Modeling and Robust Trajectory Tracking Control for a Novel Six-Rotor Unmanned Aerial Vehicle

Abstract

Modeling and trajectory tracking control of a novel six-rotor unmanned aerial vehicle (UAV) is concerned to solve problems such as smaller payload capacity and lack of both hardware redundancy and anticrosswind capability for quad-rotor. The mathematical modeling for the six-rotor UAV is developed on the basis of the Newton-Euler formalism, and a second-order sliding-mode disturbance observer (SOSMDO) is proposed to reconstruct the disturbances of the rotational dynamics. In consideration of the under-actuated and strong coupling properties of the six-rotor UAV, a nested double loops trajectory tracking control strategy is adopted. In the outer loop, a position error PID controller is designed, of which the task is to compare the desired trajectory with real position of the six-rotor UAV and export the desired attitude angles to the inner loop. In the inner loop, a rapid-convergent nonlinear differentiator (RCND) is proposed to calculate the derivatives of the virtual control signal, instead of using the analytical differentiation, to avoid “differential expansion” in the procedure of the attitude controller design. Finally, the validity and effectiveness of the proposed technique are demonstrated by the simulation results.

1. Introduction

In the past years, unmanned aerial vehicles (UAVs) are broadly concerned and applied. They provide tremendous advantages over manned operations like searching and rescuing, remote inspecting, and surveillance as well as in civil use such as disaster monitoring, aerial photographing, and transmission lines inspection. The merit of UAVs is maximized for the practical use in the places where are dangerous and difficult to approach.

UAVs are classified into two categories, fixed and rotary wing types. In practical operations, fixed-wing UAVs have been used for years in routine surveillance missions, but they lack the hovering flight capability. So, a kind of rotary wing type UAVs is focused to offer the possibility of vertical take-off and landing (VOTL), omnidirectional flying, and hovering performance. Rotary wing type UAVs are classified into multirotor type (such as quad-rotor and six-rotor), coaxial helicopter, traditional helicopter, and so forth. quad-rotor has the simplest mechanical structure among them. Moreover, its simple design and relatively low-cost feature make it an attractive candidate for swarm operations, a field of ongoing research in the UAV community [1–5].

However, quad-rotor also has some potential defects such as smaller payload capacity, lack of hardware redundancy, and anticrosswind capability. For this reason, we introduce, in this paper, one configuration of a six-rotor UAV (the structure as shown in Figure 1) composed of six rotors to solve these problems of quad-rotor. The main characteristics of the six-rotor UAV are increased payload capacity and stability in the windy and hardware redundancy. And the six rotors and propellers of the UAV are distributed on both sides of the fuselage symmetrically. So, the configuration of the six-rotor UAV is similar to an insect. The detailed structure, features, and purpose of the six-rotor UAV can be seen in [6, 7].

Figure 1 

The structure of six-rotor UAV and the associated frames.

Compared with the quad-rotor, the super characteristics of the six-rotor UAV are as follows.

(1) Increased Payload Capability. By the actual flight experiment, the six-rotor UAV has a payload capacity of approximately 0.9 kilogram, the largest of all the quad-rotor with the same mechanical size.

(2) Increased Stability in the Wind. The six-rotor UAV weights approximately 1700 grams and has the increased thrust of 6 counter-rotating rotors which allows it to be much more stable in windy conditions. The lager size and weight of the six-rotor UAV make it a very stable UAV to fly.

(3) Hardware Redundancy and Fault Tolerance. Six-rotor UAV could remain stable flighting while some of the six rotors are broken. If one or two rotors are broken, the flight controller with fault diagnosis and fault-tolerant can implement fault diagnosis and fault tolerant flight control immediately, and other rotors will compensate the reduced lift caused by the broken rotors to keep the stability of the six-rotor UAV. It can effectively improve the reliability and safety of the UAV and reduce the probability of crash. As for the quad-rotor, it will crash inevitably in the above situations.

At the same time, in this paper, the translational dynamics and rotational dynamics with aerodynamic moments, external disturbance, and parameter uncertainty of the six-rotor UAV are also developed. And, a second-order sliding-mode disturbance observer (SOSMDO) is proposed to reconstruct the disturbance of the rotational dynamics.

Next, a robust trajectory tracking control strategy forcing the six-rotor UAV to track the desired trajectory accurately is designed. Considering the under-actuated and strong coupling characteristics of the six-rotor UAV, the closed loop flight control system is treated as a nested double-loops structure system, where the outer loop indicates the relationship between translation and rotation movement, and the inner loop denotes the attitude control loop. Meanwhile, in the outer loop, a position error PID controller is developed, of which the task is to compare the desired trajectory with real position of the six-rotor UAV and construct the desired attitude angles to the inner attitude control loop. In the control methods of nonlinear system, fuzzy variable structure control [8], network-based feedback control [9], dynamic inversion [10], feed linearization and sliding mode control [11], and adaptive output feedback control [12] have been widely used. Backstepping control design [3–5] has also become an effective approach for designing controller because of its simplicity. The rotational dynamics of the six-rotor UAV satisfies the strict feedback form, so backstepping technique can be used to design the attitude controller. However, the analytic derivative expressions of virtual control variables are usually overly complicated or unknown especially for the uncertain systems, which limits the backstepping technique in practical applications. Therefore, to overcome this drawback, a rapid-convergent nonlinear differentiator (RCND) is proposed to extract the ideal angular rate differential command signals and without calculating the virtual control signal derivative analytically, and to decrease the dependent degree on the analytic model. In succession, an attitude controller of the six-rotor UAV based on disturbances compensation, RCDN and backstepping is designed. And the robust terms of the attitude controller can reduce the effect of disturbance reconstruction errors on the tracking capability of the six-rotor UAV. Finally, the validity and effectiveness of the proposed SOSMDO and trajectory tracking strategy are demonstrated through various simulation experiments.

The whole paper is organized as follows. Section 2 presents the detailed mathematical model of the six-rotor UAV. Section 3 presents a second-order sliding-mode disturbance observer for disturbance reconstruction. The robust trajectory tracking control strategy is proposed in Section 4. The position controller and attitude stability controller are designed, respectively, in this section. In Section 5, simulation results of the six-rotor UAV trajectory tracking are compared and discussed. And finally, the conclusion of the proposed robust trajectory control strategy is presented in Section 6.

2. Mathematical Modeling

In order to simplify the modeling of the six-rotor UAV and therefore make the controller design easier, several reasonable assumptions are made.

Assumption 1. Six-rotor UAV is rigid. Then, the nonlinear dynamics can be derived by using Newton-Euler formulas.

Assumption 2. The configuration of the six-rotor UAV is symmetrical with respect to the axes 𝑜𝑥, 𝑜𝑦, and 𝑜𝑧.

Assumption 3. The height between the rotors and the plane of the six-rotor UAV is ignored.

Firstly, two frames have to be defined: a body-fixed frame (𝔹-frame) and an earth-fixed inertial frame (𝔼-frame). Let 𝔹={𝑜𝑥𝑦𝑧} denote the body-fixed frame whose origin 𝑜 is at the center of mass of the six-rotor UAV and 𝔼={𝑜_𝑔𝑥_𝑔𝑦_𝑔𝑧_𝑔} denote the earth-fixed inertial frame, as shown in Figure 1. Therefore, under Assumption 3, the structure of the six-rotor UAV can be simplified as shown in Figure 2.

Figure 2 

The simple structure of six-rotor UAV.

In Figure 2, 𝑙₁ denotes the length of OB, and OE, 𝑙₂ denotes the length of AM, DM, CN, and FN; 𝑙₃ denotes the length of OM and ON, and 𝛼 represents the included angle between AM and OM.

2.1. Translational Dynamics

Modeling the rigid body dynamics aims at finding the differential equations that relate system outputs (position and orientation) to its inputs (force and torque vectors). The equations of motion for a rigid body of mass 𝑚∈𝑅 and inertial matrix 𝐉∈𝑅^3×3 subject to extern force vector 𝐅∈𝑅³ and torque vector 𝜏^𝑏∈𝑅³ are given by the following Newton-Euler formulas [14], expressed in the body-fixed frame 𝔹: 𝑚̇𝐉̇𝐕+Ω×𝑚𝐕=𝐅,Ω+Ω×𝐉Ω=𝜏^𝑏, (1) where 𝐕=(𝑢,𝑣,𝑤)^𝑇 and Ω=(𝑝,𝑞,𝑟)^𝑇 are the linear and angular velocities in the body-fixed frame, respectively. The translational force 𝐅 combines gravity, main thrust, and other body forces components. Under Assumption 2, the total inertial matrix 𝐉∈𝑅^3×3 of the six-rotor UAV is a diagonal positive definite constant matrix expressed in the body-fixed frame 𝔹.

Using Euler angles parametrization and “𝑍𝑌𝑋” convention, the airframe orientation in space is given by a rotation matrix 𝐑(𝜙,𝜃,𝜓) from the body-fixed frame 𝔹 to the earth-fixed inertial frame 𝔼, where 𝐑∈𝑆𝑂(3) is given by𝐑=cos𝜃cos𝜓sin𝜃cos𝜓sin𝜙−sin𝜓cos𝜙sin𝜃cos𝜓cos𝜙+sin𝜓sin𝜙cos𝜃sin𝜓sin𝜃sin𝜓sin𝜙+cos𝜓cos𝜙sin𝜃sin𝜓cos𝜙−cos𝜓sin𝜙−sin𝜃cos𝜃sin𝜙cos𝜃cos𝜙, (2)where 𝜉=(𝜙,𝜃,𝜓)^𝑇 denotes the vector of three Euler angles.

By considering this transformation between the body-fixed frame 𝔹 and the earth-fixed inertial frame 𝔼, it is possible to separate the gravitational force from other force and write the translational dynamics in the earth-fixed inertial frame as follows: ̇̇𝐩=𝐯,𝐯=−𝑔𝐳_𝑒+1𝑚𝐑𝐅^𝑏, (3) where 𝐩=(𝑥,𝑦,𝑧)^𝑇 and 𝐯=(𝑥̇,𝑦,̇𝑧)̇^𝑇 are the position and velocity of the six-rotor UAV in the inertial frame, respectively. 𝑔 is the gravitational acceleration, 𝐅^𝑏=(0,0,𝑇)^𝑇 is the resulting force vector in the body-fixed frame (excluding the gravity force) acting on the airframe, and 𝐳_𝑒=(0,0,1)^𝑇 is a unit vector expressed in the earth-fixed inertial frame.

2.2. Rotational Dynamics

In the following, we will use Euler angles parametrization to express the rotational dynamics in an appropriate form for control design.

Let us first write the kinematic relation between Ω and 𝜉̇: 𝜉̇=𝐖Ω, (4) where the Euler matrix 𝐖 is given by 𝐖=1sin𝜙tan𝜃cos𝜙tan𝜃0cos𝜙−sin𝜙0sin𝜙sec𝜃cos𝜙sec𝜃 (5) and det(𝐖)=sec𝜃. Thus, when the pitch angle satisfies 𝜃≠(2𝑘−1)𝜋/2, 𝑘∈𝑍 the matrix 𝐖 is invertible.

Therefore, from (4) and the second equation of (1), the rotational dynamics of the six-rotor UAV, used for flight controller, is given by 𝜉̇=𝐖Ω+𝐃_𝜉,(𝐉+Δ𝐉)Ω̇=−[Ω×(𝐉+Δ𝐉)Ω]−𝐆_𝑎+𝜏^𝑏+𝐝, (6) where (6) expresses the attitude angle loop and angular rate loop of the six-rotor UAV, respectively. And 𝐃_𝜉 is an external disturbance of the attitude angle loop. Δ𝐉∈𝑅^3×3 denotes an uncertain part of the inertial matrix, which is caused by variations of particular payloads from a nominal one. 𝐝=[𝑑_𝜙,𝑑_𝜃,𝑑_𝜓]^𝑇 denotes the aerodynamic moments and external disturbance. And the vector 𝐆_𝑎 expresses the gyroscopic torque which is given by 𝐆_𝑎=𝐽_𝑟Ω×𝐳_{𝑒𝑖=3,4,5}𝜔_𝑖−_𝑖=1,2,6𝜔_𝑖, (7) where 𝐽_𝑟 and 𝜔_𝑖 are, respectively, the rotor inertia and the rotor speed.

For the simplicity of this study, the rotational dynamics (6) for the six-rotor UAV with aerodynamic moments, external disturbance, and parameter uncertainty input is rewritten as 𝜉̇=𝐖Ω+𝐃_𝜉,Ω̇=−𝐉⁻¹(Ω×𝐉Ω)−𝐉⁻¹𝐆_𝑎+𝐉⁻¹𝜏^𝑏+𝐃_Ω, (8)(9) where 𝐃_Ω=𝐉⁻¹[𝐝−Δ𝐉Ω̇−(Ω×Δ𝐉Ω)] is called the composite disturbance of the angular rate loop, (9), which includes aerodynamic moments, external disturbance, and parameter uncertainty.

The expressions of 𝐅^𝑏 of (3) and 𝜏^𝑏 of (9) are detailed in the next section.

2.3. Aerodynamics Force and Torques

The six-rotor UAV is under-actuated mechanical system with six degree of freedom (DOF) and four main control inputs. It is characterized by three main control torques 𝜏^𝑏=(𝜏_𝜙,𝜏_𝜃,𝜏_𝜓)^𝑇 and one main control force 𝐅^𝑏=(0,0,𝑇)^𝑇. The collective lift 𝑇 is the sum of the thrusts generated by the six propellers in the free air: 𝑇=⁶_𝑖=1𝑓_𝑖=𝑏⁶_𝑖=1𝜔²_𝑖. (10)

At the same time, the reactive torque caused by air drag is also generated by each propeller and given by 𝑄_𝑖=𝑑𝜔²_𝑖. Thus, the total reactive torque by the six propellers is given by 𝑄=_𝑖=3,4,5𝑄_𝑖−_𝑖=1,2,6𝑄_𝑖=𝑑_𝑖=3,4,5𝜔²_𝑖−_𝑖=1,2,6𝜔²_𝑖, (11) where 𝜔_𝑖 is the rotor speed. The parameters 𝑏 and 𝑑 are positive proportionality constants relative to the density of air, the shape of the blades, the number of the blades, the chord length of the blades, the pitch angle of the blade airfoil, and the drag constant. And it is reasonable to assume that the scalars 𝑏 and 𝑑 are constants when the six-rotor UAV is performed at low speed in free air [4].

Therefore, according to Figure 2 and (11), we can easily obtain that the airframe torques generated by the six rotors are given by 𝜏^𝑏=𝜏_𝜙𝜏_𝜃𝜏_𝜓=𝑏𝑙₂sin𝛼𝜔²₄+𝜔²₆−𝜔²₁−𝜔²₃+𝑏𝑙₁𝜔²₅−𝜔²₂𝑏𝑙₂cos𝛼+𝑙₃𝜔²₁+𝜔²₄−𝜔²₃−𝜔²₆𝑑𝜔²₃+𝜔²₄+𝜔²₅−𝜔²₁−𝜔²₂−𝜔²₆. (12)

In order to facilitate the computation of the real control inputs, that is, 𝜔_𝑖, (𝑖=1,2,…,6), (10) and 𝜏^𝑏 put together𝑇𝜏_𝜙𝜏_𝜃𝜏_𝜓=𝑏𝑏𝑏𝑏𝑏𝑏−𝑏𝑙₂sin𝛼−𝑏𝑙₁−𝑏𝑙₂sin𝛼𝑏𝑙₂sin𝛼𝑏𝑙₁𝑏𝑙₂𝑏𝑙sin𝛼₂cos𝛼+𝑙₃0−𝑏𝑙₂cos𝛼+𝑙₃𝑏𝑙₂cos𝛼+𝑙₃0𝑏𝑙₂cos𝛼+𝑙₃𝜔−𝑑−𝑑𝑑𝑑𝑑−𝑑²¹𝜔²₂𝜔²₃𝜔²₄𝜔²₅𝜔²₆=𝐌𝜔. (13)

3. SOSMDO Design

Sliding mode disturbance observer (SMDO) has advantages of easy realization, simple design, rapid convergence, and so forth. Besnard et al. [15] uses SMDO to reconstruct the disturbance of the mathematical model of quad-rotor. But in [15], the disturbance reconstruction terms all contain symbolic function, and it will cause discontinuity of the disturbance compensation and chattering phenomenon. To solve this problem, Zeng et al. [16] design an SMDO based on super-twisting algorithm to reconstruct and compensate the disturbance of a missile system. The algorithm can make the high frequency switching part hidden into the higher derivative of the sliding mode variable, and thus the chattering phenomena can be weakened. However, the reconstruction terms of the algorithm still contain symbolic function, and the algorithm is relatively complicated.

In view of this, in this paper, a second-order sliding mode disturbance observer (SOSMDO) to provide continuous disturbance compensation term is introduced which makes use of the integral of a symbolic function to reconstruct the disturbance. Compared with the algorithm in [16], the proposed SOSMDO has the advantage of implementing easily, designing process simply, and inhibiting chattering phenomenon further.

First, in order to make the SOSMDO design easy, a couple of reasonable assumptions are given as follows.

Assumption 4. The states 𝜉 and Ω of the system (8) and (9) are accessible for measurement.

Assumption 5. Partial derivatives with respect to time of the disturbances 𝐃_𝑖, 𝑖=𝜉,Ω are continuous and bounded, and there exists a known bounded constant 𝐶_𝑖 such that 𝛼_𝑖(⋅)_∞=sup_{[)𝑡∈0,∞}𝜕𝐃_𝑖𝜕𝑡≤𝐶_𝑖(𝑖∈𝜉,Ω). (14)

Theorem 6. Consider the rotational dynamics of the six-rotor UAV (8) and (9) under Assumptions 4 and 5; consider the following second-order sliding mode dynamics: 𝐬_𝑖=𝑥_𝑖−𝑧_𝑖,𝐳̇_𝑖=Γ_𝑖𝑠_𝑖+𝜑_𝑖+𝐃_𝑖,𝐃̇_𝑖=𝜒_𝑖sign𝑠_𝑖, (15) where 𝑖∈{𝜉,Ω}, 𝐬_𝑖=𝑠_𝑖1𝑠_𝑖2𝑠_𝑖3^𝑇∈𝑅³ is an auxiliary sliding vector, Γ_𝑖=diag{𝑘_𝑖1,𝑘_𝑖2,𝑘_𝑖3}∈𝑅^3×3 is a positive definite matrix to be designed if 𝑖=𝜉, 𝜑_𝜉=𝐖Ω and if 𝑖=Ω, 𝜑_Ω=−𝐉⁻¹(Ω×𝐉Ω)−𝐉⁻¹𝐆_𝑎+𝐉⁻¹𝜏_𝑎. 𝐃_𝑖 is the disturbance reconstruction value, and sign(𝑠_𝑖)=sign(𝑠_𝑖1)sign(𝑠_𝑖2)sign(𝑠_𝑖3)^𝑇.
When we choose 𝜒_𝑖>𝐶_𝑖, the auxiliary sliding vector 𝐬_𝑖 and its derivative 𝐬_𝑖̇ converge to origin asymptotically.

Proof. Substituting (8) and (9) into (15), respectively, we have 𝐬_𝑖̇=−Γ_𝑖𝐬_𝑖+𝐃_𝑖−𝐃_𝑖𝑖∈{𝜉,Ω}. (16)
And under Assumption 5, taking the time derivative of (16) and substituting it into (15), it yields that 𝐬_𝑖̈+Γ_𝑖𝐬_𝑖̇=−𝜂_𝑖sign𝑠_𝑖, (17) where 𝜂_𝑖∈[𝜒_𝑖−𝐶_𝑖,𝜒_𝑖+𝐶_𝑖].
Select the candidate Lyapunov function as 𝑉_𝑖=³_𝑗=112𝑠_𝑖𝑗,𝑠̇_𝑖𝑗𝐐_𝑖𝑗𝑠_𝑖𝑗,𝑠̇_𝑖𝑗^𝑇+𝑝_𝑖𝑗𝜂^𝑠𝑖𝑗₀sign𝑠_𝑖𝑗𝑑𝑠_𝑖𝑗, (18) where 𝐐_𝑖𝑗=^{𝑞𝑖𝑗1𝑞𝑖𝑗3}_{𝑞𝑖𝑗3𝑞𝑖𝑗2} is a positive definite matrix to be designed and it satisfies 𝑞_𝑖𝑗1𝑞_𝑖𝑗2>𝑞²_𝑖𝑗3,  𝑞_𝑖𝑗1=𝑘_𝑖𝑗𝑞_𝑖𝑗3, 𝑞_𝑖𝑗2=𝑝_𝑖𝑗, 𝑞_𝑖𝑗2𝑘_𝑖𝑗>𝑞_𝑖𝑗3.
Then, substituting the positive definite parameter matrix 𝐐_𝑖𝑗 and (17) into the time derivative of (18) yields that 𝑉̇_𝑖=³_𝑗=1𝑠̇_𝑖𝑗,𝑠̈_𝑖𝑗𝐐_𝑖𝑗𝑠_𝑖𝑗,𝑠̇_𝑖𝑗^𝑇+𝑝_𝑖𝑗𝜂𝑠̇_𝑖𝑗sign𝑠_𝑖𝑗=³_𝑗=1𝑞_𝑖𝑗1𝑠_𝑖𝑗𝑠̇_𝑖𝑗+𝑞_𝑖𝑗2𝑠̇_𝑖𝑗+𝑞_𝑖𝑗3𝑠_𝑖𝑗−𝑘_𝑖𝑗𝑠̇_𝑖𝑗−𝜂sign𝑠_𝑖𝑗+𝑞_𝑖𝑗3𝑠̇²_𝑖𝑗+𝑝_𝑖𝑗𝜂𝑠̇_𝑖𝑗sign𝑠_𝑖𝑗=³_𝑗=1−𝑘_𝑖𝑗𝑞_𝑖𝑗2−𝑞_𝑖𝑗3𝑠̇²_𝑖𝑗−𝜂𝑞_𝑖𝑗3𝑠_𝑖𝑗sign𝑠_𝑖𝑗. (19)
From (19), we can get the conclusion that if 𝑠_𝑖𝑗≠0 or 𝑠̇_𝑖𝑗≠0, then 𝑉̇_𝑖<0 is true. Thus, the auxiliary sliding vector 𝐬_𝑖 and its derivative 𝐬_𝑖̇ of the SOSMDO converge to origin asymptotically.
This ends the proof.

Remark 7. Compared with the disturbance reconstruction based on SMDO in [16], the proposed SOSMDO in this paper, the disturbance reconstruction vectors are continuous, and the high frequency switching part is hidden into the first-order derivative of the sliding mode variable. Therefore, the SOSMDO can remarkably restrain the chattering phenomenon.

4. Robust Trajectory Tracking Control Strategy

To give a clear idea of the robust trajectory tracking control strategy proposed in this paper, a control block diagram is depicted in Figure 4. The outer loop controller employs PID control methodology to compare the desired trajectory with real position of the six-rotor UAV and construct the desired attitude angles to the inner attitude control loop. In the inner loop, a robust attitude controller based on disturbance compensation and RCND is designed in which uses a nonlinear differentiator to calculate the derivative of the virtual control signal, instead of using the analytical differentiation.

4.1. Translational Control

Define the position tracking error 𝐩_𝑒 of the translational dynamics given by (3) as 𝐩_𝑒=𝐩_𝑐−𝐩=𝑥_𝑐−𝑥,𝑦_𝑐−𝑦,𝑧_𝑐−𝑧^𝑇, (20) where the vector 𝐩_𝑐=(𝑥_𝑐,𝑦_𝑐,𝑧_𝑐)^𝑇 is the desired smooth trajectory.

A sliding mode error function is defined as follows [17]: 𝐫=𝐩̇_𝑒+Λ𝐩_𝑒, (21) where Λ is a diagonal positive definite design parameter matrix. Then, (21) is a stable system so that 𝐩_𝑒 is bounded as long as the controller guarantees that the filtered error 𝐫 is bounded. In fact, it is easy to show that one has [17] 𝐩_𝑒≤‖𝐫‖𝜆_min(Λ),𝐩̇_𝑒≤‖𝐫‖, (22) where 𝜆_min(Λ) is the smallest eigenvalue of the matrix Λ.

Note that 𝐩̇_𝑒+Λ𝐩_𝑒=0 defines a stable sliding mode surface. The function of the controller to be designed is to force the system onto this surface by making 𝐫 small. The parameter Λ is selected for a desired sliding mode response 𝐩_𝑒(𝑡)=𝐩_𝑒^−Λ𝑡𝐩_𝑒(0). (23)

We now focus on designing a controller to keep ‖𝐫‖ small. Then, the error dynamics becomes 𝑚𝐫̇=𝑚𝐩̈_𝑒+𝑚Λ𝐩̇_𝑒. (24)

Then, we can obtain 𝑚̇𝑥̈𝐫=𝑚𝑦̈^𝑐_𝑐𝑧̈_𝑐𝑥̈𝑦̈𝑧̈−𝑚+𝑚Λ𝐫−Λ𝐩_𝑒. (25)

Submitting the second equation of (3) into (25), we have 𝑚𝐫̇=𝑚𝐩̈_𝑒−𝐔−𝐦_𝑔+𝑚Λ𝐫−𝑚Λ²𝐩_𝑒, (26) where 𝑇𝑇,𝐦𝐔=sin𝜃cos𝜓cos𝜙+sin𝜓sin𝜙sin𝜃sin𝜓cos𝜙−cos𝜓sin𝜙𝑇cos𝜃cos𝜙_𝑔=00.−𝑚𝑔 (27)

Then, the virtual control input 𝐔=(𝑢_𝑥,𝑢_𝑦,𝑢_𝑧)^𝑇 which stabilizes the tracking error dynamics (26) is designed as 𝑈=𝑚𝐩̈_𝑒−𝑚Λ²𝐩_𝑒−𝐦_𝑔+𝐊_𝑟𝐫+𝐊_𝑖^𝑡₀𝐫𝑑𝑡^′, (28) where 𝐊_𝑟 and 𝐊_𝑖 are diagonal positive definite matrices to be designed.

Now, let us computer the control force vector magnitude and the desired attitude angles for tracking the desired trajectory. The virtual control input 𝐔=(𝑢_𝑥,𝑢_𝑦,𝑢_𝑧)^𝑇 calculated by the (28) is the force vector components along axes that are needed for tracking some desired trajectory. Thus, the components of 𝐔 are given by 𝑢_𝑥=𝑇sin𝜃_𝑐cos𝜓_𝑐cos𝜙_𝑐+sin𝜓_𝑐sin𝜙_𝑐,𝑢_𝑦=𝑇sin𝜃_𝑐sin𝜓_𝑐cos𝜙_𝑐−cos𝜓_𝑐sin𝜙_𝑐,𝑢_𝑧=𝑇cos𝜃_𝑐cos𝜙_𝑐. (29)

According to (29), the required total thrust 𝑇 and the desired attitude angles 𝜙_𝑐 and 𝜃_𝑐 can be proposed as 𝑇=𝑢²_𝑥+𝑢²_𝑦+𝑢²_𝑧𝜙_𝑐𝑢=arcsin^𝑥sin𝜓_𝑐−𝑢_𝑦cos𝜓_𝑐𝑢²_𝑥+𝑢²_𝑦+𝑢²_𝑧𝜃_𝑐𝑢=arctan^𝑥cos𝜓_𝑐+𝑢_𝑦sin𝜓_𝑐𝑢_𝑧, (30)(31) where 𝜓_𝑐 is the desired yaw angle given by as an external reference command.

Theorem 8. Consider the translational dynamics of the six-rotor as described in (3) with a control strategy shown in Figure 3 and control input given by (28) and (30). And suppose that [𝐊_𝑟−𝑚Λ]>0. Then, the error 𝐫 and the position tracking error 𝐩_𝑒 are uniformly ultimately bounded (UUB).

Figure 3 

Block diagram of the robust trajectory tracking control strategy.

(a)   Vertical helix flight

(b) Horizontal rectangle flight

Figure 4 

The 3D diagram for real flight trajectory (solid line) and desired trajectory (dashed line).

Proof. Using (28) into (26) yields the closed-loop error dynamics 𝑚𝐫̇=−𝐊_𝑟−𝑚Λ𝐫−𝐊_𝑖^𝑡₀𝐫𝑑𝑡^′. (32)
Choose the candidate Lyapunov function as 1𝐿=2𝐫^𝑇1𝑚𝐫+2^𝑡₀𝐫^𝑇𝑑𝑡^′𝐊_𝑖^𝑡₀𝐫𝑑𝑡^′, (33) and the time derivative of 𝐿 with respect to time is given by 𝐿̇=𝐫^𝑇𝑚𝐫̇+𝐫^𝑇𝐊_𝑖^𝑡₀𝐫𝑑𝑡^′. (34)
Substituting (32) into (34), we obtain 𝐿̇=−𝐫^𝑇𝐊_𝑟−𝑚Λ𝐫≤−𝐊_𝑟−𝑚Λ_min‖𝐫‖²≤0, (35) where ‖𝐊_𝑟−𝑚Λ‖_min is the minimum possible norm of different parameter values of (𝐊_𝑟−𝑚Λ).
Then, we conclude that the error 𝐫 is UUB. Moreover, according to the definition of 𝐫 given by (21), we have 𝐩_𝑒≤‖𝑠+Λ‖⁻¹‖𝐫‖≤𝜆_min(Λ)⁻¹‖𝐫‖, (36) where 𝜆_min(Λ) is the smallest eigenvalue of Λ.
Thus, from (36), we prove that the position tracking error 𝐩_𝑒 is also UUB.

4.2. Rotational Control

4.2.1. RCND Design

Using the approach of combination of linear and nonlinear differentiator to design the RCND under the introduction of perturbation parameter which is the design idea of the RCND. In order to weaken the chattering of differential estimation, a perturbation parameter is introduced into nonlinear term under the proposed differentiator robustness, and a linear term with perturbation parameter is utilized to accelerate the convergence speed of the differentiator and restrain the high frequency noise interference.

Define sig(𝐱)^𝑚=[sig(𝑥^𝑚₁),…,sig(𝑥^𝑚_𝑛)]^𝑇, where sig(𝑥^𝑚₁)=|𝑥_𝑖|^𝑚sign(𝑥_𝑖), 𝑚>0,  𝐱∈𝑅^𝑛,and sign is a symbolic function.

To give a clear idea of the RCND design procedure the following step is given.

Lemma 9. Consider the nonlinear system in the form of 𝐱̇₁(𝑡)=𝐱₂(𝑡),𝐱̇₂(𝑡)=−𝐑¹₁𝐱₁(𝑡)−𝐑²₁sig𝐱₁^{𝜀/(2−𝜀)}(𝑡)−𝐑¹₂𝐱₂(𝑡)−𝐑²₂sig𝐱₂^𝜀(𝑡), (37) where 𝐱₁(𝑡), 𝐱₂(𝑡)∈𝑅^𝑛 are the states of the nonlinear system. 𝐑¹₁=diag{𝑟¹₁₁,…,𝑟¹_1𝑛}, 𝐑²₁=diag{𝑟²₁₁,…,𝑟²_1𝑛}, 𝐑²₂=diag{𝑟²₂₁,…,𝑟²_2𝑛}, 𝐑¹₂>0 and 0<𝜀<1 are the parameters to be designed.
For any initial bounded 𝐱₁(0) and 𝐱₂(0), the system (37) converges to origin in a limited time. This means that there exists a time constant such that 𝐱₁(𝑡)≡0,𝐱₂(𝑡)≡0∀𝑡≥𝑡_𝑓. (38)

Proof. Based on (37), we choose the following Layapunov function: 1𝐕=2𝐱^𝑇₁𝐑¹₁𝐱₁+𝐱^𝑇₂𝐱₂+^𝑛_𝑗=1𝑟²_1𝑗(2−𝜀)2𝑥_1𝑗^2/(2−𝜀). (39)
The time derivative of 𝑉 along the trajectories of (37) is given by 𝑉̇=𝐱^𝑇₁𝐑¹₁𝐱₂+𝐱^𝑇₂−𝐑¹₁𝐱₁−𝐑²₁sig𝐱₁^{𝜀/(2−𝜀)}(𝑡)−𝐑¹₂𝐱₂−𝐑²₂sig𝐱₂^𝜀+^𝑛_𝑗=1𝑟²_1𝑗sig𝑥_1𝑗^{𝜀/(2−𝜀)}𝑥_2𝑗≤−𝐑¹₂𝐱^𝑇₂𝐱₂−^𝑛_𝑗=1𝑟²_2𝑗𝑥_2𝑗^1+𝜀<0. (40)
Under the bounded initial values 𝐱₁(0) and 𝐱₂(0), the solution of the system (37) can be calculated by [18]. In addition, from the solution of (37) and (40) and LaSalle’s Invariant Principle, we conclude that the system (37) is asymptotically stable.
In order to further prove that 𝐱₁=0, 𝐱₂=0 is the global unique equilibrium point of (37); it is needed to use Theorem 3.2 of [19]. And we can obtain that the system (37) uniformly converges to the origin in a limited time.
In order to extract the virtual control derivative signal without tedious calculation in procedure of the backstepping attitude controller design, a rapid-convergent nonlinear differentiator (RCND) is designed as follows, according to Lemma 9and Theorem 1 of [20]: 𝐳̇₁=𝐳₂,𝜅²𝐳̇₂=−𝐑¹₁𝐳₁−𝜈−𝐑²₁sig𝐳₁−𝜈^{𝜀/(2−𝜀)}−𝐑¹₂𝜅𝐳₂−𝐑²₂sig𝜅𝐳₂^𝜀, (41) where 𝐳₁,𝐳₂∈𝑅^𝑛 are the states of the differentiator, 𝜈∈𝑅^𝑛 is a known vector field, and it has almost continuous second derivative everywhere. 𝜅=diag{𝜅₁,…,𝜅_𝑛}>0, 0<𝜀<1 are, respectively, the perturbation parameter and constant to be designed.
It is crucial to stress that, according to Lemma 9 and (41), we conclude that ‖𝐳₁−𝜈‖→0, ‖𝐳₂−𝜈̇‖→0 come into existence in a limited time.

Remark 10. The chattering phenomenon of the differential estimation of the proposed RCDN caused by the symbol function is fully weakened by introducing the perturbation parameter. At the same time, the dynamic performance of differentiator is improved by introducing the linear part. Furthermore, because of the combination of linear differentiator and nonlinear differentiator, the proposed RCDN has the characteristics of strong robustness and rapid convergence.

4.2.2. Attitude Controller Design

To give a clear idea of the controller design procedure, the following steps are given.

Step 1. The task of this step is to design an ideal virtual angular rate command under the attitude angle loop (8) existing disturbance 𝐃_𝜉.
Consider the attitude angle loop dynamics (8) and define the attitude error as follows: 𝐙_𝜉=𝜉−𝜉_𝑐, (42) where 𝜉_𝑐=(𝜙_𝑐,𝜃_𝑐,𝜓_𝑐)^𝑇 are the desired attitude angles which are computed by (31).
Based on (8), the derivative of 𝐙_𝜉 is given by 𝐙̇_𝜉=𝐖𝐙_Ω+Ω^𝑑+𝐃_𝜉−𝜉̇_𝑐, (43) where 𝐙_Ω=Ω−Ω^𝑑 is the angular rate tracking error, Ω^𝑑 is the ideal angular rate command, and it will be defined later.
We design the virtual angular rate command Ω^𝑑 as Ω^𝑑=𝑊⁻¹−𝐂_𝜉𝐙_𝜉−𝐃_𝜉+𝑟_𝜉+𝜉̇_𝑐, (44) where 𝐂_𝜉 is a diagonal positive definite matrix to be designed, 𝐃_𝜉 is the reconstruction value of the unknown disturbance 𝐃_𝜉, 𝑟_𝜉 is a robust controller used to reduce the effect of reconstruction error on the control performance, and its structure is given by 𝑟_𝜉=−𝜏_𝜉𝐙_𝜉, (45) where 𝜏_𝜉 is a positive constant to be designed.
We choose the Lyapunov function 𝑉₁=(1/2)𝐙^𝑇_𝜉𝐙_𝜉. Then, based on (43), (44), and (45), the time derivative of 𝑉₁ is given by 𝑉̇₁=𝐙^𝑇_𝜉𝐖𝐙_Ω+Ω^𝑑+𝐃_𝜉−𝜉̇_𝑐=𝐙^𝑇_𝜉−𝐂_𝜉𝐙_𝜉+𝐖𝐙_Ω+𝐃_𝜉−𝐃_𝜉+𝑟_𝜉≤𝐶⁻_𝜉𝐙_𝜉²+𝐙^𝑇_𝜉𝐖𝐙_Ω+𝐞_𝐃𝜉²4𝜏_𝜉, (46) where 𝐶⁻_𝜉=𝜆_min(𝐂_𝜉) is the smallest eigenvalue of 𝐂_𝜉, 𝐞_𝐃𝜉=𝐃_𝜉−𝐃_𝜉 is the reconstruction error of the disturbance of the attitude angle loop, 𝐙^𝑇_𝜉𝐖𝐙_Ω is a coupling term, and it will be disposed in the next step.
Thus, from (46), we conclude that attitude angles 𝜉 can track the desired attitude angles 𝜉_𝑐 when ‖𝐙_Ω‖→0 and ‖𝐙_𝜉‖ are bounded.

Step 2. The task of this step is to design control torque 𝜏^𝑏 and achieve Ω asymptotical tracking Ω^𝑑 under the angular rate loop (9) existing composite disturbance 𝐃_Ω. At the same time, consider that the differential of the ideal virtual angular rate command can not be measured.
Taking the time derivative of the angular rate tracking error 𝐙_Ω=Ω−Ω^𝑑 and substituting (9) into it, yields that 𝐙̇_Ω=𝐉⁻¹(−Ω×𝐉Ω)−𝐉⁻¹𝐆_𝑎+𝐉⁻¹𝜏^𝑏+𝐃_Ω−Ω̇^𝑑. (47)
In order to achieve 𝐙_Ω stability and eliminate the coupling term of (46), the control torques are designed as 𝜏^𝑏=𝐉−𝐂_Ω𝐙_Ω−𝐖^𝑇𝐙_𝜉−𝐉⁻¹(−Ω×𝐉Ω)+𝐉⁻¹𝐆_𝑎+Ω̇^𝑑−𝐃_Ω+𝑟_Ω, (48) where 𝐂_Ω is a positive definite parameter matrix to be designed, 𝐃_Ω is the reconstruction value of the unknown composite disturbance 𝐃_Ω, and Ω̇^𝑑 and Ω^𝑑 are, respectively, the angular rate differential estimation vector (the output) and the input of the RCDN. Similar to 𝑟_𝜉, 𝑟_Ω is also a robust controller, and its structure is given by 𝑟_Ω=−𝜏_Ω𝐙_Ω, (49) where 𝜏_Ω is a positive constant to be designed.
Aiming at (47) and considering (43), we choose the candidate compound Lyapunov function 𝑉₂=𝑉₁+(1/2)𝐙^𝑇_Ω𝐙_Ω. Then, taking the time derivative of 𝑉₂, and according to the (46)–(49) and Young inequality, we have 𝑉̇₂=𝑉̇₁+𝐙^𝑇_Ω𝐉⁻¹(−Ω×𝐉Ω)−𝐉⁻¹𝐆_𝑎+𝐉⁻¹𝜏_𝑎+𝐃_Ω−Ω̇^𝑑=𝑉̇₁+𝐙^𝑇_Ω−𝐂_Ω𝐙_Ω−𝐖^𝑇𝐙_𝜉+Ω̇^𝑑−Ω̇^𝑑+𝐃_Ω−𝐃_Ω+𝑟_Ω≤_𝑖=𝜉,Ω𝐶⁻_𝑖𝐙_𝑖²+𝐞_𝐃𝑖²2𝜏_𝑖+𝐞_𝑓²2𝜏_Ω, (50) where 𝐶⁻_Ω=𝜆_min(𝐂_Ω) is the smallest eigenvalue of 𝐂_Ω, 𝐞_𝐃Ω=𝐃_Ω−𝐃_Ω is the reconstruction error of the disturbance of the angular rate loop, and 𝐞_𝑓=Ω̇^𝑑−Ω̇^𝑑 is the estimation error of the RCDN.

4.2.3. Stability Analysis

Before proceeding with the stability analysis of this section, we recall a useful existing lemma presented in [21].

Lemma 11. If there exist constants 𝛼₁, 𝛼₂, 𝛼₃, 𝛾₁∈𝐾_∞, 𝜍₁>0 and a smooth function 𝑉: 𝑅^𝑛→𝑅⁺ such that the arbitrary bounded input 𝑢 satisfy 𝛼₁(‖𝑥‖)≤𝑉(𝑥)≤𝛼₂𝑥,𝑉̇(𝑥)(‖‖)≤−𝛼₃(‖𝑥‖)+𝛾₁‖𝑢‖_[0,𝑡)+𝜍₁. (51) Then, the system 𝐱̇=𝑓(𝐱,𝐮) is input to state practically stability (ISpS) in the interval [0,𝑡).

Theorem 12. Given the rotational dynamics of the six-rotor UAV as described in (8) and (9) with a control strategy shown in Figure 3 and the SOSMDO given by (15), the attitude angle loop controller given by (44) and the angular rate loop controller given by (48), the tracking errors 𝐙_𝜉 and 𝐙_Ω are input to state practical stability (ISpS) with respect to 𝐞_𝐃𝜉, 𝐞_𝐃Ω, and 𝐞_𝑓 in the interval [0,𝑡). This means that there exist a 𝐾𝐿 function 𝛽₁, a class 𝐾_∞ function 𝛼₁, 𝛾₁, and 𝛾₂ such that 𝛼₁𝐙_𝑖≤𝛽₁𝐙_𝑖𝑡₀,𝑡+𝛾₁𝑒_{𝐷[𝑡0,𝑡)}+𝛾₂𝑒_𝑓²_[𝑡0,𝑡)𝑡≥𝑡₀, (52) where ‖𝑒_𝐷‖=max_𝑖=𝜉,Ω(‖𝑒_𝐷𝑖‖²).

Proof. According to (50), we have 𝑉̇₂≤−Υ𝑉₂+𝜍₁𝑒_𝐷+𝜍₂𝑒_𝑓², (53) where Υ=min_𝑖=𝜉,Ω(𝐶⁻_𝑖), 𝜍₁=max_𝑖=𝜉,Ω(1/2𝜆_𝑖), 𝜍₂=1/2𝜆_Ω.
From Theorem 6, it is known that there exists a constant 𝛿>0 which makes ‖𝑒_𝐷‖≤𝛿 established. Therefore, we can get the conclusion that the Lyapunov fuction 𝑉₂ satisfies the Lemma 11. And thus, the validity of Theorem 8 is proved.

5. Simulation Results and Discussion

In this section, in order to verify the validity and efficiency of the technique proposed in this paper, simulations of two typical trajectory tracking tasks are performed on Matlab/Simulink.

The six-rotor UAV model parameters are taken as listed in Table 1.

Table 1 

Six-rotor UAV model parameters.

In view of the limitations of real measuring device, the sampling time is fixed to Δ𝑡=0.02 s. In the simulation, the initial positions and attitude angles are 𝐩₀=(0,0,0)^𝑇, 𝜉₀=(0,0,0)^𝑇, respectively, and so are linear and angular velocities, respectively. And we choose the composite disturbance input 𝐃_Ω(𝑡)=[0.5cos(𝑟),0.5sin(0.4𝑡),0.5⋅cos(𝑝)+0.3]^𝑇.

Experiment 1. The first task is to track a vertical helix trajectory which is defined as 𝑥_𝑐(),𝑦=0.5cos0.5𝑡_𝑐(),𝑧=0.5sin0.5𝑡_𝑐=1+0.1𝑡,𝜓_𝑐=0.1rad. (54)
The position controller and attitude controller parameters in simulations are fixed at 𝐊_𝑟=diag{0.5,0.5,0.5}, 𝐊_𝑖=diag{0.2,0.2,0.2}, Λ=diag{1,1,1}, 𝐂_𝜉=diag{2,2,2}, 𝐂_Ω=diag{5,5,5}, 𝜏_𝜉=15, 𝜏_Ω=45. The SOSMDO parameter is chosen as Γ_Ω=diag{1.2,1.0,1.5}, and the RCND parameters are chosen as 𝜅=diag{0.01,0.15,0.4}, 𝐑¹₁=4𝐈_3×3, 𝐑²₁=1.5𝐈_3×3, 𝐑¹₂=𝐑²₂=0.5𝐈_3×3.

Experiment 2. The second task is to track a horizontal rectangle trajectory given by 𝑥_𝑐()()()()(),𝑦=0.8𝑡−5fsg𝑡,5,10+4fsg𝑡,10,15+0.820−𝑡fsg𝑡,15,20_𝑐()()()()(),𝑧=0.6𝑡−10fsg𝑡,10,15+3fsg𝑡,15,20+0.625−𝑡fsg𝑡,20,25_𝑐=0.6𝑡fsg(𝑡,0,5)+3fsg(𝑡,5,30),𝜓_𝑐=0.1rad, (55) where fsg is an interval function and is expressed as fsg(𝑥,𝑎,𝑏)=(sign(𝑥−𝑎)+sign(𝑏−𝑥))/2.
In Experiment 2, the position controller parameters, attitude controller parameters, SOSMDO parameter, and the RCND parameters are chosen as same as Experiment 1.
Simulation results are presented in Figures 4, 5, 6, 7, 8, 9, and 10 which depict the 3D flight trajectory, the trajectory tracking errors of 𝑥, 𝑦, 𝑧 orientation, the response of the attitude angles, control thrust and torques, disturbance reconstruction values and estimation values of the angular rate derivative, respectively.

(a)  Vertical helix flight

(b) Horizontal rectangle flight

Figure 5 

Trajectory tracking errors of 𝑥, 𝑦, 𝑧 orientation.

(a)  Vertical helix flight

(b) Horizontal rectangle flight

Figure 6 

Attitude angles response (solid line) and the desired attitude angles for tracking the different reference trajectory (dashed line).

(a)   Vertical helix flight

(b) Horizontal rectangle flight

Figure 7 

Control thrust.

(a)  Vertical helix flight

(b) Horizontal rectangle flight

Figure 8 

Control torques.

(a)  Vertical helix flight

(b) Horizontal rectangle flight

Figure 9 

Disturbance reconstruction values (solid line) and the real disturbance (dashed line).

(a)  Vertical helix flight

(b) Horizontal rectangle flight

Figure 10 

Angular rate derivative estimation values using the proposed method in this paper (solid line) and the method in [13] (dashed line).

Figures 4 and 6 depict the performance of trajectory tracking and the attitude angles response, respectively. They verify the proposed control strategy which has the same efficiency to track different trajectory types. In addition to this, as expected, the proposed trajectory tracking control strategy is capable of making the six-rotor UAV with aerodynamic moments, external disturbance and parameter uncertainty input track the desired trajectory in a satisfactory way, seeing Figures 4 and 5 and guaranteeing the attitude angles tracking the desired attitude angles in an appropriate scope, and seeing Figure 6. At the same time, from Figure 5, we can evidently see that the maximal trajectory tracking error does not exceed 0.2 meter which is allowed in the actual flight. So, the actual flight trajectory with disturbance compensation closely follows the desired trajectory even under the composite disturbances. So, it is obvious that the control law performs better with the disturbance reconstruction and compensation under the composite disturbances inputs.

From Figure 9, we can get the conclusion that the proposed SOSMDO can reconstruct the disturbance accurately. Thus, the correctness and effectiveness of the proposed SOSMDO are verified through this result.

Moreover, compared with the method of [13], the proposed RCND in this paper has the characteristics of faster convergence and stronger robustness which can be demonstrated by Figure 10.

6. Conclusions

(1)Aiming at the potential inadequacies, such as smaller payload capacity, lack of hardware redundancy, and anticrosswind capability of the quad-rotor, a novel six-rotor UAV and its mathematical model are developed in this paper. An SOSMDO is also proposed to reconstruct the disturbance of the rotational dynamics. The correctness and effectiveness of the proposed SOSMDO are verified by the simulation results.(2)The proposed trajectory tracking control strategy is capable of making the six-rotor UAV track the desired trajectory in a satisfactory way. The maximal trajectory tracking error does not exceed 0.2 meter, which is allowed in the actual flight. And the results also show that this control strategy can make the six-rotor UAV track the desired attitude angles very well.(3)Compared with the approach of [13], the proposed RCND in this paper has the characteristics of faster convergence and stronger robustness.(4)Based on the proposed control law, how to design the flight controller for the six-rotor UAV and achieve the objective of practical flight is our future works.

Acknowledgments

This study was cosupported by Specialized Researched Fund for the Doctoral Program of Higher Education of China (no. 20113218110013), Funding of Jiangsu Innovation Program for Graduate Education (no. CXLX11_0200), the National Natural Science Foundation of China (no. 60674100), the Industry-academic Joint Technological Innovations Fund Project of Jiangsu Province, China (no. BY2012018), and the Natural Science Foundation of Shandong Province, China (no. ZR2012FQ030).