm, shoot the ball a dozen or so times, recording how far it went. With that, you can calculate how much velocity your shooter has at that rpm. If you plan on having a static rpm with varying angles and distances, you shouldn't need any more data beyond what you have. If you plan on varying your rpm as well, you will need to take more data to see how your velocity varies with rpm.

These balls are heavy enough, that I don't feel wind resistance will be too critical at 57' (maximum length you will never reach) but I could easily be wrong. Have your team member with the best arm go outside and just whip the thing. You should be able to see lift if it occurs. And if there is significant resistance, you should be able to observe that as well.

Re: Ball Trajectory Planning
 

This. I would see how far I could get with fitting a line to a set of data points.

Re: Ball Trajectory Planning
 

The physics degree in me says this is a very interesting question to pursue.

The engineer in me says 2 things:
1) speed is a factor in how much drag comes into play. I don't think you're going to be approaching baseball speeds unless you're going for the robot that parks in the alley, receives the pass and chucks it full court (if you are--bravo!).

2) this math would only matter after you have a mechanism that is VERY repeatable with the shot. I would worry about that error first.

Re: Ball Trajectory Planning
 

To at least provide some useful info, a paper in 2002 from a prof at the University of Waterloo proposed:

45 degrees + half the elevation from the shot origin to the basket.

Take it for what it's worth. I tend to believe him.

Re: Ball Trajectory Planning
 

Well, really it depends on how much power your throwing mechanism has. If it throws hard you'll have to put much more arch on it so that it goes in instead of bouncing off of the backboard. On that same note, if it throws too light, you simply can't make a basket. You will have to perfect the mechanism before you can do any math towards which trajectory your throw it.

I personally plan on doing research on how catapults and what not determine where what they throw will land. Sometimes they can be very accurate though they tend to throw odd shapped things which makes it more difficult to guess.

If at all possible, I'd like to be able to accurately throw a ball from the other end of the field and make it in though this will require very precision mechanics which I'm not sure the guys running the machines around here can do. One thing is off and all my code doing the angle estimations will do no good.

Well, good luck to you!

Re: Ball Trajectory Planning
 

I'm a senior year mechanical engineer student mentoring my former high school trying to do these calculations.

so a few things, Drag is a function of v^2, i would say that unless you are using a large air cannon it is pretty much MOSTLY negligible

Coefficient of restitution will probably be the greatest loss of energy you will encounter

if you want to tackle this i would say, construct a free body diagram, do some energy calculations, maybe some work calculations. balance using Newtonian mechanics and then try to come up with something feasible.

Good luck, and remember to just have some fun

Re: Ball Trajectory Planning
 

Spin can have a major effect on the path. Also a ball with spin has very different interaction when bouncing off the wall as we noticed today.In 2006 we choose top spin. After some simple testing today the opposite may be better.

Re: Ball Trajectory Planning
 

For what it's worth, my first impressions as a mathematician and former basketball coach... (and as a mentor who was around for Aim High)

Do some math first. You want to work out trajectories that bring the ball down in the center of the rim. The amount of arc is a function of the height above or below the rim and the distance to the rim. If you just start building prototypes you will likely make several designs which have fatal flaws. A little math ahead of time can avoid some of the mistakes.

Second, have someone with some skill at free throws shoot a few with this ball. Shoot some video (from a variety of angles) of the shots to give you some idea about trajectories. You might also want to watch a good free throw shooter shoot some shots with a regular basketball.

There balls deform a lot more than a basketball or baseball. That deformation will vary greatly depending on the mechanisms you use for shooting. This will have a lot of effect on the air resistance and flight characteristics of the ball. So there will be no getting around doing a lot of testing with prototypes.

The (horizontal) angle at which you are shooting at the rim will matter. In a theoretical world where your shooter is always the same, this isn't the case. But when you are shooting at an actual hoop, you are going to want to shoot as close to straight on as you can. There will be shots that are a little too long. If you are shooting close to straight on a decent percentage of the shots are going to bounce off the backboard and come back to the rim. This means you want to err a little bit on the side of too far rather than too short in your shooting.

Figuring out how quickly you can shoot without having the ball interfere with one another is another important thing to figure out. If you shoot too quickly you are likely to have one or two balls bounce out off of a preceding ball.

Re: Ball Trajectory Planning
 

Keep it simple, for the purpose of what most teams want to accomplish (shooting), stuff like drag, density and other factors can be set aside if you want to just shoot. If you have a general physics background, apply that math because even with calculating exactly how much speed, height, etc. needed to make a shot, remember we live in a imperfect world. If you want to have reasonable success and want to use math and physics, use the basics of physics formulas and you'll be fine. Otherwise build and test. Remember we only have 6 weeks to get a robot ready for a competition.

Re: Ball Trajectory Planning
 

Hi all; thanks for all the feedback. I just got off with phone with Rhett Allain (http://www.wired.com/wiredscience/author/rhettallain/) - he is a physics professor and the friend-of-a-friend. He feels that the Magnus Effect can be effectively ignored in these foam basketballs because of their size, density, and low RPM of spin. Likewise, while air resistance will certainly be a measurable amount, it's exceptionally hard to calculate and also likely to be within the inconsistency of the machine itself in the best case scenario. This also means temperature, pressure, etc. can all be ignored.

Looks like you guys were right and I was wrong - no need to over-complicate it. This is good news for us though as the physics problem is now just one of projectile motion and nothing too nasty.

-Mike

Re: Ball Trajectory Planning
 

Ive worked on finding a good formula to make a good curve but to no avail. I think im just going to make a chart when our launcher is working what maps out various distances and initial velocities/theta. This way you can just either create a curve in labview or just store the data on the crio to have the code refer to

Re: Ball Trajectory Planning
 

Thats great news, I'm glad you went ahead and did a lot of research and posted it all here.

The information you are giving may be very vital to some of the games played this year!

Re: Ball Trajectory Planning
 

I hear from some of my basketball friends that you want to aim for the square on the backboard, and that will guarantee you a basket.

Assuming you are shooting a basket straight on, could you just calculate the normal angle of the plane that would give you the most area between the uppermost point of the backboard and outward most point of the hoop that would give you the greatest chance of scoring?

Re: Ball Trajectory Planning
 

I've been doing some basic physics on shooting the ball with near-optimal trajectory (still working on the equation for optimal trajectory) and have found that in some situations, essentially a full-field shot, ball initial velocities will need to exceed 10 m/s without accounting for air resistance (or spin, for that matter). As such speeds, wouldn't air resistance come into play?

Honestly, I'm doing this for fun without expecting accuracy, but I'd love to get as accurate as possible.

Ether, if you happen to be around, I'd love to get your input on this problem, since you know pretty much everything.

Re: Ball Trajectory Planning
 

Heh, thanks. I'm trying to get this whole cooperation thing though it goes against my extremely competitive nature :)

Re: Ball Trajectory Planning
 

You don't want to aim for the square on the backboard in general in basketball if you are shooting from a distance from in front of the rim. You aim for the backboard (and banking the shot in) when you are in close to and below the basket, particularly from the side. For shots from more than a few feet basketball players are taught to aim for the hoop itself and not the backboard.

Re: Ball Trajectory Planning
 

ooooh got it, thank you!

Re: Ball Trajectory Planning
 

http://hyperphysics.phy-astr.gsu.edu...traj.html#tra7

This should do work for you! Gives you everything you need to know. As long as you know the speed at which the ball is launched and the angle it will tell you how far it will launch. With some manipulations with the equation you can determine much other stuff also.

Re: Ball Trajectory Planning
 

The size and weight of the basketballs is listed in the rules in 2.2.11

here:
The Basketball is a Size 4 Compact foam basketball (Gopher Deluscious™ Foam Basketball, Item # 71-502) with an outer circumference of 25 in. and approximate weight of 11.2 oz.

Re: Ball Trajectory Planning
 

Ah, thanks, I totally missed that!

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All of our foam balls weigh ~10 ounces. That's gonna be a problem :(

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I am growing more and more convinced that any kind of consistent shooter will be rendered useless by the thoroughly inconsistent balls. :mad:

Re: Ball Trajectory Planning
 

Oh, here are some calculators for finding ball exit velocities and angles I put in a different thread.

Forum post

Re: Ball Trajectory Planning
 

This is all great and well, but unless they give us the equation somewhere to be used in our code, this is essentially rendered useless without creating a reference table for the robot. I'm working on getting the equation with some professors right now though.

Re: Ball Trajectory Planning
 

Using the viscosity of the air, I'm able to estimate the resistive force on the sphere (since it is a sphere, it makes this calculation much simpler instead of using the drag calculations.. which is okay, but i just prefer this one).. at 10m/s you will encounter significant drag.

Here's an Octave/Matlab script that does the estimation (working with iterative solution... no calculus involvement yet.. which seemed rather difficult to do)

https://bitbucket.org/ultimatebuster...12/bbairdrag.m

Re: Ball Trajectory Planning
 

Diameter is 25"/Pi
Weight is 302 grams +/- about 2% (based on an unreasonable sample size of 3)
Density can be calculated, and is not necessarily uniform.

Also consider that the surface WILL change as the ball gets used in a tournament.

The Magnus effect is NOT trivial nor can it be ignored, if your launcher imparts spin greater than about 2 or 3 spins per second. That spin also has a significant effect upon how the ball will bounce after it hits the backboard.
Right. Use the match for a first order estimate, and go empirical from there.

BUT, understanding all those variables will come in handy when you try to solve any problems with variability of your trajectory.

Re: Ball Trajectory Planning
 

Ball diameter ~ 8"
Hoop diameter ~18"
That is some serious leeway for close shots, but a little tighter percentage-wise for shots from the top of the key and beyond.

What I would do, and will attempt to for my kids, is implement the solution to this set of equations:

y = height difference from launcher to hoop
x = horizontal distance from launcher to hoop center
a = angle above horizontal for launcher
t = time in air
g = accel of gravity

y = v*sin(a) - 0.5g*t^2
x = v*cos(a)*t

If you know y, and g, you can substitute and solve to eliminate t. Then the two variables are x and a. The analytic solution is, well, ugly, and you'll need a computer to assist, but if you enter values it is a bit easier.

Re: Ball Trajectory Planning
 

I made this for my team (you need GeoGebra to run it), which basically simulates projectile motion of the ball (ignoring all other factors). You can drag around V to shift the muzzle velocity and firing angle, as well as Height and Backboard to modify distances. Everything is considered relative to the point of release. There are obvious problems with it, but it does provide a nice model for the basic concepts here.

Re: Ball Trajectory Planning
 

Simple Ballistic Calc

http://inceptus.org/calc.html

Re: Ball Trajectory Planning
 

Wow, that GeoGebra thing is pretty sweet!

Re: Ball Trajectory Planning
 

I believe the ambiguity around the significance of the drag on the basketball vs. the baseball has to do with the density, size and speeds that they are being thrown at.

As an illustrative example imagine a two bowling balls dropped off a tall building. the first weighs 16 pounds, but the second has been hollowed out to weigh only 6 ounces. The exterior surfaces are the same. Neglecting wind resistance, both should hit the ground at the same time since gravity accelerates at -32ft/s^2, however in practice the heaver ball will land first. The aerodynamics of both balls are the same, but the added mass of the heavier ball will over come more drag.

With that said, I tend to agree with others here, that it's good to use math and physics, but my approach would be to keep the math as simple as possible, and build a mock up to see how well your physical results match your theoretical ones.

Re: Ball Trajectory Planning
 

I missed this thread earlier when I was looking for places to post my spreadsheet, so I made a new thread instead. I hope it doesn't fracture the conversation too much. :yikes:

Re: Ball Trajectory Planning
 

I agree that teams won't be approaching baseball speeds but a baseball is also significantly more dense than the basketball in the KOP.

I don't have any math to back myself up but 6 years of playing dodgeball with 6-8 year old kids at a summer camp tells me that even at low speeds drag effects the trajectory of a foam ball. I am sure that people involved in 2006 would back up this anecdotal evidence.

Re: Ball Trajectory Planning
 

But, it's NOT a sphere. The little bumps? They change the boundary layer behavior and aerodynamic effects.

This both increases the effective diameter of the ball and changes every other aerodynamic factor. The bumps will act to actually REDUCE overall drag by reducing pressure drag significantly; but it increases skin friction drag which increases the affect of most aerodynamic forces, including magnus effects. (same way that the little holes in a golf ball work... increase range, make it harder to remain accurate).

Just saying... if you want to be ultra accurate, you're forgetting some stuff.
And to anybody that thinks it won't end up coming down to testing and evaluation... well... good luck with that. Most of these simple equations are made with some extreme aerodynamic simplifications that will introduce an error of 10-25% in your calculations anyways.

Drag WILL be important. Magnus effect MIGHT be (depends on your launching mechanism). Being able to adjust your scaling factors (you should definitely have these) on the fly, mid-match, will probably be a nice thing to have.


I'd give you the math... but you either wouldn't understand it, or already know it.

Re: Ball Trajectory Planning
 

I think the general agreement by most is that the math is VERY complex if you include all the contributing factors, but in reality it's not that important in the long run. I think teams should definitely put thought into the trajectory of their throwers, but anything beyond simple kinematic equations is going to be wasted effort. I think the quote from Ian Curtis put it best,
Rather than worry about every little detail that could potentially throw off a throw, people need to realize that this game, by it's nature, is going to be full of variables. Heck, the balls are going to get sliced up, no doubt, so that right there is going to throw off any calculations of drag. The best thing to do is to do basic calculations to get a good idea, then build a shooter that is consistent, and try to get as accurate as you can but accept the fact that there will always be percent error. The old expression goes "measure with a micrometer, mark with chalk, cut with an ax."

Re: Ball Trajectory Planning
 

I'd have to fervently agree with the sentiment here.

But, some people still like to do the math. So I thought I'd hand over a few more tidbits of information like the word "boundary layer" to open a world to as much math as they could possibly want (and the realization that all the math in the world can't describe how air behaves)

Re: Ball Trajectory Planning
 

See this thread: http://www.chiefdelphi.com/forums/sh...ad.php?t=99485

Re: Ball Trajectory Planning
 

I cooked this program up a while ago... more like more than a year ago, and I'm not sure how it still works, but it does. Based entirely on the metric system, uses meters/kilograms/seconds/degrees/jouiles/etc std SI units.

I planned on making it able to work backwards using given variables but never got that far. Feel free to edit, compile and post. Executable in the zip, scource in the .c file.

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Make a sticker we can put on the side of the shooter giving you some "props" for the research?

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My thought is to build a (reasonably) consistent shooter and then trying to create an "auto mode" (similar to autonomous) where the operator, once he has the robot in range of the targeting system, could let go of the stick and put the bot into auto so it can position itself and shoot the ball... is that possible??

Re: Ball Trajectory Planning
 

This is why we believe the majority of this year's challenge is a programming one.
No successful basketball player has one type of shot making ability from different areas of the court or with defense.
However, our robots are limited to being of one type, thus making it a bigger challenge for the programming folks vs. everyone else.

Re: Ball Trajectory Planning
 

Definitely. Top teams will likely use this. You could do a more simple method (but possibly less effective) and instead of having the code drive, just have it alert the driver when they are in the correct spot.

I can't imagine having a wonderfully repeatable 3 point shooter and then having drivers try to eyeball it from across the court.

Re: Ball Trajectory Planning
 

Our team didn't even consider drive controlled shooting as an option. Imagine the driver with little dials for pitch, yaw, velocity, etc... They would never get anything done on the field.

Re: Ball Trajectory Planning
 

throwing a dodgeball as hard as you can is much faster than shooting from the key. Also, I think we can agree that the density of this year's ball is much higher than a dodgeball.

Re: Ball Trajectory Planning
 

There is a big difference between me throwing a traditional dodgeball (playground ball) as hard as I can and a 6 or 7 year old throwing a foam ball (simmilar in density to this years ball) as hard as he can (as described in my anecdote). The latter I am sure is much closer to the speeds we will be seeing in this game, and again I have observed the effects of drag on the ball.

Re: Ball Trajectory Planning
 

This is exactly what we did in 2006, and made it to the finals solely based on autonomous. In teleop we used the same function but because of defense we only able to score in teleop in a couple of matches. Where auto aiming will pay off is from anywhere where you can be incontact with they key, or close to the key. I have a feeling if you're a few inches off the key, a robot will be hesitant to interact with you.

In 2006 we were shooting 10 balls in sequence for about the same range. We shot at about 8 or 9 for 10. This year is a tougher shot, but you are only shooting 2 or 3 balls so you will be able to be more consistent by taking one shot at a time.

My hope is to also make it better by providing feedback, so when the robot takes a shot I can provide feedback as to if it was too far, short, left, or right. Allowing it to learn and adjust similar to a player would if it had to take a similar shot again. Also it could auto adjust to a location it knows a little better. Similar to a player shooting from a familiar location, or a foul shot.

Re: Ball Trajectory Planning
 

Good programming and STABILITY are the keys.

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Ok, I’m going to table the debate of whether drag and magnus are significant in order to clarify the second point I was trying to make: who cares?

remember the science lesson when you learned about accuracy vs. precision (repeatability)?

all the calculations discussed are concerned with accuracy; however, your FIRST concern should be building a shooter with very good precision!! drag and magnus won't cause bad precision.
drag and magnus depend on speed and spin. If you build a shooter that always has exactly the same speed, angle, and spin, then there will be an "accuracy" error, but the ball will land in the same place each time. a few tweaks and you're golden.

Re: Ball Trajectory Planning
 

Because teams will be shooting from different parts of the field requiring different speeds/angles/what have you.

Drag calculations can be very nonlinear even at speeds we may see in this competition.

Re: Ball Trajectory Planning
 

Your goals determine which is more important. A very precise robot is easier to defend, and more susceptible to problems. An accurate shot in this game has the opportunity to land in one of the other baskets. If you're shooting from the key, I'd definitely agree with you that precision is critical, however from the open court, putting it in a vicinity may be better than being precise.

Shooting for THE hoops as opposed to A hoop is not necessarily a bad strategy.

Re: Ball Trajectory Planning
 

I see. I guess I hadn't considered that teams would have the goal of being able to shoot from anywhere on the court. I was figuring for being able to shoot from the key, and from 1 particular spot on the opposite side of the court.

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You don't have to choose between precision and accuracy.

I'm not sure I understand what you mean

I'm assuming you mean a shooter that has GOOD accuracy and BAD precision MIGHT make it, where one with BAD accuracy and GOOD precision, WON’T make it. I agree, but I’m not encouraging the latter robot be built.

That’s completely fine if this is your strategy, but why worry about drag/magnus? Say your software automatically compensates your expected landing spot by 2 feet; however, mechanically your margin of error is 10 feet. What’s the point there?

Re: Ball Trajectory Planning
 

Samuel, Thanks for posting this. Very neat way to visualize the basic trajectories involved this year. I was moving the backboard value and got a y-displacement doing it that altered the paths. Whoa, says I. Backboard distance should not alter the path, should it? Then I went on a search for a formula using y(backboard) - fruitlessly. Turns out that theta is defined as an angle involving backboard as a vector. I changed theta's definition to be angle(V) and things look more reasonable now. But in case I missed something (neophyte geogebra-ist and all) I hope you can explain why you had theta defined the way you did.

Re: Ball Trajectory Planning
 

This pretty much sums up where we're headed. Design a consistent shooter (arc and distance) and shoot from the same spot (top of the key) with an auto-aimer with a swivel head (that will adjust for position of the target) and take a shot. I'm not sure we can design a program that will auto-adjust and "learn" as the tournament progresses... sounds very complicated to me, although we may have to manually adjust some parameters as the balls become more worn. Good luck!!

Re: Ball Trajectory Planning
 

Did I hear someone say something about this problem being at the University level?

I think 2D kinematics will be sufficient, but hey, maybe that's just trying to build for an ideal world. I'll post this link in the other thread to make sure that people see it.

Re: Ball Trajectory Planning
 

My team was actually planning on having a camera track the target, direct the turret to that target, use a laser rangefinder to determine distance, and have the turret adjust velocity and angle to adjust for that range. So, yes, some teams were planning on being able to shoot from anywhere.

We want to be able to get balls in the vicinity of the hoops from our player station, across the field, and were planning on a 70fps+ muzzle velocity. does that seem fast enough to range 55'?

Re: Ball Trajectory Planning
 

With a intitial velocity os 12m/s teams were able to shoot at a higher target from half court in 2006 with a ball of similar density, 70f/s should get you the whole way across the field. Keep in mind it will not be easy to shoot that far accuratly, and the faster you shoot the more effect drag has on the ball.

Re: Ball Trajectory Planning
 

If you do the math, at 45-degrees, neglecting drag and air resistance, you would need to be going about 40ft/s to shoot 50ft. Probably going to need a little more to hit that top hoop.

I really don't think the variable here is the drag though. You can calculate for that, sure. But after the first match, the majority of balls are going to be ripped, torn, and not appropriate for your calculations.

Keep the calculations simple, like everyone is saying. Going for a consistent shooter is the best you can really do. Doing the 2d physics is plenty because after that you are throwing in variables that are going to change after every match.