Question

Imagine that a GPS device measures your position at every \(s\) seconds. The positions are stored as \((x, y)\) coordinates in a file src/files/pos.dat with the an \(x\) and \(y\) number on each line, except for the first line which contains the value of \(s\). First, load \(s\) into a float variable and the \(x\) and \(y\) numbers into two arrays and draw a straight line between the points (i.e., plot the \(y\) coordinates versus the \(x\) coordinates). The next task is to compute and plot the velocity of the movements. If \(x(t)\) and \(y(t)\) are the coordinates of the positions as a function of time, we have that the velocity in \(x\) direction is \(v_{x}(t)=d x / d t\), and the velocity in \(y\) direction is \(v_{y}=d y / d t .\) Since \(x\) and \(y\) are only known for some discrete times, \(t_{k}=k s, k=0, \ldots, n-1\), we must use numerical differentation. A simple (forward) formula is \(v_{x}\left(t_{k}\right) \approx \frac{x\left(t_{k+1}\right)-x\left(t_{k}\right)}{s}, \quad v_{y}\left(t_{k}\right) \approx \frac{y\left(t_{k+1}\right)-y\left(t_{k}\right)}{s}, \quad k=0, \ldots, n-2\) Compute arrays vx and vy with velocities based on the formulas above for \(v_{x}\left(t_{k}\right)\) and \(v_{y}\left(t_{k}\right), k=0, \ldots, n-2 .\) Plot vx versus time and vy versus time. Name of program file: position2velocity.py.

Short answer

Load data, plot positions, compute velocities using numerical differentiation, and plot velocities.

Step-by-step solution

Read and Parse the File

Open and read the file 'src/files/pos.dat'. The first line of the file contains the value of \(s\), which determines the time interval between position measurements. Load this value into a float variable \(s\). The subsequent lines contain \((x, y)\) coordinates for each measured position, which should be stored in two separate arrays: one for \(x\) coordinates and another for \(y\) coordinates.

Plot the Position Data

Use a plotting library like Matplotlib to plot the \(y\) coordinates against the \(x\) coordinates. This will create a visual representation of the path based on the collected GPS positions. Ensure both axes are labeled appropriately and add a title to the plot for clarity.

Compute Velocities

For each coordinate list, compute the velocities \(v_x\) and \(v_y\) using numerical differentiation. Use the formulas: \(v_{x}(t_k) \approx \frac{x(t_{k+1}) - x(t_k)}{s}\) and \(v_{y}(t_k) \approx \frac{y(t_{k+1}) - y(t_k)}{s}\) for \(k = 0, \ldots, n-2\). Store these computed velocities in arrays \(vx\) and \(vy\). Note that the length of these velocity arrays will be one less than the original position arrays.

Plot the Velocity Data

Use the calculated velocity arrays \(vx\) and \(vy\) to create two separate plots. Plot \(vx\) against time on one graph and \(vy\) against time on another. Time can be represented as \(t_k = ks\). Label the axes appropriately and include titles for each plot to indicate that they represent velocities in the \(x\) and \(y\) directions, respectively.

Key concepts

GPS Data Analysis

GPS data analysis involves extracting meaningful information from the positional data recorded by a GPS device. Imagine a scenario where your position is tracked at specific time intervals. These positions can be represented by coordinates

• The dataset consists of (x, y) coordinates recorded at regular time intervals denoted by \( s \).
• Analyzing the raw data helps understand movement patterns and paths based on these coordinates.
• By plotting the position data, one can visualize the journey or path taken over time.

For effective analysis, load the recorded time interval \( s \) into a variable, and store the coordinates in separate arrays. Then, utilize plotting libraries like Matplotlib to draw a line between all points. This simple visualization enables us to perceive the trajectory one has traveled.
Such analysis can be crucial for applications like navigation, sports tracking, or geographic studies, providing a digital map of physical movements.

Python Programming

In this scenario, Python serves as a powerful tool to process and analyze GPS data. Python's libraries facilitate tasks such as reading files and plotting data.
Here's how you can employ Python for GPS data analysis:

• Read the data file: Start by opening and reading the GPS data file using Python's file operation functions. This includes reading the value of \( s \) from the first line and parsing the subsequent lines for coordinate data.
• Storage in arrays: Utilize Python's list or array structures to store \( x \) and \( y \) coordinates fetched from the file. This structure helps in easy manipulation and processing later on.
• Data visualization: Leverage libraries like Matplotlib for plotting. Create clear and labeled graphs that represent the path traveled based on the positional data.

The code skills required include reading and parsing data, handling lists and arrays, and utilizing plotting libraries.
With Python, these tasks become streamlined and manageable, making it an excellent choice for handling GPS-based computations.

Velocity Calculation

Calculating velocity from GPS data involves numerical differentiation, particularly since the data represents discrete time intervals.
This process allows us to determine how fast something is moving over time.The velocity in each direction (\(v_x\) for x, \(v_y\) for y) is computed using simple formulas:

• The formula \(v_{x}(t_k) \approx \frac{x(t_{k+1}) - x(t_k)}{s}\) calculates velocity in the x-direction.
• Similarly, \(v_{y}(t_k) \approx \frac{y(t_{k+1}) - y(t_k)}{s}\) handles y-direction velocity.

This numerical differentiation approach requires knowing the positions at specific time intervals \(t_k = ks\).
Velocity calculation can provide deep insights into speed dynamics during the observed period.
Finally, plotting these velocities (\(vx\) and \(vy\)) against time can visualize how the speed changes over the recorded time, offering a deeper understanding of motion dynamics.