Drones

U NMANNED F LYING O BJECTS

7.3 Drones

The term drone is very versatile. It is used in diverse fields (e.g., melittology–specialized entomology, film and television, science fiction, science, and technology) and names (e.g., music, bands, albums, and songs).

Perhaps the two fields that may be related in several ways are melittology, the science of the study of bees, and science and technology. A “drone” in melittology refers to fertile male ants (not the female workers or the male soldiers) that mate with the females to expand the kingdom of ants. It is also a name for similarly functioning bees who neither sting nor gather nectar but mate with an unfertilized queen and guard hives from predators. They travel from hive to hive mating with the receptive queens, not the virgin ones [153]. The virgin ones are the most active and fly away if they are disturbed, for fear of execution, and they are harder to spot. A virgin queen is easily accepted in a new hive while a mated one is at high risk of being killed by the workers. The mating takes place in the sky as they fly. Recall the fueling of a jet plane on a mission by the Boeing 747. The hovering bee with all its flight control mechanisms is like a UAV—in fact, it is more correct to say the opposite, since the bees came into existence long before UAVs (and to answer that old riddle—it is the egg that came before the chicken, for crocodiles were the earlier inhabitants of the Earth who laid eggs).

Examples of different types of UAVs are unmanned combat aerial vehicles also known as drones (to carry airborne weapons such as missiles);

multi-rotors, also known as multi-copters (to carry people or ammunitions and which are radio-controlled); quadcopters (a subset of multi-rotors), also known as quadrotor helicopters (to carry out projects in the military, photography, journalism, art, sports, and research); passenger drones (to move people); delivery drones (to transport goods and packages); and agriculture drones (to monitor and increase crop growth and production).

All of these flying objects have common flight mechanics. For them to ascend, descend, hover, and move in the desired direction, and to demonstrate the full range of capability of flying objects such as roll (moving about the longitudinal axes), pitch (moving about the lateral axis), and yaw (moving about the vertical axis), they are to satisfy the physics laws of linear and angular motions as well as balance of forces. Recall the earlier discussion on the balance of forces related to flying objects and fixed-wing aircraft. The drones are flying objects with rotors as the main mechanism to provide the thrust required to make the aforementioned motions possible. Each rotor interacts with the surrounding air by means of a propeller, which is either

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single-bladed or multi-bladed. A multiple rotor drone may be designed so that the direction of the rotation of the rotors or the relative location of the axes of rotations may vary (e.g., twin-synchropter, twin-coaxial, twin-lateral, twin-tandem, triple, and quadruple helicopters).

Consider the case of a single-copter—that is, assume there is only one rotor (rotor 1) versus four as shown in Figure 64. The thrust is generated by the rotational motion of the rotor blades. The resisting forces are the weight and the drag. If the balance of the forces (equivalent components of drag, weight, and thrust) is nonzero, the drone will accelerate in the direction where the force balance vector is pointing. This can be ascent, descent, or horizontal movement. Once the new desired velocity along a linear path is established, the balance of forces will become zero, since the acceleration is zero.

Think about this as a violin player who is to deliver Sergei Rachmaninoff’s Suite in D-Minor with a touch of Chopin and Liszt [154]. Although each musician is to exercise his own harmonic ingenuity, the music is open to interpretation, and it cannot be fully molded into the musical uniqueness that Rachmaninoff stands for. The violin player is free to introduce new harmonious arrangements while respecting the integrity of the original piece.

FIGURE 64 Quadcopter (drone) diagram along with the applicable forces in a horizontal orientation (drawings created using Solid Edge CAD tool).

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Now assume that the number of players is increased—a string quartet. The team is the violist, cellist, and two violinists. In this scenario, the additional players are to take their cues from the lead violin player (acting also as the conductor) in order to create a harmonious piece. Each musician may play the same piece of music in different arrangements or tempo. Although one may assume that they have practiced the piece for many long hours and have the notes memorized by heart, their ensemble may not sound as pleasant as intended in the original work—some may call that a “modern” piece. For this team to work flawlessly, they must have practiced alone and together to address the technical challenges as well as the synchronization of the efforts—the single effort in addition to the collective one. That has occurred by the means of the communication and feedback mechanism between the team. They may vary the tempo to emphasize the feeling, introduce more vigor in some places. No matter what the ultimate objective is, the collaboration of the parts is a must. The same applies to the drone operation.

The balance of the forces and torques must produce the desired motion;

the rotors are to work with each other correctly through carefully designed control and feedback mechanisms in order to make the required changes at the right time to follow the desired path.

Among the multi-rotors, fixed-wings, single-rotor helicopters, and fixed-wing hybrid Vertical Take-off and Landing (VTOL) drones, multi- rotors are the most popular due to their small size, high maneuverability, and agility. These qualities make them well-suited for small-area mapping;

however, limited endurance and speed make them unsuitable for large- area mapping, that is, one lasting over 30 min—the approximate life of their battery. The fixed-wing type is capable of flying for up to 16 hrs, carrying larger payloads, and employing gas engines. On the other hand, they are not as maneuverable, cannot hover over a point, and require more space for landing. Other considerations that affect their usage are associated cost, whether the collected data requires extensive processing, and even whether they are more prone to be attacked by birds of prey (e.g., wedge-tailed Australian Eagles also known as bunjil).

The single-rotor helicopter has one rotor with a slower spinning speed and larger blades, which makes it a more efficient option compared to a multi-rotor drone. For the same reason, a quadcopter drone is more efficient than a hexa-rotor drone. Nevertheless, the structure of a single-rotor helicopter is more complex and is associated with higher maintenance and operation costs. The VTOL aircraft combines the efficiency of a fixed-wing with the agility of a helicopter by adding a tilting rotor mechanism. There

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are also drones with swiveling wings to which propellers are attached. They are vertical at takeoff and then become horizontal in level flight. A tail sitter, sitting on its tail on the ground, is another possible configuration.

Figure 64 shows a multi-rotor drone, a Micro-Aerial Vehicle (MAV), in a horizontal orientation (for hovering, ascending, or descending while stationary relative to ground), and Figure 65 shows the same drone tilted (for translating relative to ground) along with the applicable forces and moments in both cases. These drones are capable of navigating inside buildings and tight spaces without the need to use a GPS system (which may not be available due to a lack of satellite signal reception); therefore, they are suitable for surveying inside and in between tall buildings, as well as underground and in tight spaces. This makes them suitable for investigating natural disaster zones and other hazardous areas. The four thrusts shown in Figure 64 and Figure 65 are to balance the four individual rotors’ weights in addition to the weight of the frame, onboard processors, communication systems, cameras or laser scanners, and batteries—the total weight—due to gravity as well as the drag force.

Assume that the weight of the drone is equally divided among the rotors.

Then, the total body weight is w (

4

1 i i

mg g m







^{), where m}i is the mass of the

FIGURE 65 Quadcop ter (drone) diagram along with the applicable forces in a tilted orientation (drawings created using Solid Edge CAD tool).

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rotor “i.” The weight is balanced by the thrust of the four rotors (FiKf i²i)—

where K _fi is a proportionality constant that considers the mass (m_i) and _i is the angular speed of the rotor. One pair of opposing rotors spins clockwise and the other spins counterclockwise in order to balance the torques. The resultant force vector and torque determine the direction of acceleration and the rate of rotation. Ignoring any wind disturbance, for the drone to hover over a location, the four thrusts are to be the same to create the opposite effect to gravity so that the resultant is zero, as in equation (121), where a is acceleration. Assuming similar-weighing rotors, the thrust is a quarter of the total weight (mg/4). If the total thrust is more or less than the weight, the drone will accelerate in the direction of the resultant force according to Newton’s second law of motion.

4

1 2 3 4

1

i D

i

F F F F F mg F ma



      



⁽¹²¹⁾

Equation (121) may be written in the form of equation (122), where y is the second derivative of the position vector (y) along the y-coordinate—

also known as acceleration along the y-coordinate. Let u₁ be the objective function representing the force (i.e., mass by the acceleration) along the y-coordinate, transverse to the hovering direction. The purpose is to obtain an objective function for linear acceleration (u₁) that minimizes the deviation from the input state variable (y) and desired location (y_desired) at a given time—in other words, you are to minimize the error, as in equation (123). The resultant moment is presented by equation (124), where I is the moment of inertia,  is angular acceleration, Mi is the reaction moment created by each rotor due to its angular rotation, and u₂ is the torque objective function.

4 2

2

1 2

1 f i i i

K mg ma md y my u

 dt

     



^{ (122)}

( )e t y t( )y_desired( )t (123)

4

1 1 2 2 3 3 4 4 1 2 3 4

1

2

. . . .

i i

xx xx

M F r F r F r F r M M M M

I I u



       

  





(124)

For the object to move in the positive horizontal direction along the y-coordinate, the resultant thrust vector due to rotor 2 is to be greater than that of rotor 4. This results in the thrust vector tilting toward the direction

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of motion and producing the motion due to the horizontal component of the resultant thrust vector. For the drone to move in any direction, there must be a component of the thrust in the direction of the motion, as in equation (125). Note that  is the rotation about the longitudinal axis (x)—

also known as roll,  is the rotation about the lateral axis (y)—also known as pitch, and  is the rotation angle about the vertical axis (z)—also known as yaw. Equation (126) is the moment equation that is based on the angular acceleration and moment of inertia in the y-z plane.

The vector form of equations (125) and (126) is presented by equation (127), showing the linear and angular accelerations along the y, z, and .

Note that the third matrix (3  3) from the left in equation (127) may be represented as a resultant rotation transformation matrix by  about the y-coordinate, which is a one-rotation matrix. It is possible to rotate the frame by a sequence of  or ; these rotational angles are also known as Euler angles. A 3  3 orthogonal matrix, which is a function of time (t), can be used to define each rotation; the total rotation may be obtained from the product of the three matrices. Orthogonality means that R(t)R^T(t) = 1.

Equations (128), (129), and (130) represent the rotations about the x axis (x, ), y axis (y, ), and z axis (z, ). The transformation representing the sequence of these three rotations is given by equation (131). Note that for the rotation to take place about an axis, the axis does not move—in other words, there is a point within that frame (body-fixed or space-fixed) whose position does not change—equation (132)—assuming that the ori ginal frame of a body at time t from p(t), which is a 3  1 vector, has rotated to q(t) by means of a rotation matrix R(t). It is possible to take a time derivative of this equation and therefore predict the velocity in the body frame (R t q t^T( ) ( ) R t R t p t^T( ) ( ) ( ) ) and space frame (q t^{ }R t p t^{   })—

where R^{T   }t R t is the angular velocity in the body-fixed frame and ( ) ^T( )

R t R t is the angular velocity in the space-fixed frame. The body-fixed frame is attached to the body; the body-fixed frame is moving with respect to a space-fixed frame (a fixed frame somewhere in the environment), as in the second part of equation (132). For a translation and rotation motion, however, the entire frame and therefore the associated points move.

 

4

1 2 3 4

1

i y Cos D y

i y

F ma my mg F F F F F



 

          

 





    

 

 (125)

4

1 1 2 2 3 3

1

4 4 1 2 3 4

. . .

.

i xx xx

i

M I I F r F r F r

F r M M M M



      

   



^{ (126)}

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¨

¨ 1

¨ 2

1 0

0 Sin 0 0

Sin Cos 0 1

0 0 1

0 0

0 1

xx

y m

g u

z m u

I

 

   

 

 

     

   

 

       

       

      

   

   

(127)

 

1 0 0

, 0 Cos Sin

0 Sin Cos

Rot x

 

 

    

  

 

(128)

 

Cos 0 Sin

, 0 1 0

Sin 0 Cos

Rot y

  

 

  

  

 

(129)

 

Cos Sin 0

, Sin Cos 0

0 0 1

Rot z

    

 

    

 

 

(130)

 

 

 

11 12 13 21 22 23 31 32 33

, , ,

R R R R Rot x Rot y Rot z R R R R R R

 

 

       

 

 

Cos Cos Cos Sin Sin

Sin Sin Cos Cos Sin Sin Sin Sin Cos Cos Sin Cos Cos Sin Cos Sin Sin Cos Sin Sin Sin Cos Cos Cos

       

 

             

 

            

 

(131) ( ) ( ) ( ) ( )^. ( ) ( )^. ( ) ( )^.

( ) ( ) ( ) ( ) ( ) ( )

T

T T

q t R t p t q t R t p t R t q t R t R t p t q t R t R t

   

    

(132)

The drones shown in Figure 64 and Figure 65 have six degrees of freedom, three translational along the x, y, and z-coordinates and three rotational about the lateral (pitch), longitudinal (roll), and vertical (yaw) axes (, , and  angles), which along with their derivatives and second derivatives determine the drone dynamics. It can be inferred then that the rotor angular velocity, the forces due to the thrust by each rotor, and the resultant force are the factors in determining the direction of the drone translation, rotation, hovering, acceleration, and deceleration. Controlling the combination of these parameters is technically challenging, since for the

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mechanical system to work as expected, the rotors are to be interconnected to adjust their reaction based on the received feedback—meaning that the combination of four rotors, which results in twenty-four outputs, needs to be adjusted simultaneously, which is not practical given the complexity of the system. A control feedback system is then to be designed for such systems.

To get a better idea of how a control system works, consider holding in your hand a string to which a small mass suspended in the air is attached, that is, a pendulum, as in Figure 66. The task of this “control system” is to move this small mass along a horizontal line, from one point to another. The desired output in this case is the mass position along the line of movement.

For example, you can specify one possible desired output as a step function of time whose value changes from 0.1 to 1 at 0.3 s—in other words, it is desired that the mass moves from position 0.1 to a new location at position 1 at 0.3 s.

If you were to simply execute this motion with your hand, without any consideration of what the mass was doing, you can imagine that, when you abruptly started moving, the mass initially lagged behind, and when you stopped the movement, the mass kept swinging like a pendulum for some time. Now consider that you are carefully watching the small mass as you move it. You observe how faithfully the mass follows the desired movement

FIGURE 66 Example of a pendulum position contro l.

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and try to adjust your hand position so as to minimize the deviation from the desired position. What you are doing is using your visual feedback system to adjust the control input (your hand position) to minimize the error between the desired and actually observed mass position.

Figure 67 illustrates the behavior of the control system described in the previous example. The figure shows a solid line with hollow circles depicting the reference signal. This is the desired output for your system (a step change in position from 0.1 to 1 at time 0.3 s). So, at 0.3 s, when this new desired position is set, if you have a feedback mechanism in place, it will immediately inform your control system—with the coolness of a robot—that the mass is off-target by the position error e_p(t). You can then generate a control signal (your hand movement speed) proportional to this error, with the proportionality constant equal to K_p.

The mass responds to your movement, and as a result its position decreases or increases over time. This position change, however, reflects neither the future nor the past response. Thus, the pendulum may overshoot or undershoot the target. To account for the future response, the rate of change of the amplitude (i.e., the derivative with respect to the time) needs to be calculated. This new feedback mechanism, which is called a derivative control, with proportionality constant K_d, is added to the previously described proportional control. It acts similarly to a dashpot (or a shock absorber) to dampen oscillations in a controlled system.

FIGURE 67 Example of response versus the time for a PID system.

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This new control system design provides both instantaneous and future-time-related feedback to the amplitude response; however, it does not consider the history of the motion. For example, if there is a persistent small error between the desired and actual position, the proportional response may be too weak to correct it. However, if we keep adding up this error over time (i.e., integrating it), it will accumulate to provide sufficient corrective action. Therefore, this feedback mechanism is to consider the past states of the system, and they are to be accumulated to reflect the effect of the temporal changes of the spatial variables to modify the present results—an integral control is introduced by the proportionality constant K_i. The sum of the new integral control input and the previous two control types now determine the destiny of the mass. Depending on the values given to these three control constants, the system response can vary from having a large overshoot (dashed line), a small overshoot (solid line), or a damped response (dashed-double-dotted line), as in Figure 67.

The previous discussion demonstrates the concept of the Proportional, Integral, and Derivative (PID) control systems. A properly designed control system combines the three types of feedback so that the objective temporal- spatial function is achieved (Figure 68). Similar control mechanisms (PID) control the rotors mounted on a rigid frame and work independently from each other. They cause increase or decrease in angular velocity of the rotors, and the result is the roll or pitch motions in the direction of the acceleration. The created thrust is normal to the rotation plane. To make the drone move horizontally, the drone is to pitch forward so that the thrust vector has a component in the direction of the motion. To make the drone stop, the angle of the pitch is to be reversed until it gets to the equilibrium state where the destination is reached.

FIGURE 68 Example of a PID control diagram for a dron e (drawings created using Solid Edge CAD tool).

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