fatima_hachem_Locating_the_Wii_remote_in_a_3_Dimensional_Field

Hachem 1
Fatima Hachem
IB Mathematics SL DP2
Locating the Wii remote in a 3 Dimensional Field
February 10, 2015

Hachem 2
Introduction:
In the year 2006, Nintendo® released the Nintendo® WiiTM and its revolutionary
controller the WiiTM remote, or wiimote (“Nintendo - Corporate Information”, n.d.). The
wiimote is a wireless controller that gives games a sense of depth by allowing players to control
their player avatar by performing the movement they wish for the avatar to perform. For
example, if the avatar wields a sword and the player swings the wiimote in the horizontal
direction, the avatar will replicate the movement, swinging the sword in the horizontal direction.
Similarly the wiimote may act as a steering wheel in a racing game such that when the player
rotates the wiimote to the right, the car makes a right turn.
Rationale
When I first played with the WiiTM console I found it absolutely thrilling how I was in
direct control of my avatar. After years of using a joystick and pressing button combinations, I
now could more actively interact with the game and really delve into the great worlds packed
with imagination. So naturally I questioned how the WiiTM could register all of the different
motions players made behind the screen. Initially I had settled for the most basic explanation,
the sensor bar and wiimote worked transmitting and receiving infrared (IR) waves. Upon deeper
investigation I discovered there was a great deal more to the registration of player movements.
As such, the objective of this mathematical exploration is to determine the conditions that
distinguish one command from another within a game. To do so, let’s first look at the
components of the wiimote.
A Breakdown of the Wiimote:
The wiimote is composed of multiple parts, the parts used to detect motion, location and
communicate between the WiiTM console and the wiimote are the accelerometer, Bluetooth
radio, and infrared camera (“How the Wiimote Works”,
2008).
Accelerometer: This component measures the change in the
wiimote’s velocity, hence acceleration, as acceleration is
defined as the rate of change in velocity, along three axes, x, y,
and z, thus allowing the game avatar’s action to correspond to
the player’s (“How the Wiimote Works”, 2008).
Bluetooth radio: This component allows the WiiTM to
recognize the wiimote as a point of reference (“How the
Wiimote Works”, 2008).
Infrared Camera: The infrared camera works together with the
WiiTM sensor bar that is sold along with the WiiTM console.
What happens here is that the WiiTM sensor bar contains 10
Light Emitting Diodes, or LEDs, with a cluster of five LED
lights on either end of the sensor bar and a space of approximately 15.1 ±
0.1 cm between them, as seen in Figure 2 (“Wiimote: IR Camera”, 2012).
Figure 1- The wiimote on the
three axes

Hachem 3
Figure 2- LED light clusters
These LED’s emit the infrared waves that the wiimote camera
placed at the very top of the wiimote (Figure 3) uses to determine
its location in space relative the sensor bar using a process known
as triangulation.
Triangulation:
Triangulation is a process used by GPS satellites to locate
objects using two or three different reference points. For example
when locating a mobile device three different cellular towers may
detect the signal and provide varying coordinates in order to
provide a more accurate location of the device. The wiimote also
uses this system in order to locate the wiimote relative to the
sensor bar. Considering each of the two clusters of LED lights is a
reference point, the wiimote can define its location relative to each
cluster, using three dimensional vectors (Rouse, 2005).
Use of Triangulation in games
Among the many games I have played on the Nintendo® WiiTM
console, one of the most enjoyable was The Legend of Zelda Skyward SwordTM. In the game the
avatar wields a sword; hence the wiimote acts as a sword. The avatar moves his sword in
correspondence to the motions the player makes with the wiimote. Taking that into
consideration, I thought that it was more like, we, the players, were drawing vectors that the
wiimote and WiiTM sensor bar used to detect and recognize motion. As such I decided to
construct three dimensional vectors to model the basic movements players perform using the
wiimote.
The game registers a variety of movements from using a shield to defend oneself to flying
a gigantic bird as though it is some kind of kite. However for our purposes I will be discussing
the use of the wiimote as a sword. First off I would like to define some terms I shall be using.
1. Slice: A straight horizontal or vertical displacement of the wiimote. This movement
should be along one single axis such as only the x or z axes. (Refer to Figure 6 for
clarification).
2. Jab: the forward thrust of the wiimote, causing displacement along the y axis.
Representing each movement as a vector on a three dimensional plan
To mathematically represent each motion as a vector on a three dimensional plane, I
took measurements of the distance from the Wii sensor bar to where I usually play with the
wiimote. (See Figures 4 and 5). Measurements were taken in all 3 directions (x, y, and z). These
measurements were then used to define the position the wiimote is usually at during game play.
We will call this position, the “origin”. Then distances were taken to measure the distance
travelled from the origin so as to illustrate the vectors induced by the change in the position of
the wiimote.
Figure 3- View of wiimote
from multiple angles
Infrared
camera

Hachem 4
Values at the Origin:
Distance along x-axis 25.5 cm Distance along y-axis: 273 cm Distance along z-axis 28 cm
Figure 4- The wiimote at the Origin Figure5- Measuring the distances in the 3 axes
Defining the Coordinate System
Figure 6- Conventional coordinate system used in relation to the wiimote
The coordinate system is mapped out by the
Wii sensor bar in relation to the position of
the wiimote as seen in Figure 6.
Obtaining the Conditions at the Origin
To obtain the x coordinate of the wiimote’s “origin”, first I measure the length of the Wii
sensor bar. Then, using a ball of yarn I drew out a perpendicular line from one side of the sensor
bar above one of the two clusters of LED lights to the position of the wiimote. Using a ruler I
measure the distance from the yarn to the center of the IR camera at the head of the wiimote.
The process was repeated with the other side of the sensor bar. The values were then computed
according to their location relative to the wiimote.
Length of Wii sensor bar= 24 cm
Distance from wiimote to LED clusters on positive x axis= 13.5 cm
Distance from wiimote to LED clusters on negative x axis= 37.5 cm
ar

Hachem 5
Calculating the x coordinate of the origin:
Method A:
𝑥 =
1
2
(24) + 13.5
𝑥 = 25.5 𝑐𝑚
Method B:
𝑥 = 37.5 −
1
2
(24)
𝑥 = 25.5 𝑐𝑚
To obtain the y coordinate of the wiimote’s standard position I measured the distance
from the point on the floor perpendicular to the location of the WiiTM sensor bar to the point on
the floor perpendicular to the location of wiimote.
y coordinate = 273 cm
To obtain the z coordinate of the wiimote’s standard position I first measured the
distance from the Wii sensor bar to the floor. I then measured the distance from my waist down.
I then subtracted the two values to figure out the vertical distance of the Wii sensor bar relative
to the wiimote.
Calculating the z coordinate of the origin:
Distance from Wii sensor bar to the floor= 131 cm
Distance from my waist the floor = 103 cm
Distance of Wii sensor bar relative to the Wii mote= 131-103=28 cm.
As we are using the coordinate system (x, y, z) the origin lies at (25.5, 273, 28) or,
in vector form [
𝒙
𝒚
𝒛
] = [
𝟐𝟓. 𝟓
𝟐𝟕𝟑
𝟐𝟖
].
This value is assumed to be exact as our concern is the change in the wiimote’s position,
hence any arbitrary numbers could have been used to indicate the position that the wiimote
generally lies within.
The Horizontal Slice:
Figure 7- The Horizontal Slice:
The horizontal slice can be represented using two different vectors and then simplifying it.
Starting at the origin [
25.5
273
28
] and moving the wiimote to the right we get a vector of[
70.5
273
28
].
Now let’s assign a name to each of these vectors. The vector at the origin will be called 𝑠⃗ and the
vector at the new position will be called 𝑎⃗.

Hachem 6
Thus:
𝑠⃗ = [
25.5
273
28
] and 𝑎⃗ = [
70.5
273
28
]
Now to find the difference between 𝑎⃗ and 𝑠⃗ we can use the rules of vector by scalar
multiplication and vector addition.
𝑎⃗ − 𝑠⃗ = [
70.5
273
28
] − [
25.5
273
28
]
𝑎⃗ − 𝑠⃗ = [
70.5
273
28
] + (−1) [
25.5
273
28
]
𝑎⃗ − 𝑠⃗ = [
70.5
273
28
] + [
−25.5
−273
−28
]
𝑎⃗ − 𝑠⃗ = [
70.5 − 25.5
273 − 273
28 − 28
]
𝑎⃗ − 𝑠⃗ = [
45
0
0
]
Set up an equation equal to the difference of
the two vectors
Exchange the (-) sign for a (+(-1))
Distribute the (-1) throughout the vector
Add vectors
Simplify
As such the resultant vector, is 45 in the x axis, which aligns with the previous definition of the
word slice. To visualize this process, I used Photoshop to illustrate a graphical model as seen in
Figure 8.
Figure 8- Graphical Representation of the Horizontal slice

Hachem 7
The Vertical Slice
Figure 9-The Vertical Slice:
It is important to keep in mind that the vertical slice is actually occurring
along the z axis and not the y.
We simply repeat the process used when calculating the Horizontal Slice.
Where 𝑠⃗ = [
25.5
273
28
] and 𝑏⃗⃗ = [
25.5
273
73
]. Here we are searching for the difference of
𝑏⃗⃗ − 𝑠⃗.
𝑏⃗⃗ − 𝑠⃗ = [
25.5
273
73
] − [
25.5
273
28
]
𝑏⃗⃗ − 𝑠⃗ = [
25.5 − 25.5
273 − 273
73 − 28
]
𝑏⃗⃗ − 𝑠⃗ = [
0
0
45
]
As such the resultant vector, is 45 in the z axis, which aligns with the
previous definition of the word slice. This process can also be graphically
modelled as seen in Figure 10.
Figure 10- Graphical Representation of the Vertical slice

Hachem 8
The Jab
Figure 11- The Jab
The jab involves the movement of the wiimote across the y axis and away from the
player’s body. Using the same methods used in computing the horizontal and vertical slices, we
can calculate the resultant vector of the jab.
Where 𝑠⃗ = [
25.5
273
28
] and 𝑐⃗ = [
25.5
244.9
28
]. Here we are searching for the difference of 𝑐⃗ − 𝑠⃗.
𝑐⃗ − 𝑠⃗ = [
25.5
244.9
28
] − [
25.5
273
28
]
𝑐⃗ − 𝑠⃗ = [
25.5 − 25.5
244.9 − 273
28 − 28
]
𝑐⃗ − 𝑠⃗ = [−
0
28.1
0
]
As such the resultant vector, is -28.1 in the y axis, which aligns with the previous definition of
the word jab. The reason why this value is negative is because the distance between the wiimote
and Wii sensor bar has decreased. This process can also be graphically modelled as seen in
Figure 12.
Figure 12- Graphical Representation of the Jab

Hachem 9
Graphically Modelling the Motion of the Wiimote
All the movements I have modelled thus far are attacks; however there were instances in
the game during which one would have to move the sword without performing an attack to
progress further. So how does the system differentiate between a motion intended to be an
attack and any other motion?
To answer this question, I modelled the IR camera in the wiimote as a particle in space,
and relating its change in displacement over time, we can determine the wiimote’s velocity at
any given instant. From there we can calculate its acceleration. Thus, depending on the rate of
change of the position wiimote’s IR camera in space, we can determine the type of command
actuated by a system.
To model the IR camera as a particle in space, I would simplify its motion to one
dimension to enable myself to draw conclusions. The complication is that I have previously dealt
with the movement of the wiimote in three-dimensions. How can I model this motion in one
dimension? As shown previously, 3 dimensional vectors are comprised of three coordinates; for
each of the slices and jab, there has been change in only one of the three coordinates. Thus I only
need to detect the change in only one of the coordinates of the vector for this investigation.
Therefore regardless which of the axes parallel to the motion I use to model the displacement
function, the outcome should be the same. As such I can treat the IR camera of the wiimote as a
particle in space.
In order to study the rate of change of a particle, I require a graph. To derive a graph I
need to measure the position of the wiimote’s IR camera at various points of time. Thus leading
me to the question: how will I measure the change in position of the wiimote over time?
Obtaining Graphical Values and Defining a Curve:
Graphs of two motions varying in speed, the magnitude of velocity which is a vector
value, were plotted. The first motion occurred at a considerably great speed, whereas the second
occurred at a much slower speed. These differences in speed were great enough for the human
eye to detect them.
These two motions are defined, and will be referred to as:
1. Attack: Displacement of the wiimote at a considerably great speed. Generally occurring
within the time span of half a second.
2. Non-Attack: Displacement of the wiimote at a considerably low speed. The minimum
time required for this motion is approximately one second.
Table 1- “Attack” Table of Values
Time
(seconds)
Displacement
from Origin
(cm)
t-axis y-axis
0.000 3.4
0.0427 5.1
0.0580 9.4
0.0855 14.7
0.124 23.7
0.138 35.7
0.166 39.8
0.194 42.7
0.210 43.6
Table 2- “Non-Attack” Table of Values
Time
(seconds)
Displacement
from Origin
(cm)
t-axis y-axis
0.00 2.40
1.40 7.40
1.98 13.6
2.08 19.3
2.65 26.1
2.76 30.5
3.37 35.9
5.16 42.6

Hachem 10
Graph 1- “Attack”
To explain the trends of the graph, we shall refer to the horizontal slice occurring in the x axis
Table 3-Trends in Graph 1 “Attack”
Region Wiimote Motion Player Motion
A
Approaches origin from a distance about
4 cm away while decelerating, until
instantaneous velocity is 0 cm/s.
The distance between the wiimote IR
camera and origin is initially decreasing,
hence the reflection across the x axis.
Player originally has “wound” themselves,
bringing the wiimote closer to the center of their
body in preparation for the sharp movement to
follow.
B
Moves away from the origin while
accelerating.
Player has thrust their arm out horizontally at
full force.
C
Point of inflection of the graph, where
velocity of the wiimote is at a maximum
and will decrease after this point.
Player has stopped transferring energy to
displace the wiimote and is instead relying on the
momentum from their initial thrust as their arm
moves farther away from them.
D
v=0; at the maximum of the graph,
where the wiimote has come to an
instantaneous halt.
Player has deliberately stopped the movement of
the wiimote at the maximum distance they can
move their arm from their body.
E Wiimote is accelerating towards origin.
Player is adducting their arm to their body to
prepare for the next attack.
s(t) = -12174x3 + 4018.8x2 - 117.14x + 3.6719
0
5
10
15
20
25
30
35
40
45
50
0 0.05 0.1 0.15 0.2 0.25
displacement(cm)
Time (sec)
Attack Motion
E
A
B
C
D

Hachem 11
Graph 2- “Non-Attack”
To explain the trends of the graph, we shall refer to the horizontal slice occurring in the x axis
Table 4- Trends in Graph 2 “Non-Attack”
Region Wiimote Motion Player Motion
A
Approaches origin while decelerating
until instantaneous velocity is 0 cm/s.
Player originally has moved the wiimote into a
comfortable position enabling them to maximise
the length of their arm when they perform the
slice in slow motion. This is because the limit that
a player can displace the wiimote is about the
length of their arm from their body.
B
Moves away from the origin while
accelerating.
Player begins to abduct their arm from their body
with a controlled speed.
C
Point of inflection of the graph, where
velocity of the wiimote is at a maximum
and will decrease after this point.
Player is attempting to maintain the initial speed
of the wiimote, hence is subconsciously
beginning to reduce the speed at which they
move the remote.
D
v=0; at the maximum of the graph,
where the wiimote has come to an
instantaneous halt.
Player has deliberately stopped the movement of
the wiimote at the maximum distance they can
move their arm from their body.
E Returns slightly to the origin.
The player has significantly reduced the speed of
their arm as their arm is nearly entirely stretched
outwards. It now loiters around the same area
moving towards their body slightly as they
prepare to resume moving their arm outwards,
hence the upcoming upwards trend.
In summary: the non-attack motion is of a higher order as the player constantly
moderates the speed at which they are moving the wiimote. The attack motion is of a lower
order as the player is less in control of the action and instead stops only when their arm has
moved too far.
s(t) = 0.2752t4 - 4.0137t3 + 17.721t2 - 14.599t + 2.4355
-10
0
10
20
30
40
50
60
0 1 2 3 4 5 6
Displacement(cm)
Time (s)
Non-Attack Motion
A
B
C
D E

Hachem 12
The following procedure was followed to obtain the graphs for the two varying motions:
1. I video-taped the attack and non-attack motions. A
screenshot from one of these videos displaying the set
up can be seen in Figure 13.
2. I decreased speed of video to observe and mark
position of wiimote at various points in time.
3. I used Photoshop to figure out what the position
would be, as seen in Figure 14.
4. I used video editing software, with a timer of the
format (hr:min:sec;millisec), to figure out the time
for each marker in the slow-motion video.
5. I used the ratio:
T(original)
T(slow motion)
, where T=total time,
to calculate the actual time in the original motion.
(See calculations below)
6. I graphed using a spreadsheet program, and defined
two functions.
Table 5- Times in “Slow-Motion”
Time Slow-Motion (seconds)
Motion T
(Original)
T
(Slow-motion)
M1 M2 M3 M4 M5 M6 M7 M8 M9
Attack 0.21 15.25 0.00 3.10 4.21 6.21 9.00 10.05 12.08 14.06 15.25
Non-Attack 5.16 8.01 0.00 2.17 3.07 3.23 4.12 4.29 5.23 8.01
*Where T=total time, and M=marker
Calculating the “Original” Time”
At M2 in “attack”: t(slow-motion)= 3.10 s, then:
T(original)
T(slow motion)
=
0.21
15.25
T(original)
T(slow motion)
=
t(original)
t(slow motion)
t(original)
3.10
=
0.21
15.25
t(original) = 0.0427 sec
At M2 in “non-attack”: t(slow-motion)= 2.17 s,
then:
T(original)
T(slow motion)
=
5.16
8.06
T(original)
T(slow motion)
=
t(original)
t(slow motion)
t(original)
2.17
=
5.16
8.06
t(original) = 1.40 sec
Where t is time and T is total time.
This method was then applied to the remainder of the markers used.
Deriving Velocity and Acceleration
Depending on the rate at which the wiimote’s position changes, a different command will
be acknowledged by the Wii system. As the “attack” command is the faster of the two
movements, there exists a minimum rate of change required to activate that command. Hence,
the “non-attack” command will have a maximum rate of change of position before the command
is registered as an “attack” command. To find the minimum of the “attack” function and
maximum of the “non-attack” function I took the derivatives of each motion’s respective
displacement function.
Figure 13- Screen shot of Set Up
Figure 14- Determining Displacement

Hachem 13
Minimum Speed of Attack
Attack: s(t) = -12174t3 + 4018.8t2 - 117.14t + 3.6719
The minimum instantaneous speed achieved during an attack as seen in Graph 1 is: 0
cm/s. Hence the conventional method of searching for minimums by equating the first
derivative to 0 and solving will not work. Instead I searched for average velocities over varying
intervals of time. This is done by drawing secants between the minimum, maximum, and point
of inflection as well as other points as seen in Table 6 below, and then solving for their
gradients.
To find max/min values:
𝑣(𝑡) =
𝑑𝑠
𝑑𝑡
=
𝑑
𝑑𝑡
− 12174𝑡3
+ 4018.8𝑡2
− 117.14𝑡 + 3.6719
𝑣(𝑡) = −36522𝑡2
+ 8037.6𝑡 − 117.14
∵ rate of change at extrema = 0
0 = −36522𝑡2
+ 8037.6𝑡 − 117.14
𝑡 =
−𝑏±√𝑏2−4𝑎𝑐
2𝑎
𝑡 =
−8037.6±√8037.62−4(−36522)(−117.14)
2(−36522)
∴ 𝑡 = 0.0157 or 𝑡 = 0.204
Looking at Graph 1 we see that the minimum occurs at 𝑡 = 0.0157, and the maximum occurs at
𝑡 = 0.204.
To find point of inflection:
Points of inflection occur when
𝑑2 𝑠
𝑑𝑡2 = 0
𝑎(𝑡) =
𝑑2 𝑠
𝑑𝑡2 =
𝑑𝑣
𝑑𝑡
=
𝑑
𝑑𝑡
− 36522𝑡2
+ 8037.6𝑡 − 117.14
= −73044𝑡 + 8037.6
0 = −73044𝑡 + 8037.6
∴ 𝑡 = 0.11
By substituting all of these values of time into the original function s(t), we can find the
displacement at each time.
s(𝑡) = −12174𝑡3
+ 4018.8𝑡2
− 117.14𝑡 + 3.6719
s(0.0157) = −12174(0.0157)3
+ 4018.8(0.0157)2
− 117.14(0.0157) + 3.6719
s(0.0157) = 2.78 cm
This process was repeated for the maximum and point of inflection.
Solving for Average Velocity:
gradientsecant =
Δ𝑦
Δ𝑥
, which is in this case: gradientsecant =
Δ𝑠
Δ𝑡
gradientsecant =
𝑠(𝑡2)−𝑠(𝑡1)
𝑡2−𝑡1
gradientsecant =
43.67−2.78
0.204−0.0157
∴ gradientsecant = 217.15 cm/s
The same method was applied to calculate all the other secant gradients.

Hachem 14
Table 6- Average Velocity of Wiimote during an “Attack” motion
While the date provides the variation of average velocities over the course of the attack
motion, distinguishing velocities between approximately 120 ≤ 𝑣 ≤ 320 cm/s as the average
velocity achieved by an attack motion, it is not conclusive. In order for a minimum velocity to be
obtained, multiple trials repeating the attack motion with different speeds should be conducted
and their average velocities from the min to the max calculated and compared to find the
threshold value of velocity required for the WiiTM to acknowledge the movement as an attack
command.
Maximum Speed of Non-Attack
Non-Attack: s(t) = 0.2752t 4 - 4.0137 t 3 + 17.721 t 2 - 14.599 t + 2.4355
To find the maximum speed that can be achieved by a non-attack without changing the
command, the value of the inflecting tangent must be found, for at the point of inflection, the
velocity of the wiimote will be at a maximum. To solve for the maximum velocity we must find
the second derivative of the non-attack function before setting it equal to zero and then solving.
𝑣(𝑡) =
𝑑𝑠
𝑑𝑡
=
𝑑
𝑑𝑡
0.2752𝑡4
− 4.0137𝑡3
+ 17.721𝑡2
− 14.599𝑡 + 2.4355
= 1.1008𝑡3
− 12.0411𝑡2
+ 35.44𝑡 − 14.599
𝑎(𝑡) =
𝑑2 𝑠
𝑑𝑡2 =
𝑑𝑣
𝑑𝑡
=
𝑑
𝑑𝑡
1.1008𝑡3
− 12.0411𝑡2
+ 35.44𝑡 − 14.599
= 3.3024𝑡2
− 24.0822𝑡 + 35.44
0 = 3.3024𝑡2
− 24.0822𝑡 + 35.44
𝑡 =
−𝑏±√𝑏2−4𝑎𝑐
2𝑎
𝑡 =
−(−24.0822)±√(−24.0822)2−4(3.3024)(35.44)
2(3.3024)
∴ 𝑡 = 2.043 or 𝑡 = 5.247
Looking at Graph 2, we realize that the inflecting tangent at t=2.043 is steeper at t=5.247.
Therefore the maximum velocity that can be achieved by a non-attack is:
𝑣(𝑡) = 1.1008𝑡3
− 12.0411𝑡2
+ 35.44𝑡 − 14.599
𝑣(2.043) = 1.1008(2.043)3
− 12.0411(2.043)2
+ 35.44(2.043) − 14.599
𝑣(2.043) = 16.31 cm/s
Interval (s) Displacement (cm)
t1 t2 s(t1) s(t2) Average velocity (cm/s) Region of Graph
0.0157* 0.11** 2.78 23.22 216.76
Around
Min–
P.O.I
0.025 0.11 3.06 23.22 119.53
0.05 0.11 6.34 23.22 281.33
0.075 0.11 12.36 23.22 310.3
0.11 0.125 23.22 28.05 322
P.O.I-
Around
Max
0.11 0.15 23.22 34.44 280.5
0.11 0.175 23.22 41.00 273.54
0.11 0.204*** 23.22 43.67 217.55
0.025 0.125 3.06 28.05 249.9
Around
Min–
Around
Max
0.05 0.15 6.34 34.44 281
0.075 0.175 12.36 41.00 286.4
0.0157 0.204 2.78 43.67 217.15
1. Find first derivative
*Minimum (Min)
**Point of Inflection
(P.O.I)
***Maximum (Max)
3. Set second derivative equal to 0
2. Find second derivative
4. Solve

Hachem 15
Average Velocity from first local Minimum to local Maximum
The method used to calculate the maximum and minimum of the attack motion by equating
the first derivative to zero and solving were used to find the local maximum and minimum of the
non-attack motion as well.
Local Minimum at: (0.49, -0.92)
Local Maximum at: (4.76, 42.86)
𝑣average = gradientsecant =
Δ𝑠
Δ𝑡
=
𝑠(𝑡2)−𝑠(𝑡1)
𝑡2−𝑡1
=
42.86−(−0.92)
4.76−0.49
∴ 𝑣average = 10.25 cm/s
The velocity value obtained from the first of these calculations is one of the maximum
velocities that could be achieved by any player. The second velocity obtained displays the overall
average velocity of the completed motion. This data is not conclusive. Multiple trials with the
motion occurring at varied speeds need to be tested so as to determine a threshold velocity at
which a motion is no longer registered as a non-attack, but an attack.
Considering the large gap between the minimum speed required for the “attack”
command and the maximum speed allowed for the “non-attack” command, this creates a “gray
area” in which either command could be activated depending on whether or not the value leans
nearer to the maximum value possible or nearer the minimum value required. This explains why
often the avatar on the screen fails to exact the commands input by the users.
The gray area as of this exploration is rather large. To reduce the range of values within
this gray area and determine the threshold velocity that must be achieved before the motion is
recognized as an “attack,” multiple trials with variations in the velocity of wiimote must be
conducted. Also, to further our understanding of the function of the wiimote we should also look
for the minimum distance that the wiimote must be displaced from the origin before any
command is recognized. Furthermore, could there be a maximum velocity that the wiimote can
recognize before it ignores the command, as in, if a player were to jerk the wiimote at a high
speed, what is the maximum speed with which they can move the wiimote before it fails to
recognize the command? This is a complication that has occurred while I played the game a few
times, hence it would be worthy of investigation.
Conclusion
Commands are actuated in The Legend of Zelda Skyward SwordTM when the WiiTM
sensor bar detects the movement of the wiimote via the exchange of infrared signals. The type of
command depends on two factors, the direction of the change in position and the magnitude of
the velocity, or the speed, with which the wiimote changes position. Using vectors, the direction
can be calculated, and depending on the direction of the resultant vector a variety of commands,
among those are the horizontal and vertical slices as well as the jab, could be actuated by the
system. The second factor determines the type of command that would occur in the direction
calculated by the vectors. Depending on the speed at which the wiimote changes its position in
space there are multiple commands that a player can activate. Generally speeds between 10.25
cm/s to 16.31 cm/s activate a “non-attack” command in The Legend of Zelda Skyward SwordTM,
whereas speeds between 120 cm/s and 320 cm/s actuate an “attack command.” The existence of
a great difference between them forms a “gray area” in which either command can be registered,
thus explaining the technical difficulties a player tends to face during a game.

Hachem 16
References
. (n.d.). . Retrieved May 8, 2014, from http://www.osculator.net/doc/_media/faq:pry-
wiimote.gif
| Nintendo - Corporate Information | Company History. (2014, January 1). | Nintendo -
Corporate Information | Company History. Retrieved May 17, 2014, from
http://www.nintendo.com/corp/history.jsp
Nintendo wiimote. (n.d.). . Retrieved May 10, 2014, from
http://gk12.cis.ksu.edu/uploads/lesson_material/file/3/wiimote.pdf
HowStuffWorks "Wii Controller ". (2010, January 27). HowStuffWorks. Retrieved May 15, 2014,
from http://electronics.howstuffworks.com/wii2.htm
How the Wiimote Works. (2008, September 24). trinalins Science Blog RSS. Retrieved May 10,
2014, from http://pagefillers.com/scienceblog/?p=36
Rouse, M. (2005, September 1). triangulation. What is ?. Retrieved May 10, 2014, from
http://searchnetworking.techtarget.com/definition/triangulation
Whitehead, T. (2012, December 24). Round Table: Let's Talk About Festive Gaming. . Retrieved
May 18, 2014, from
http://images.nintendolife.com/news/2012/12/round_table_time_for_a_festive_chat1/attach
ment/8/large.jpg
Wiimote: IR Camera. (2012, October 3). . Retrieved May 18, 2014, from
http://wiibrew.org/wiki/Wiimote#IR_Camera

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