AIAA 2010-3530
AIAA Infotech@Aerospace 2010 20 - 22 April 2010, Atlanta, Georgia
Gain Scheduling Based PID Controller for Fault Tolerant Control of a Quad-Rotor UAV Alaeddin Bani Milhim1 and Youmin Zhang2* Concordia University, Montreal, Quebec, H3G 1M8, Canada Camille-Alain Rabbath3 Defence Research and Development Canada, Valcartier, Quebec, G3J 1X5, Canada
In this paper, in view of the advantages of widely used Proportional-Integral-Derivative (PID) controller and gain scheduling control strategy in aerospace and industrial applications, a control strategy by using gain scheduling based PID controller is proposed for fault tolerant control (FTC) of a quad-rotor Unmanned Aerial Vehicle (UAV). The nonlinear dynamic equations of motion of the quad-rotor UAV are firstly derived based on the Newton’s second law. PID controllers under fault-free and several different fault situations are then designed and tuned to control the quad-rotor UAV model under normal and different faults flight conditions. Each PID controller uses the speed and the orientation of each propeller for stabilizing and hovering motion of the quad-rotor UAV. Based on a decision variable (used as scheduling variable in this case) associated with a fault detection and isolation module, switching of each pre-tuned PID controller can be made in real-time and on-line. Simulation results under different fault scenarios have shown the effectiveness of the proposed approach for a six-degree-of-freedom nonlinear quad-rotor UAV model.
Nomenclature M x, y, z φ ,θ , ψ Ixx, Iyy, Izz g
Fg
= = = = = =
mass of the vehicle x, y, and z position of the vehicle, respectively roll, pitch, and yaw angle of the vehicle, respectively mass moment of inertia about x, y, and z axis acceleration of the gravity gravitational force in the earth frame
Fgb
= gravitational force in the vehicle frame
cd ( x , y , z ) = drag coefficients in the x, y, and z direction Fd ( x , y , z )
= drag forces in the x, y, and z direction
FT ( x , y , z ) = thrust forces in the x, y, and z direction R(1,2,3,4) (x, y ) ( x% , y% ) Π v(x,y,z) L, M, N
= the orientation angles of each propeller = current position of the vehicle = = = =
commanded position of the vehicle transformation matrix velocity in the x, y, and z direction, respectively the moments produced by thrust force around x, y, and z axis, respectively
1
Master Student, Department of Mechanical and Industrial Engineering,
[email protected]. Associate Professor, Department of Mechanical and Industrial Engineering,
[email protected], Senior Member, AIAA & IEEE, *Corresponding Author. 3 Defence Scientist, DRDC - Valcartier, 2459 Pie-XI Blvd. North,
[email protected]. 1 American Institute of Aeronautics and Astronautics 2
Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by the copyright owner.
I. Introduction
U
nmanned Aerial Vehicles (UAVs) are remotely operated or autonomous aircrafts or remotely piloted vehicles. These vehicles have been used in applications such as fields of security, management of natural risks, intervention in hostile environments, management of ground installations, agriculture, and military1-3. UAVs make it possible to gather information in dangerous environments without risk to flight crews. UAVs are classified into two categories: fixed- and rotary-wing types. The rotary-wing type UAVs are more advantageous than the fixed-wing type UAVs in the sense of Vertical Taking-Off and Landing (VTOL) capability, omnidirectional flying, and hovering performance, and can be divided into quad-rotor and single-rotor type shapes4-5. The advantages of the quad-rotor UAV versus single-rotor helicopters are that the quad-rotor does not require mechanical linkages to vary rotor angles of attack as they spin. The use of four rotors allows each individual rotor to have a smaller diameter than the equivalent single-rotor UAV, allowing them to store less kinetic energy during flight and thus reduces the damage caused by the rotors hitting any objects. By enclosing the rotors within a frame, the rotors can be protected during collisions. The most famous example for the quad-rotor UAV is the Draganflyer quad-rotor helicopter as shown in Fig. 1. This quad-rotor UAV is a radio-controlled four-rotor helicopter manufactured by Draganfly Innovations Inc6. Although it can fly simply, stabilization of its hovering is still a challenge2. The controlling of quad-rotor UAVs has been widely researched in a number of studies7-11. The quad-rotor helicopter of the Stanford Testbed of Autonomous Rotorcraft for Multi-Agent Control II (STARMAC II) has been caused by moments that affect altitude control, and thrust variation that affects attitude control. The results of this work have proven the insufficiency for accurate trajectory tracking of the vehicle12. The Omni-directional Stationary Flying Outstretched Robot OS4 is a quad-rotor project developed for fully autonomous operation. The main Figure 1. Draganflyer: one well-known example of the objective of this flying robot is the development and quad-rotor UAV. implementation of an active control system for this 3,5 vehicle . Although many control algorithms have obtained good results in previous studies, those studies have primarily focused on the control of the vehicle by taking into account mainly the velocity of the propeller. Fault tolerant control (FTC) is the ability of a system to be able to continue its operation in the event of a failure in some of its components. Over the last three decades, the growing demand for safety, reliability, maintainability, and survivability has drawn significant research in fault tolerant control14-15. Modern UAVs need to be designed to achieve the desired performance under both normal and fault conditions. Fault tolerant control systems for small UAVs should not significantly increase the number of actuators or sensors needed to achieve a safer operation16. Fault tolerant control can be classified into passive and active. Passive FTC can tolerate a predefined set of faults by using a specially-designed fixed controller while active FTC relies on fault detection and diagnosis (FDD) process to monitor system performance and to detect and identify faults in the system and the controller is reconfigured on-line and in real-time14-18. The objective of this work is to present the dynamic model of the quad-rotor UAV and to propose a gain scheduling based fault tolerant control strategy in which each local controller is designed by a Proportional-Integral-Derivative (PID) controller. For achieving better control performance, each PID controller consists of controllers for control of orientation of the propeller and for control of speed of the rotor. PID controller gains under each fault condition are tuned off-line based on assumed fault conditions and switched on-line based on the fault detection information provided by a fault detection algorithm in a framework of gain-scheduling control. In this paper, a high-fidelity model of the quad-rotor UAV is firstly derived in Section II. In Section III, PID controllers are tuned to control the 2 American Institute of Aeronautics and Astronautics
UAV under different fault conditions. There are two subsections in this section: the first one shows the control of the orientation of each propeller, and the second one shows the control of the speed of the propellers. In Section VI, faults are injected to the actuators of the UAV and simulation responses of the UAV with tuned parameters of PID controller under different fault situations are analyzed. Finally, conclusions are given in Section VII.
II. Equations of Motion of the Quad-Rotor UAV A quad-rotor UAV is an aircraft without a pilot whose lift is generated by its four rotors. It contains four propellers, with two of them forming a pair of propellers. The first pair is located on the x-axis and rotated clockwise and the second pair is located on the y-axis and rotated counterclockwise as shown in Fig. 2. Generally, its motion is controlled by varying the lift forces produced by the propellers19. Each propeller produces both a thrust and torque from its center of rotation2. The gyroscopic effects and the aerodynamic torques tend to cancel and these four propellers do not have a swash plate20. The equations are presented about the body frame (X, Y, Z). For any point of the airframe expressed in the earth-fixed frame (XE, YE, ZE), one can write the transformation matrix as:
⎡ cθcψ Π = ⎢⎢ c θ s ψ ⎢⎣ − s θ
Figure 2. Quad-rotor UAV configuration. The earth-fixed frame (XE, YE, ZE) is used to specify the location of the vehicle. The vehicle body frame (X, Y, Z) and roll ( φ ), pith ( θ ), and yaw (ψ ) angles are used to specify the vehicle orientation.
− cφ s ψ + s φs θ s ψ c φc ψ + s φs θ s ψ sφcθ
s φs ψ + c φs θ cψ ⎤ − s φ c ψ + c ι s θ s ψ ⎥⎥ ⎥⎦ c φc θ
where s ( φ , θ , ψ ) = sin ( φ , θ , ψ ), c ( φ , θ , ψ ) = cos ( φ , θ , ψ ). The equations of motion of the quad-rotor UAV at inertial coordinate system is given as4: x⎤ ⎡ && ⎡ Fx ⎤ m ⎢⎢ && y ⎥⎥ = Π ⎢⎢ Fy ⎥⎥ ⎢⎣ && ⎢⎣ Fz ⎥⎦ z ⎥⎦
(1)
(2)
Fx , Fy, , Fz are the net forces in an inertial system represented in the body coordinate system and (x, y, z) represent the location of the vehicle in the body frame and П is defined as Eq. (1). It can be seen that the accelerations in the earth body frame are obtained by multiplying the forces by the transformation matrix. The main forces acting on the body are the propeller thrusts, drag forces and gravitational forces. The direction of the propeller thrust is perpendicular to the surface of rotating and its value is proportional with the speed of rotation3. The propeller is driven by a DC motor mounted on the stepper motor located at the end of a crossing body frame. By the rotation of the stepper motor, the orientation of the propeller will take effect. To
Figure 3. Propeller 2 with certain orientation. This figure shows the propeller 2 with angle (R2) along x-axis. 3 American Institute of Aeronautics and Astronautics
find the projection of the forces on each axis, we need to multiply the thrust force by the sine or cosine of the angle that the propeller is tilted. Figure 3 shows the orientation of propeller 2 and the components of the thrust force. The drag force is proportional with the square of the velocity on each axis by a coefficient of drag as shown in following equation:
⎡ c dx v x 2 ⎤ ⎡ Fdx ⎤ 1 ⎢ ⎢F ⎥ = 2 ⎥ ⎢ dy ⎥ Π ⎢ c dy v y ⎥ ⎢ c dz v z 2 ⎥ ⎢⎣ Fdz ⎥⎦ ⎣ ⎦
(3)
Since the velocity of each axis is presented in earth frame, we need to divide it by the transformation matrix (1) to get its component on the body frame. The main force acting on the quad-rotor UAV is the gravity force1 equaling to the total mass of the quad-rotor UAV multiplied by the acceleration of the gravity as following: Fg = M ⋅ g
(4)
Its direction is toward the ground (negative z-direction of the earth frame). By dividing the gravitational force by the transformation matrix, we can calculate the effect on each axis on the body frame. ⎡ ⎤ ⎡ F gb1 ⎤ − F g sin θ (5) ⎢ ⎥ ⎢ ⎥ F gb = ⎢ F g c o s θ sin φ ⎥ = ⎢ F gb 2 ⎥ ⎢ Fg cosθ cos φ ⎥ ⎢ F gb3 ⎥ ⎣ ⎦ ⎣ ⎦ The equations of the forces affected to the quad-rotor UAV are:
F x = F dx + F Tx + F gb 1
(6)
F y = F dy + F Ty + F gb 2
(7)
F z = F d z + FT z + F g b 3
(8)
By the derivatives of the acceleration, we can get the velocity and then the location of the vehicle: t
ξ& =
∫ ξ&&d τ ξ =
ξ& 0
ξ&0 and ξ 0
,
t
∫ ξ&d τ
ξ0
⎡x⎤ w h e r e ξ = ⎢⎢ y ⎥⎥ , ⎢⎣ z ⎥⎦
are the initial conditions of the velocity and position, respectively.
The roll equations of motion are:
⎡ I xx ⎢ 0 ⎢ ⎢⎣ 0
0 I yy 0
0 ⎤ ⎡ φ&& ⎤ ⎡ L ⎤ ⎢ ⎥ 0 ⎥⎥ ⎢ θ&& ⎥ = ⎢⎢ M ⎥⎥ I zz ⎥⎦ ⎢⎣ ϕ&& ⎥⎦ ⎢⎣ N ⎥⎦
(9)
Note that the drag force was neglected in computing the moment. This force was found to cause a negligible disturbance on the total moment over the flight regime of interest12. t
η& =
∫ η&&d τ , η =
η& 0
η& 0
and
t
⎡φ ⎤
η0
⎢⎣ψ ⎥⎦
∫ η& d τ , where η = ⎢⎢θ ⎥⎥
η 0 are the initial conditions of the angular velocity and Euler angles rotational components, respectively.
4 American Institute of Aeronautics and Astronautics
III. Controllers Design A set of PID controllers are designed for trajectory tracking control of the quad-rotor UAV under normal and assumed different fault conditions. Each PID controller consists in two control channels: control of orientation of each propeller and control of speed of each motor. A. Control of the orientation of each propeller We use the orientation primarily to change the yaw angle, which is the angle between a quad-rotor UAV’s heading and a reference heading, to get the commanded angle. This is the yaw controller. The criterion of employing yaw control depends on the distance, which is denoted as d, between the current position ( x , y ) and commanded position ( ~ x,
~ y ) , where the distance d is expressed as the following equation: d =
xe
2
+ ye
2
(10)
where xe and ye are defined as:
xe = x − ~ x y e = y − ~y Then, following yaw control will be activated when this distance is larger than a specific value γ, d > γ, where γ is a positive constant. Assume γ = 3.
x% = { x% 1 , x% 2 , x% 3 , K , x% n } y% = { y% 1 , y% 2 , y% 3 , K , y% n } where n is the number of the command points. Then assume the distance between the command points is
λ =
(~ xn − ~ x n − 1 ) 2 + ( ~y n − ~y n − 1 ) 2
(11)
The importance of the above part of the controller is coming from the distance λ. In other words, we can neglect this part if the distance λ is less than γ. Also, we can divide the yaw controller into two sub yaw controllers: one of them represents the controller for the pair at the x-axis and the other is for the pair at y-axis. The pair on the x-axis will orient dependently on the commands coming from the yaw controller, which is the Proportional-Differential (PD) controller output, and they orient in the opposite direction as shown in Fig. 4. In other words, when the distance (d) is larger than the specific value (γ), the yaw angle will be updated by a new angle (ψ d ) which is equal to the angle of the trajectory; otherwise the yaw angle does not be modified. After that, a PD controller is proposed before sending the command to the actuators. The procedure to updating yaw angle is shown as:
Figure 4. The direction of orientation of the propellers on the x-axis. This view is taken toward positive y-axis.
If d > λ { ψ = ψd ;
ψ
Otherwise =ψ ;
} where 5 American Institute of Aeronautics and Astronautics
Δy x − x , Δ y = ~y − y , Δ x = ~ Δx The pair on the y-axis will be adjusted by a stepper motor that takes command from yaw controller as a constant value (υ) which is the step of the stepper motor, otherwise the stepper motor is still with zero rotation. The direction of each stepper motor in this pair is rotated in the same direction of each other to allow the quad-rotor UAV to be capable of rotating.
ψ
d
−1
= tan
The equations of the orientation controller are:
R1 = R3 = ±(k pψ (ψ d −ψ ) + k dψ (ψ& ))
(12)
R2 = R4 = v
(13)
where kpψ and kdψ are the parameters of the PD controller and v is the step of the stepper motor.
Now, we can get the forces and moments components as described in the following equations:
F T x = F 2 s in R 2 + F 4 s in R 4
(14) (15)
FT y = F1 sin R1 − F3 sin R 3
(16)
4
∑
FT z =
Fi c o s R i
i=1
M
Tφ
= l2 F2 c o s R 2 − l4 F4 c o s R 4
(17)
M
Tθ
= l1 F 1 c o s R 1 − l 3 F 3 c o s R 3
(18) (19)
M
Tψ
=
4
∑
li Fi s in R i
i =1
li is the length between the propeller and the Center Of Gravity (COG) and i=1, 2, 3, and 4 for each propeller. B. Control of the speed of each motor We control the speed of motor mainly to reach the desired altitude and to get the right direction. This part of controller has three sub-controllers as follows: a) Z-Controller. b) Roll Controller. c) Pitch Controller. In the following, we can propose a PID controller for each sub-controller21. a) Z-controller: The purpose of this controller is to achieve the desired altitude by applying a PID controller on the altitude. The output of this controller should be divided by four to generate the equivalent force on each propeller, hence stabilize the quad-rotor UAV. The equation of this controller is shown as: C 1 = k pz ( z d − z ) + k dz ( − z& ) + k iz
∫ (z
d
− z)
where zd is the desired value, and kpz, kdz, and kiz are the parameters of PID controller. b) Roll Controller:
6 American Institute of Aeronautics and Astronautics
(20)
To move into y-direction, it requires the quad-rotor UAV to roll around the x-axis as shown in Fig. 5. To obtain the roll rotation, we have to generate imbalance forces in the pair on the y-axis. This imbalance force has to rise above the inertial forces opposing the rotation. The roll angle ( φ ) is approximated as the first-order system of (yd – y). In other words the roll angle is output of PD controller of y as following relation:
φd = k
py
( y d − y ) + k dy ( − y& )
(21)
where yd and φ d are the desired values, kpy and kdy are constant values. The roll controller will stabilize the roll angle to create the y-directional motion at its desired value if the gains are well-chosen22. c)
Pitch Controller:
The pitch controller is the same as the roll controller, except that the controller will be applied on the pitch angle (θ) as shown in Fig. 6 which means the rotation about y-axis, to create the motion along the x-axis. This relation is:
Figure 5. Roll rotation. This rotation allows the vehicle to move in the y-direction.
θ d = k px ( x d − x ) + k dx ( − x& )
(22)
where xd and θd are the desired values, kpx and kdx are PD controller gains. The rotation of the body yaw axis is applied to obtain the components of roll and pitch by using this transformation matrix:
⎡ cos( ψ ) Rψ = ⎢ ⎣ sin( ψ )
− sin( ψ ) ⎤ cos( ψ ) ⎥⎦
As mentioned before, if the criterion of using orientation is satisfied, the yawing controller is proposed to get the hovering motion of the quadrotor UAV. If the criterion is not satisfied, roll and pitch controllers are proposed to keep the hovering motion of the quad-rotor UAV.
Figure 6. Pitch rotation. This rotation allows the vehicle to move in the x- direction.
Two PD controllers will be applied on roll and pitch angles according to the following relations:
C 2 = k p φ ( φ d − φ ) + k d φ ( − φ& ) C = k (θ − θ ) + k ( − θ& ) 3
pθ
d
dθ
(23) (24)
where C2 and C3 are the outputs of the controllers, and kpф, kdф, kpθ, and kdθ are the parameters of the controllers. The output of each controller should be divided by two, with opposite signs to distribute the input controller on each propeller in each pair, to guarantee the balanced forces. Opposite signs imply that each propeller in the pair rotates in the opposite direction of the other. The equations of the distribution to generate the forces are: 1 1 F1 = C1 − C3 4 2 7 American Institute of Aeronautics and Astronautics
(25)
F2 =
1 1 C1 − C 2 4 2
(26)
F3 =
1 C 4
(27)
F4 =
1 1 C1 + C 2 4 2
+
1
1 C 2
3
(28)
These forces are sent to the propeller which represents the speed of the propeller that is exactly the speed of the DC motor. In this paper, we focus on the controller to produce the needed forces. Interested readers can refer to Ref. 3, 11, 13 for the information about the DC motor configurations. The equations of the controller are mainly presented through the Eqs. 12, 13, 20, 23, and 24.
IV. Fault Modeling and Gain Scheduling of PID Controllers The fault types considered in this paper is partial loss of effectiveness (LOE) in actuators, i.e. DC and stepper motors in the quad-rotor UAV. Generally, reconfigurable controller (switching controller in this paper) should be designed to maintain the stability and acceptable performance of the system when a fault occurs. Under the assumption that a fault detection and diagnosis module can detect and identify the fault quick and correctly, this fault decision information can be used for switching pre-tuned PID controllers for obtaining best performance for each considered fault case. It is based on this consideration, the gain scheduling based PID fault tolerant control has been proposed in this paper by exploiting the advantages for PID controllers and gain scheduling control strategy14. Assume that proper controller gains of the PID controllers have been found for the normal and different fault situations, these values can be stored in a parameter table which will be picked up based on a gain scheduling variable based on the different fault types and magnitudes. When different faults occur, the PID controllers will be switched to the corresponding fault case. This method works best for the system having predictable fault cases with predetermined parameters of the controller23. For the quad-rotor UAV, there are eight inputs: four forces represent the DC motors for each propeller and the other four components of R represent the orientation of each propeller. To analyze the effects of each fault case based on the designed PID controller for each fault and robustness to unforeseen fault cases, different fault cases are presented and tested as follows: 1.
The effect of the faults on all DC motors
In this subsection, the fault is applied to all the DC motors together with the same percent and at the same time. Figure 7 shows the response of the UAV when the percent of the fault is 50%. For this case, if controller gains of the Z-controller to be chosen as twice as in normal case, the new parameters can accommodate the fault effects with a good tracking performance as shown in Fig. 7. x-component of the path
9
100
100 50 0 -50 -100
50 0 -50 -100 -150
-150 -200
10
z-axis position
150
z-component of the path
y-component of the path
150
without fault with fault with fault & new parameters
y-axis position
x-axis position
200
0
20
40
60
80
time
(a)
100
120
140
-200
8 7 6 5 4 3 2 1 0
0
20
40
60
80
100
120
140
0
20
40
60
80
time
time
(b)
(c)
100
120
140
Figure 7. 50% of fault on all DC motors. The solid blue line shows the response of the vehicle without fault, the dashed red line presents the response of the vehicle with 50% fault while the dashed green one showing the results when the gains of the controller have been modified. 8 American Institute of Aeronautics and Astronautics
When the fault is applied for all the DC motors of the quad-rotor UAV, the vehicle is going far away from the desired path but as shown in Fig. 7, the vehicle is robust for the altitude without modification. Since the Z-controller contains all DC motors effect, the partial losses of the motors can adapt by re-tuned gains of the Z-controller. 2.
The effect of fault on each pair of motors
Figure 8 shows the response of the vehicle with a 70% fault. It also shows the response of the vehicle when the parameters of the pitch command, PD controller for x position of the vehicle, the constants of Eq. (22) have been modified to the half value. This change reduces the dependence of the control on the first and third motors. The system can maintain acceptable performance for a fault up to 80%. x-component of the path 40 20 0
20 0
-40
-40
-60
-60 -80
40
-20
-20
0
20
40
60
80
100
120
140
-80
0
20
40
60
80
100
z-component of the path
10
60
z-a xis p o sitio n
60
y-component of the path
80
without fault with fault with fault & new parameters
y-a xis p o sitio n
x-axis position
80
120
8
6
4
2
0
140
0
20
40
60
80
time
time
time
(a)
(b) Figure 8. 70% of fault on the (1, 3) DC motors.
(c)
100
120
140
In the following figure, 90% of a fault is applied and the new pitch controller is 0.01 of the original one. When the fault level is increasing and the performance of system with nominal PID controller becomes even worse. However, based on the fault-tolerant control, the vehicle can follow its desired path. x-component of the path
0 -200 -400 -600 -800 -1000
0 -200 -400 -600 -800
-1000 0
20
40
60
80
100
120
140
800
600
400
200
0 0
20
40
60
80
100
120
140
0
20
40
time
time
(a)
z-component of the path
1000
200
y-a xis p o sitio n
x-axis position
200
y-component of the path
400
without fault with fault with fault & new parameters
z-a xis p o sitio n
400
60
80
100
120
140
time
(b) Figure 9. 90% of fault on the (1, 3) DC motors.
(c)
The responses of the vehicle with 60% loss of control effectiveness applied on the second and fourth DC motors are presented in Fig. 10. By reducing the roll controller gains in half, it allows the vehicle to follow the desired path as shown in Fig. 10. With the fault level reaches up to 80%, the controller can still provide acceptable performance. x-component of the path
50 0 -50
-100 -150
100 50 0 -50
-100
0
20
40
60
80
time
(a)
100
120
140
-150
0
20
40
60
80
100
120
z-component of the path
10
z-a xis p o sitio n
100
y-component of the path
150
without fault with fault with fault & new parameters
y-a xis p o sitio n
x-a xis p o sitio n
150
140
8
6
4
2
0
0
time
(b) Figure 10. 60% of fault on the (2, 4) DC motors. 9 American Institute of Aeronautics and Astronautics
20
40
60
80
time
(c)
100
120
140
y-a xis p o sitio n
x-a xis p o sitio n
800
y-component of the path
1000
without fault with fault with fault & new parameters
800
Figure 11 shows the result when the applied fault is 90%. 600
400
200
0
z-component of the path
10
z-a xis p o sitio n
x-component of the path
1000
600
400
200
8 6 4 2 0
0 0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
-2
140
0
20
40
60
80
time
time
time
(a)
(b)
(c)
100
120
140
Figure 11. 90% of fault on the (2, 4) DC motors. Figure 11 shows the result when the applied fault is 90%. 3.
The effect of the faults on each individual DC motor
When a 10% fault is applied on the first DC motor, the responses for fault-free case, fault case with nominal PID controller, and the fault case with reconfigured PID controller are shown in Fig. 12. After the fault occurrence, the nominal pitch controller is switched to reconfigured controller gains based on the scheduling variable from FDD module. As the fault becomes bigger, the response will be negative until it reaches a certain percentage equal to 15%. At this fault level, the simulation cannot be carried out. This certain percentage can be defined as critical fault level. x-component of the path
0
20
40
60
80
100
120
0
-50
140
0
20
40
60
80
100
120
z-component of the path
10
z-a xis p o sitio n
0
-50
y-component of the path
50
without fault with fault with fault & new parameters
y-a xis p o sitio n
x-a xis p o sitio n
50
8
6
4
2
0
140
0
20
40
60
80
time
time
time
(a)
(b)
(c)
100
120
140
Figure 12. 10% of fault on the first DC motor. The response of the second motor for a 10% of fault is shown in Fig. 13. The critical fault level for this motor is 11%. x-component of the path
50 0 -50
-100
100 50 0 -50
-100
-150
-150
0
20
40
60
80
100
120
140
-200
0
20
40
60
80
100
120
z-component of the path
10
150
z-a xis p o sitio n
100
-200
y-component of the path
200
without fault with fault with fault & new parameters
150
y-a xis p o sitio n
x-a xis p o sitio n
200
140
8
6
4
2
0
0
20
40
60
80
time
time
time
(a)
(b)
(c)
Figure 13. 10% of fault on the 2nd DC motor.
10 American Institute of Aeronautics and Astronautics
100
120
140
The responses of the vehicle for the fault occurred in the third motor are similar to those when the fault is applied to the first motor, and the same for the fourth and second motors, due to the symmetry of the quad-rotor. Simulation results show also that the second and fourth motors are less sensitive to the fault. 4.
The effect of fault on the first or third stepper motor
Figure 14 shows the response of the vehicle under 70% fault applied on the first stepper motor at 10 seconds. Proportional and derivative gains of the PD orientation controller are modified to (0.5, 5) instead of (1, 1). This modification can tolerate the fault up to 90%. x-component of the path
20
0
20 0
-40
-40
-60
40
-20
-20
-60 0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
z-component of the path
10
z-a xis p o sitio n
40
y-component of the path
60
without fault with fault with fault & new parameters
y-a xis p o sitio n
x-axis position
60
8
6
4
2
0
140
0
20
40
60
80
time
time
time
(a)
(b)
(c)
100
120
140
Figure 14. 70% of fault on the first stepper motor. The response of the vehicle with applying fault on the third stepper motor is similar to the response of the vehicle when applying fault on the first stepper motor. 5.
The effect of fault on the second or fourth stepper motor
Figure 15 shows the response of the vehicle when the percent of the applied fault on the second stepper motor is 70% and the controller can be held until 85% fault. We have used other paths to show the effect of the fault on the second and fourth motors. x-component of the path
10 5 0 -5
-10
-20
15 10 5 0 -5
-10
-15 0
50
100
150
15 10 5 0 -5
-10
-15 -20
z-component of the path
20
z-a xis p o sitio n
15
y-component of the path
20
without fault with fault
y-a xis p o sitio n
x-a xis p o sitio n
20
-15
0
50
100
150
-20
0
50
100
time
time
time
(a)
(b)
(c)
150
Figure 15. 70% of fault on the second stepper motor. The responses of the vehicle when the applied fault on the fourth stepper motor are similar to the responses when the applied fault on the second stepper motor. The second and fourth stepper motors show more fault tolerance capability than the first and third motors.
V. Conclusions We have proposed a method to control the quad-rotor UAV based on a gain scheduling based PID controller. The equations of motion were obtained based on the Newton’s second law. PID controller in each control channel consists in two parts: the first part controls the orientation of the propeller, and the second part controls the speed of each propeller. This type of design for controllers increases the performance of the quad-rotor UAV in tracking the desired trajectory and increases fault tolerance capability and reliability of the UAV. The proposed controller has 11 American Institute of Aeronautics and Astronautics
been implemented in a nonlinear six-degree of freedom simulation model of a quad-rotor UAV and good tracking performances have been obtained through different commanded trajectories. Based on the proposed gain-scheduled fault tolerant PID controller design, advantages (for example, robust properties, easy design and implementation) of PID controller can be employed and the drawback of lack of on-line auto-tuning of PID controller gains is avoided by using the gain scheduling control strategy with wide range of faults being able to be handled.
Acknowledgments This work is partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through a Strategic Project Grant (STPGP 350889-07) and an NSERC Discovery Project Grant.
References 1
Castillo, P., Lozano, R., and Dzul, A., Modelling and Control of Mini-Flying Machines, London: Springer, 2005, pp. 1-15. 2 McKerrow, P., “Modelling the Draganflyer Four-rotor Helicopter,” in Proceedings of the IEEE International Conference on Robotics and Automation, pp. 3596- 3601, 2004. 3 Canttea, C., Chin, J., Mehrabian, S., Montejo, L., and Thompson, H., “Quad-rotor Unmanned Aerial Vehicle,” Report, Columbia University, New York, 2007. 4 Park, S., Won, D.H., Kang, M. S., Kim, T.J., Lee, H. G., and Kwon, S. J., “RIC (Robust Internal-loop Compensator) Based Flight Control of a Quad-rotor Type UAV,” in Proceedings of the IEEE/RSJ International Conference, 2005, pp. 3542-3547. 5 Bouabdallah, S., Murrieri, P., and Siegwart, R., “Design and Control of an Indoor Micro Quad-rotor,” in Proceedings of the IEEE Int. Conf. on Robotics and Automation, 2004, pp.4393-4398. 6 DraganFlyer – http://www.rctoys.com/draganflyer3.php. 7 RCtoys, http://www.rctoys.com/. 8 Bouabdallah, S., Design and Control of Quadrotors with Application to Autonomous Flying, Master Thesis, Aboubekr Belkaid University, Tlemcen, Algeria, 2007. 9 Soumelidis, A., Gaspar, P., Bauer, P., Lantos, B., and Prohaszka, Z., “Design of an Embedded Microcomputer Based Mini Quadrotor UAV,” in Proceedings of the European Control Conference, 2007, Kos, Greece. 10 Castillo, P., Albertos, P., Gracia, P., and Lozano, R., “Simple Real-time Attitude Stabilization of a Quad-rotor Aircraft with Bounded Signals,” in Proceedings of the 45th IEEE Conference on Decision & Control, 2006 San Diego, CA, USA. 11 Camlica, F., Demonstration of a Stabilized Hovering Platform for Undergraduate Laboratory, Master Thesis, Istanbul Technical University, Ankara, Turkey, 2004. 12 Hoffmann, G.M., Haung, H., Waslander, S. L., Tomlin, C. J., “Quadrotor Helicopter Flight Dynamics and Control: Theory and Experiment,” AIAA Guidance, Navigation, and Control Conference, Hilton Head, South Carolina, August 2007. 13 Bouabdallah, S., Noth, A., and Siegwart, R., “PID vs LQ Control Techniques Applied to an Indoor Micro Quadrotor,” in Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, 2004, pp. 2451-2456. 14 Zhang, Y. M., and Jiang, J., “Bibliographical Review on Reconfigurable Fault-tolerant Control Systems,” IFAC Annual Review in Control, Vol. 32, No. 2, 2008, pp. 229-252. 15 Mahmoud, M., Jiang, J., and Zhang, Y. M., Active Fault Tolerant Control Systems: Stochastic Analysis and Synthesis, London: Springer, 2003. 16 Ducard, G., Fault-tolerant Flight Control and Guidance Systems: Practical Methods for Small Unmanned Aerial Vehicles, London: Springer, 2009, pp. 1-20. 17 Patton, R. J., "Fault-tolerant Control: the 1997 Situation," in Proceedings of the IFAC Symposium on Fault Detection, Supervision and Safety for Technical Processes (SAFEPROCESS '97), pp. 1033-1055, Hull, UK, August 1997. 18 Zhang, Y. M., “Active Fault-tolerant Control Systems: Integration of Fault Diagnosis and Reconfigurable Control” (Invited Plenary Lecture), The 8th Conf. on Diagnostics of Processes and Systems, Slubice, Poland, Sept. 2007; Fault Diagnosis and Fault Tolerant Control, J. Korbicz, K. Patan, and M. Kowal (Eds), Academic Publishing House EXIT, Warszawa, 2007, pp. 21-41. 12 American Institute of Aeronautics and Astronautics
19
Das, A., Subbarao, K., and Lewis, F., “Dynamic Inversion with Zero-dynamics Stabilization for Quadrotor Control,” IET Control Theory and Appllications, 2009, Vol. 3, No. 3, pp. 303-314. 20 Castillo P., Lozano R., and Dzul A., “Stabilization of a Mini Rotorcraft Having Four Rotors,” IEEE Control Systems Magazine, 2005, Vol. 25, No. 6, December 2005, pp. 45-55. 21 Kendoul, F., Lara, D., Fantoni-Coichot, I., and Lozano, R., “Real-Time Nonlinear Embedded Control for an Autonomous Quadrotor Helicopter,” Journal of Guidance, Control, and Dynamics, Vol. 30, No. 4, July-August 2007, pp. 1049-1061. 22 Fantoni, I., Zavala, A., and Lozano, R., “Global Stabilization of a PVTOL Aircraft with Bounded Thrust,” in Proceedings of the 41st IEEE conference on Decision and Control, Las Vegas, USA, December 2002. 23 Shin, J. Y., “Gain-scheduled Fault Tolerance Control under False Identification,” AIAA Guidance, Navigation, and Control Conference and Exhibit, Aug. 2005.
13 American Institute of Aeronautics and Astronautics
