Quadcopter Simulation

This project aims to visualize a Quadcopter's motion (roll, pitch, yaw, and hover) in 3D space by adjusting the speed of four motors in Python. The project consists of three main parts: motor speed adjustment window, dynamic calculation, and 3D visualization.

Table of Contents

• Installation
• Component
• User Guide
• Demos & Result
• Conclusion
• Future plan
• References

Installation

Pygame

1. Open Visual Studio Code.
2. Go to the terminal or open a new terminal.
3. Copy and paste the following command:

   pip install pygame

4. Press Enter and wait for the download to complete.

OpenGL

1. Open Visual Studio Code.
2. Go to the terminal or open a new terminal.
3. Copy and paste the following command:

   pip install PyOpenGL
   pip install PyOpenGL_accelerate

4. Press Enter and wait for the download to complete.

Numpy

1. Open Visual Studio Code.
2. Go to the terminal or open a new terminal.
3. Copy and paste the following command:

   pip install numpy

4. Press Enter and wait for the download to complete.

Component

Motor Slider

In motor slider UI have 4 component

1. The text shows the speed of all 4 motors in rads/s units. The lowest value is 0 and the highest is 1,000 rads/s
2. Slide bar is used only for adjusting the speed of individual motors in order of Left is the least value. The right is the most valuable.
3. Text box is used to override the speed of all motors to have the same speed.
4. Enter button is used to confirm the override of the motor speed after entering the speed into the text box. The value will not be changed immediately. You must press the Enter key under the text box or press Enter on your keyboard.

The speed is adjusted by interacting with the UI in one way or another. The motor speed value is stored through An array variable with 4 fields, going from index 0 as the 1st motor to index 3 as the 4th motor.

There are 2 main libraries used:

1. os, which is a library that increases the convenience of cross-program development. platform by allowing the program to have access to certain functions so that it can be used with the operating system of the computer
2. Pygame, which is a library for developing games using the Python language as the main language The creator has applied it for user interaction through UI and simulation.

Dynamic Calculation

1. Quadcopter Degree of Freedom

- Reference Frame

Set the system axis as shown in the picture.

Set Frame 0 to be the Inertial Frame for reference and Frame Base to be the Frame of the Quadcopter's center of mass (CM). The Quadcopter's Pose can be specified with position (x, y, z) and Orientation via Euler angle (Roll (ϕ), Pitch (θ), Yaw (ψ)) or q=(x,y,z). ,ϕ,θ,ψ) The quadcopter speed can be specified as follows: $\dot{q} = \left(\dot{x}, \dot{y}, \dot{z}, \dot{\phi}, \dot{\theta}, \dot{\psi} \right)$

- Translation

For a quadcopter, the force that occurs is only one force in the $Z\left(F_{z}\right)$ axis. which serves to increase-decrease the Altitude of the Quadcopter. The said force is the sum of the force from all 4 propellers (Thrust) which is directly proportional to the Angular velocity squared.

$$thrust \propto w^2$$ $$thrust = kw^2$$ $$F_{z} = k\left(w_{1}^2 + w_{2}^2 + w_{3}^2 + w_{4}^2\right)$$

When k is Lift constant

Because there is only one force in the Z axis, we want to control the direction in the X and Y axes as well. Therefore, Rolling and Pitching are used to help create the force in the Y axis, respectively.

- Rotation

The thing that will allow the quadcopter to be able to roll, pitch, and yaw is from the torque that occurs in each axis.

$$\tau_{x} = lk\left(w_{4}^2 - w_{2}^2\right)$$ $$\tau_{y} = lk\left(w_{3}^2 - w_{1}^2\right)$$ $$\tau_{z} = b\left(w_{1}^2 - w_{2}^2 + w_{3}^2 - w_{4}^2\right)$$

When b is Drag Constant

2. Motors control algorithm

- Motor mixing algorithm

To control the Quadcopter's rolling pitching and Yawing, it can only be controlled through the speed of the individual motors. which can be written out as a relationship as follows

$$motor 1 = thrust_{cmd} - pitch_{cmd} + yaw_{cmd}$$ $$motor 2 = thrust_{cmd} - roll_{cmd} - yaw_{cmd}$$ $$motor 3 = thrust_{cmd} + pitch_{cmd} + yaw_{cmd}$$ $$motor 4 = thrust_{cmd} + roll_{cmd} - yaw_{cmd}$$

- Angular velocity of base frame

We can find the Angular velocity $\left(\vec{w_{b}}\right)$ of the point CM with respect to Frame 0 from

$$\vec{w_{b}} = \begin{bmatrix}\dot{\phi} \\ 0 \\ 0 \end{bmatrix} + R_{x}^T \left(\phi\right)\begin{bmatrix} 0 \\ \dot{\theta} \\ 0 \end{bmatrix} - R_{x}^T \left(\phi\right)R_{y}^T \left(\theta\right)\begin{bmatrix} 0 \\ 0 \\ \dot{\psi}\end{bmatrix}$$ $$\vec{w_{b}} = \begin{bmatrix} w_{bx} \\ w_{by} \\ w_{bz} \end{bmatrix} = \begin{bmatrix} 1 & 0 & -S_{\phi} \\ 0 & C_{\phi} & C_{\theta}S_{\phi} \\ 0 & -S_{\phi} & C_{\theta}C_{\phi} \end{bmatrix} \begin{bmatrix} \dot{\phi} \\ \dot{\theta} \\ \dot{\psi} \end{bmatrix}$$

When $R_{n}$ is Rotation matrix

3. Quadcopter Dynamics

- External force and torque

From the above, we can write external force and torque in matrix form as follows.

$$\vec{F}_{ext} = R_{rpy} \begin{bmatrix} 0 \\ 0 \\ k \displaystyle\sum_{i=1}^4 w_i^2 \end{bmatrix} - \begin{bmatrix} A_{x}\dot{x} \\ A_{y}\dot{y} \\ A_{z}\dot{z} \end{bmatrix}$$ $$$$ $$\vec{t}_{ext} = \begin{bmatrix} t_{x} \\ t_{y} \\ t_{z} \end{bmatrix} = \begin{bmatrix} t_{\phi} \\ t_{\theta} \\ t_{\phi} \end{bmatrix} = \begin{bmatrix} lk\left(w_4^2 - w_2^2\right) \\ lk\left(w_3^2 - w_1^2\right) \\ b\left(w_1^2 + w_2^2 + w_3^2 + w_4^2\right)\end{bmatrix}$$

When $A_{n}$ is damper

$$R_{rpy} = R_{z}\left(\psi\right)R_{y}\left(\theta\right)R_{x}\left(\phi\right)$$

- Equation of Motion (EOM)

How to solve equations to find The equations of motion of the quadcopter are therefore applied by the Euler-Lagrange Method, where the Lagrangian (L) is the difference between the kinetic energy (K.E.) and the gravitational potential energy (P.E.).

$$K.E. = \frac{1}{2}m\left(x^2+y^2+z^2\right)+\frac{1}{2}\left(I_{x} w_{bx}^2+I_{y} w_{by}^2+I_{z} w_{bz}^2\right)$$ $$P.E. = mgz$$ $$L = K.E. - P.E.$$

Then we can find the equation of motion by

$$\frac{\mathrm d}{\mathrm d t}\left(\frac{\mathrm \partial}{\mathrm \partial \dot{q}_{k}}\right) - \frac{\mathrm \partial L}{\mathrm \partial q_{k}} = \Gamma_{k} = \begin{bmatrix} F_{ext} \\ \tau_{ext} \end{bmatrix}$$

3D Visualization

The visualization part involves drawing the Quadcopter on the screen and updating its position based on differential values of X, Y, Z, roll, pitch, yaw.

The libraries used for this visualization are Pygame for display creation and OpenGL for graphics rendering.

System Architecture Diagram

Over all diagram

Diagram in focus part

User Guide

1. Within the Jupyter Notebook file, there are a total of 3 sections, consisting of:

• Section 1: Importing relevant packages into the project.

• Section 2 : The part where various parameters of the quadcopter are adjusted, using SI units. The adjustments should be made before clicking the "Run" button.

• Section 3 : Main loop witch operating at a frequency of 100 Hz.

2. After clicking 'Run All,' the window as shown in the picture will appear.

After adjusting the speed of each motor successfully, press 'Start Sim' to enter the Visualization page.

3. When visualization is complete, Program will shut down automatically.

Demos & Result

Examples

- Example 1: falling and spin about z-axis

Fallin.mp4

To simulate rotation around the z-axis and falling along the z-axis, start by adjusting the speed of all four motors to achieve a balanced condition calculated from

$$ω = \sqrt{\frac{mg}{4k}}$$

From the set parameters, you have obtained ω = 620.61 rad/s. Next, adjust the speed of motors 1 and 3 to decrease to 600 rad/s.

- Example 2: hover and spin about z-axis

Hover.mp4

To simulate rotation around the z-axis and ascending along the z-axis, start by adjusting the speed of all four motors to achieve a balanced condition. Then, decrease the speed of motors 1 and 3 to 645 rad/s.

Validation

For the validation of dynamics, as we do not have a real quadcopter, we can only compare the computed values between Python and MATLAB. This comparison will be divided into two parts: visualization and the values obtained from the calculations.

- Animation compare with MATLAB

spin about z-axis

Spinz.mp4

In this test, we will compare the rotation around the z-axis using the same set of parameters for both simulations, running for a total of 1 second (100 time steps). From the clips, it can be observed that the rotational motion around the z-axis in both simulations has similar characteristics.

spin about x-axis

Spinx.mp4

In this test, we will compare the rotation around the x-axis using the same set of parameters for both simulations, running for a total of 1 second (100 time steps). From the clips, it can be observed that the rotational motion around the x-axis in both simulations has similar characteristics.

- Dynamic compare with MATLAB

Spin about z-axis and hover

The obtained values are close, and by examining the results at step 100, it can be seen that the position value from Python is 0.0373, while the value from MATLAB is 0.0367. The discrepancy may arise from rounding differences in the calculations between Python and MATLAB.

Conclusion

Simulating the motion of a quadcopter using Python can be done using various libraries available in Python, such as NumPy, Matplotlib, and others, to model the dynamics of the quadcopter and visualize the results graphically. Comparing the results to MATLAB may show some differences depending on the simulation methods and parameters used in each system. However, by choosing appropriate parameters and using accurate simulation methods, you can achieve results that closely match those in MATLAB.

To enhance the reliability of your testing, you can experiment with refining and adjusting the parameters of the Python simulation to obtain results that closely align with the MATLAB outcomes. Testing that involves comparing results with MATLAB is a good step to verify the accuracy and reliability of your simulation.

Keep in mind that since there is no real-world quadcopter for direct comparison, testing can only be done by comparing results within the simulation environments, such as Python and MATLAB.

Future plan

• Improve GUI and visualization.
• Testing with real quadcopter.
• Real-time speed adjustment and display of results.

References

• [1] Lebedev, A. (2013). Design and Implementation of a 6DOF Control System for an Autonomous Quadrocopter (Master's thesis). Julius Maximilian University of Würzburg, Faculty of Mathematics and Computer Science, Aerospace Information Technology, Chair of Computer Science VIII, Prof. Dr. Sergio Montenegro.
• [2] DRONE OMEGA, 2020, What is a Quadcopter Explained Thoroughly [Online], Available: droneomega.com [02/11/23]
• [3] Pranav Bhounsule, 2020, Robotics Lec25,26: 3D quadcopter, derivation, simulation, animation (Fall 2020) [Online], Available: YouTube [02/11/23]
• [4] MATLAB, 2020, Drone Simulation and Control, Part 1: Setting Up the Control Problem [Online], Available: YouTube [02/11/23]
• [5] Kanishke Gamagedara (2021). Plotting 3D Objects with Matplotlib. Github. https://github.com/kanishkegb/pyplot-3d
• [6] P. Wang, Z. Man, Z. Cao, J. Zheng and Y. Zhao, "Dynamics modelling and linear control of quadcopter," 2016 International Conference on Advanced Mechatronic Systems (ICAMechS), Melbourne, VIC, Australia, 2016, pp. 498-503, doi: 10.1109/ICAMechS.2016.7813499.
• [7] Ahmad, F., Kumar, P., & Patil, P. P. (2018). Modeling and simulation of a quadcopter with altitude and attitude control. Nonlinear Studies, 25(2), 287–299.