The modeling and cascade sliding mode control of a moving mass-actuated coaxial dual-rotor UAV

In this study, an attitude control scheme based on a three-track moving mass control mechanism is proposed to address the problems of the overcomplicated rotor components, low service life, and low reliability of coaxial dual-rotor unmanned air vehicles (UAV). The motion and aerodynamic models of a moving mass-actuated ducted coaxial dual-rotor UAV are derived. The rotational dynamic characteristics of a moving mass-actuated UAV (MAUAV) with different slider positions and mass ratios are analyzed. An attitude controller based on backstepping sliding mode control is designed to address the nonlinearity and uncertainty of the MAUAV rotation. Based on this, we developed a position controller using cascade sliding mode control. The simulation results demonstrate that the designed attitude controller can achieve a settling time of 1.438 s in the unit-step response and a steady-state error of less than 5% in the sinusoidal attitude-tracking experiment. Additionally, the designed position controller exhibited a better trajectory-tracking effect under different levels of gust disturbance than that of a linear quadratic regulator control-based position controller.


Introduction
A coaxial dual-rotor unmanned air vehicle (UAV) is an aircraft that realizes yaw control by balancing the torque difference between its upper and lower rotors. 1It is widely used in agriculture, forestry, multimedia, traffic control, transportation, and other civil and military fields owing to its strong maneuverability, high flexibility, and ability to vertically take off and land in a narrow space without requiring dedicated runways or ejection frames. 2 In particular, the flight characteristics of fixed-point hovering and low-speed flights make coaxial dual-rotor UAVs advantageous for high-precision query tasks. 3,4urrently, the traditional coaxial dual-rotor aircraft, which relies on the cyclic pitch of the rotor, changes the lift distribution on the propeller-disk plane to alter the resultant force and moment on the fuselage. 5However, this control method is somewhat limited.Complicated variable-pitch mechanisms and rotors with opposite rotation directions cause more severe mechanical wear, which results in a shorter service life, more frequent maintenance, and consequently, higher maintenance costs. 6Moreover, for a system in which only the lower rotor can pitch periodically, the lower rotor becomes susceptible to interference from the wash flow of the upper rotor. 7This results in a lower forward thrust, poorer performance, and shorter flight duration.
In summary, the cyclic pitch of the rotor restricts the existing coaxial dual-rotor UAVs from further improving their service life, flight endurance, and performance.Therefore, a new control scheme is required to effectively address these problems.
Moving mass control (MMC) solves this problem by providing an attitude control method that relies on changing the centroid position. 8This method was first applied to the attitude control of reentry vehicles. 9Subsequently, it has become widely used for the attitude control of spacecraft, fixed-wing aircraft, and multi-rotor aircraft.Li et al. 10 conducted a comprehensive review of the development and applications of MMC.Its mechanism can be placed completely inside an aircraft and is independent of the aerodynamic force, thereby allowing it to reduce air resistance on the fuselage and eventually improve flight duration.Erturk et al. 11 applied MMC to the attitude control of a fixed-wing UAV, they removed the steering rudders on the wings, thereby making the attitude control independent of the aerodynamic force.Haus et al. 12,13 used MMC to replace the rotor rotational speed control to mitigate the difficult attitude control of fuelpowered quadrotors, thereby proving the feasibility of MMC for rotorcrafts.Darvishpoor et al. used the linear quadratic regulator (LQR) to control the pitch and roll channels of a coaxial dual-rotor UAV with a moving mass system. 14They completed an 8-shaped trajectory tracking, which proved that the MMC had sufficient rotorcraft maneuverability.
This study proposes an attitude control scheme based on a three-track moving mass control mechanism.The main contributions of this study are as follows: (1) A three-rail MMC mechanism is designed with a simpler structure, less space occupation, and lighter material weight than that of the four-track scheme. 14Using batteries instead of counterweights as sliders in the control mechanism can increase UAV endurance.An external ducted shell is designed to suppress the tip vortex and reduce the tip energy loss of the rotor system.In addition, it can avoid damage caused by severe collisions and improve safety during debugging.(2) We prove the uncertain nonlinearity of the MAUAV system by analyzing the influence of the slider position and mass ratio on body motion.(3) An attitude controller based on the backstepping sliding mode control (BSMC) was designed by exploiting the sliding mode control advantages of fast response and strong robustness in an uncertain nonlinear system. 15On this basis, we develop a position controller using the cascade sliding mode control (CSMC).In the trajectory-tracking simulation experiment, we prove that the position controller using the CSMC exhibits less error and a stronger anti-wind disturbance than those of the position controller using the LQR under different gustdisturbance levels.

Description of the controlled object
As shown in Figure 1, the moving mass control-based ducted coaxial dual-rotor UAV proposed in this study uses a ducted shell on the outside to increase energy efficiency and protect the internal rotor system.As shown in Figure 2, the three belt-roller structures inside the duct form the MMC mechanism of the UAV.Three batteries with spring wires are fixed to the three belts, and the rollers are driven by servo motors.The MMC mechanism relies on the movement of batteries driven by belts to change the center of gravity (CoG) of the UAV, and their movement directions are not in the same plane.This ensures that the unique positions of batteries are determined by the CoG of the UAV.There are two rotors near the duct entrance and exit, which are coaxial and rotate in opposite directions.The movement of the battery sliders changes the CoG of the UAV, thereby altering the resultant moment of lift and gravity.This allows us to control the pitch and roll angles.The yaw angle can be controlled by adjusting the rotational speeds of the upper and lower rotors and changing the reaction torque of each rotor.

Definition of the coordinate system
To describe the motion of the UAV relative to the ground and the motion of the MMC mechanism, the ground O g À X g Y g Z g and body O b À X b Y b Z b coordinate systems are introduced, as shown in Figure 3.
In the ground coordinate system, we define the origin as a point on the ground, O g X g as the axis of the horizontal plane pointing east, O g Y g as the axis of the horizontal plane pointing north, and O g Z g as the vertical upward axis.In UAV research, the ground coordinate system is typically regarded as an inertial system.In the body coordinate system, the origin is defined as the centroid position of the  UAV without sliders, and the initial direction of the coordinate axis is parallel to the ground coordinate system.
The transformation relationship between the body and ground coordinate systems can be described by the pitch f, roll θ, and yaw ψ angles.The transformation matrix C g b ðf, θ, ψÞ from the body to the ground coordinate system can be obtained by rotating f, θ, and ψ around the O b Z b , O b Y b , and O b X b axes of the body coordinate system, respectively, from the initial state: cosψ Àsinψ 0 sinψ cosψ 0 0 0 1 C y ðθÞ ¼ (3) The flight speed of the UAV in the ground coordinate system can be expressed as V X , V Y , V Z , and its three-axis components in the body coordinate system can be expressed as u, v, w.The conversion matrix in (1) can then be used to establish the following relationship: The three-axis components of the UAV rotational angular velocity in the body coordinate system are represented by p, q, r.Its relationship with the Euler angle velocity can be expressed as follows: Formula ( 6) can be rewritten as follows: The transformation matrix C ω used to transform the angular velocity in the body coordinate system into the Euler angle velocity can then be expressed as 1 sin f tanθ cos f tanθ 0 cosf Àsinf 0 sin f secθ cos f secθ (8)

Kinematics model
The UAV model is simplified into a system containing only two parts: the MMC mechanism and UAV body.The quality of the MMC mechanism is concentrated on the sliders.The masses of the entire system, MMC mechanism, and UAV body are set as m S , m C , and m B , respectively.The corresponding centroid position vectors in the ground coordinate system are P S , P C , and P B , respectively.According to the definition of the centroid of the particle system, we determine that The mass and position of a single slider are represented by m i C and P i C , then The MMC mechanism mass ratio in the UAV can be expressed as follows: Therefore, the system centroid position can be expressed as follows: Time derivatives of P S , P C , and P B are denoted as V S , V C , and V B , respectively.Based on the theory of the momentum of a particle system, the following relationship exists: Using R C to denote the position vector of P C relative to P B in the ground coordinate system, the following relationship exists: Using V to represent the angular velocity of the UAV in the ground coordinate system, the motion of P C relative to P B can be decomposed into translational motion and fixedaxis rotation as follows: The velocity of P C relative to the ground coordinate system can be expressed as Joining formulas ( 13) and ( 16) results in the following: The moving speeds of P S , P C , and P B in the body coordinate system are represented by v S , v B , and v C , respectively.r C represents the position vector of P C relative to P B in the body coordinate system.ω represents the rotational angular velocity of the UAV in the body coordinate system, then can be expressed as follows:

Dynamic model
In the body coordinate system, we can decompose the MAUAV motion into translational motion and rotation around the centroid.When studying the centroid motion of the MAUAV, we can regard the whole UAV system as a particle system.The motion of the system centroid position P S conforms to the centroid motion law, and the particle system centroid acceleration is only related to the particle system resultant force.When studying the MAUAV rotation around the centroid, we can regard the entire UAV system as a rigid body.The rotation around the centroid conforms to the moment of momentum theorem, and the rotation angular acceleration is only related to the resultant moment of the rigid body.
Translational motion's dynamic model.The resultant force of the UAV in the body coordinate system is expressed as The derivative of v S with respect to time can be written as follows: where Joining formulas ( 20) and ( 21) gives the following: The relationship between the acceleration and resultant force of the UAV in the body coordinate system can be expressed as follows: Dynamic model of rotating around the centroid.A dynamic equation for the rotation of the UAV around the centroid was also established in the body coordinate system.According to the moment of momentum theorem, the UAV angular momentum H and resultant moment M ¼ ½L, M, N T have the following relationship: where H ¼ J Á ω, and J denotes the moment of inertia of the entire UAV system rotating around its centroid P S .The derivative of the angular momentum with respect to time can also be expressed as follows: Joining formulas (26) and (27) gives the following: Rearranging the above formula gives r S ¼ ½x S , y S , z S T denotes the position vector of P S relative to P B in the body coordinate system.r i ¼ ½x i , y i , z i T denotes the position vector of a single slider position P i C relative to P S in the body coordinate system.J 0 B represents the moment of inertia of the UAV body as it rotates around P B .J B represents the moment of inertia of the UAV body as it rotates around P S .J C represents the moment of inertia of the MMC mechanism as it rotates around P S .According to the parallel-axis theorem, we can express J B and J C as where E 3 represents the third-order unit matrix.Then, J, the moment of inertia of the entire UAV system as it rotates around the system centroid P S , can be expressed as follows: The derivative of J with respect to time can be expressed as Force and moment Gravity.With its constant downward vertical direction, gravity, G, continuously affects the MAUAV during its flight.Thus, we can write According to the conversion relationship between the ground and body coordinate systems, gravity can be expressed as follows: Because gravity always passes the center of mass of the entire UAV system, the moment of gravity is constantly zero.
Rotor lift.This study provides a method for modeling a ducted rotor system.This method combines the momentum and blade element theories to establish an aerodynamic model of a ducted coaxial dual-rotor UAV.The number of calculations was small, and the accuracy was high.
According to the blade element momentum theory, 16,17 when the rotor system's far-section upstream velocity is V 0 and the rotor speed is V u , the rotor induction velocity at the position r from the spindle is v u .However, the duct increases the induction speed of the upper rotor to a certain extent; 18,19 thus, the influence coefficient η was used.The induction speed v u can be expressed as follows: The existence of the duct made the flow-tube contraction less obvious, thereby causing the lower rotor to be in the washing flow of the upper rotor.When the induction velocity of the upper rotor is v u , the distribution of the lower rotor induction velocity v l can be expressed as follows: where R denotes the blade radius, σ is the blade solidity, a S denotes the slope of the airfoil lift curve, and θ i is the local installation angle of the rotor blade.Here, i can be u or l, which indicates that it is located on a different rotor.f i p is the Prandtl factor, which is used to correct the blade tip energy loss.The calculation process is as follows: where φ i ¼ ðV 0 þ v i Þ=rV i is the inflow angle under the assumption of having a small angle.
The above process for calculating the induction velocity requires the initial value of v i0 .The induction velocity at each moment is calculated iteratively.The initial value can be determined using momentum theory. 16,17The rotor lift was equal to the momentum-change rate of the fluid in the slipstream model.Thus where ρ is the air density.The values of T u and T l can be measured using a ducted coaxial dual-rotor lift test bench.
In the suspended state, V 0 ¼ 0. According to blade element theory, 16,17 the lift of the blade element is only affected by its airfoil, incoming flow velocity V, and incoming flow attack angle α i ¼ θ i À φ i .When the rotor chord length is c and the drag coefficient is C d , the lift dL and drag dD acting on the blade element section of the rotor can be expressed as follows: The incoming flow velocity V can be expressed as If the inflow angle of the blade element section is φ i , and the number of blades is B, the differential forms of the blade element lift F i rotor and torque Q i rotor in the body coordinate system can be expressed as follows: The lift F i rotor and torque Q i rotor provided by a single rotor can be obtained by the following integral: The lift F rotor and torque Q rotor provided by the ducted coaxial dual-rotor system are expressed as follows: Resultant force and moment.Because the body of the MAUAV is small, and the rotor uses light materials, we can ignore the body air resistance and the gyroscopic effect of the rotor during flight.Accordingly, the UAV centroid acceleration becomes mainly determined by F b .The angular acceleration of the UAVaround the centroid is primarily determined using M. The rotor lift is parallel to O b Z b in the body coordinate system O b À X b Y b Z b and passes through the origin O b .Therefore, we determined the lift moment using the UAV centroid position r S and rotor lift F rotor .Thus

Dynamic characteristics
For the MAUAV attitude control system, the characteristics of the aircraft itself, the output of the control mechanism, and the various characteristics of the controlled quantity have important reference significance in the control system design.The changes in M, J, and _ ω are used to reflect the dynamic characteristics of the MAUAV.Additionally, we analyzed the influence of different slider positions r i C ¼ ½x i C , y i C , z i C and mass ratio μ C values on the motion of the UAV.

Moment characteristics
When the UAV adjusts the attitude of the pitch and roll channels, the resultant moment that causes it to rotate around the centroid is as follows: where r S satisfies the following relationship: Joining formulas (46) and (47) gives the following: where Therefore, in the UAV attitude adjustment, M is related to μ C , Àr C , and F rotor .

Moment of inertia characteristics
Considering the rotation around the O b X b axis of the body coordinate system as an example, according to (32), the moment of inertia of the UAV rotation around the O b X b axis can be expressed as follows: We can infer that the moment of inertia rotates around a particular axis of the body coordinate system and can be expressed as a function of μ C and r C .Therefore, this moment of inertia continues to vary during the attitude adjustment, which demonstrates the uncertainty of the MAUAV's attitude control.As shown in Figure 4, as μ C and y C increase, the rate of J X gradually accelerates.

The BSMC-based attitude controller
Assuming that the mass distribution of the UAV body is symmetrical and that this symmetry change is small when the MMC mechanism works, that is, J XY ¼ 0, the angular acceleration of the pitch, roll, and yaw channels in (29) can be simplified to Because the moment of inertia J continuously varies during attitude adjustment, the angular acceleration described in (51) can be linearized to an uncertain system: Rearranging the above gives where F represents the total uncertainty, and jFj ≤ F. ΔA and ΔB represent uncertain parts of the system parameters.
dðtÞ represents the interaction between the angular velocities, as shown in the third term of equation ( 54).We can express the total uncertainty in (51) as follows: where J X ð0Þ, J Y ð0Þ, and J Z ð0Þ denote the values of J X , J Y , and J Z , respectively, when r C ¼ 0. For the MAUAV attitude control system, , and F r ¼ maxjF r j.
BSMC 20 can be used for the system in (53).The tracking error z 1 , virtual control term z 2 , and switching function ρ can be defined as follows: where c > 0, and k > 0.
We define the Lyapunov function as follows: After derivation, we obtain The controller can be designed as follows: where h > 0, and β > 0.
We substitute (58) into (57) to obtain the following: where The values of h, c, and k are then adjusted such that jPj > 0 to ensure that _ V ≤0.We can express the attitude controller output as the expected moment, which can be expressed on the pitch and roll channels as follows: The expected moment on the yaw channel can be expressed as By substituting (61) into (48), the desired centroid position of the MMC mechanism is obtained as follows: : The slider movement model of the MMC mechanism is relatively simple, and b r C tracking can be realized using a cascade proportional integral derivative (PID) control.
By substituting b N into (62) and the expected rotor lift b F rotor into (44), the constraint conditions of the expected rotational speeds b ω u and b ω l can be obtained as follows: The attitude controller design described above was derived from the simplified model in (51).J XY cannot be constantly zero in a simulation experiment or in actual attitude control.However, owing to the robustness of the sliding mode control, the parameter F represents the total uncertainty in (53) and can be appropriately increased to resist the interference of each axial control moment caused by J XY ð0Þ ≠ 0 with the angular acceleration of the other axes.

CSMC-based position controller
The acceleration of the MAUAV in the ground coordinate system can be described as a second-order linear system: where x ¼ ½x 1 , x 2 T denotes the position and velocity of the MAUAV in the ground coordinate system.B ¼ 1=m S , and m S is the quality of the MAUAV system.Δ is the deviation between the body and system centroid accelerations caused by the operation of the MMC mechanism, and jΔj ≤ D.
For systems in (65), the control rate can be designed as follows: where e ¼ x 1 À x d , c > 0, and k > 0.
For the Lyapunov function, such as V ¼ s 2 =2, we obtain (67) The output of the position controller is expressed as the expected force in the ground coordinate system: where X , Y , and Z represent the position of the MAUAV in the ground coordinate system.The desired attitude angle of the MAUAV can be expressed as Moreover, the desired rotor lift in the height control can be expressed as The calculation results of (69) and (70) can be used as inputs for (61) and (64), thereby forming the CSMC.

Simulation experiment and analysis
The effectiveness of the controller on the uncertain nonlinear system of the MAUAV was verified through attitude-tracking and trajectory-tracking simulation experiments conducted on the designed coaxial dualrotor MAUAV attitude and position controllers.We conducted a comparative analysis of the effects of the proposed CSMC and those of the LQR.Moreover, we recorded the trajectory of the centroid of the MMC mechanism in the body coordinate system to verify whether the designed MMC mechanism could completely control the fuselage attitude without touching the maximum stroke of the sliders.In the trajectorytracking experiment, the system was disturbed by Asinð2tÞ signal to simulate gusts. 21The maximum wind pressure was 72.9 NÁm À2 , which was equivalent to a strong breeze.The overall MAUAV parameters are listed in Table 1.The maximum stroke of the sliders was 0.25 m; thus, the maximum value of kr C k 2 was 0.0833 m, and the maximum sliding speed was 0.7 mÁs À1 .

Attitude tracking
Unit-step response experiment.In the unit-step response experiment for the attitude angle, we compared the controller output when the sgn and tanh functions were used as the switching functions.The sliding mode switching term using the sgn function causes jitter in the control output when approaching a sliding surface. 20To eliminate this effect, the sgn function in ( 61) and ( 62) can be replaced with the fuzzy switching function, which is designed as follows: The membership degree of input and output is shown in Figure 6.
The Mamdani algorithm and area center method are used to obtain a clear output ∂k i .The approximation coefficient is set as τ, and the approximation result of the switching gain can then be expressed as follows: The controller unit-step response results for the two functions in the pitch channel are shown in Figures 7  and 8, respectively.We observe that the control effects of the two functions are similar (settling time t sgn =1.274 s, t tanh =1.438 s); however, the controller output y C corresponding to the fuzzy switching function does not cause jitter.In the subsequent   simulation experiments, the controller sliding mode switching term uses a fuzzy switching function.
Sinusoidal attitude tracking.Considering that the change in angular acceleration described in (51) can be influenced by the variation of the angular velocities of the two other axes and the moment of inertia, Figure 9 shows the attitude angle curves and attitude controller errors when tracking f d ¼ 0:61sinðt þ π=3Þ, θ d ¼ 0:61sint, and ψ d ¼ 0:61sinðt À π=3Þ simultaneously.Additionally, 0.61 rad is approximately 35°.From Figure 8, we can observe that the designed attitude controller can achieve a steady-state error of less than 5% (in the steady-state process, maxðe f Þ ¼ 0:0156, maxðe θ Þ ¼ 0:0223, and maxðe ψ Þ ¼ 0:0156) in the case of mutual interference between the angular velocities of each axis.
Figure 10 shows the centroid trajectory of the MMC mechanism during the attitude-tracking process.We observe that the centroid trajectory eventually changes periodically because the target pose is a periodic function.The duration of the aperiodic process was 1.735 s, which indicated that the designed controller could make the centroid converge to the optimal control position quickly.
Figure 11 shows the position changes of the three sliders on their tracks p 1 , p 2 , and p 3 .Each track is 0.25 m long, and each slider position is at a large distance from both ends of its track, thereby indicating a potential control margin.

Trajectory tracking
In the trajectory-tracking experiment, we compared the control effects of the CSMC and LQR and tested their wind-disturbance resistance abilities.According to Darvishpoor, Roshanian and Tayefi 14 the design of the LQR controller is based on the simplified model of the following equation: Figure 12 shows the MAUAV trajectory when tracking X d ¼ sinð0:25tÞ, Y d ¼ sinð0:5tÞ, and Z d ¼ 0:1 using the CSMC and LQR control.The trajectory of the CSMC trajectory almost coincided with the target trajectory, whereas that of the LQR always deviated from it.As shown in Figure 13, the CSMC can reduce the tracking error to approximately zero, whereas the LQR method cannot effectively eliminate this error.
Figure 14 shows the changes in the positions of the sliders under the two control algorithms during trajectory tracking.When using the CSMC, the slider positions can converge to the optimal control position faster, which indicates that the CSMC algorithm can easily achieve stability in actual control, and it consumes less power.
We applied a perturbation signal with a form of Asinð2tÞ to the system to simulate gusts of wind.The   value of A refers to the wind pressures of a moderate breeze, fresh breeze, and strong breeze.Figure 15 shows the trajectory-tracking curves using the CSMC and LQR at different gust levels, and the error curves and average errors relative to the target trajectory are shown in Figure 16.Under the same gust disturbance, the average trajectory-tracking error using the CSMC was always less than that using the LQR control.
To simulate the irregular external force or sensornoise interference in a real flight process, Gaussian white noise with a standard deviation varying from 0.    MAUAV is always within an acceptable range, and the MMC mechanism can operate well with the white-noise interference, with a standard deviation of 3. Thus, the anti-interference capability of the CSMC reached the expected goal.

Conclusion
In this study, a control scheme with an MMC mechanism is proposed for the attitude control of a coaxial dual-rotor UAV, and it achieves the following preliminary results: (1) A three-rail MMC mechanism with a simpler structure, less space occupation, and lighter material weight than that of a four-track scheme was designed.Using batteries instead of counterweights as sliders in the control mechanism increased UAV endurance.
(2) We analyzed the rotational dynamic characteristics of the MAUAV for different slider positions and mass ratios.In addition, the uncertainty and nonlinearity of the MAUAV attitude control were proved.(3) We designed a BSMC-based attitude controller based on the rotational dynamic characteristics of the MAUAV.Furthermore, a position controller using a CSMC was developed.The simulation results demonstrated that the designed attitude controller could achieve a settling time of 1.438 s in the unitstep response and a steady-state error of less than 5% in sinusoidal attitude tracking.(4) The trajectory-tracking experiment showed that CSMC enabled the trajectory-tracking error of the UAV to be close to zero under a no-interference condition.Additionally, the designed position controller had a smaller average error under different levels of gust disturbances than that of the LQRbased position controller.
However, some issues still require further study.Although the CSMC algorithm exhibits superior control effects and the fuzzy switching function effectively eliminates the jitter of control output, the state trajectory of CSMC still needs to cross the sliding mode surface frequently, resulting in the state trajectory of the sliding mode control is not optimal.Therefore, in future, we will study the optimal state trajectory of MAUAV based on deep reinforcement learning.

Figure 3 .
Figure 3. Ground and body coordinate systems.

Figure 5
Figure 5 shows the angular acceleration of the UAV corresponding to different μ C and y C values in the hovering state.As shown in the figure, _ p increases with increasing μ C and y C .The increase rate of _ p gradually accelerated with an increase in y C , but slowed down with an increase in μ C .The angular acceleration response characteristics demonstrate the nonlinearity of the attitude control of the MAUAV.

Figure 4 .
Figure 4. Distribution contour of the body's moment of inertia about x-axis J X at different slider mass ratios μ C and coordinates of the centroid on the y-axis y C .

Figure 5 .
Figure 5. Distribution contour of the angular acceleration about x-axis _ p at different slider mass ratios μ C and coordinates of the centroid on the y-axis y C .

Figure 7 .
Figure 7. Controller output using the sgn switch function.

Figure 6 .
Figure 6.Membership degree of input and output.

Figure 9 .
Figure 9. Attitude angle curves and the attitude controller errors.

Figure 8 .
Figure 8. Controller output using the fuzzy switching function.

Figure 11 .
Figure 11.Position changes of the three sliders on their tracks.

Figure 14 .
Figure 14.The MMC mechanism centroid trajectory during trajectory tracking.

Figure 12 .
Figure 12.Comparison of the two controller trajectorytracking results.

Figure 13 .
Figure 13.Error curves relative to target trajectory of the two controllers.

6 to 3
was applied to the model when tracking the step signals.The tracking results and slider movements of the MMC mechanism are illustrated in Figure 17.The figure shows that the disturbance displacement of the

Figure 16 .
Figure 16.Error curves and average errors relative to target trajectory.

Figure 17 .
Figure 17.When tracking the step signal, white noise with standard deviation of 0.6-3 was applied to the model.

Figure 15 .
Figure 15.Trajectory-tracking curves of two controllers at different levels of gusts.

Table 1 .
Overall parameters of the MAUAV.