Attitude control of small unmanned four-rotor helicopter based on adaptive inverse control theory/Keturiu rotoriu mazo bepilocio sraigtasparnio padeties kontrole remiantis adaptyviaja inversine kontroles teorija.

Jin-song, Li ; Xi, Cao

1. Introduction

Small unmanned four-rotor helicopter (four-rotor) is a kind of noncoaxial, multirotor, dished vehicle with vertical take off and landing (VTOL) ability. Due to the complexity, strong coupling and sensitivity effects on the environment of four-rotor's dynamic model, the controller must have high quality of robust and adaptive. According to these requirements, a lot of control theory have been proposed, including Backstepping [1], LQG [2], ADRC [3] and adaptive sliding mode [4]. This paper propose a new method that applying the adaptive inverse control (AIC) theory to attitude stabilization control. The AIC theory uses output difference of real object and its co-input model to drive the inverse model. This model can generate a filtered noise and interference. And the ultimate input is the difference between previous one and this filtered signal.

Sections 2-3 construct the experimental platform hardware and dynamic model of four-rotor; Section 4 conducts the structure of AIC controller; Section 5 gives the final result of applying AIC theory to four-rotor attitude control experimental.

2. Experimental platform hardware system of four-rotor

The experimental platform for four-rotor is shown in Fig. 1.

[FIGURE 1 OMITTED]

The experimental platform has orthogonal glass-fiber pipes structure, 4 Hi-Model brushless motors and 4 rigid plastic rotors. When the rotors on X-axis rotate clockwise, the other rotors on Y-axis will rotate counterclockwise simultaneously so that the anti-torque can be cancelled out. In the process of flight control, changing the speed of all rotors equally at the same time will cause the up-and-down motion of the four-rotor. Increasing the speed of one rotor meanwhile equally decreasing the one belongs to the same group (each two rotors on the same axis is called a group), the pitch and rolling motion can be accomplished. In addition, the yaw motion will be genera-ted by increasing the speed of one group while decreasing the speed of the other.

The hardware of this system consists of power unit, inertia measurement unit (IMU), airborne GPS navigation and positioning unit, wireless communication unit, height measurement unit, motor speed measurement unit and embedded microcontroller unit. The detailed devices of each module are shown in Table.

3. Dynamic model of four-rotor

Two coordinate systems are set up to describe the dynamic model of four-rotor, which are shown in Fig. 2.

[FIGURE 2 OMITTED]

E(X, Y, Z) is an absolute ground coordinate system which relatives to a stationary reference point. The center of frame, says O, is chosen as the origin. The velocities and displacements of the four-rotor are measured in this system. And the body coordinate system B(x, y, z) is a system whose origin is the geometric center of four-rotor. The pitch angle [theta], roll angle [phi] and travel angle [psi] (Euler attitude angles) are described in this system.

According to Newton-Euler equation [5-8], we can get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where m is the mass of model; [F.sub.x], [F.sub.y], [F.sub.z], are the components of the lift force belong to each axis. Air resistance is assumed to be proportional to the speed of model, with a coefficient [k.sub.1]. Therefore, [k.sub.1][??], [k.sub.1][??], [k.sub.1] is the air resistance which is opposite to the speed vector. Furthermore, there is a relationship as shown in Eq. (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [F.sub.1], [F.sub.2], [F.sub.3], [F.sub.4] are the lift forces of each rotor, which are proportional to square of rotation speed; [A.sup.BE] is defined as a transfer matrix between body coordinate system and the inertia one. Rearrange Eqs. (1) and (2), the displacement dynamic equation can be described by Eq. (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Similarly, the angular velocities p, q, r can be described by Euler angels [psi], [phi], [theta] which is shown in Eq. (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](4)

According to Newton-Euler equation, we can get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](5)

where l is the distance from the center of the model to the center of any of the rotors (the action point of lift force); [lambda] is a scale factor between z axis torsional torque and the lift force; [I.sub.x], [I.sub.y], [I.sub.z] are the rotating moments of the four-rotor reference to axis x, y, z, respectively. Therefore, ([I.sub.y] - [I.sub.z])qr, ([I.sub.z] - [I.sub.y])rp, ([I.sub.x] - [I.sub.y])pq reflect of the gyroscopic effect of the model; and [-I.sub.R]q ([-[omega].sub.1] + [[omega].sub.2] - [[omega].sub.4] + [[omega].sub.3], [-I.sub.r]P([- [omega].sub.1] + [[omega].sub.2] - [[omega].sub.4] + [[omega].sub.3] the gyroscopic effect of the rotors.

In order to take further derivation, some more variables are defined: [u.sub.1] is defined as the sum of [F.sub.1], [F.sub.2], [F.sub.3], [F.sub.4]; [u.sub.2] is the resultant moment of the rotors which generate the roll angle, while [u.sub.3] is the resultant moment of the rotors which generate the pitch angle; finally, [u.sub.4] is defined as the travel moment due to adjusting the rotor speed, which is proportional to the lift force. So there is a matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Taking [I.sub.x] = [I.sub.y] into consideration and neglecting the gyroscopic effect of the rotors. Rearrange Eqs. (4)-(6), the dynamic equations of the Euler angles can be described by Eq. (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Therefore Eqs. (3), (7) are the dynamic equations of the four-rotor.

4. Design of AIC controller

AIC is successfully utilized in many fields [9]-[13], and especially suitable for the control objects having multivariable, nonlinear, strong coupling and interference sensibility. The completed structure of controller is shown in Fig. 3, which was in the published paper of authors [14]. Under the ideal forward and inverse model condition, the unique antiinterference structure can make the transfer function, the ratio of output and the noise from sensors, approach to zero. Which means the noise and interference can be effectively restrained at the output [15, 16]. Due to this characteristic, the AIC theory can be applied to antiinterference and attitude control of four-rotor.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

According to the form of the dynamic equations, the flight control of the four-rotor is divided into four independent channels. They are roll angle control channel, pitch angle control channel, travel angle control channel and height control channel. The structure of four-rotor control system based on AIC is shown in Fig. 4.

Where AIC1, AIC2, AIC3, AIC4 is in roll angle, pitch angle, travel angle and height control channel, respectively.

Least mean square (LMS) algorithm is a typical control algorithm of AIC method. Since simulation results show that, for the model presented in this paper, LMS algorithm has a slow convergence rate and easily divergent. Therefore, N-LMS algorithm is used to identify the parameters of the controller and the reference model. The iterative formula is shown in Eq. (8)

[W.sub.k+l] = [W.sub.k] + [mu] [e.sub.k] [X.sub.k]/[[gamma] + [[parallel] [X.sub.k][parallel].sup.2]] (8)

where [X.sub.k], [W.sub.k] and [e.sub.k] are the output vector of filter, desired output, weight vector and error; [gamma] > 0 is a positive, which is small enough to confirm the step is bounded; 1 > [mu] > 0 is the weight coefficient. Taking the roll angle control channel for example, the iterative formula is

[[[phi].sub.k+1] = [[phi].sub.k] + [mu][[phi].sub.k] /[[gamma] + [[parallel][[ PHI].sub.k][parallel].sup.2]] (9)

Similar for the other three models, the rotation speed of the rotors is described by Eq. (10).

The rotation speed of rotors is controlled by processor so that the lift force, further the flight attitude, can be stabilized.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

5. Attitude control experiment of four-rotor

In actual experiment, the length and width of Four-rotor are equal, L = W = 0.54 m; height H = 0.15 m; l = 0.24 m; m = 0.725 kg. Sunplus microcontroller SPMC75F2413 is used to generate the PWM to control the brushless motor. The lift force is measured by high-accuracy electronic balance (force sensor).

According to the measurement, the relationship between lift force F and PWM can be described as shown in Eq. (11)

PWM = 1105 - 261 x F + 22.23 x [F.sup.2] (11)

Through three-line pendulum, the moment can be calculated by Eq. (12)

J = [mga.sup.2]/4[[pi].sup.2]b [T.sup.2] (12)

where a is the distance from suspension line to body of four-rotor, b is the length of suspension line. And T is the swing period. Therefore, the rotating moment is obtained as

[I.sub.z] = [mga.sup.2]/4[[pi].sup.2]b [[T.sub.z.sup.2] = 0.0664 kg [m.sup.2]

[I.sub.x] = [I.sub.y] = [mga.sup.2]/4[[pi].sup.2]b [[T.sub.xy.sup.2] = 0.0479 kg [m.sup.2]

where a = 0.512 m, b = 1.2 m, [T.sub.z] = 1.33 s, [T.sub.xy] = 0.777 s.

And the relationship of [u.sub.i] and [F.sub.i] are described by Eq. (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Then combined with Eq. (11), [u.sub.1] ~ [u.sub.4] can be easily calculated. Moreover, using these values [[omega].sub.i] can be obtained by Eq. (10).

Setting [k.sub.1] = 2.703 x [10.sup.-4]. Then, at time 4 s, the rotation speed of the rotors can be calculated as [[omega].sub.1] = 30.7667 r/s, [[omega].sub.4] = 31.1167 r/s, [[omega].sub.2] = 30.8167 r/s, [[omega].sub.3] = 30.8627 r/s, ([u.sub.4]/[lambda] = ([F.sub.1] + [F.sub.4] - [F.sub.2] - [F.sub.3])/4 , [lambda] can be divided out so that there is no value set for it).

And the values sampled by Hall sensors are [[omega].sub.1] = 30.6167 r/s, [[omega].sub.4] = 30.9267 r/s, [[omega].sub.2] = 30.7447 r/s, [[omega].sub.3] = 30.8627 r/s. The errors are in 0.5%, which meet the requirement of the experiment. In this experiment, the average values of these two o are used as the samples of rotation speed and a desired result was obtained. Since the rotation speed has a great influence on lift force, and the Euler angles. Therefore, the flight attitude control can be accomplished by rotation speed regulation.

The control experiment was conducted on the platform, which was proposed in Section 2. The initial state was: roll angle - 1.3 rad, pitch angle - 1.3 rad and travel angle - 5.2 rad. In the process, all these angles were turned to 0 rad. The stable attitude state of the four-rotor is shown in Fig. 5.

[FIGURE 5 OMITTED]

In this experiment, the attitude is measured by IMU. Then, under the control of ARM7, the sampled signals are transmitted to PC through wireless module. After analyzing, the effect of AIC control can be obtained and the input signal can be compensated by the feedback.

6. Conclusions

1. Though the experiment, it can be concluded that the steady-state error of Euler angles, which is caused by the sensor noise can be limited in a small interval by AIC method.

2. It also can show the robust of AIC method. Moreover, AIC is proved to suitable for the control of the four-rotor which has the requirement of stability and rapidity.

http://dx.doi.org/ 10.5755/j01.mech.18.1.1288

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Li Jin-song, Eng. Training Center, Shanghai Jiao Tong University, Shanghai 200240, China, E-mail: [email protected]

Cao Xi, School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China, E-mail: [email protected]

Received March 23, 2011

Accepted February 02, 2012

Table
Detailed devices of each module

Module Devices

IMU UZZ9001+KMZ41
 ENC-03RC
 LIS302DL(302D)
GPS navigation Dagama SG-959
Wireless communication APC802-43
Height measurement URM05 (ultrasonic)
Rotor speed measurement Hall sensor
 A1101
Central processor ARM7