Ladder Game Paper

 A Quantitative
Analysis the Carnival Ladder Game Using My Cellphone

Katelyn L. Chu

Saint Francis High School, Mountain View, California

September 16th, 2019

Abstract

In this paper, I use accelerometers and gyroscopes built into cell phones to analyze the ladder game at the carnival. The objective of the game is to climb up a rope ladder attached at only two ends without flipping over. I have been very successful at climbing the ladders at Great America, Santa Cruz Boardwalk, and most recently Canobie Lake Park. The purpose of this paper was to delve deeper into how I was successful. I determined that guidelines such as taking steps with the right arm and left foot (or vice versa) at the same time were fairly helpful. They were useful in enabling me to minimize my twist, or rotation around my center of mass. I determined my maximum twist to be +/- 2 radians/sec. And while finding the ideal speed proved difficult to hypothesize, I determined it to be around 2 seconds per step, or fast enough so my muscles wouldn’t get too tired but slow enough so I could still climb carefully.

 

Introduction

I have been interested in cancer research for several years now and have discussed cancer related topics on my biochemistry blog.  [1] Also during this time, I attended various summer programs such as Rosetta Institute’s Molecular Biology of Cancer Workshop at Berkeley and Stanford’s EXPLORE Lecture Series on bioengineering, bioinformatics, and genetics. Each program inspired me to do my own hands-on project and publish my results in a public forum. [2] [3]  This last summer I attended a program about Molecular Oncology in Boston.  I have not decided what type of project I would like to do yet relating to the material I learned during the program, however, some of my new friends and I went on a fun weekend outing at Canobie Lake Park, New Hampshire.[1] It was here where I was introduced to Feller’s Folly Ladder Game. “Test your balance in making it to the top of the ladder. Ring the bell and get a HUGE PRIZE!”   My friends challenged me to make it to the top.   Little did they suspect that I have been training and winning prizes on carnival ladder games for years.   I have been donating the large stuffed animals to children at the Lucile Packard Children’s Hospital and have even made a YouTube tutorial for the carnival ladder game. [4]   In any case, after I won, I tried to show my friends how to make it up the ladder.  Most of my explanation was qualitative as is my YouTube tutorial.   On my way back to Boston, I was on my cell phone texting friends trying to think how to explain how to win more quantitatively.  As I stared at my phone, I realized that the phone has built in sensors like accelerometers and gyroscopes and could possibly help put some numbers and analysis in my explanation.   This is a result of my investigation[2].

What are Accelerometers and Gyroscopes ?

In a nutshell, accelerometers and gyroscopes are sensors that detect movement by inertia.  This is different than radar guns used by police or closed current loops that detect cars at a stop light.   These inertial sensors are placed in or on devices or items and can sense if it is picked up, moved, rotated, or dropped.

Accelerometers are sensors that measure translation. They can measure the acceleration along the X, Y and/or Z direction depending on the orientation of the sensor.  If there are three orthogonal accelerometers in a device, each accelerometer will report the acceleration component in a given direction for any arbitrary translation.  The translation components are denoted by the solid arrows in Figure 1.

Gyroscopes are sensors that measure rotation or “twist” around an axis.  Similar to accelerometer example, if there are three orthogonal gyroscopes in a device, each gyroscope will report the “twist” component along a particular axis for any arbitrary rotation.  The rotational twist component can be seen by the dashed arrows in Figure 1.  By using both types of sensors in a device, things like orientation, position, and/or velocity can be calculated.

Figure 1 – Accelerometer and Gyroscope directions

Why are these Sensors in my Cell Phone ?

There are many ways accelerometers and gyroscopes are used in a cell phone.  They measure motion, vibration, tilt, incline, rotation, etc.   It is how my cell phone knows if I rotated the screen from portrait to landscape view and automatically adjusts the screen for me.   When my dad plays his race car driving game, the sensors allows the phone to be used as a steering wheel.  The sensors “sense” the orientation of the phone and how fast he is rotating it to adjust for curves and how much he is shaking it when he crashes in the game (he is a terrible race car driver… but a great dad 😊).   When my mom takes her phone on a hike with her friends, the accelerometers and/or gyroscopes are able to count her steps and reports them to a fitness app.   The sensors are even used during GPS applications.  When my phone loses the GPS signal from the GPS satellites, like when you go thru a tunnel or if the signal is blocked by skyscrapers in a big city like San Francisco, accurate positioning systems fill in the gaps with data from these sensors since it knows how fast you are moving, what direction you are moving, and if you are changing direction.   There are sometimes other sensors in your phone like magnetometers (for compass-like measurements), barometers (that measure air pressure or altitude), and sometimes even fingerprint sensors for security.  In any case, there are some apps that allow the user to see the raw data from these sensors, record them to a file, and send them to computer for analysis. For my experiment, I downloaded the Sensor Kinetics Pro App by Innoventions, Inc.[3]   It is inexpensive, allows family sharing so I can install it on several phones, and allows me to record the data to a file for me to analyze.

The Carnival Ladder Game

The carnival ladder game is a game that can be won with skill rather than by luck. The object of the game is to climb using your hands and feet from the bottom of the ladder to the top of the ladder and ring a bell. The difficult part is that the ladder is at an incline and very unstable as it is only secured by two points (usually swivels) indicated by the red arrows in Figure 2.  The key to winning the game is all about maintaining control over your center of gravity[4]. Easier said than done, right ?

Figure 2 – The Carnival Ladder Game

The center of gravity is a point where the weight of the body is concentrated, and the resultant force is zero. In other words, it is the point of balance.  For most people, it is around their belly button.

The carnival ladder game is all about balance.  The general guidelines for making it up the ladders are: placing your feet wide apart from each other will give you a better control over your center of gravity, each of your moves should counterbalance the other.  For example, when your right foot moves up, your left arm should follow to create a kind of equilibrium between both sides of the ladder.  I suppose technically you could also reach forward with both your right and left hands simultaneously to maintain equilibrium with respect to the centerline of the ladder but it seems like it would be much more difficult.  As I mentioned previously, I have trained on the ladder game for several years after my dad and I built a carnival ladder setup in our backyard to practice on.

The Experiment

For this experiment, I will attach a cell phone to the center of the ladder using bungee cords and a cell phone to my approximate center of gravity by way of a elastic belt.  Each phone will have the previously mentioned sensor app running.  This way I can compare the motion data between my center of mass relative to the ladder’s center of mass.

I will record my climbs using two different video cameras.  One camera will videotape my progress from afar and another camera (a GoPro) I will wear on my head so you can see what I am looking at as I climb.

Boundary Conditions

In order to understand and interpret the data correctly, we need to understand orientation of the phones.   We can do this by examining the boundary conditions.  Since the ladder is fixed at two points shown by the red dots in Figure 3, we can assume that the ladder should have minimal translation along the axis, Y, since the ladder is made of rope stretched between the two points.  The ladder’s center of mass should not move much in this direction.   It is much easier to move in directions X and Z since most carnival rope ladders sag a little (not pulled completely tight) which allows the ladder to move a bit in these directions.  Using the same understanding, we should expect that the ladder’s gyroscope data should show minimal rotation along two of the axes, X and Z, with most of the rotation occurring around the axis, Y, defined by an imaginary line connecting the two points of the ladder, as shown by the green rotation arrow in Figure 3.   Any small deviation from these boundary conditions can be possibly be explained by either non-ideal conditions like sagging rope or slight misalignment of the cell phone relative to the ladder.

Figure 3 – Ladder Axes

Similar boundary conditions apply to the accelerometers and gyroscopes in the cell phone attached to my center of mass. There will be much more variation however as I am not rigidly attached to the ladder and when I fall, my center of mass can freely go in any direction.   Also, note that when the cell phone is placed in my belt, the orientation of the cell phone’s accelerometers and gyroscopes will be different than that of the ladder since the cell phone is placed lengthwise along my belt.    This can be accounted for during the data analysis.

There are two different coordinate systems that the sensors on my belt sees.   When I am standing vertically, walking around, and just before I step onto the ladder, the sensors report X’, Y’, Z’ coordinates. Once I step on the ladder and put my hands and feet on the steps and my body is relatively parallel to the ladder, the sensors will report X, Y, Z coordinates.  This is a 3-dimensional coordinate rotation and can be solved using Matlab or various other programs.   I was able to find a helpful image [5] on internet that visually explains 3-dimensional coordinate rotation.  See Figure 4.   3-Dimensional coordinate rotation is used widely in robotics and 3D computer graphics and there are various user created routines and Matlab codes that are publicly available to perform this transformation.

Figure 4 – 3-dimensional coordinate rotation – see Reference [5]

Accelerometer/Gyroscope Orientation

The phone app, Sensor Kinetics, will report 3 accelerometer values (X, Y, Z) or 3 gyroscope values (X, Y, Z) with respect to time.   In a vacuum, we could use the boundary conditions above to determine the orientation.   The other way is to play around with the phone with the app running. For example, when I drop the phone from a short height like 1 cm (my phone is too precious to drop it any higher), I can see the Z-accelerometer change.   If I move it slightly length-wise, the Y-accelerometer changes.   When I move it width wise, I see the  X-accelerometer value change.   This leads to the directions shown in Figure 5.   Again, playing with the phone with the app running, I find out:  rotation around z is yaw, rotation around y-axis is roll, and rotation around x-axis is pitch.

Figure 5 – Accelerometer and Gyroscope orientation on phone

 

To avoid confusion and to make the analysis more understandable, I will refer to Figure 6 for the ladder game orientation.  The direction up the ladder will be known as “Towards A” or basically the Y-axis.   The side to side motion will be called “Sway”.  

Figure 6 – Ladder Game Orientation

Setup

For this experiment, I attached a cell phone to the center of the ladder using bungee cords and a cell phone to my approximate center of gravity by way of an elastic belt.  I paid particular attention in how the phones were oriented since the app will be measuring values with specific orientations, i.e. vectors.   I also attached a GoPro camera to my head and videotaped myself with a camcorder climbing up the ladder.   I did this since I want to correlate and analyze the sensor data as I make my way up the ladder and see what happens before I fall and what happens during a successful run.

Since the app on both phones will record sensor data independently and I will most likely be starting them at slightly different times, I need to create a “sync” signal to synchronize the videos and the two different sensor data sets.   After I start all the devices, I will hold both cell phones and jump up and down.   This should cause a measurable spike in both data sets and can be synchronized to the videos easily by noting the timestamp on the video data.

Since the app only will not allow me to record accelerometer and gyroscope measurements concurrently, I will do a total of six climbs.   A set of three climbs to collect accelerometer data and a set of three climbs to collect gyroscope data.   Each set of climbs will consist of me: falling off below the ladder’s center of gravity (fall #1), falling off above the ladder’s center of gravity (fall #2), and making it up the ladder completely achieving the goal (success!).   This is illustrated in Figure 7.   After that, I will compare and analyze the data sets and attempt to reach a conclusion.

Figure 7 – Climbs: Fall #1, Fall #2, and Success

     Results

Below are the plots of the accelerometer and gyroscope data after the climbs.  See Figure 8 and Figure 9.   I adjusted the datasets to be consistent with the phone orientations that was attached to the ladder and the one that I was wearing.   In the figures, I highlight both falls and when I make it up the ladder successfully.   I am also including two videos here:

I superimposed the GoPro video and the sensor data on top of the video of the climbs.  I used the previously mentioned “sync” signal to the synchronize everything.    Since the app data provides the sensor data as a file, I created a Matlab script to plot the data point by point over time to simulate real time data so that we can better visualize what the data means as I do the climbs.   I added comments to Figure 8 and Figure 9 showing the milestones and observations.

Figure 8 – Accelerometer Data

 

 

 

 

Figure 9 – Gyroscope Data

 

Discussion

As I mentioned previously, the accelerometer data is separately recorded from the gyroscope data.   This is a limitation of the app because it does not record both accelerometer data and gyroscope data simultaneously.   This could be possible in a future update of the app.   In any case, we will not be able to line up the accelerometer data side by side with the gyroscope data.   It should be okay if we can look at each data set individually. 

If we understand the boundary conditions, as previously discussed, “Sway” should not only be seen as that ladder’s X-axis translation but also ladder’s Y-axis twist.  And if I fall, I will fall in the Z-direction or “Down”.    Note here, that although I will be falling straight down defined by gravity, we will see accelerometer components in “Towards A” or Y-axis due to the previously mentioned 3-dimensional coordinate transformation. Also, most likely I will see a “Sway” component right before I fall.  This is because when I fall, I do not fall cleanly through the ladder, but as you see in the videos the ladder ‘twists’ and ‘sways’ when I fall off the side of the ladder.

Figure 10 shows only the “Sway (X) component of accelerometer data attached to the ladder and attached to me.  We can see the periodic left-right motion, highlighted in the red dotted boxes, in both sets of data as I successfully move up the ladder.  This is due to the slight left-right shift of my center of mass as I switch my hands and feet.  This can be verified in the video.  Note the magnitude of the periodic left-right motion.  They are small motions with a period of about ~4 seconds in duration.  This would be considered the pace in which I can climb successfully, i.e. transition from right arm/left leg to left arm/right leg or vice versa approximately every 2 seconds.

 Figure 10 Accelerometer data – Sway (X) component only

Figure 11 shows the other accelerometer data in the other directions.  While the data shows a similar pattern before Fall #2 and success, it is less pronounced and the pattern is not apparent before Fall #1.  This is expected since the accelerometer components in these directions should be small as explained previously and that distance traversed before Fall #1 is not far enough to generate the periodic shift in center of gravity.

Figure 11 – Accelerometer data other than Sway (X)

 

If we examine the gyroscope data in Figure 9, we can see much larger values in ladder gyroscope data, primarily during the falls, in comparison to  values of the gyroscope attached to me.  The values in between falls seem to agree in magnitude.   This makes sense.   I would expect that when I am on the ladder, both the ladder’s and my center of mass should experience similar twisting force.   I would expect the discrepancy should come during or right after I release the ladder as I fall off.   Since the ladder is much lighter than me, it can twist and turn much more than me and with higher magnitude (thank goodness!!).

Figure 12 only shows the gyroscope data in the Sway (X) direction.  One thing to note here is the tight distribution roughly +/- 2 radians/second in the gyroscope attached to me.  I highlighted this by the two dashed lines.  In other words, when I am on the ladder and between the falls, there is not much twisting of my center of mass along the ladder’s centerline.  For the most part, the times it falls beyond this range is when I fall off or when I am not even on the ladder, like right before I am climbing on or when I am reattaching the phone to the ladder.   This gives us a maximum range of values allowed before it becomes too difficult to stay on the ladder.

Figure 12 – Gyroscope data in Sway (X) direction

 

Conclusion

In conclusion, I have shown using a cell phone’s built-in accelerometers and gyroscopes can be used to analyze the fundamentals behind the ladder game. While some of the measurements confirmed general guidelines like “when taking a step with your right foot, your left arm should follow in order to maintain equilibrium between both sides of the ladder”, a maximum range of a person’s twist (along the ladder centerline) was observed. This range is +/- 2 radians/sec.  When going outside this range, it becomes difficult to stay on the ladder.  Using the accelerometer measurements, I was able to calculate my pace at about 2 seconds per step using the left hand/right foot and right hand/left foot method.   Of course there are many other different ways of climbing up the ladder successfully, but I have been able to use this method consistently at different places (carnivals, amusement parks, and boardwalks) and won the big prize.

 

 

 

 

 

Bibliography

[1]

K. Chu, “Katelyn Chu’s Biochem Blog,”
2019. [Online]. Available: https://katelynchu.wordpress.com/category/cancer/.

[2]

K. Chu, “The effect of UV light on the
CaCl2/PEG 3350/DMSO transformation buffer for CRISPR Cas9-edited E.
coli,” Young Scientist Journal, Accepted for publication,
2019.  [updated] Published February 17, 2020.  https://ysjournal.com/the-effect-of-uv-light-on-the-cacl2-peg-3350-dmso-transformation-buffer-for-crispr-cas9-edited-e-coli/

[3]

K. Chu, “A 3D Cellular Automata Cancer Stem
Cell Model using Matlab and App Designer,” Young Scientist Journal, p.
Accepted for publication, 2019.

[4]

K. Chu, “Stuffed Animals for Lucile Packard
Children’s Hospital,” 2019. [Online]. Available:
https://katelynchu.wordpress.com/stuffed-animals-for-lucile-packard-childrens-hospital/.

[5]

“Stack Overflow Public Forum,” 2015.
[Online]. Available: https://stackoverflow.com/questions/31285396/change-from-one-cartesian-3d-co-ordinate-system-to-another-by-translation-and-ro.


[1]
Canobie Lake Park, New Hampshire  https://www.canobie.com/

[2]
I am doing this investigation more for fun and I may be a little more informal
than I normally would be if I were planning to have this published in some peer
reviewed journal.  However, I am planning to just publish this on my blog since
it is easier to incorporate video and animation.

[3]
Sensor Kinetics Pro Version 2.1.2 by Innoventions, Inc., Apple App Store

[4]
For this paper, I will use “center of mass” and “center of gravity” interchangeably.
For most objects and circumstances they are pretty much the same if we assume
the gravitational field across the object is uniform.