Figure Eight Motion Drive




About: I'm a total gearhead. I love working on cars, airplanes, rockets, sailboats, whatever. Cars i've owned and/or rebuilt: 1966 Pontiac Grand Prix, 1977 Lotus Esprit, 1979 Fiat X1/9 & Strada, 1970 Ford Bronco...

Back in the 90s I was very interested in Micro Aerial Vehicles (MAVs).

I researched hummingbirds and bumblebees in hover flight and discovered their wings flapped in a figure eight motion. This motion is similar to how you would move your arms and hands when trying to stay afloat in water.

So I researched all kinds of mechanisms and designs that could provide this motion and was surprised with the complexity of every system.

I decided to figure out my own mechanism and started running simulations using a kinematic dynamic software package called Mechanica Motion. NOTE: Mechanica is now ProEngineer which can be found at

For my first simulation i was thinking that a figure eight looks like two circles touching each other. Based on that i created a two gear system with two connecting rods. I arranged the gears to first provide a circle as they both spun at the same speed. I then thought if I spun one of the gears twice as fast as the other i should see two perfect lobes. Well i didn't. I got something close to a lopsided figure eight with one side having a convex shape. So i played with phasing of the gears and still no perfect figure eight. A colleague of mine happened to be visiting and asked me to spin the two gears at the same speed. I told her i already tried that and proceeded to show her. It was a truly eureka moment as the mechanism traced a really small figure 8! It was all in the phasing.

So after a few optimizations the perfect figure eight motion was accomplished with 4 basic parts, ie 2 counter rotating gears coupled with two connecting arms.

I was so excited, applied for a patent and two years later the US Patent & Trademark Office gave me the saddest news of my life. The mechanism had already been patented ( #2,775,899 ) in 1957 by C.L. Vagneur .

During the anxious two years i researched features like adding a second pair of connecting rods to flip the "wing" during the figure eight motion.

I looked at other applications like rowing & bike pedaling mechanisms, hair braiding, ceiling fans, wind energy device, etc . I researched the web on all things figure eight and found it was a common motion for polishing, cleaning and mixing.

Anyway, after discovering Instructables I thought it would be great to share all this info for anyone looking for that perfect figure eight motion drive!

Sabri Sansoy

The steps are more of slide presentation.

Step 1: Frame by Frame Trace

Frame by Frame analysis showing how the point where the two connecting rods are connected traces the figure 8. The two gears must counterrotate at the same speed.

Step 2: Front View Diagram

Each connecting arm must be attached at the outer rim of each gear (points C & D) and then attached to each other at pont P

l = length of connector arm which should equal twice the radius of the gear
r = radius of circle
x = distance from origin

Step 3: Motion Graphs

The top left graph shows the Horizontal Position vs Time of point P in the previous slide.
The top right graph shows the Vertical Position vs Time.
The bottom left graph shows the Velocity Magnitude vs Time.
The bottom right graph shows the Acceleration Magitude vs Time

Step 4: Predicted Motion Vs Empirical Results

NOTE: This is not an Apples to Apples comparison.

The right side graph depicts real measurements taken from high speed photography of a fly or bee in forward motion. (Reference to follow)

The left side of the graph shows the velocity of point P for the various positions along the figure 8.

Just what was interesting was how the two patterns are somewhat similar.

Step 5: Reduced to Practice

Using a Lego robotics kit it was really easy to assemble a working prototype.

Step 6: Variations on a Theme - Smaller Eight

x is greater than r
l equal to twice x

Step 7: Variations on a Theme - Droopy Eight

l is greater than twice x
x is greater than r

Step 8: Variations on a Theme - Smiley Eight

x is greater than r
l is less than twice x

Step 9: Variations on a Theme - Lopsided Eight

The length of connecting arm l1 is longer than connecting arm l2.

Step 10: Drive Shaft Configuration

As long as the phasing between the two counter rotating arms can be maintained any variation of the mechanism can work, depending on the loads, natural frequecies, etc. of course :)

Step 11: Making the Wing Flip

Hummingbirds and bumble bees flip their wings during the back and forth motion.

Part of the challenge was to figure out how to do this with my figure eight mechanism.

After some thought and analysis, the solution was to add a second pair of connecting arms (8 & 9) right behind the primary set of arms (6 & 7).

Point 10 represents the leading edge of the wing and point 11 represents the trailing edge of the wing.

When i first modeled this in the simulation package the mechanism locked up.

After further investigation it was discovered that the distance between 10 and 11 vary thru the course of the figure 8 motion.

So the fix is to put some sort of slider joint between the two.

What's not shown here is the side view that shows the second pair of connecting arms are offset ,ie sit on top of, from the first set of connecting arms. The two pairs of arms would collide otherwise.

Step 12: Micro Aerial Vehicle

Here's one concept for a micro aerial vehicle. This more closely represents how a hummingbird or bumblebee would fly. Another option not shown here is to attach the wings directly to the drive, in other words points 2 would be attached at the connector arms and not the frame 3.

Step 13: Animations

Figure Eight Motion Drive Demonstration - Micro Aerial Vehicle

Figure Eight Motion Drive Demonstration - Bike Pedaling

Figure Eight Motion Drive Demonstration - Bike Pedaling with Legs

Figure Eight Motion Drive Demonstration - Horizontal Mixer

Figure Eight Motion Drive Demonstration - Horizontal Mixer

Figure Eight Motion Drive Demonstration - Sculler



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    19 Discussions


    6 years ago on Step 2

    What's the exact phasing of C and D to get a perfect figure eight?

    1 reply

    Reply 6 years ago on Introduction

    My question too. Measuring Step 6 figure 1 with a screen tool reads: C 28º above and D 28º below horizontal.

    Any chance of posting a close up of the slider mechanism in the above photo?


    9 years ago on Introduction

    It takes a genius to design this concept. Great work!


    Reply 10 years ago on Introduction

    I made these animations about 10 years ago using an animation package called Lightwave from NewTek.
    Basically, I exported the motion data out of Mechanica Motion and imported it into Lightwave.
    There is a free animation package you can download today at .

    It has inverse kinematics (IK) which should allow you to make the same mechanism and any of your own design.

    More info about IK can be found at


    10 years ago on Introduction

    Thank you! Great instructable. I know this has nothing to do with the actual subject of the instructable, but the last video is a great explanation for how to drive a boat using a single oar. I had always wondered how single-oar boats worked after seeing one on television. Aside from being an interesting instructable on a fascinating topic, the progression of videos at the end makes it simple to extract both the method for propelling such a boat along with the reason that one would use that particular method. Great job!


    10 years ago on Introduction

    AMAZING! AWESOME! (what more I can say?) Thanks and congratulations!


    Reply 10 years ago on Introduction

    I had begun to build a prototype but i was so crushed when the patent office gave me the news i gave up on this for 10 years. I just checked out your awesome ornithopter instructable. i'm so busy at work but it would be great if you made a figure 8 ornithopter!


    Reply 10 years ago on Introduction

    thanks, ill defiantly consider it. but isn't the figure 8 only good for hovering on a flying vehicle? or could you possible use linear motors to move the cranks to change from hovering to accelerating


    Reply 10 years ago on Introduction

    I like to use the analogy of a helicopter in hover. Its rotor blades create a downward force just as the figure eight drive. a helicopters forward motion is achieved when you tilt the blade system forward. in theory you should be able to do the same thing with tilting forward the figure eight mechanism. Also in hover both the bumble bee and hummingbird assume the rear bottom down approach, ie improving the center of gravity from a stability standpoint. exactly how its depicted in this image
    Lift the rear portion up and off you go into the forward direction....maybe..its only in i do need to build a prototype.


    Reply 10 years ago on Introduction

    good analogy. a prototype would be a big help in proving the viability of the design, and if you need any help just ask as im sure a good amount of people here would be interested. however i do suggest building the boat idea first just as a proof of concept and ease of design. in fact, i may just build one myself :)


    10 years ago on Introduction

    All those avi files... If you uploaded them to YouTube or similar, then embedded them here, you would get a lot more views.

    5 replies

    Reply 10 years ago on Introduction

    (Oh, and if reality is as promising as those videos, you should definitely build a prototype!)