Math/Physics in FRC

With many of our team members being juniors next year, we would like to use a lot more math in our robot design. This year, we did some to figure out our elevator and some gear ratios, but we have never really focused on math when designing. We don’t have a lot of kids on the team, and even less alumni/mentors. We do have a machinist who comes in and helps us sometimes, but it’s not a day to day thing.

We do have kids on the team who have taken/are taking physics, as well as some advanced math, so we definitely have the possibility of including math in our design and construction, but as of right now it is very minimal.

What kind of math/physics do other teams do in order to improve their robot design?
How much math do teams do?
Is math/physics more of just a backup, or are mechanisms designed around that knowledge?


I recommend searching through the numerous examples of Ether’s work hosted here at CD. A great example is his ballistics math, which is really fun to implement and test! Practice by making measurements of drag and lift coefficients this off-season and tuning a model with a new prototype shooter! Surely 2020 will be a shooting game! Get going!

Ether has posted many other great examples, check them all out!


If you don’t start designing your mechanisms by doing the physics to figure out how they’ll work, you’re basically just guessing. Calculating geometries, trajectories, gear ratios, wheel speeds, etc is all math and physics.

Pretty much all of our mechanical students understand the basic high school physics needed for designing simple mechanisms. Some of the older students have already taken physics in school when they join, and the younger students that haven’t are taught by the older students. Those students who are really interested in the mechanical engineering will learn more advanced physics and engineering, up to college level. Learning how to actually design things properly instead of just saying “that looks good” and making it takes a ton of math and physics.

I can’t say for sure what math/physics students in the other subteams learn and use on my team, but there are certainly tons of applications. Programming uses a ton of math, and lots of advanced programming uses linear algebra (college level math). Scouting can use statistics and linear algebra to analyze data. Electronics uses physics that a student might learn in high school (AP Physics C: E&M), and more advanced physics if you want to get into the internals of the components. Strategy can use advanced statistics to predict how games will play out after kickoff, and to generate the best match strategies based on scouting data.

One advantage of having a team of students only grades 10-12 is that even the new students come in with some more advanced courses.


Shooters and parabolic arc is one often used and referred to.

Plotting exit height, angle, velocity vs. target window and whether using an APEX style shot (2014, 2016 target for tower window) vs. “three-pointer” arcing into a flat target like 2012 and 2016.
Calculating the target exit velovity vs. the gear ratios required to hit motor speeds.
Calculating system rotational kinetic energy vs. energy for projectile. Hint, if it is a back-spin style shooter, the rotational kinetic energy can be on the order of the linear kinetic energy.

A really interesting one can be robot designed geartrain ratio vs. top speed vs. time to distance vs. target travel distance. For full field, sprints, you might want a higher top speed. For games like 2012 that often involve short sprints under 1/4 length of field, a lower top speed, but faster acceleration can often be a better configuration.

A real simple piece of math is often discussing matters related to weight. Say you are using 1x1 aluminum box. It can be good to compare how many holes you would have to drill in order to equal reducing gauge to 1/16th wall. I have found that to be an eye opener for many people.


An arm driven by an air cylinder wants you to use some math and physics. The cylinder is usually mounted to make a 3rd class lever. Calculate the mechanical advantage, and calculate the force needed at the end of the lever arm and the amount of force the cylinder would have to provide to achieve that force. Then you can select a cylinder bore. Then you can calculate how much air each stroke uses. This example is nice since none of the pieces is really complicated by itself.



There are two kinds of engineering: the real kind is grounded in math and physics.

The other kind is practiced by the likes of Wile E. Coyote, and generally doesn’t end well.


It usually ends up pretty well for the road runner (or Bugs), the audience, and the studio…

Seriously, totally agree with you and Ari. Basically, every dimension, gear ratio, and motor selection should be based on some sort of calculation or at least objective comparison with some alternatives. Sometimes this is as simple as making a cardboard shape and figuring out where a pivot point needs to be in order to reach two objectives, more often it’s using JVN’s calculator (or at least the methods in it) to figure gear ratios, and sometimes it runs to full blown calculus.


Few from this year:

(1) Our robot had an arm on the front to lift the front end of the robot onto the 2nd level. The arm was about 8 inches long and was turned by a gear attached to the arm. That gear was turned by a smaller gear that was driven a Versaplanetary gearbox. Vex has a chart someplace about how much load is too much on a VP gearbox. Figure out the size of the gear on the arm, the smaller gear, the gear ratio in the VP and what motor to use.

(2) Once the front end of our robot was on the 2nd level, we used a pneumatic “butt lifter” to lift the tail end of our robot up so we could get onto the 2nd level. Figure out the bore size of the cylinders.

(3) An arm is driven by a sprocket connected to a chain. The sprocket has 74 teeth and the arm has about a 90 degree travel. In addition to the sprocket driving the chain, you have a third sprocket on the chain, that sprocket will be mounted to a potentiometer that you’re planning to use to measure the angle of the arm. That sprocket is McMaster part 2737T119. Do you buy a 1-, 3-, 5- or 10- turn potentiometer?

We used the math in this presentation to simulate elevator speeds / response times when selecting a gear ratio.

If you are talking math on gears I currently go through this again with a couple of newcomers - mostly freshmen

its very helpful if you “roll” your own gears.

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I second all of the above, but haven’t yet seen anyone mention power. If you want to be good in FRC, you don’t just need to be able to do the thing reliably, you need to be able to do the thing reliably and faster than other robots do the thing.

Slow and reliable is better than fast and unreliable most of the time, but fast and reliable is always better.

P = Fv = (tau)(omega)


Tagging on to @pfreivald, a lot of calculations center around work/energy, especially when selecting pneumatic cylinders or springs. If you first calculate the amount of work you need the cylinder or spring to do, then double it for a safety/inefficiency margin, you can quickly winnow down the reasonable candidates to do the job.
Energy equals:

  • [average] force times distance (special case: weight times height)
  • Torque times angle (in radians)
  • Pressure times change in volume
  • mass times speed squared, halved

I agree about the energy equations, but I generally find power equations more useful. For most things in FRC, it’s not just important that you can do the thing but how long it takes to do the thing. Power = Energy / Time. If you can lift a weight a certain distance, but it takes the whole match to do so, that’s not very effective.

As others have said gear ratios are one of the most useful uses of math in FRC. Too much torque makes you slow, not enough torque makes you useless.

Another useful math application is pneumatic volume estimations, where you take the volume of your cylinders, the psi you will use them at and the estimated match usage to determine how many/what kind of air tanks you need on the robot.

Also using math, game theory and what you know about how your events/regionals play to determine where you should focus your limited resources on during build season.

I have the exact same viewpoint as you. I love using math and physics in design, and being able to study the motions of things through CAD analyzation, and would love to get my hands on some topology software.

As the lead CAD and design, I have noticed that on our team (we are also a small team), we usually focus on the design, and hope the physics play out right. I feel that most team’s mechanisms aren’t that precise that they need high level math application for reassurance. Ex. I have never seen something like a FBD (free body diagram) in FIRST. Of course, there are specific mechanisms where you have to some calculations however. Things like pneumatics, really.

If you CAD, the option for math is all around you, although CAD does most of it. You can always use trigonometry in vision targeting as well. That was our main use of math. For example if you are working with a piston for a latch, you can find out where holes should go for your desired motion based off of the stroke.

I also felt the need to revive this as we get closer to build season. Also, I share the same view.


There’s some slightly complicated physics in continuous elevators that free body diagrams can help people understand. Also, simple physics equations like V^2 = 2a∆X come in useful sometimes.

As a Mechanical Engineer, I love Free Body Diagrams! Your specific comment about not seeing them used in FRC prompts me to add the the following “top of the mind” examples of where we (2767) have used them recently:

  1. 2018: FBD of robot with a climbing partner robot on our ramp to determine the required fulcrum point for the climb winch rope so that the ramp tipped up before leaving the ground (preventing partners from being dumped off).
  2. 2019: FBD of robot climbing to Hab 3 using a suction pad–to determine the required area of the suction pad as a function of vacuum depth and reach over the Hab lip.
  3. Every year: FBDs of spur gears in mesh is used to describe how gear ratios work, leading to extensions regarding pulleys/sprockets, and discussions such as trading torque for speed, etc… We also use them to show the power of separating bearings on shafts in order to support moment/side loads and demonstrate mechanism leverage/acceleration.

Note that although we perform the calculations regarding these FBDs with students, we don’t set our designs based on the minimum values calculated. We use generous safety factors to cover all of the wear, tear and general mahem conditions encountered in FRC. The point of the calculations is to properly visualize the loadings, apply academic concepts to the real world, and garner a feel for how robust (or NOT!) a design is likely to be.


Same here. The first place several team members ever saw one was early in my drivetrain class under the heading of “how to keep the robot from falling over”.

I probably still wouldn’t get how the torque actuated strafe modules work without one.

Here’s a great Blue Alliance blog post I recently ran across. It’s from 2013, so there is no Neo or Falcon or even 775pro insight, but there are some great fundamentals covered here:

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