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Sensor Fusion Calculator

Several sensors, several answers. This combines them into one estimate weighted by how much each can be trusted, and tells you when they disagree more than they should.

Inputs

One row per sensor, all measuring the same thing in the same units. The error column is how far that sensor is typically off, which you get by taking a few readings of a known value and seeing how much they scatter.

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Results

Combined estimate

—

Its expected error

—±

Best single sensor

—

Better than that alone by

—%

Do they agree?

—

Most out of line

—

If you just averaged them

—

Sources

—

How much each one counts

SourceReadingError ±WeightOff the combinedIn its own error terms

Trust one sensor less

Trust one sensor less
1×

Each sensor's reading with the error you gave it, and the combined estimate underneath. A sensor counts for one over its error squared, so halving a sensor's error makes it four times as important. Slide the error on the sensor furthest from the rest: trust it more and the combined estimate is dragged towards it, trust it less and the others take over.

How this is calculated

Averaging sensors treats them as equals, which throws away the best thing you know: some of them are better than others. A tracking wheel accurate to a millimeter should not be outvoted by a GPS that wanders by three centimeters.

Weight by confidence, not by count

The right weighting is by the inverse of each sensor's variance, which is its error squared:

weight = 1 ÷ error² combined = Σ(weight × reading) ÷ Σ(weight) its error = √(1 ÷ Σ(weight))

Squaring is what makes this decisive. A sensor three times more accurate than another gets nine times the say. That is not arbitrary; it is the combination that minimizes the expected error of the result, and no other weighting does better.

Combining beats every individual sensor

The result is always at least as good as your best single source, and usually better. Even a mediocre sensor adds a little information, so long as its error is honest and independent of the others.

That last condition matters. Two encoders on the same slipping wheel are not independent: they will agree with each other and be wrong together, and this maths will report false confidence. Independence is what makes fusion work, so prefer sources that fail in different ways.

Disagreement is information

If a sensor sits several of its own error bars away from the combined answer, one of two things is true: the sensor has failed, or the error you gave it is too optimistic. Either way it is worth knowing mid-match, and it is the cheapest fault detection available.

The usual response is not to discard the outlier automatically, but to stop trusting the fused answer until the sources agree again.

Getting the error figures right

Everything here depends on those numbers, and a sensor given an unrealistically small error will dominate the result and drag it wrong. Measure them: put the robot at a known position, take twenty readings from each source, and use the spread. Guessing them defeats the purpose.

Sources & assumptions

No VEX data is used. This is inverse-variance weighting, the standard result for combining independent estimates, applied to your own numbers.

Assumes the sources are independent and unbiased, and that the error figures are realistic. It cannot detect a sensor that is consistently wrong in the same direction, because a steady bias looks identical to a correct reading from the outside.

This is the static case. A real robot also has motion between readings, which is what a Kalman filter adds on top of exactly this calculation.

    Every source the site uses, and when it was checked

    Save this run, and compare

    Keeps what is on screen so you can change something and see both sides of the change. Saved in this browser only, never uploaded.

    Make a notebook page

    Turns this calculation into a dated engineering notebook page: the setup, data, results, figures, working and sources, numbered and cited. You add the goal, the decision and the reflection in your own words, plus photos and sketches. Print it, paste it into Google Docs, or download a Word file or slides.

    My notebook

    Notebook page

    More
    Open My notebook

      This page

      Our own layouts

      A layout remembers how a page is set up: its type, its system, which parts print and which of your tables it uses. It never remembers a word, a number or a photo.

      Your team kept on this device
      Add someone to the team list
      Your words never written for you

      These are the parts a judge or teacher reads to see your thinking, so the page leaves them to you. Leave a box empty to get lined space for writing by hand. Start lines with "- " for a list. The notes under each box only count and point things out; they never change what you wrote.

      Rough notes

      Rough notes

      Rough notes

      Rough notes

      Rough notes

      Rough notes

      Rough notes

      Rough notes

      Rough notes

      The word counts can be measured against a target your team sets on My notebook. The reading level is Flesch-Kincaid grade level = 0.39 × (words ÷ sentences) + 11.8 × (syllables ÷ words) − 15.59. Syllables are counted by vowel groups, so the grade is an estimate of the school year the writing reads at. It says nothing about whether the writing is any good.

      One part at a time, with a clock. At the end of practice, while you remember it.
      How you tested your details

      Fill in what applies. These print as a small table under the method.

      Before and after

      Compare this calculation with a run you saved earlier on this tool (under “Saved runs”). The page shows what changed and by how much.

      A line through the data from your recorded table

      Pick two measured columns from the same table. The page plots them and draws the least-squares line, only across the range you measured.

      Robot log a CSV from the robot's SD card

      Load a CSV your robot saved. Pick the time column and up to three columns to plot, and type their units. Only those columns go on the page; the file stays on this device.

      Two versions' trials your data, compared

      Pick a column of recorded trials for each version, from this page or another saved page. The page shows both averages and a 95% range for the difference. The numbers are copied in when you pick them, so later changes to the other page don't change this one.

      Options compared your scores

      Score each option 1 to 5 on each criterion, and weight each criterion 1 to 5 by how much it matters. The page only multiplies and adds.

      Photos and sketches stay on this device

      Up to 4. Paste a screenshot with Ctrl+V, drop pictures here, or pick them. Photos are shrunk before they’re kept, and an unsaved page keeps them in its draft. Use Mark up to draw arrows, labels, dimensions and numbered callouts on a photo. Mark one photo Before and one After to print them side by side.

        Test plan written before testing

        Write this before you test: the question, what you expect with a number, and what result would make you drop the idea.

          Build steps our build

          One row per step, in order, so someone could rebuild from this page.

            Parts used our parts list

            Lengths in holes are also shown in inches between the end holes. Weights are only added up where you type them.

              Plan the cuts: packs every piece of one part (sizes in holes, or inches on the hole grid) onto whole sticks of the stock length you type.

              Ways it could fail our scores

              List the ways this could fail. Score how likely (1 to 5) and how bad (1 to 5); the page multiplies them and puts the biggest risks first.

                Designed against built our measurements

                What the CAD or the plan said, and what you measured on the robot. The tolerance is yours.

                  Changes from the last version our changes

                  What changed since the last version, why, and which result it was meant to move.

                    That page’s settings, results and first photo are copied in and set beside this page’s. The result each change should move is looked up on both pages.

                    Tasks our plan

                    What needs doing, who, and by when. Open tasks carry to the next meeting page until someone ticks them done.

                      The pages behind this idea our evidence

                      The pages in this notebook that show the idea being tried, measured or changed. A judge reading the claim can turn straight to them.

                        Time spent our hours

                        Who worked on this and for how long.

                          Session log what we did, as we did it

                          Notes taken during a session. The time comes from the clock when the note was taken; every word is yours.

                            Size limit does it fit?

                            Type the starting size limit from the current game manual, and the robot's size as designed or measured, in inches. The site doesn't store the limit, because it changes with the game.

                            What broke from your breakage log

                            The breakage log is on My notebook. Copy it onto this page as a table, as it stands now; later entries don't change this page.

                            CAD model which version this page is about

                            Paste a link to the model (Onshape, Fusion or anything else) and the name you use for this version. For Onshape, the page reads from the link whether it points at a saved version or a workspace that keeps changing. Nothing is fetched.

                            Your code
                            Comments notes for the team, not the page

                            A teammate or coach can leave a note here. Comments stay beside the page, travel with it in exports and links, and don't print unless you tick them under Include.

                              Sources you used besides this tool

                              Videos, other teams, manuals, people. The tool’s own sources are cited for you.

                                Sign-off
                                Include

                                Press a ? to see what an option is for, or read them all on How the notebook tools work.

                                What “The calculator’s figures” is for

                                The calculator’s own diagrams and charts, numbered as figures and drawn from your numbers. Judges look for pictures that explain a design.

                                What “How it was calculated” is for

                                The formulas and steps the calculator used, so a reader can check the arithmetic for themselves.

                                What “The calculator’s warnings” is for

                                Things the calculator flagged, such as a motor near stall. They’re quoted and credited to the calculator, so they never read as your conclusion.

                                What “The calculator’s notes” is for

                                The calculator’s plain-language notes on the result. Off at first, because they read like conclusions, and the conclusions on a page should be yours.

                                What “Lined space” is for

                                Ruled lines under any part you haven’t written yet, so a printed page can be finished by hand.

                                What “Writing space as” is for

                                The blank space for handwriting as ruled lines, a dot grid or quarter-inch squares. Dots and squares suit sketches and hand-drawn graphs.

                                What “A summary under recorded data” is for

                                The number of readings, the average, the spread, the lowest and the highest under each table you recorded, worked out from your numbers. It shows every run counted, not just the best one.

                                What “Words used” is for

                                A small box with the site’s one-line definitions of glossary terms the calculator used, for readers who don’t know them. It never looks at your own writing.

                                What “Robot at the time” is for

                                The Robot Profile as it was when the page was first saved, so weeks later you can tell which version of the robot a test was run on.

                                What “Long tables in full” is for

                                Long calculated tables printed complete on an extra page, instead of every few rows on the page itself.

                                What “Fit on one sheet” is for

                                Smaller type and shorter tables, aiming for one printed sheet. Use it when every entry in your notebook should be a single page.

                                What “Wider left margin” is for

                                A wider left edge, so the page can be hole-punched or bound without losing any words.

                                What “Larger type, for reading aloud” is for

                                Bigger print with more room between lines, for the team member who reads the page to a judge. It overrides Fit on one sheet.

                                What “Our rubric scores” is for

                                Your own scores against the rubric, on a sheet of their own, for checking a page before a judge does.

                                What “When this page was first saved” is for

                                The day the page was first saved, printed beside its date. It shows the notebook was kept while the work happened.

                                What “Teammates’ comments” is for

                                Prints the comments teammates left on this page at the end, marked as not part of the page. Off at first, because comments are notes for the team, not for judges.

                                What “Show every table and block” is for

                                Shows every table and block in the editor, not only the ones this type of page usually uses. It changes the editor, not the printed page.

                                Rubric check

                                  This checks what is on the page, not how good it is. Clear, specific writing in your own words is what earns the top marks.

                                  Preview, letter size

                                  Write it now

                                  Where this goes next

                                  Explore further on the full map

                                  Where the answers on this page can be used, and which tools could have given you the numbers it asks for.

                                  Comes after

                                  Odometry Calibration Calculatorgives you your position errorIMU Driftgives you your position error

                                  You are here

                                  Sensor Fusion Calculator

                                  Where your answer goes

                                  Odometry Calibration Calculatorfix the scaling on your wheels if they are the sensor out of lineuses your position error againLine Tracker Thresholdset a clean line-sensor threshold before trusting that reading

                                  a number carries across the next question it raises

                                  Keyboard shortcuts

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                                  Copy the results
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                                  Copy a link to these exact numbers
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                                  Print or save as PDF

                                  Letter keys do nothing while you're typing in a box.