[3]

Introduction

[1] Drones have  evolved  from the unmanned  aircraft vehicle  (UAV)  drones that were  first developed as military technology in the 1970s. Today's drones, specifically consumer drones, are marketed  for  aerial  videography  and  racing  and,  in  industry, use  of drones  is  explored by companies such as Amazon with shipping and Intel with entertainment as seen in the opening ceremony of the 2018 Winter Olympics.

In this project, you will explore drone control which is focused on adjustment of pitch, roll, and yaw axes to achieve desired orientation as well as vertical and horizontal motion (see Figure 1). Pitch drives the quadcopter forward/back, roll makes the quadcopter fly sideways left/right, and yaw – the rotational directional – is the deviation of the heading of the quadcopter right/left. The coordinate system shown in Fig. 2 is fixed to the body such that the xb axis points forward on the drone and the zb axis points downward. This is referred to as a “body frame” coordinate system.

Figure 1: Illustration of roll, pitch, and yaw rotations of a quadcopter [2].

Figure 2: Free Body Diagramfor the 3 DOF Hover Drone Model showing three rotation

angles and three azis of translational motion [2].

During flight, a quadcopter experiences several scenarios. Exclusive directional controls are used for maneuriving up/down, forward/backward, and left/right, as well as for maneuvers that require angular adjustment such as take-off and forward motion (pitch), capturing sideline footage of a football play (roll), and panorama aerial shots (yaw). Combinations of controls are used to combat wind gusts and take advanced cinematic shots that rely on synchronous directional and rotational movement. Automatic control of these combination shots is used in modern videography drones which can follow people and objects, as depicted in Figure 3.

(a)  Take-off and landing                     (b) Combating a gust of wind

(c)        Combined controls used tofollow the movement of a subject Figure 3. Example maneuvers of drones in action.

Project Goals

The goals of this project are to model a quad-copter, design controllers for it, and simulate its behavior under different conditions (and if possible test your controller on a real system).   In this phase of the project we will model the full system and simulate it. In the final phase we will design and test controllers for a subset of states, including pitch, roll, and yaw motion.

Basic Theory

We will employ a simple, linear dynamic model of the Quanser 3 DOF Hover Drone that is available to be used for testing our control laws.  We will plan to control the pitch, roll, and yaw axes of the 3 DOF Hover Drone with smooth control signals that do not saturate the actuator amplifiers. First, we will define a state-space representation (dynamic model) of the open-loop system, that can be used to simulate the system response.  In the second phase of the project, we will design various state feedback controllers that satisfy our design specifications.  Finally, we plan to implement the designed controller on the 3 DOF Hover Drone system to evaluate its actual performance.

Modeling Rotational Degrees of Freedom

A free body diagram of the quadcopter that highlights the three rotational degrees of freedom (DOF) is shown in Figure 2. There are four rotors that can be controlled independently in this multi-input-multi-output system. The three angles of interest are:

Roll: rotating about the xb axis

Pitch: rotating about the yb axis

Yaw: rotating about the zb axis

Note that positive direction for each angle is taken to be counter-clockwise when viewed down it’s associated axis toward the origin of the body frame.

When a positive voltage is applied to any of the motors, a positive thrust force is generated causing lift. There are four forces created, denoted by Ff, Fb, Fl, Fr (Ff = front, Fb = back, Fl = left, and Fr = right, and which correspond to F3, F1, F2, F4 in Figure 2, respectively). These forces control the angles of roll and pitch. For example, when the front motor force is greater than the back-motor force, then the system will rotate about the yb  axis, changing the pitch-angle. Similarly, the roll angle increases when the thrust force from the right motor is larger than that of the left motor.

The dynamics are modeled with the following equation.

J  = ∆F ∗ l                                                                     (1)

In this case, θ is the angle of the pivot about a given axiz, l is the distance between the propeller motor and the axis, J is the moment of inertia about the axis, and ΔF is the differential force-     thrust. The equations for pitch and roll are created from equation 1.

Jp p  = Kf l(Vf  − Vb )

Jr r  = Kf l(Vr  − Vl )

(2)

(3)

where Kf represents the thrust-force constant, V is the control voltage applied to one of the       motors (Vf = front, Vb = back, Vl = left, and Vr = right), and Jr and Jp refer to roll and pitch axis mass moments of inertia, respectively.

The yaw axis control is caused by the difference in torques exerted by the two clockwise and two counter-clockwise rotating propellers.

Jy y  = Kt (Vr  + Vl ) − Kt /Vf  + Vb 0

where Kt  represents the thrust-torque constant and Jy  refers to the yaw mass moment of inertia. The yaw axis rotation is a function of all four voltages.

Modeling Translational Degrees of Freedom

In a hovering state, when the quadcopter is level (rotors are spinning parallel to the ground) the   weight of the vehicle is supported by the combined thrust of the rotors (T) and this determines     their nominal speed.  To cause the quadcopter to change its height, the total vertical thrust can be adjusted by changing the motor speeds simultaneously (increase speed to accelerate upward by a defined thrust, or decrease speed to accelerate downward).  One can relate the total thrust to        acceleration along the vertical axis (see Fig. 2) as:

z̈c  =                                                                      (5)

where m is the vehicle’s mass and g is acceleration of gravity. For small angles, the body-fixed z- axis, zb, and the global vertical axis, zc , are similar, so the acceleration in Eq. 5 is approximately  the z-axis acceleration.

Horizontal motion in the xb and yb axes can be achieved by decreasing the speed of the rotor in  the intended direction while increasing the speed of the opposite rotor.  This causes a tilt in the  quadcopter such that its top is tilted slightly toward the desired direction of motion.  For small   angles, the resulting acceleration in either the xb or yb directions due to this type of offset motion can be written as:

ÿb  = ge$                                                                                                           (7)

Tasks to be Completed

Given the information supplied about the system, you are to:

1.   Sketch free body diagrams of the quadcopter in tilted configurations (e.g. horizontal views that show the appropriate off-axis rotors) and use force sum equations (Newton’s second law) along with basic geometry and the necessary information from Eqs. 1-4 to show that Eqs. 5-7 are reasonable small-angle approximations of the translational governing equations.

2.   Using the quadcopter angles and body-fixed coordinates defined above and their derivatives as the states, derive the state equations for the system.  Use motor voltages as the inputs and use the angles and translational positions as outputs. Clearly show the A, B, C, and D matrices. Leave all terms in variable form. Write your state vector in the following form: [xb, yb, zb, qr, qp, qy, derivatives]T

3.   Substitute the model parameter values into the state equations and show them in numeric form.

4.   Find the eigenvalues of the system and explain why they make sense given the physical nature of the system.

5.   Assuming the system starts in an equilibrium state of level hovering, numerically simulate     the system's open-loop response to observe these four separate cases: a (small) step input in   motor voltage that will cause a) roll, b) pitch, c) yaw, and d) vertical motion.  Show plots of  the results of the important states.  Explain why the resulting motion is as expected, given the system dynamics and eigenvalues.

6.   Create input functions (you can do this heuristically – that is, by trial and error) to the motors that cause the quadcopter to do the following:

a.   Increase height by 1 meter and come to rest (hovering) at that new height,

b.   Move to and maintain a pitch angle of 5 degrees.

Simulate these cases and show plots of the results of the important states.  Explain any   unexpected behavior, or motions apart from these specific degrees of freedom.  For case b, you don’t have to do this but consider if it is possible to cause the quadcopter to          maintain this pitch angle during stationary hovering.

To be submitted:

Submit a report (.pdf) for this project that contains a step-by-step discussion that leads me          through your work on parts 1-6, in the order stated above.  For each part, state the problem        statement and then provide your answer, including derivations, schematics (hand-written           derivations and schematics are fine), code, results, response plots, and explanations.  Do not use an appendix – instead provide all relevant material in each section.  In a separate file, provide    your Matlab code (.m script) or other code, e.g. Python if you don’t use Matlab.

Information for the Quadcopter

The following information is provided by the maker of the quadcopter system (Quanser) [3]:

% Motor Rotor Moment of Inertia (kg.m^2)

Jm = 1.91e-6;

% Moving Mass of the Hover system (kg)(total mass of the system)

m_hover = 2.85;

% Mass of each Propeller Section = motor + shield + propeller + body (kg)

m_prop = m_hover / 4;

% Distance between Pivot to each Motor (m)

l = 7.75*0.0254;

% Propeller Force-Thrust Constant found Experimentally (N/V)

Kf = 0.1188;

% Propeller Torque-Thrust Constant found Experimentally (N-m/V)

Kt = 0.0036;

% note: front/back motor are counter-clockwise (negative torque) and

% left/right motor are clockwise (positive torque).

%

% Equivalent Moment of Inertia of each Propeller Section (kg.m^2)

Jeq_prop = Jm + m_prop*l^2;

% Equivalent Moment of Inertia about each Axis (kg.m^2)

Jp = 2*Jeq_prop;

Jy = 4*Jeq_prop;

Jr = 2*Jeq_prop;

(Note that I’m not sure that I agree with the equation for Jy, considering the equations for Jp and Jr.  It makes sense that Jp = Jr, but not necessarily that Jy is twice Jp or Jr.  The manufacturer       seems to be saying that the motor (and rotor) mass moments of inertia (Jm) are the same about    each axis, which I would not expect to be the case. Nonetheless, these are the equations that they have provided, and we don’t have more detailed information about Jm, so we’ll assume they are  correct.)