CC BY
26
UDC 621.3.07
https://doi.Org/10.20998/2074-272X.2021.5.03
Z.A. Gulshan, M.Z.H. Ali, M.S. Shah, D. Nouman, M. Anwar, M.F. Ullah
A ROBUST CONTROL DESIGN APPROACH FOR ALTITUDE CONTROL AND TRAJECTORY TRACKING OF A QUADROTOR
Introduction. Unmanned aerial vehicles as quadcopters, twin rotors, fixed-wing crafts, and helicopters are being used in many applications these days. Control approaches applied on the quadrotor after decoupling the model or separate altitude control and trajectory tracking have been reported in the literature. A robust linear Hw controller has been designed for both altitude control and circular trajectory tracking at the desired altitude. Problem. The ability of the quadrotor system to hover at a certain height and track any desired trajectory makes their use in many industrial applications in both military and civil applications. Once a controller has been designed, it may not be able to maintain the desired performance in practical scenarios, i.e. in presence of wind gusts. Originality. This work presents the control strategy to ensure both altitude control and trajectory tracking using a single controller. Purpose. However, there is a need for a single controller that ensures both altitude control and trajectory tracking. Novelty. This paper presents a robust H„ control for altitude control and trajectory tracking for a six degree of freedom of unmanned aerial vehicles quadrotor. Methodology. Multi input multi output robust H„ controller has been proposed for the quadrotor for altitude control and tracking the desired reference. For the controller validation, a simulation environment is developed in which a 3D trajectory is tracked by the proposed control methodology. Results. Simulation results depict that the controller is efficient enough to achieve the desired objective at minimal control efforts. Practical value. To verify that the proposed approach is able to ensure stability, altitude control, and trajectory tracking under practical situations, the performance of the proposed control is tested in presence of wind gusts. The ability of the controller to cater to the disturbances within fractions of seconds and maintaining both transient and steady-state performance proves the effectiveness of the controller. References 16, table 1, figures 9. Key words: Hx controller, six degree of freedom quadrotor, unmanned aerial vehicle, attitude regulation, nonlinear system, robust control.
Вступ. Безтлотт лжальш апарати, таю як квадрокоптери, двороторш апарати, апарати з нерухомими крилами та гелжоптери сьогодт використовуються у багатьох сферах застосування. У лiтературi повiдомляeться про тдходи до керування, застосоват на квадрокоптерi тсля вiд'eднання моделi або окремого контролю висоти та вiдстеження траекторп. Надшний лтшний регулятор H„ був розроблений як для контролю висоти, так i для вiдстеження круговой траектори на потрiбнiй висотi. Проблема. Здаттсть квадрокоптерноi системи зависати на певтй висотi та вiдстежувати будь-яку бажану траекторю робить ix застосування можливим у багатьох сферах як у вшськових, так i в цивыьних цыях. Розроблений контролер може не тдтримувати бажаш характеристики у реальних умовах, тобто за наявностi поривiв втру. Оригтальтсть. У цт роботi представлена стратегiя керування, яка забезпечуе як контроль висоти, так i вiдстеження траектори за допомогою одного контролера. Мета. Однак кнуе потреба в единому контролерi, який забезпечуе як контроль висоти, так i вiдстеження траектори. Новизна. У цш статтi представлено надшний регулятор H„ для контролю висоти та вiдстеження траектори для шести ступетв свободи безпыотних лтальних апаратiв. Методологя Для квадрокоптера запропоновано багатовxiдний багатовиxiдний надшний контролер H„ для контролю висоти та вiдстеження бажаного курсу. Для перевiрки контролера розробляеться середовище моделювання, в якому тривимiрна траекторiя вiдстежуеться за запропонованою методологiею керування. Результата. Результати моделювання показують, що контролер е досить ефективним для досягнення бажаноi мети при мтмальних зусиллях контролю. Практична щншсть. Щоб переконатися, що запропонований пiдxiд здатний забезпечити стабтьтсть, контроль висоти та вiдстеження траектори в реальних ситуацях, параметри запропонованого контролю перевiряються за наявностi поривiв втру. Здаттсть контролера усувати порушення протягом кыькох секунд i тдтримувати як переxiднi, так i стабыьш показники доводить ефективтсть контролера. Бiбл. 16, табл. 1, рис. 9. Ключовi слова: H„ контролер, квадрокоптер з шктьма ступенями свободи, безпшотний л^альний апарат, регулювання позищ!, нелшшна система, надшне керування.
1. Introduction. Unmanned Arial Vehicles (UAV) like fixed wing crafts, quadcopters, and helicopters have found applications in several domains [1]. Amongst these, quadcopters and helicopters are commonly used as UAVs due to their hovering ability. These systems possess nonlinear and coupled dynamics, which leads to the challenges in their autonomous control. The dynamics of helicopter can be approximated by a laboratory setup namely Twin Rotor Aerodynamic System (TRAS) [2]. Like helicopter, TRAS has two rotors main and tail rotors. Though in helicopter, main rotor is able to tilt in order to execute forward motion [3].
Control design for UAVs is a difficult task because of the coupling and nonlinearities involved in their mathematical models. In autonomous applications, trajectory tracking is one of the most basic and important tasks. Other equally important scenarios involve e.g.,
hover control. In [4], four independent proportionalintegral-derivative (PID) controllers with independent inputs for control have been designed to achieve the objective of trajectory tracking. Here, real value type genetic algorithm has been used to tune the controller parameters in order to reduce total error and control efficiency. System performance index is used as a fitness function here. In [5], hover control problem is addressed by a control structure that involves feedback and feed forward control. Four impulse input shaper is used for feed forward and PID controller with acceleration feedback input is used for the tracking controller. In [6], robust PID based dead beat control scheme is proposed. As PID controller does not contain the model information, so for small dynamic systems, it performs reasonably well but for higher order system it may lead to oscillations. To
© Z.A. Gulshan, M.Z.H. Ali, M.S. Shah, D. Nouman, M. Anwar, M.F. Ullah
counter these problems, different model-based controllers have been reported in the literature. In [7] an linear-quadratic regulator (LQR) controller is designed. In [8], an integral sliding mode controller using necessary and sufficient conditions has been proposed for the uncertainty handling and trajectory tracking of the quadrotor. A nonlinear PID control strategy is discussed in [9] for the trajectory tracking of the quadrotor. The method proposed here is a novel contribution of nonlinearity in the conventional PID controller based on frequency domain design. A model free control approach is used in [10] for the quadrotor trajectory tracking. Control structure adopted here is based on the internal-external-control-loop structure. Controller uses the conventional sliding mode approach for control effort generation. A super twisting slide mode approach has been used in [11] for trajectory tracking with the nonlinear sliding surface. Experimental validation of the proposed control architecture has also been included. A neural network based self-tuning control structure using double derivative action with proportional control is proposed in [12] for trajectory tracking of the quadrotor UAV. Fractional slide mode control is also proposed for the quadrotor in [13].
The single-input single-output (SISO) linear time invariant (LTI) controllers like proportional-derivative (PD), proportional-integral (PI) and PID require the decoupling of the multi-input multi-output (MIMO) model of the quadrotor which takes the extra computational effort, i.e. one needs to design a de-coupler to obtain multiple SISO models from a single MIMO model of the quadrotor and then design the SISO controller for each model. This may also cause slowing the closed-loop response of the system when connected with hardware and thus can be resulted in an increased control effort and poor tracking. However, the MIMO LTI controllers like LQR, Linear Quadratic Gaussian (LQG), and Hx controllers tend to have a larger control effort while trajectory tracking for a quadrotor.
Nonlinear controllers on the other hand require that the desired trajectory be twice differentiable, i.e. 1st and 2nd derivatives of the reference trajectory are required to design the nonlinear controller like slide mode, flatness based, backstepping, twisted slide mode and twisted backstepping controllers. Thus, in case of a fast-changing input trajectory, i.e. reference trajectory with the sharp edges, the value of its 1st and 2nd derivatives become sufficiently high leading to the instability of the closed-loop system. So, the controller designed using these approaches ensures the trajectory tracking only for the smooth reference trajectories and hence the tracking of a circular trajectory, even in 3D is possible with these nonlinear approaches is possible. But the trajectories with sharp edges, i.e. square and triangular trajectories can't be tracked. Also, the nonlinear controller based the system dynamics has the equal computational complexity to that of its nonlinear model and thus for the high nonlinear and more complex system like quadrotor, the computational complexity is increased which may cause slowing the closed-loop response when connected to the hardware for experiments. While the nonlinear controller designed through heuristic and intuitional approaches like neural
network and fuzzy logic, one needs to have the complete information and knowledge about the system behavior and design the set of rules to design a controller ensuring the desired performance. One drawback of using these approaches is once the designer misinterprets the system behavior, intentionally or unintentionally, the resulting controller will lead the instability of the closed system. So, one need to have complete information about the system and consider all the ambiguities, disturbances in each operating zone, and understand the system performance under every possible condition. This requires a lot of experience and experimentation.
Aim and objectives of the paper. Motivated by the issue mentioned above a single multi-input multi-output linear time invariant controller has been designed that can ensure both altitude control and trajectory tracking in presence of wind gusts and measurement noise. The main contribution of this work is design of a single robust controller for flight control of the quadrotor, i.e. make it fly to achieve a certain altitude and then track the desired trajectory.
Following objectives are met to reach the stated aim of the research:
• a MIMO Hm controller design for the quadrotor model;
• application of designed controller in feedback with the quadrotor model for altitude control and trajectory tracking;
• introduction of wind gusts and measurement noise to verify the controller performance.
Rest of the paper is organized as follows: Section 2 describes the control oriented mathematical modeling of quadrotor. Section 3 briefly discusses the control design approach proposed in the paper. Discussion on the simulation results are presented in Section 4 and, finally, conclusions are drawn in Section 5.
2. Mathematical modeling of the quadrotor. Quadrotor consists of four arms bearing equal weights and length with a DC motor embedded in each of them to achieve the desired motion in a three-dimensional space. The inertial reference frame of a quadrotor system can be seen in Fig. 1.
Left
Front
Right
JfXc
N
Fig. 1. Inertial reference frame of a quadrotor
c
z
c
Rotor angular positions of the motors can be denoted by i2i where i represents the motor on the zth arm of the quadrotor and i = 1, 2, 3, 4. The angular velocities are denoted by c. Rear and front rotors of the quadrotor revolve in counter-clock-wise-direction by angular speeds co1 and c2, thus generate the thruster torques r2 and r1 respectively. While motors 2 and 3 rotate in clock-wise-direction generating torques r3 and r4.
For hovering the quadrotor at a specific height, clock- and counter clock-wise-torques are desired to be the same and thus rotating the respective motors at a same speed, i.e. balancing the body weight of the quadrotor, is the main concern. Since the motor operate as to generate the thruster torques in opposite direction, there is no imbalance of the quadrotor reaction torque, i.e. there is zero imbalance in reaction torque. Roll, pitch and yaw angles, while considering the angular positions of the quadrotor body, are denoted by < 6 and \ respectively. Roll angle is created by increasing (decreasing) the speed of motor associated with right propeller and decreasing (increasing) that of the left one to make the body roll or turn along its own axis.
Throttle is achieved by rotating all the motors (propellers) in same direction at a same speed. Pitch angle is associated with the angular speed or front and rear motors and yaw movement can be generated by moving the front-rear and left-right propellers. If the speed of one pear, described earlier, is increasing that of the other should be reducing to generate the yaw movement. Three orthogonal movements in space and same number of orthogonal movements of the quadrotor offer six degree-of-freedom (DOF). The state-space of the quadrotor is given in (1). For the detailed study of the mathematical model, reader is referred to [14].
< 1 0 0 0 0 0 " <
6 0 1 0 0 0 0 6
\ 0 0 1 0 0 0 \
x 0 0 0 A- 0 0 x
y 0 0 0 0 Ay 0 y
z 0 0 0 0 0 Az _ z
' 0 " "0"
0 0
0 0 F
+ --
0 0 m
0 0
g _ 1
(1)
Equation (1) represents the state-space model of the 6-DOF quadrotor. However, the detailed state-space model can be represented by the state-space equations given in (2)-(13) [14]
xi = X2 ; (2)
x2 ="
l • k C2 - c|)
x6 =
x3 = x4; l • k(el -C2);
Iyy
x5 = x6 ;
12 2 2 2 I b C -C2 + C3 -C4J
1zz x7 = x8 ; x8 = Axx1 ; x9 = x10 ; x10 = Ayx3 ; x11 = x12; — (c
VV1 *
(5)
(6)
(7)
(8)
(9)
(10) (11) (12)
-¿12 = AzX5 -g +—(12 +®32 +®4); (13)
m x '
where c represent the angular velocities of the motor on ith arm of the quadrotor; g is the gravitational acceleration; l, b, k, m denote the arm length, coefficient of left and right drag, and mass of the quadrotor respectively; Ixx, Iyy and Izz denote the moments of inertia in x, y and z axis respectively.
The set of state equations written in (2) - (13) are converted into state-space. The state-space matrices of the system are written as follows:
A =
Ix
(3)
" 0 1 0 0 0 0 0 0 0 0 0 0"
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0
A- 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0
0 0 Ay 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 1
0 0 0 0 Az 0 0 0 0 0 0 0
- 0 0 0 0 -
0 -1 • k/l-x 0 l • k/1 xx
0 0 0 0
-1 kl1 yy 0 l • kjl yy 0
0 0 0 0
B = b/lzz - b/l zz b/lzz - bi zz
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
k k k k
(14)
; (15)
4
C =
(16)
100000000000 001000000000 000010000000 000000100000 000000001000 000000000010
The inputs, states, and outputs of the systems are given in (17)-(19), respectively
2 2 2
a>i a>2 ®3
2
a>4
r ;
V V x x y y
.TT
(17)
zr ;(18)
(19)
y = [< e y x y
The angular positions of the motors at quadrotor arms can be given as
Q = Km • JN V
(20)
where V is the input voltage for the ith motor and oi = D.i, rest of the parameters and their corresponding values are given in Table 1.
Table 1
Parameters of the quadrotor
Parameter Value Parameter Value
Ax 0,45 Ixx, kg-m2 0,357-10-5
Ay Iyy, kg-m2
Az Izz, kg-m2
k 2,98-10-6 m, kg 1,316
g, m/s2 9,8 l, m 0,5
This model of the quadrotor is used to design the robust Ex, controller for the flight control. A brief discussion about the controller design is given in the following section.
3. Robust control design. Controller is placed with the system for controlling the plant according to desired parameters and responses. Main objective general configuration of plant P (to be controlled) with the controller K is shown in Fig. 2.
Fig. 2. General plant with controller configuration
Objective is to minimize the norm of transfer function from w to z and the design problem is to find controller gain K based on v which gives u as control signal to the plant which minimize the closed loop norm from w to z. The generalized configuration will then be represented as [15]:
= P(s)
" P„(s ) P12 (s ) _ P21 () P22 ()_
u = K(s)v . (22)
The linear fractional transformation is
z = Fl (P,K)w , (23)
where
F (P, K) = Pn + P12 K (1 - P22 K)"! P21. (24)
3.1. Ha controller.
The Hx optimal control problem is to find all stabilizing controllers K that minimize [15]
\\Fi (P, K) = maxo( (P, K )-(jo(). (25)
w
The Hm norm has several interpretations in terms of performance. One is that it minimizes the peak of the maximum singular value of F(P(jo), K(jo)). In practice, it is usually not necessary to obtain an optimal controller for the Hm problem, and it is often computationally (and theoretically) simpler to design a suboptimal one (i.e. one close to the optimal ones in the sense of the Hm norm). Let 7min be the minimum value of overall stabilizing controllers K. Then the Hm sub-optimal control problem is: given a Y> 7mm, find all stabilizing controllers K such that
IF ^ K )l=<r. (26)
If we desire a controller that achieves rmin, to within a specified tolerance, then we can perform a bisection on r until its value is sufficiently accurate. The above result provides a test for each value of r to determine whether it is less than rmin or greater than rmin.
Two methods are there for Hm controller design: the transfer function shaping approach and the signal-based approach. In the former, Hm optimization is used to shape the singular values of specified transfer functions over frequency. The maximum singular values are relatively easy to shape by forcing them to lie below user defined bounds, thereby ensuring desirable and widths and roll-off rates. In the signal-based approach, we seek to minimize the energy in certain error signals given a set of exogenous input signals [16]. The latter might include the outputs of perturbations representing uncertainty, as well as the usual disturbances, noise, and command signals. Both two approaches will be considered again in the remainder of this section. In each case we will examine a problem and formulate it in the general control configuration.
A difficulty that sometimes arises with Hm control is the selection of weights such that the Hm optimal controller provides a good trade-off between conflicting objectives in various frequency ranges. Thus, for practical designs it is sometimes recommended to perform only a few iterations of the Hm algorithm. The justification for this is that the initial design, after one iteration, is like a H2 design which does trade-off over various frequency ranges. Therefore, stopping the iterations before the optimal value is achieved gives the design H2 flavor which may be desirable.
z
w
v
u
u =
4. Results and discussions. For the controller validation, a simulation environment is developed in which a 3D trajectory is tracked by the proposed control methodology.
Figures 3, 4 show the tracking of x and y coordinates. It can be observed from plots that controller reaches and stay on desired x and y coordinates with minimum estimation error.
x, m
Circular Trajectory Tracking
10 15 20 25 30 35 40 Fig. 3. Tracking of x coordinates
Tracking in Y-axis
-10 -5 0 5 10
Fig. 5. Tracking of planar trajectory
motion. If quadrotor tracks the desired trajectory under excessive roll and pitch motion, then stability of motion cannot be guaranteed. Actually, it is the best possible case that quadrotor tracks the desired trajectory with minimum roll and pitch angle.
The tracking of desired attitude is displayed in Fig. 6, the plot shows that initially at starting point quadrotor start motion with greater pitch and roll angle but after 1 s of flight it achieves attitudes close to zero. These observation increases the confidence on controller performance and real time implication of designed control scheme. At last the most important observation is the control efforts calculated by controller. In existing case the control efforts are the angular velocities of four motors.
z, m
Altitude Control
10 15 20 25 30 35 40 Fig. 4. Tracking of y coordinates
Initially it is assumed that, quadrotor is hovering at zero altitude. Then it is desired that it gains 10 m of height with slowly increasing altitude value as shown in Fig. 5.
After gaining the desired height it is aimed that quadrotor moves in a circular trajectory having radius of 10 m as shown in the Fig. 5. From Fig. 5, 6, it can be observed that initially tracking error is high but few seconds later controller achieves the desired height. Quadrotor reaches desired altitude within 5 s, which is reasonable performance. Alter gaining desired altitude quadrotor tracks the specified trajectory with minimum estimation error. From Fig. 5, the tracking of circular trajectory can be seen. Despite the tracking of desired trajectory, the most important is that quadrotor maintains its stable attitudes. Stability of attitudes mean that quadrotor do not observe the excessive roll and pitch
t, s
0 5 10 15 20 25 30 35 40 45 50 Fig. 6. Tracking of desired altitude
Figures 7-9 show the control effort plots. From the plots it can be seen that controller calculates smooth control inputs for motors with no chattering. Initially large fluctuations in motor speed can be seen but after 2 s they gains the steady value that are consistent with the trajectory tracking and attitude tracking of the quadrotor.
m
3-D Trajectory Tracking
5
y, m
x, m
-10 -10 Fig. 7. Tracking of 3D trajectory
Angular positions of the quad rotor
t, s
15
20
25
30
35
500 450 400 350 300 250 200 150 100 50
rad/s
Fig. 8. Attitudes of quadrotor Motor angular velocities
1 2 3 4 5
Fig. 9. Quadrotor motor speeds 5. Conclusions.
A robust Ню single Multi input multi output linear time invariant controller for six degree of freedom quadrotor has been designed that can ensure both altitude control and trajectory tracking in presence of wind gusts and measurement noise. Quadrotor is first lifted to the height of 10 m and after achieving the desired altitude, it
starts moving in circular path which keeping roll and pitch angles at 0°. Roll, pitch and angles are shown in results and discussion section. From the plots it can be seen that controller calculates smooth control inputs for motors with no chattering. Initially large fluctuations in motor speed can be seen but after 2 s they gains the steady value that are consistent with the trajectory tracking and attitude tracking of the quadrotor. In future, same work can be applied to the quadrotor by considering the wind gusts.
Conflict of interest. The authors declare that they have no conflicts of interest.
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Received 13.06.2021 Accepted 28.08.2021 Published 26.10.2021
Zulfiqar Ali Gulshan1, Engineer, MS, Muhammad Zulqarnain Haider Ali2, Engineer, MS, Muhammad Shahzaib Shah2, Engineer, MS, Danish Nouman2, Engineer, MS, Mehwish Anwar2, Engineer, MS, Mian Farhan Ullah3, PhD Scholar, Lecturer,
1 Department of Electrical Engineering, Fast University CHT-FSD campus, CHT-FSD, Islamabad, Pakistan, e-mail: [email protected]
2 Department of Electrical Engineering,
University of Engineering and Technology, Taxila, Pakistan, e-mail: [email protected]; [email protected]; [email protected]; [email protected];
3 Department of Mechatronics Engineering, Wah Engineering College, University of Wah, Quaid Avenue, Wah Cantt, Rawalpindi District, Punjab 47040, Pakistan.
e-mail: [email protected] (Corresponding author)
How to cite this article:
Gulshan Z.A., Ali M.Z.H., Shah M.S., Nouman D., Anwar M., Ullah M.F. A robust control design approach for altitude control and trajectory tracking of a quadrotor. Electrical Engineering & Electromechanics, 2021, no. 5, pp. 17-23. doi: https://doi.org/10.20998/2074-272X.202L5.03.
