Adaptive Fuzzy Sliding Mode Control for a Model-Scaled Unmanned Helicopter

This paper presents a novel Adaptive Fuzzy Sliding Mode Controller (AFSMC) for a model-scaled unmanned helicopter as real nonlinear plant. First, in order to efficient control law design, the nonlinear model of the helicopter is reformulated as an affine nonlinear system. To do this aim, a Dynamic Inverter (DI) is introduced as a bijective function. The proposed DI is used to interconnect the helicopter actuators’ main inputs to the helicopter dynamic inputs. Then, AFSMC is designed to control it, and the asymptotic stability of the closed loop system is proved using Lyapunov stability theorem. To verify the merits of the proposed controller, it is compared with traditional sliding mode control system. Simulation results confirmed that the controller as a robust and stable control method has desired controlling performance and well cope with the undesirable chattering phenomenon.


Introduction
The term Unmanned Aerial Vehicles (UAVs) refer to unpiloted aerial vehicles that are remotely piloted or autonomously controlled for the performance of a predefined flight operation.UAVs are divided into two main configurations, fixed-wing and rotorcraft or in particular unmanned helicopters [1].In recent years, autonomous unmanned helicopters have attracted considerable interests from both research and industry.Comparing to the fixed-wing counterpart, they have significant useful characteristics of hovering, vertical take-off, and landing (VTOL), low-altitude cruise, and low-velocity flight [2].These features make them suitable for a wide range of military and civilian applications [3].The unmanned helicopter is a multi-input and multi-output (MIMO) unmanned aerial vehicle (UAV).This helicopter can be considered as an underactuated system with high nonlinearities, and its nonlinear structure consists of high coupling between its dynamics and uncertainties caused by unmodeled dynamics, parameter uncertainty http://www.ispacs.com/journals/jfsva/2016/jfsva-00356/International Scientific Publications and Consulting Services and external disturbances [4].Therefore, the controller design is a challenging concept in both the theory and experimental implementation.Model-scaled helicopters control methods consist of two main types of control approaches, linear and nonlinear control.Linear control methods, such as PID [5] or LQR [6], are designed based on a linear helicopter model.However these linear controllers are uncomplicated and reliable, lack of robustness is a major defect of them.Most of linear controllers that have been applied to unmanned helicopters are based on the  ∞ approach [7][8][9].The main advantage of the  ∞ approach is its robustness in the presence of model uncertainties and disturbances.Although the capability of linear control methods in performing maneuvers in a hover or low velocity regimes, have been proved, they only effective when the states of the unmanned helicopter system are near the equilibrium points.In order to overcome these deficiencies, many nonlinear control methods are designed and utilized, such as backstepping [10][11][12], feedback linearization [13][14], model predictive control [15][16], sliding mode control [17][18][19][20][21] and adaptive control [22][23].Among them, one of the most effective for underactuated systems is backstepping, a recursive technique based on Lyapunov stability analysis, but this method needs the exact nonlinear model and it is impossible to obtain the exact model of the unmanned helicopter system.Sliding mode control as a robust control method is able to keep robustness and stability against uncertainties and disturbances.However, the traditional sliding mode control uses unsmooth sign functions, which would affect helicopter control inputs.Lastly, intelligent control methods, such as fuzzy logic and neural networks (NN), have been combined to nonlinear control methods.In [24] a fuzzy basis fuzzy networks augmented adaptive backstepping in order to trajectory tracking control of a small-scale helicopter and a nonlinear adaptive neural network control for same purpose used in [25].In [26] an adaptive control optimized by neural networks applied to an unmanned helicopter.Although, how to choose an appropriate fuzzy rule base and guarantee the closed loop system stability is a challenging problem.And also, in most NN based cases, the inevitable learning procedure degrades the transient performance in the presence of disturbances and entails a higher computational time for the larger sized neural networks.Specifically, the concept of robustness has been a subject matter in intelligent control techniques for UAVs control.Model-free and learning based methods such as neural networks [27], fuzzy logic [28] and human-based learning techniques [29][30], also used for control of unmanned helicopters.The main characteristic of these control methods is that although the dynamical model of the helicopter is not needed, they require several flight tests and trials in order to train the system.The following paper is organized as below: Section 2 explained the unmanned scaled model helicopter equations, and a bijective function as the Dynamic Inverter.Adaptive Fuzzy Sliding Mode Controller (AFSMC) is described in section 3, and the use of it to control the plant is detailed in section 4. Finally, several simulation, comparison, and analysis are clarified in section 5 and the conclusion is illuminated in section 6.

Model-Scaled Unmanned Helicopter Description
The main aim of this research is to control a model-scaled unmanned helicopter that is explained in this section.For this purpose, a novel adaptive fuzzy sliding mode controller is introduced.The proposed method is a robust, adaptive, and stable controller, which its advantages are proved during simulations.

Reference Frames and Coordinates
To describe the kinematics, dynamics, rotation, and transformation of the plant versus the earth, we should define two frames: (i) The earth reference frame, "Base Frame" that is fixed to the earth where its origin, , is located at a fixed point.Its x-axis is to the north, z-axis points to the vertical up and finally, the y-axis is specified through the right-hand rule.(ii) Body reference frame, "Body Frame", that is attached to the http://www.ispacs.com/journals/jfsva/2016/jfsva-00356/International Scientific Publications and Consulting Services helicopter fuselage with its origin,   , located at the center of gravity of the helicopter.The   axis is to the helicopter head, the   points to the upright and   , again, is specified using the right-hand rule.The body frame rotates and transforms exactly with the helicopter.Axes of these two coordinates are illustrated in Fig. 1.Also, some assumptions are inevitable to be considered, (i) The helicopter fuselage is assumed to be completely rigid.(ii) The motion of earth and earth effects are neglected, here.(iii) Gravity acceleration, inertia matrix and the mass are supposed to be known constants values.For simplicity, all utilized parameters used in the sequel are summarized in Table 1.
Here, it is assumed that Skew symmetric, rotation and inertia matrix of helicopter model are described below.In which  and  are shorts for (.) and (.), respectively.Without loss of generality, we omit disturbance, , in continue until the simulation.
External force and torque that are the results of the main and tail rotors are detailed below, with respect to the Table 1  (2.9) Because of the physics of helicopter, the above equations can be simplified, for more details about this approximation, please see [26] and [31][32][33].This simplification leads to  = [0,0,   ]  (2.10)   (2.12) In addition, Thrusts' expressions are approximated into precise and reliable equations, as below [33].Let, subscript  stands for  or , for each main or tail rotor.Also, the main or tail rotors' counteractive torques can be written as: Using this nonlinear cascade model, the "Actual Control Vector" which contains all the real actual control inputs, is: However, using this "Actual Control Vector", Eq. (2.17), the total helicopter system model cannot be able to reform into an "affine" nonlinear equations system, because the state equations of resulted system are complex.Thus, we redefined a new vector entitled "Virtual Control Vector", such that there are two important properties; first, using this new vector, the system can be modeled as an affine system; second, these two control vectors can be transformed into each other easily.Affine systems have important properties, as instant; there is a variety of control design methods for this type of nonlinear systems, so it is a suitable representation.Thus, we select the control vector below instead, as the controller output. = [  ,  1 ,  2 ,  3 ]  (2.18)In which [ 1 ,  2 ,  3 ]  = .
Considering  as the control vector, we can transforming the helicopter model into a nonlinear affine system and so the controller design process will be much easier and more flexible.Definition 2.1.For an affine nonlinear system with the following equations: In which  is an output and [ 1 ,  2 , … ,   ]  is an input vector.We can rearrange it into Which, Eq. ( 2. 19-b) is again an affine nonlinear system, with y as an output and  as an input, and  = ( 1 ,  2 , … ,   ) is an injective surjective function (bijective) from   to u.Then considering the bijection of , it is possible to design a control law for Eqs.(2.19) with affine systems controller design methods, and then compute   with the inversion of  = ( 1 ,  2 , … ,   ).
This process doesn't affect stability and performance of the system.The change in control vector makes a need of using "Dynamic Inverter" in the real control cycle.Proposed Dynamic Inverter maps the "Virtual Control Vector" into the "Actual Control Vector", as a bijective function.In the other word, by the Dynamic Inverter a unique  leads to only one unique   that makes  at the first, as it seen in Fig. 3. http://www.ispacs.com/journals/jfsva/2016/jfsva-00356/International Scientific Publications and Consulting Services Controller for affine system > 0, and the only real  is then equal to: Eq. 18 Eq. 20 Eq. 17 Eq.16 Eq. 11 Eq. 12 Eq.20 Eq. 14 The helicopter model equations of motion, as kinematic and affine nonlinear dynamic equations are detailed in this section.Next section will explain the proposed controller to control the affine nonlinear systems.

Adaptive Fuzzy Sliding Mode Controller
In this section the concept of proposed controller will be formulated, and utilized to the unmanned model helicopter's position control.

Affine Nonlinear Systems Control Using Traditional Sliding Mode Control
Consider a typical  ℎ order affine nonlinear system as:  () = () + () + () (3.29)In which,  = [, , ̇, … ,  (−1) ]  is the state vector, (. ) and (. ) are nonlinear equations.External disturbance is considered as (), that |()| < , and  is a positive constant.In this paper model-scaled unmanned helicopter that is selected as the under control plant, is a second order affine nonlinear system and so, the Eq.If the system is not affected by uncertainties, then the model will be reliable.Here, it is assumed that the exact value of the function (. ) is unknown and it is proposed to use fuzzy approximation instead.

Fuzzy Approximation
Based on global approximation theory, fuzzy systems with singleton fuzzifier, center average defuzzifier, and product inference engine, are global approximator and can approximate every function in the domain.
The fuzzy system output is: In which,     (  ) is the membership value of   related to    .Let the  ̂ = [ ̅  1 ,  ̅  2 , … ,  ̅   ]  is the parameters that must be approximated, and Then  ̂(|  ) =  ̂ () The fuzzy membership function is selected by expert, and clearly all the states must be measureable and known.
Based on global approximation theory, a set of optimal parameters exist that results in minimizing the approximation error.

Indirect adaptive fuzzy sliding mode control
Adaptive control methods are fundamentally divided into two major aspects.In one hand, if the adaptive law directly adapts the controller parameters by the concept of control performance indices and error, it is considered as a "direct adaptive method".In the other hand, there are "indirect adaptive methods", for which, a model-based controller will be designed at the first, assuming the model is accurate enough, and then the system model's parameters must be adapted such that the controller can work well enough.http://www.ispacs.com/journals/jfsva/2016/jfsva-00356/International Scientific Publications and Consulting Services Here, the proposed method is indirect adaptive, because the controller is first designed by a hypothetical model of plant.In fact, the model is uncertain and inaccurate, and we propose the use of fuzzy approximation in model.Control law is presented in Eq. (3.34), and fuzzy approximation is defined in Eq.The suggestion of  ≥   + , where || <   , cause  ̇< 0. Thus, the proposed control law can stabilize the closed loop system.

Model-Scaled Unmanned Helicopter Proposed Control
The affine second order nonlinear system model of a model-scaled unmanned helicopter dynamic system is described in Eq. (2.26).In which, () is assumed to be a known function of helicopter rotations and inertia.In addition, (, ) due to the presence of the differentiation  ̂̇(), is complicated and under effect of uncertainties, thus it is assumed unknown.Fuzzy system is utilized to approximate this function.Proposed control scheme's block diagram is introduced in Fig. 5  As Fig. 5 illustrates, the adaptive law adapts the fuzzy approximator using the state variables and sliding surface.After, the approximated value of function  is return to the model based control law, and the control signal is computed very fast.It is considerable that the control loop, the approximation loop, and the adaptation loop can be run at the different frequencies.

Simulation Results
The merits of the proposed controller are confirmed by several simulations.Table 2 summaries the value of utilized parameters in this simulations.Here, to check the flight control performance, we use a desired flight path [22], which is described below: if  ≤ 10 sec (5.49) Flight time length is 2 minutes, and the start point is considered to be an origin on the earth.The desired trajectory is an 8-shaped curve.
The initial position of the helicopter is set to be [3, −2, 0] m and the initial yaw angle (0) = 0, where the starting point of the desired trajectory is set to be [0, 0, 0].Actual wind blows to the helicopter as an external disturbance, between 40sec and 80sec.For the ease of formulation, we omit the wind disturbances, but in simulation results, it is applied to system as below: (5.50)As it seen in Fig. 6, the proposed controller is effective in trajectory tracking control for a model-scaled unmanned helicopter and the tracking errors are bounded.3D trajectory tracking in Fig. 7 illustrates that the trajectory tracking objective is accomplished with the proposed controller.For defined comparison between traditional sliding mode and the proposed controller, the position tracking error is illustrated in Fig. 8.As it is mentioned there is not any significant difference between the results of these methods.To find out the efficiency of proposed AFSMC method, the control efforts of traditional SMC and proposed method are shown in Fig. 9.Although the tracking performances are a bit similar, the control signals are far different.Unwanted oscillations with high-frequency is one of the major problems in sliding mode controller implementation, that is called chattering phenomenon.In many practical control systems, (e.g.aircraft control), in order to providing continuous and smooth control signal, chattering avoidance is very important, for instance, helicopters can't move back and forth with high frequency.As it seen in simulation results, chattering in the control inputs is significantly reduced due to the applying proposed control method.The control effort of SMC controller is very rough; with this frequency, there is not any actuator to react.The helicopter motors cannot response to this control signal.We can limit the control signal, from the frequency point of view, for example through a low pass filter, (as the actuators are), the control effort becomes more smoothly; but the closed-loop system will suffer from unstability.Numerical indices are compared in Table 3, that are defined in Table 4.

Conclusion
This paper aims to introduce a novel adaptive fuzzy sliding mode controller, to control a model-scaled unmanned helicopter.First, the helicopter's model is proposed as an affine nonlinear equation system, and then this affine nonlinear system is utilized to design the controller.Stability of closed loop control system is proved using Lyapunov theorem.Adaptive laws also are derived considering Lyapunov stability theorem.In this paper we use the affine model in formulations without need of linearization.In addition, a dynamic inverter is introduced to make the helicopter, affine.The novel controller's performance is compared with traditional sliding mode control method.The merits and advantages of the proposed controller are confirmed during simulations.An important property of the proposed controller is the smooth control efforts.

Figure 1 :
Figure 1: A simple draw that illustrates the model-scaled unmanned helicopter coordinates

Figure 5 :
Figure 5: The block diagram of proposed control scheme for control a model-scaled unmanned helicopter

Figure 6 :
Figure 6: Position tracking of the proposed AFSMC

Figure 7 :
Figure 7: Position tracking of the proposed AFSMC

Figure 8 :
Figure 8: Tracking Error, using traditional sliding mode and the proposed controller

Figure 9 :
Figure 9: Control efforts of both traditional sliding mode and the proposed controller

Table 1 :
Utilized Parameters External

resultant forces 𝑄 ≜ [𝐿, 𝑀, 𝑁] 𝑇 External resultant torques 𝑚 The gross mass 𝐽 Inertia matrix 𝐼 𝑥𝑥 , 𝐼 𝑦𝑦 𝑎𝑛𝑑 𝐼 𝑧𝑧 The moments of inertia around the Base Frame axis 𝐼 𝑥𝑧
Vertical and horizontal distances from tail rotor hub to the center of gravity of fuselage   ,   The thrusts generated by Main and tail rotor   ,   Thrust coefficients of main and tail rotors   ,   Counteractive torques of main and tail rotors   ,   Main rotor's coefficients of longitudinal and lateral stiffness   ,   Torque coefficients of main and tail rotors   ,   Longitudinal and lateral flapping angles   ,   Collective pitches of main and tail rotors  1 ,  3 ,   ,   Constant uncertain aerodynamic parameters (i=m or t) http://www.ispacs.com/journals/jfsva/2016/jfsva-00356/International Scientific Publications and Consulting Services 2.2.

Kinematic and Dynamic Model of Unmanned Helicopter
According to the notations established in Table1, and from Newton-Euler equations, the dynamic model of this helicopter can be described as Eqs.(2.1) -(2.4).

Table 3 :
Numerical performance indices