A study on the modeling of a hexacopter

2.1 Modeling coordination

To keep track of the hexacopter, two different coordinate frames are used to represent the position and orientation in three dimensions—the earth frame (Oxyz) and the body frame(O_bX_bY_bZ_b), as seen in Figure 1. The earth frame is a fixed frame used as an unmoving reference. For example, if the user wants to define a path that the hexacopter should follow, then that path would be represented in the earth frame. The body frame axes are aligned with the sensors, which is convenient when reading sensor data and controlling the angular orientation (attitude). As seen in Figure 1, the body frame in this configuration is defined as having the X axis lie between the arms of motor 1 and motor 6, the Y axis lie along the arms of motor 2, and the Z axis pointing upward. The value represents the distance from a given motor to the axis of rotation and should be the same for every motor. The x axis is assumed to be the positive forward direction for vehicle movement. For clarity, the rotation conventions are shown in Figure 2.

Figure 1:

System coordination

Figure 2:

Axis labels and conventions

The angular position of the body frame with respect to the inertial one is usually defined by means of the Euler angles: the roll φ, pitch θ and yaw ψ angles. According to the aerospace rotation sequence, the rotation of an aircraft is described as a rotation about the z axis (yaw) and then a rotation about the y axis (pitch) followed by a rotation about the x axis (roll). Each rotation is made on the basis of a right-handed system and in a single plane.

For the sake of simplicity, let us denote the inertial position vector and the Euler angle vector by means of ξ and η , respectively.

In the body frame, the linear velocities are determined by V_B and the angular velocities by υ, respectively.

Using these three rotations, a composite rotation matrix can be created, which can transform the motion of the aircraft from the body frame to a new reference frame. The resulting rotation matrix transforms the rotations from the body frame with respect to the inertial frame and can be found using matrix multiplication. In Equation (3), s and c represent the sine and cosine functions, respectively; R is the rotation matrix that transforms the rotations from the body frame to the inertial frame.

As a consequence, the transformation matrix from the inertial frame to the body frame is R^-1 = R^T.

As shown in [1], the transformation matrix for the angular velocities from the inertial frame to the body frame is W_η.

Further, the transformation matrix from the body frame to the inertial frame is W_η^-1.

Then the transformation laws are

2.2 Quaternions

In the above section, it is pointed out that the rotation of a rigid body in space could be represented by Euler angles; however, the singularity of the transformation laws in Equation (6) leads to the adoption of a new parameterization—the quaternion—with the aim of describing the orientation of the aircraft with respect to the fixed earth frame [2][3].

The quaternion representation is based on Euler’s rotation theorem, which states that any rigid body displacement, where a point is fixed, is equivalent to a rotation. Therefore, if α is the rotation angle about the unit vector u , it is possible to define a quaternion as

whose components describe the three-degree orientation. Unlike the Euler angles, quaternion rotations do not require a set of predefined rotation axes because they can change its single axis continuously. Owing to the fact that the method of rotation around an arbitrary direction has only one axis of rotation, the degrees of freedom cannot be lost; therefore, a gimbal lock cannot occur. Otherwise, a quaternion can be defined by four components describing a three-dimensional rotation. Thus, the quaternion elements must satisfy a constraint equation called the normality condition, i.e.,

where q_i = q_i(t ) is assumed for the sake of readability. The transformation of the translational velocity representation from the body frame to the inertial one can be expressed by

where Q is the following matrix:

As for the matrix R, Q is orthogonal; therefore, Q^-1 = Q^T. For the angular velocities, the involved transformation can be written as

where the matrix S depends on quaternion components as follows:

On the other hand, it is possible to consider the transformation matrix depending on the angular velocity components, thereby obtaining the link between quaternions and their derivatives with respect to time, which are

In conclusion, it is remarked that the advantage of considering the quaternion reference is twofold because it avoids critical positions, and, thanks to the linearity of the coefficients of the transformation matrix, it is also numerically more efficient and stable compared to traditional rotation formulation.

2.3 Mass moment of inertia matrix

The mass moment of inertia of an object (J) plays a similar role in the rotational motion to the role that mass plays in translational motion: the mass moment of inertia determines how the rotational velocity is affected by the applied torque. This of course depends not only on the mass of the object but also on how the mass is distributed around the rotation axis. It is important to note here that the hexacopter is assumed to be perfectly symmetric about the xy and z axes and to have its center of mass at the geometric center of the arms. With these assumptions, the matrix J_b in Equation (15) becomes a diagonal matrix (note that this is related to our choice of the x- and y-axis positions). The J_xx and J_yy terms are also taken to be identical owing to this symmetry.

2.3 Thrust

The motors' thrust is the driving force behind all hexacopter maneuvers and is thus integral to control design and simulation. The thrust, T_S, provided by a single motor/propeller system can be calculated as Equation (16)

where C_T is the lumped parameter thrust coefficient that pertains to the individual motor/prop system, and $$ \stackrel{\mathrm{\text{̅}}}{\mathrm{\omega }}$$ is the angular velocity of the rotor. The total thrust T provided by the motor/prop system is a force perpendicular to the X–Y plane of the body frame in the positive Z direction and is formulated by Equation (17)

2.4 Torque

In theory, an increase/decrease in the speed of the six rotors independently will create torques around the x y and z axes, thereby creating roll, pitch, and yaw rotations. The torque is force multiplied by a distance, and the rotors will affect the total rotation about a certain axis differently, depending on the distance from the center of gravity. Figure 3 shows the lengths and angles of the arms to the relative distance from the center of gravity, which is the distance from the rotor to the axis of rotation.

Figure 3:

Hexacopter rotor distances to center of gravity

By decreasing w₁,w₂,w₃ and increasing w₄,w₅,w₆ a positive roll moment is produced as Equation (18)

By decreasing w₁,w₆ and increasing w₃,w₄ , a positive roll moment is produced as Equation (19)

In order to understand the effect of the motor on the yaw, the torque force of the motor/prop system must also be determined and can be done in a similar fashion to that of the thrust tests. The related lumped parameter is Equation (20)

where τ is the torque created by the motor, and C_τ is the torque coefficient for the motor/prop system. This torque provides a force that acts to yaw the system about the z axis.

By decreasing w₁,w₃,w₅ and increasing w₂,w₄,w₆, a positive roll moment is produced as Equation (21).

The rotation of the propellers produces a gyroscopic effect that is expressed as Equation (22).

where J_r is the rotational inertia of the propeller, and ω_r =- ω₁ + ω₂ - ω₃ + ω₄ - ω₅ + ω₆ is the overall propeller speed.

3.1 Translational dynamics

The force acting on the hexacopter is provided in the body frame, and the force required for the acceleration of mass and the centrifugal force are equal to the gravity and the total thrust of the rotors as follows: Equation (23).

where R^TG is the gravitational force acting in the body frame, and G = [0 0 -g]^Tis gravitational force in the inertial frame.

In the inertial frame, the centrifugal force is nullified. Thus, only the gravitational force and the magnitude and direction of the thrust contribute to the acceleration of the hexacopter as Equation (24).

From Equations (1), (3), and (17), Equation (24) is equivalent to Equation (25).

or in quartation form

3.2 Rotational dynamics

In the body frame, the external torque, τ_B = [τ_φτ_θτ_ψ]^T, is equal to the torques due to the angular acceleration of the inertia, $$ J_b\stackrel{\text{̇}}{\upsilon }$$ the centripetal forces, υ ☓ (J_bυ) and the gyroscopic forces, τ_gyro as Equation (27).

After some algebra, the following is obtained:

Equation (28) can be written as Equation (29)

Once the angular velocity has been evaluated, the angular acceleration in the inertial frame can be easily deduced as

Finally, by combining Equation (26) and Equation (30), the differential dynamics equation of the hexacopter system can be obtained as

4.1 Altitude control

The goal of the altitude controller is to keep the hexacopter at a reference value by using altitude and altitude-velocity measurements. This is an attitude-command-only system. In other words, there is no control system to track the position. Instead, the controller only tries to track the attitude (φ, θ, ψ) and altitude (z) commands using PID controllers.

A cascaded Position → Velocity PID control structure was implemented according to Figure 4, where the outer loop used a PID controller in which the controller’s output signal was calculated on the basis of the altitude reference, altitude measurements, and velocity measurement as Equation (32).

Figure 4:

Altitude controller

This desired velocity was used by the inner loop, where a PI controller calculated the required change in the throttle as Equation (33).

The simulation conditions are summarized in Table 1–3. The reference signal is listed in Table 1, in which (φ, θ, ψ) are in units of degrees and the altitude (z) in reported in meters. The initial conditions of given system and the PID controllers' parameters [8][9] are listed in Table 2. Table 3 summarizes the parameters of a certain model of a hexacopter.

Table 1:

Reference signals

Table 2:

Initial conditions and PIDs parameters

Table 3:

Hexacopter's information

The simulation results for altitude control are shown in Figure 5.

Figure 5:

Altitude control results

It is obvious that all altitude control purposes were achieved by the proposed PID controllers. All parameters consisted of roll, pitch, yaw, and the high flight of the hexacopter reached the set points after a controlling time. Figure 6 shows the reaction of the motors during the simulation time. At 0 s, the speeds of all motors increased at same rate to generate the required thrust to lift hexacopter from the initial position of 0 m to a set point of 8 m. At 10 s, w₁,w₂,w₃ increased, and w₄,w₅,w₆ decreased to produce a negative roll moment, which decreased the roll to a set point of -10°, as seen in Figure 7.

Figure 6:

Speed of motors

Figure 7:

Speed of motors when changing roll

Similarly, Figure 8 and 9 show the behavior of the motors at points where the pitch and yaw and changed.

Figure 8:

Speed of motors when changing pitch

Figure 9:

Speed of motors when changing yaw

4.2 Horizontal control

The horizontal controller used the roll and pitch angles to change the horizontal position and horizontal velocity.

Similar to altitude control, a cascaded Position → Velocity controller was used again, as seen in Figure 10. This time, the outer position loop used a PID controller that outputs a desired velocity on the basis of the current position error and velocity measurement as Equation (34).

Figure 10:

Horizontal controller

The inner velocity loop uses a PD controller to calculate the desired roll and pitch angles from the velocity error using Equation (25). Here, the PD controller is used for the sensitive response to the possible future values of the error based on its current rate of change.

The position and velocity control utilizes the earth frame; thus, the roll and pitch set points have to be rotated by the yaw rotation matrix because the hexacopter operates in the body frame.

In this simulation case, the information of the hexacopter model and the initial conditions were the same as previous simulation work. The reference signals in this simulation were a point in the inertial frame, which is summarized in Table 4. Table 5 summarizes the parameters of the PID controllers, which are proposed for this case of horizontal control. The parameters applied to the inner-loop PD controllers are listed in Table 6.

Table 4:

Reference point information

Table 5:

outer PIDs parameters

Table 6:

Inner PD parameters

Figure 11 shows the tracking reference-point simulation results. After 10 s, the hexacopter almost reached the set point (10, 5, 5) in the inertial frame, meaning that the tracking task was achieved. Figure 12 shows the tracking error during the simulation time. The tracking error decreased with time and almost reached zero around 10 s when the tracking task was achieved.

Figure 11:

Tracking of reference-point

Figure 12:

Tracking error

The behavior of the motors during tracking is shown in Figure 13. From the initial point, the horizontal controller sent a command to increase the speeds of all six motors to generate thrust to lift the hexacopter and reach the set point. The hexacopter changed its Euler angles to move from the initial point to the set point. Figure 14 shows the details of the motors’ reaction. In this figure, the hexacopter changes its roll angle by increasing or decreasing the speed of motors 1, 2, or 3 and motors 4, 5, or 6 respectively. These simulation results demonstrated that horizontal control is achieved with the mentioned PD and PID controllers.

Figure 13:

Speed of motors during trajectory tracking

Figure 14:

React of motors in some interval during tracking time