Quadcopter Kinematics and System Dynamics

Introduction

At first glance, the mathematics and physics needed to model a quad-rotor can be quite discouraging especially if you have never taken a Linear Algebra or Calculus class. I’ll admit, modelling a quad-rotor is by no means an easy undertaking but it is doable and very satisfying once you successfully derive all the equations. In this blog post I will do my best to boil down the key concepts so that they are more palatable. However it is assumed that you have some knowledge of differential equations.

Motivation

When Dealing with a complex dynamic system it is often necessary to simulate the system before physically building it. Doing so will help you better understand how your system behaves under various conditions.

For you to simulate a dynamic system you first need to formulate a model that encapsulates information about all the forces that interact with the system and how they interact with the system. To formulate the model it is very important to go back to first-principles and work your way upwards.

Let’s get into it

Forces in action

The characteristic curves presented in figure-1 show that the relationship between thrust (F) and rotor speed( ω) is approximately quadratic. In figure-1,  drag-momentum and thrust are shown to be directly proportional to the square of angular velocity. The constants of proportionality eqn (20) and eqn (21) can be derived experimentally, these two quantities  vary  depending on the motor & propeller combination.

rotor physics

figure 1

Inertial-and-body-fixed-frame-of-quadrotor.png
figure 2 (Illustrated By : Peter Hubinský)

In the illustration shown above, we can clearly see that the torque generated by motors 1,2,3 and 4 results in the generation of thrust. The thrust generated by the quad-rotor’s motors is parallel but opposite in direction to the quad-rotor’s weight. The interaction of these forces is captured in equation-1 as shown below.

eqn (5).png
equation-1

In figure-1 the relationship between rotor-speed and thrust is shown to be approximately quadratic.  Equation-2 captures the empirically deduced relationship between these two quantities.

As previously stated, the constant of proportionality  eqn (20) can be derived experimentally. The details of the experimental setup needed to derive this quantity will be outlined in a later blog-post.

eqn (6)
equation-2

Substituting equation-2 into equation-1 yields equation-3 as shown below

eqn (7)
 equation -3

When the system is in a state of mechanical-equilibrium i.e hovering at a fixed altitude above the ground , each motor produces thrust that is equivalent to 1/4 of the quad-rotor’s total weight. This is mathematically expressed in equation-4.

eqn (2)

 

                                                                                                                                                                                                   equation-4

Given equations 5 and 6, it is now undeniably clear that the quad-rotor will never take-off if the total thrust produced is less than or equal to the quad-rotor’s total weight.

note: The negative sign on the right hand side is used to show that the weight of the quad-rotor is a vector that acts in the opposite direction to the force applied by the motors’ thrust.

eqn (3)

                                                                                                                                                                                                equation-5

The culmination of all the derivations performed is a simplistic equation that offers a glimpse into the way that thrust and weight interact in the context of a quad-rotor.

eqn (1).png

                                                                                                                                                                                                  equation-6

In the next post, feedback control will be discussed in a 2 dimensional context but in later posts this will be extended to 3 dimensions.  I chose to do it this way because I want to break everything down as much as possible for all my readers who want to get down to the nitty-gritty details of quad-rotor control and dynamics.

 

Previous Posts in this series

https://saucy-code.com/2019/05/15/the-engineering-method/

https://saucy-code.com/2019/05/15/autonomous-quad-copter-project-announcement/

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