Dynamic Modeling-based Flight P-PD Controller Applied to a Quadrotor

Abstract

In this paper, we describe performances of P-PD controllers in the quadrotor system with steady-state error compensation by adding a corrective term to the system input. A decentralized control system using P-PD controllers was successfully implemented on a quadrotor platform. We also presented the results of a mathematical modeling analysis for control the quadrotor and experimental results for each response performance according to the heading reference value in accordance with the mathematical modeling and P-PD controller design. A control experiment with the real system was implemented for the test platform, and the results were evaluated and compared.

1. Introduction

There has been ongoing research and efforts toward commercialization of unmanned aerial vehicles (UAVs) in various fields of application such as logistics services, disaster relief, surveillance, and entertainment, in accordance with the remarkable progress that has recently been achieved in the miniaturization and performance of UAVs[1][2].

This paper presents a mathematical model for the construction of a system structure and control method in order to implement remote control technology for application to a quadrotor helicopter in development, which is the ultimate objective of this study. The quadrotor UAV in development will be used in actual tasks such as gathering information during disasters or for security as a small autonomous flying robot. Two studies were conducted to identify a design solution in order to achieve this purpose[3][4].

The first involved the design of a small electronic circuit with powerful processing power centered around an inertial measurement unit (IMU) and a microcontroller unit (MCU), and the second involved the design of a control algorithm and an operational program optimized for computational speeds in order to implement the two requirements of autonomous control and task completion (Fig. 1). Fig. 3 shows the system used for the experiment. The quadrotor’s flops have been removed for safe experimentation.

Fig. 1 Flight-capable prototype of quadrotor

2. Mathematical Modeling

This section introduces the mathematical model of the quadrotor-type multi-rotor helicopter for flight tests. This model is basically obtained by representing the quadrotor as a solid body evolving in three dimensional space. The flexibility of the blade due to high-speed rotations was not considered in the dynamics of the six electric motors mounted on the vehicle.

First, the force applied to the body can be expressed as follows.

\(F = d ( m v _ { B } ) / d t + v \times ( m v _ { B } )\)       (1)

Here,  is the angular velocity and can be expressed as a vector: [p q r]T . The mass m is a constant.

Each rotor i has an angular velocity i, and generated a thrust force, i.e., f [0 02], i i together with a constant force k. Therefore, the driving force T can be expressed by B the following equation, which has the dimension of T [00T]T. B

\(T = \sum _ { i = 1 } ^ { 6 } f _ { i } = f \sum _ { i = 1 } ^ { 6 } \omega _ { i }\)       (2)

Furthermore, the full driving force that considers the gravitational force that acts on the body can be expressed as follows.

\(F = Q ^ { T } F _ { g } + T _ { B }\)       (3)

Here, Q is the quaternion, which is the operator value for the rotation calculated in a 3D space, and the equation proposed in an existing paper[2] was cited.

Furthermore, the total external moment M sum each angular momentum, and the change rate of the angular momentum H I is applied. Therefore, the moment applied to the quadrotor can be expressed as follows.

\(M = d ( I v ) / d t + v \times ( I v )\)       (4)

Thus, each rotor generates the following torque force, which includes the angular velocity and acceleration components around the rotation axis of the rotor.

\(\tau _ { M _ { i } } = b \omega _ { i } ^ { 2 } + I _ { M _ { i } } c _ { i } ^ { i }\)       (5)

Here, bis the drag constant, and IM is the inertia moment of rotor i.

The values of the roll, pitch, and yaw can be expressed in the following equation using the fi and Mi components in relation to the geometric structure and full-frame of the quadrotor-type helicopter.

       (6)

Here, l is the distance from each rotor to the quadrotor’s center of gravity, and & i appears as the differential value of  (t) in i relation to time t (i.e., d(t)/dt). i Consequently, the equation to determine the rotational kinematics of the quadrotor is as follows.

\(I 1 \& z \times ( I v ) + \Gamma = \tau _ { B }\)       (7)

Here,  is the gyro force, and B is the external torque. & in the resulting mathematical equation can be expressed as follows.

\(I \delta I ^ { - 1 } ( \left[ \begin{array} { c } { p } \\ { q } \\ { r } \end{array} \right] \times \left[ \begin{array} { c } { I _ { x x } p } \\ { I _ { y y } q } \\ { I _ { z z } r } \end{array} \right] - I _ { \Gamma } \left[ \begin{array} { c } { p } \\ { q } \\ { r } \end{array} \right] \times \left[ \begin{array} { l } { 0 } \\ { 0 } \\ { 1 } \end{array} \right] \omega _ { \Gamma } + \tau )\)       (8)

Finally, the angular acceleration in the inertial frame of reference when the angular velocity is derived can be expressed as q&&d(S)/dt .

3. Mechanism Design

ABS resin was used for the hexagon shaped base and material of the vehicle, and the top board for mounting was printed with a 3D printer using CAD data.

A brushless motor and a 15-inch propeller were mounted at the end of each rotor, and the center was equipped with an ARM processor, electronic circuits, and a battery. An attachable 2-axis gimbal was mounted at the bottom for video equipment as shown in Fig. 2.

Fig. 2 Circuit system overview

Research was conducted on the production and flight control of a multi-rotor electric helicopter with industrial applications that include services with the highest user demand such as information gathering through aerial photography and monitoring, and lightweight logistics transport, in order to develop the multi-quadrotor-type electric helicopter proposed in this study. To specify the flight position, commercial GUI was used as shown in Fig. 3. The specifications of the produced quadrotor are as follows.

Fig. 3 Adupilot GUI interface

It is possible to operate the vehicle as a flying robot in a remote control form through manual controls (manual mode) and in an automatic piloting equipment-based autonomous control form (autonomous mode). A camera is attached to the gimbal mounted on the flying robot, and is designed to allow aerial photography at a particular time.

It is possible to operate the flying robot to move in a specific direction by monitoring the status of the vehicle from the ground through a two-way wireless communication module and navigator

An open source mission planer platform is used for the flying robot, in which specific flight paths to a certain location can be saved in the navigator as remote commands in the autonomous mode. This is a flat GUI that enables the user to fly the drone while setting multiple points as turning points.

4. Experiment and Results

As Flight Simulation, the previous section proposed a mathematical model for a quadrotor helicopter. Therefore, the experimental characteristics in relation to the positions in flight (i.e., location and yaw) based on the proposed model were observed.

Fig. 4 presents the MATLAB-based simulation results in relation to the x, w, and z location and yaw angle of the flight simulation. The decoded input signal for the flight direction from the remote controller on the ground was set as the input in the simulation, and as seen in Fig. 4, the results of the step response according to the proposed mathematical model showed that the vehicle entered a steady state in 6 sec.

Fig. 4 Position and angle in Matlab with mathematical model

As Flight Experiments, in Fig. 5, we can see the selected waypoints as well as the quadrotor trajectory, plotted in real-time. We

can clearly observe that the quadrotor passed successfully through all the waypoints. The accuracy of this waypoint navigation is confirmed by Figs. 5. The reference position and orientation are tracked with high accuracy.

Fig. 5 Test field and quadrotor trajectories

From the third graph of Fig. 6, we can also see that the height, estimated by the altimeter sensor, is more accurate than the one estimated by GPS. Despite low-cost sensors, we have achieved an accurate position and orientation control even during relatively low-speed flight (3m/s).

Fig. 6 Position (x, y, z) and orientation (ψ, θ, φ) of the quadrotor. The dotted lines represent the desired trajectories

Furthermore, the take-off and landing maneuvers have been achieved autonomously.

It is interesting however, to note that position control is more accurate at low forward speeds as shown in Fig. 6 (about 50cm maximum error). Indeed, the thrust variation created by different angles of attack at varying forward speeds and wind conditions causes a disturbance that pushes the vehicle above the reference height. The controller takes some time to reject these disturbances because of the time delay in thrust.

5. Conclusion

This study analyzed a stabilization control algorithm for a mini-rotorcraft having four rotors, which is one step more advanced than the existing quadrotor design, as well as the flight motions based on the rotation of each rotor for flight. We also presented the results of a mathematical motion analysis of the wing direction and roll and pitch control required to operate the aircraft to follow a planned route remotely using a PC on the ground.

The proposed P-PD control method has been successfully applied to the quadrotor system, and the experimental results have shown that the controller performs satisfactorily. Therefore, they showed the possibility of using this controller for lateral directions with robust responses to disturbances.