Motion Model

Coordinate System

When collecting data, the phone is placed in the back right pocket, facing screen-in. The phone's internal coordinate system is oriented as showing in Figure 1.

The directions of motion in relation to the student taking the data are as follows:

The x-dimension is lateral (side to side) motion
The y-dimension is vertical (up and down) motion
The z-dimension is forward and backward

Figure 1: Illustration of phone coordinate system [8]

Description of Motion

We decided to analyze front stance for our proof of concept because it is the most basic. In front stance, the feet are shoulder width apart and the stance is two shoulder-widths long. The front knee is bent, while the back leg is as straight as possible without the knee locked. The feet are pointed forward and the hips are not be rotated. The center of gravity falls closer to the front foot than the back foot.

To move in front stance, step forwards with the back foot to bring the feet together then back out to shoulder width apart, tracing an arc with the foot that is moving. [3][4][5] In Figure 2 below a single step is shown, with the left foot remaining still and the right foot stepping forward.

Figure 2: Graphic showing the key frames of moving in front stance

This movement has two degrees of freedom. The hips should not move in the vertical y-axis, only rotate in the x- and z-axes. The main component of movement should be forwards in the z-axis, as well as a sinusoidal oscillation in the x-axis caused by the feet moving from shoulder width apart then together then back to shoulder width apart.

Motion in x dimension

Figure 3a: Predicted x-position and x-acceleration over time.

Figure 3b: Frequency plot of a(x), the acceleration in the x-dimension.

In the x dimension (side to side) sinusoidal movement is caused by the feet being brought together and then back apart. However, as the sensor is attached to one leg only side of the sinusoidal movement is present; the left foot would create the other half of the wave and only one foot moves at a time as illustrated in Figure 3a.

The second derivative of a sinusoidal wave is just the reflection of the wave over the x-axis. This sinusoidal wave in predicted acceleration over time can actually be distinguished in Figure 3b, the frequency plot of the actual data. In the frequency domain, this sine wave shows up as the vertical spike at 1 Hz that is taller than most of the other frequencies.

On the algorithm page we will explain how and why we compared the magnitude of the frequency of movement to the noise of other frequencies.

Motion in y dimension

Figure 4a: Predicted y-position and y-acceleration over time.

Figure 4b: Frequency plot of a(y), the acceleration in the y-dimension.

In the y dimension (up and down) there should be no movement and no acceleration as hips should remain level without rising or falling. A constant acceleration of 0 should appear at the frequency of 0 Hz, so it is promising that the frequency plot of the collected data shown in Figure 4b shows the main frequency present as 0 Hz.

On the algorithm page we will explain how and why we compared the magnitude of the zero frequency to the average noise of all other frequencies.

Motion in z dimension

Figure 5a: Predicted z-position and z-acceleration over time.

Figure 5b: Frequency plot of a(z), the acceleration in the z-dimension.

In the z dimension (forward and back) there should be forward motion with pauses in between steps. There should be a part of the step where the student is getting up to speed and another part with a negative acceleration of the student stopping. The important part of the graph shown in Figure 5a is actually the pauses between the steps. To ensure that motion is smooth and does not waste energy, the motion should be what in oscillatory motion is called critically damped. A step should get to its end as soon as possible without overshooting and the student having to regain their balance.

On the algorithm page we will explain how and why we compared the magnitude of the zero frequency to the next most prevalent frequency to determine if the student is stopping their steps cleanly.

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