A corridor-based flight mode transition strategy for agile ducted-fan tail-sitter UAV: Altitude-hold transition

Zihuan CHENG, Hailong PEI

School of Automation, South China University of Technology, Guangzhou 510640, China

KEYWORDS

Abstract As an attractive transition approach, the altitude-hold transition is a special type of super-maneuvering and the vertical/horizontal flight mode transition that an agile aircraft conducts at fixed altitude.However, it is still challenging to implement an autonomous control of the altitude-hold transition while the existing optimal transition planning methods cannot avoid an evident altitude change during the transition process.This paper proposes a corridor-based flight mode transition strategy and presents a successful flight demonstration of the altitude-hold transition on a small ducted-fan tail-sitter unmanned aerial vehicle.In the proposed corridor-based methodology,we model and analyze the transition corridor, concentrate on the dynamic characteristics of the altitude-hold transition, and emphasize that a valid transition trajectory should be governed by its transition corridor.The identified transition corridor reveals that for a given velocity trajectory,the solution for the corresponding trajectories of pitch angle and thrust is unique.Based on this,the transition trajectory generation problem is addressed simply on the velocity-acceleration plane.Furthermore, a proper flight control scheme is devised to track the generated transition trajectories.Finally, the effectiveness of the proposed method is verified through practical flight tests, in which the altitude change is less than 1.1 m during the entire transition course.

1.Introduction

The fixed-wing Vertical Take-Off and Landing (VTOL)Unmanned Aerial Vehicle(UAV)has attracted a rapidly growing interest in the late decade.Compared to the traditional rotorcrafts, fixed-wing VTOL UAVs have significant advantages of long endurance and high cruising speed, which makes them more competent in long-range missions such as ocean mapping and monitoring.1Among all the novel VTOL concepts,the tail sitter is a special configuration that the transition between vertical and level flight is conducted through rotating the whole body.Without any extra tilting mechanism such as the tilt-rotor,this configuration is more structurally simple and reliable.However, the traditional tail sitter with open tractor propellers cannot achieve satisfactory control performance under near-hovering conditions.2Alternatively, the ductedfan tail sitter is developed to overcome this inherent defect which replaces the conventional tractor propellers with pusher ducted fans.Integrated with outlet control vanes, the ductedfan propulsion/driven system contributes to excellent agility and control performance at low airspeed and high Angle of Attack (AOA), making ducted-fan tail sitter a promising VTOL type in the next generation.3

Conventionally, a tail sitter conducts the vertical-to-level(or level-to-vertical) flight mode transition through jumping up with an inevitable ascending of altitude,4during which the AOA keeps small and the attitude control effectiveness maintains sufficient.4–7For a ducted-fan tail sitter, attributed to the super-maneuverability provided by the ducted-fan thrust-vectored system, it becomes possible to perform the flight mode transition with a fixed altitude.8In this paper,we use the term altitude-hold transition to describe this special transition course, which is defined as: a flight mode transition during which the attitude of the aircraft shifts from vertical to level (or level to vertical) while the flight direction keeps level and the altitude maintains unchanged.

With the development of advanced control techniques, the altitude-hold transition has received a growth of attention within the UAV community.9Less ascending in height means less energy consumption during landing.Furthermore, since it is more flexible on the conversion between hovering and cruising flight,the altitude-hold transition enables an agile UAV to operate in more cluttered environments.However, the altitude-hold transition is still one of the most challenging courses to execute.Many researchers try to reduce the height change of a transition process through optimal transition planning strategies.Banazadeh and Taymourtash10used a gradient-based algorithm to generate optimal transition trajectories for a thrust-vectored tail sitter.Oosedo et al.11proposed an optimal transition trajectory planning strategy considering two different cost functions.Levin et al.12proposed an optimal transition method for an agile fixed-wing UAV based on motion primitives.Li et al.13presented a successfully practical flight test for the transition of a quadrotor tail sitter.The transition optimization was modeled by nonlinear programming and solved by direct collocation method.Yang et al.14proposed a robust optimal transition control scheme for a traditional tail sitter with two open tractor propellers and control vanes,which was effective in dealing with aerodynamic uncertainties.Hong et al.15proposed a trigonometric series-based method to produce smooth transition trajectories with an arbitrary order of smoothness.Nevertheless, the overall altitude changes in these excellent works are still obvious during the transition.Some others did realize the altitude-hold transition in simulations, such as Kubo,16Maqsood17and Naldi18et al.But the dynamic models they used are,to some extent,simplified, which might not truly reflect the real world.As a matter of fact, successfully practical flight demonstrations of the altitude-hold transition are still rarely found in open publications, especially for autonomous UAVs.

Essentially speaking,the reason why the altitude-hold transition is so difficult to implement is that the transition trajectory cannot be arbitrarily chosen.In our earlier explorations,we found that if the transition trajectory was inappropriately selected, the aircraft would be stabilized into some intermediate regimes(such as 45°of pitch with a low forward speed)that the desired transition course no longer continued.This phenomenon has also been observed and pointed out by Smeur et al.19in their tail-sitter control research.It seems that all the feasible transition trajectories are restricted into some pathways.

This pathway is called the transition corridor.Earlier investigations on the transition corridor or the transition characteristics of VTOL aircraft only focus on the trim conditions of the entire flight envelope, in which the transition process was regarded as a motion around the equilibrium point.20–24However, the transitional flight is actually a dynamic process with large acceleration/deceleration that the operating points of the dynamic system might deviate far away from the equilibrium.In some latest works, researchers tried to extend the existing concept of transition corridor from computing the equilibrium to dynamic analysis.Zhong and Wang25made an elaborate computation and comparison of the static and dynamic characteristics for the transition process of a small tail-sitter UAV.Yang et al.26studied the dynamic transition corridor and proposed a transition planning strategy with successfully practical flight verifications.Yet till now,there are no thorough studies on the transition corridor of the altitude-hold transition.

Another problem is the flight control.Basically, there exist two different methodologies for constructing a transition control scheme.One is to characterize the transition as an attitude shifting process, in which the transition planning module directly generates transition trajectories for both the attitude and the thrust with an attitude tracking control executed by the flight controller.27This approach is reliable and easy to implement but is open-looped in velocity control, which is not suitable for an altitude-hold transition.The other method is to consider the transition as a position/velocity guiding and tracking problem,in which both the reference trajectory generation and tracking control on the attitude channel are accomplished through a full-envelope flight controller.28In our earlier attempt, we also employed a full-envelope flight controller to implement an altitude-hold transition on a ductedfan tail-sitter UAV.8However, the accumulation of velocity tracking error still led to a maximum altitude change of 2.7 m.

Our investigations on the tail-sitter transition subject started from the designs and implementations of fullenvelope flight controllers of different ducted-fan UAVs.29–31In Ref.32, we encountered serious control saturation problem during the backward transition, in which we tackled the transition control problem by equivalently re-arranging the altitude-hold trajectory to an altitude-ascending one.This motivated us to study the transition corridor on finding an admissible trajectory to maintain the altitude.8

This paper is an extension and improvement of our previous work of Ref.8.In this paper, we study the altitude-hold transition of a small ducted-fan tail-sitter UAV ‘‘SHW09”,and propose a corridor-based flight mode transition strategy.In the designed methodology, we first model the six-Degreeof-Freedom (6-DOF) dynamic system of the aircraft.By formulating the longitudinal planar motion and relevant system constraints, we compute the transition corridor of an altitude-hold transition process considering the horizontal velocity, the horizontal acceleration/deceleration, the pitch angle and the thrust.Feasible transition trajectories are subsequently generated within the restriction of the identified transition corridor, consisting of the velocity trajectory, the pitch trajectory and the thrust trajectory.In the meanwhile, we design a proper flight controller which is capable of tracking both the velocity and attitude trajectories.Furthermore, practical flight test results are conducted in order to verify the effectiveness of the proposed method.

The main contributions of this paper are summarized as follows: (A) We present a specific modeling and analysis of the transition corridor of the altitude-hold transition for‘‘SHW09”.Besides the steady flight equilibrium,the transition corridor mainly focuses on the dynamic process with respect to the horizontal acceleration/deceleration.(B) We propose a corridor-based trajectory generation and flight control scheme,ensuring that the entire transition process is governed by the transition corridor.(C) Through the proposed corridor-based flight mode transition strategy, the altitude-hold transition is successfully implemented on the ‘‘SHW09” platform.

The rest of this paper is organized as follows:Section 2 presents a specific modeling of the 6-DOF dynamic system of the aircraft.Section 3 elaborates the modeling,computational process, and analysis of the transition corridor for the altitudehold transition,based on which a transition trajectory generation method is proposed.Section 4 devises a corridor-based flight control scheme and Section 5 exhibits the results of practical flight tests.Section 6 draws the conclusion.

2.Ducted-fan tail-sitter platform and system modeling

As depicted in Fig.1, the ducted-fan tail-sitter platform‘‘SHW09” is configured with one single pusher ducted-fan thrust-vectored propulsion/driven system, comprising one shrouded fan and six outlet control vanes.To control the aircraft, the required thrust and driven moments are generated from a combination of the rotational speed of the fan and proper deflections of the six control vanes.Ailerons are also utilized for the convenience of pilot-manipulations in the early stage of flight experiments.Since the thrust-vectored system is capable of controlling the aircraft in most of the cases, we intend to remove these ailerons in the future prototypes.Contrast to the conventional fixed-wing layout, the Center ofGravity (CG) of the UAV is located behind the aerodynamic center of the wing so that the aircraft is more agile to the pitch motion.Important features of‘‘SHW09”are listed in Table 1.

In this section, we present a specific modeling of the dynamic system of the aircraft, including the 6-DOF motion,the aerodynamic forces and moments, and the control inputs.

2.1.6-DOF dynamic system

Definitions of reference frames are illustrated in Fig.2.The inertial frame is denoted as ｛ XI,YI,ZI｝ defined by northeast-down.The body frame is denoted as ｛ XB,YB,ZB｝ with its origin located at the CG of the vehicle.To our preference,0°of roll and pitch corresponds to the hovering condition.The 6-DOF motion of the vehicle can then be formulated by applying Newton and Euler’s Law of Motion:

Fig.1 Photo of ‘‘SHW09” ducted-fan tail-sitter UAV and illustration of altitude-hold transition.

Fig.2 Illustration of coordinates and definitions.

where all the components will be discussed in the next part.

We use [u,v,w]Tto denote the airspeed vector in the body frame and Ω(rad/s) to denote the rotational speed of the fan.Then, the airspeed V, the AOA α, the slide-slip angle β, the local AOA of the duct αd,and the advance ratio J are given as

where D is the fan diameter.

2.2.Ducted-fan system

Forces and moments on the ducted-fan thrust-vectored system comprise the control inputs generated from the ducted fan and the other aerodynamic effects.An important variable is the outflow speed of the duct.According to Ref.33,by considering that the outflow is parallel to the fan axis with no swirl in the wake,we can derive an analytic expression for the speed of this air flow, denoted as Ve:

where T,w,σd,ρ,Afdenote the thrust, the inflow speed along ZBaxis, the expansion ratio of the duct, the air density, and the fan disc area, respectively.

Immersed in this outflow, two important aerodynamic effects are hence generated:

where Acv,CLcvdenote the area and the lift line slope of one single control vane, respectively;δcv=[δcv1,δcv2,δcv3,δcv4,δcv5,δcv6]Trepresents the deflection angles of control vanes; BF,BMare mapping matrices,given as

where l1,l2are lever arms illustrated in Fig.2.

(2) The anti-torque effect Mstaon the stators that counteracts the fan torque:

where nstais the number of stators;Asta,CLstadenote the area and the lift coefficient of one stator, respectively.

2.3.Aerodynamic forces and moments on wing and fuselage

The aerodynamic forces and moments exerted on the wing and the fuselage comprise the lift, the drag,the side force, the roll,pitch and yaw moments, denoted as L,D,Y,MX,MY,MZrespectively.

where Awf,b,denote the effective area,the wingspan,and the average chord of the wing, respectively; δadenotes the deflection of aileron.

It is worth pointing out that there is a strong aerodynamic interaction between the wing and the ducted fan since they are located very closely.35The lift coefficient CL, the drag coefficient CD, and the pitching moment coefficient Cmin Eq.(14)are functions of both the AOA α and the advance ratio J.Results from Computational Fluid Dynamics (CFD) on these coefficients of ‘‘SHW09” are shown in Fig.3.

Consequently, the aerodynamic forceand momentcan be summarized in the body frame as

where [lcX,0,lcZ]Tis the location of the aerodynamic center in the body frame.

2.4.Acquiring aerodynamic coefficients

In this paper, the relevant aerodynamic coefficients are acquired from CFD.Multiple Reference Frame (MRF) technique is adopted to handle the rotating fan,in which the whole flow field is decomposed into two domains: the rotating domain and the stationary domain.The rotating domain is a thin cylinder covering the flow field around the fan while the stationary domain contains the wing, the duct, the boundary wall of the rotating domain, and the rest of the flow field.The whole computational region has 3.1 million nodes with 20 layers of prismatic/hexahedral elements for the boundary layers for all walls.The Shear Stress Transport (SST)k-ω model is employed to simulate the turbulence for its good accuracy in the simulations of low Reynolds number boundary layer flow with adverse pressure gradient.The second-order upwind is adopted as the spatial discretization scheme.For nondimensionalization, the remote flow velocity is fixed at 10 m/s, and different fan rotational speeds correspond to different advance ratios.

Fig.3 Wing/power interaction of ‘‘SHW09”.

3.Transition analysis and trajectory generation of altitude-hold transition

In this section, we first model and compute the transition corridor for the altitude-hold transition of‘‘SHW09”,followed by the generation of transition trajectories based on the identified transition corridor.

3.1.Preparations

3.1.1.Longitudinal planar dynamics

Considering the fact that the flight mode transition is essentially a pitch motion in the longitudinal plane(XB-ZBplane),we can simplify the 6-DOF dynamics Eq.(1) into a planar motion dynamics in terms of constructing the transition corridor.For this longitudinal planar motion, we can make some rational assumptions:(A)The environmental wind speed is set to zero so that the airspeed is equal to the ground speed; (B)The slide slip angle,the roll angle,and the angular rates on roll and yaw channels are 0°; (C) The velocity normal to the XB-ZBplane is 0 m/s.

3.1.2.Features of altitude-hold transition

Based on Eq.(18), the altitude-hold transition can be characterized with some specific constraints.First,the altitude during an altitude-hold transition is unchanged such that:

In order to further simplify Eq.(18), another important constraint is that the resultant pitching moment is always considered as being balanced:

This assumption is made based on the following reasons:(A) Although the pitch angle is changing, i.e.,q≠0, the variation of pitch angular rate is usually small,whether the adopted pitch trajectory is a linear one or an optimized one.This phenomenon is widely revealed in the existing transition studies.(B) From the force model in Section 2, a sudden change in the pitch angular rate has little influence on FB.

We also assume that the pitch angular rate is not so large that the pitch damping effect is negligible:

3.1.3.Boundary conditions

The boundary conditions for computing the transition corridor are given as

In conditions Eq.(22), the highest fan rotational speed is Ωh=1555 rad/s (14855 r/min, 2G thrust for static condition)while the lowest fan rotational speed is determined as Ωl=0?3Ωh.We herein adopt a non-zero Ωlto ensure that the ducted-fan thrust-vectored system is always working.The maximum vane deflection is set as δm=30?, associated to its stall limit.For the sake of nondimensionalization, these boundary conditions can be transformed to

3.2.Transition corridor

Firstly, we employ a particular variableto describe the collective action of vane 2,3,5,6, given as

In the context of the altitude-hold transition, it is worth noticing that the characteristic of fixed altitude and the neglection of wind speed together imply that:

Therefore, there is no intrinsic difference whether we refer to the speed and the attitude as the horizontal velocity and the pitch angle or the airspeed and the AOA.This statement is valid throughout the rest of this section.

The solution in Fig.4(a)provides a first glance of the transition corridor for the altitude-hold transition, which helps identify the factors establishing each bound of this region.In the next part, we will make a more accurate calculation of these bounds and present reliable details of the transition characteristics.

Fig.4 3D demonstration of transition corridor for altitude-hold transition of ‘‘SHW09”.

3.2.1.Trim condition

In Fig.5, we first present the equilibrium for steady flight,which is obtained by setting=0.This trim condition provides a baseline for both the forward transition and the backward transition.Since the resultant moment is balanced by Eq.(20), the negative, which reflects a nose down driven moment,implies that the external aerodynamic moment causes nose up,and vice versa.3D demonstrations of the trim condition are also exhibited in Fig.4.

As shown in Fig.5, from 0 to 13 m/s, the trim pitch angle varies from 0°to-70°(AOA from 90°to 20°).In this range of flight regime,the wing is already stalled,and the aircraft is subject to a strong nose down moment from the duct, which assists in accelerating the aircraft and requires positiveto balance.Conversely, as the forward speed increases to larger than 13 m/s,the wing stall gradually recovers,and the external aerodynamic moment alters from nose down to nose up that adds a deceleration trend on the vehicle.This explains why we design the CG behind the aerodynamic center.Otherwise,the aircraft will subject to a rather strong nose down moment from the wing and the duct, making it difficult for the vehicle to conduct a backward transition through pitching up.

Fig.5 Results for trim condition.

3.2.2.Forward transition

During the vertical-to-level forward transition, the aircraft continuously pitches down and accelerates from hovering to cruising.In Fig.6, we exhibit the forward transition corridor,indicating to what extent the aircraft can perform the acceleration flight.The equilibriums for steady flight are also marked with solid curves in Fig.6.

During the forward transition, the control moments are sufficient in balancing the exogenous aerodynamic moments,and the admissible acceleration is dominated by the thrust capacity.As shown in Fig.6, the bound of forward transition is calculated from the boundary condition=1,named as the thrust induced bound.At hovering status, through pitching down to - 60° with full thrust (2mg), the aircraft reaches its maximum horizontal acceleration by 17 m/s2.It is worth pointing out that this is not a steady status.It only describes a theoretically possible case for obtaining a large instantaneous horizontal acceleration at=0 m/s.At low speed and high AOA,both the thrust and the pitch angle play important roles in the determination of horizontal acceleration.As the forward speed increases, the forward transition corridor gets narrower and narrower due to the increasing wing lift.After 15 m/s,since the attitude is nearly level and the gravity is mostly balanced by the wing lift,the required attitude at each fixed velocity point tends to be unchanged no matter how large the acceleration is.At cruising flight of 20 m/s, the maximum admissible acceleration reduces to 6 m/s2.

Fig.6 Forward transition corridor for altitude-hold transition.①Thrust induced bound.

3.2.3.Backward transition

During the level-to-vertical backward transition, the aircraft decelerates from cruising to hovering through pitching up.Results for the backward transition corridor are exhibited in Fig.7.However, the bound of backward transition is much more complicated than that of the forward transition, which contains 3 connected segments resulting from 3 different boundary conditions.The deceleration process is hence partitioned into 3 stages.

Stage 1.Cruising to stall.

From cruising to 14 m/s, resembling that in the forward transition case, the pitch angle at a fixed velocity point stays almost invariant and the deceleration is mainly accomplished via lowering the thrust.Unlike the acceleration process, we cannot create a relatively large horizontal deceleration simply by sending a corresponding negative thrust demand to the ducted fan.As a matter of fact, the electric ducted-fan system cannot generate a large negative thrust simply through controlling the fan rotational speed without a thrust reverser.Thus,there is a limit for the largest deceleration which actually depends on the exogenous aerodynamic drag.As a result, it can be seen that the backward transition corridor in this flight regime is much narrower than the forward transition corridor.At vIh=15?7 m/s, the largest admissible deceleration is only-1.76 m/s2.

Fig.7 Backward transition corridor for altitude-hold transition.①Thrust induced bound.②Vane induced bound.③AOA induced bound.Dashed arrow line with red cross: Impractical transition trajectory.

In this stage,the bound of backward transition is calculated from the boundary condition Ω-=0?3,also named as the thrust induced bound (Bound ①in Fig.7), which results in two branches.One corresponds to vIh?[14,20] m/s and the other is at vIh?[14,16?4] m/s.The former one represents a normal level flight condition that the AOA is small and the horizontal deceleration is executed through lowering the thrust.The latter one represents another special case that the aircraft is operating at high AOA, higher than the stall angle that the lift coefficient also reduces to a small one.However, as illustrated in Fig.7(a)by the dashed arrow line with red cross,some regions covering this case are actually unreachable.In other perspective, for instance, if we want to have a rapid pitching up from cruising to a flight regime of α=40?, the transition trajectory will cross some region out of the transition corridor so that the altitude-hold transition no longer holds and an ascending of altitude is inevitable.

Stage 2.Stall to Low-speed.

From 14 m/s to 9.3 m/s, as the AOA increases, the wing begins to stall and the external aerodynamic moment alters from nose up to nose down.An increase of thrust is demanded to compensate the loss of wing lift while the deceleration motion is fulfilled by pitching up, in which the required pitch angle is greater than that of the equilibrium.Since the AOA is getting higher, it is now possible to obtain a large deceleration from the aerodynamic drag.

In this stage,the critical task for an altitude-hold transition is to balance the external aerodynamic moment.The bound of backward transition is hence calculated from the boundary condition=1, named as the vane induced bound (Bound②in Fig.7).Similar to that in Stage 1, there also exist two branches:?[9?3,14] m/s and?[9?3,13?2] m/s, implying another unreachable region.

Stage 3.Low-speed to hovering.

As the horizontal velocity decelerates to lower than 9.3 m/s,the external nose down moment becomes no longer crucial.Both the thrust and the vane deflection are sufficient in performing the expected flight maneuvering.Large deceleration can be achieved via pitching backward (θ>0?).In this stage,the admissible deceleration is bounded by the limit of AOA,while the bound of backward transition is calculated from α=120?, named as the AOA induced bound (Bound ③in Fig.7).

3.2.4.Discussion

By comparing the forward transition corridor in Fig.6 with the backward transition corridor in Fig.7, some interesting results could be found.In general, the acceleration capacity of the ducted-fan UAV is greater than the deceleration capacity.This can be intuitively seen that the forward transition corridor in Fig.6 is relatively wider than the backward transition corridor in Fig.7,especially in the region of level flight.However,the factors that cause this discrepancy are different in different flight regimes.In stage 1 of the backward transition,the main reason that the aircraft cannot acquire a large magnitude of deceleration is that we cannot make the thrust as negatively large as desired.So, the largest admissible deceleration is dependent on the aerodynamic drag, or in other words, the aerodynamic feature we design.In stage 2 of the backward transition, it is available for the aircraft to pitch up at high AOA and obtain a large deceleration from the aerodynamic drag.The main problem is that the aircraft will subject to a large pitching down moment from the duct in this flight regime.Therefore, the bound of backward transition in this stage describes to what extent the aircraft could hold a flight regime to maintain the desired deceleration.By enhancing the control effectiveness of control vanes, such as enlarging the vane surface, the aircraft can counteract larger exogenous moments and hence broaden the backward transition corridor.

It is easy to implement a forward altitude-hold transition.The aircraft just need to pitch down continuously and increase the thrust.However,to perform a backward transition with an unchanged altitude, the aircraft must conduct different strategies at different flight regimes.First,it should keep the attitude around level and lower the thrust to decelerate, with a slight pitching up to fix the altitude tracking error.Second, when stall happens,the aircraft should rapidly pitch up and increase the thrust to compensate the loss of wing lift in balancing the gravity.

3.3.Transition trajectory generation

As the transition corridor being identified,it is now possible to perform the altitude-hold transition.Any transition trajectory selected in obedience with the transition corridor will lead to a successful altitude-hold transition.In the proposed corridorbased methodology for transition trajectory generation, we first define the initial and final states for an altitude-hold transition process.Then we find a curve connecting the initial states and the final states within the transition corridor.

3.3.1.Modification of backward transition corridor

As mentioned in Section 3.2.3, there exist unreachable regions at?[14,16?4]∪[9?3,13?2] m/sin the backward transition corridor.This can be clearly viewed from Fig.7 (a) that the admissible decelerations in these regions are not connected to a reachable region along theaxis.Since these regions exist only theoretically but cannot be practically reached through a continuous transition trajectory,we should make a modification on the backward transition corridor to get rid of these impractical regions.A simple approach is to add some extra bounds to the backward transition corridor for the deceleration:

The corresponding bounds for the pitch angle and the thrust can then be calculated from Eq.(18).Consequently,as depicted in Fig.8, with all the unreachable regions being sealed, it is ready to plan a reliable transition trajectory.

3.3.2.Trajectory optimization

According to Section 3.2, only two independent variables are required for characterizing the transition corridor for an altitude-hold transition.In this paper, we choose to plan the transition trajectory on theplane, based on which unique trajectories for the pitch angle and the thrust can be calculated from Eq.(18).This simplifies the optimization procedure and significantly increases the computational efficiency.Aiming to minimize the energy cost during an altitude-hold transition, the objective function is selected as

where tfis the final time when the transition terminates.J*is available for both the forward and backward transition.

To ensure a successful altitude-hold transition, the statesshould be governed by the transition corridor, which are denoted as

(1) Forward transition

For the forward transition, the initial and final states are chosen as

Fig.8 Transition trajectory.

For the sake of safety in practical flight tests and taking into account model uncertainty, we put an extra constraint to the horizontal acceleration:

Based on this,a transition period should be no less than 4 s.In this paper, the final time is empirically chosen as tf=6 s.

Therefore, the forward transition trajectory can be generated from the following optimization problem:

(2) Backward transition

The initial and final states for the backward transition are chosen as

For the same consideration of that in the forward transition planning procedure and according to the backward transition feature shown in Fig.7(a), we also put another constraint to the horizontal acceleration:

Analogously,the final time is chosen as tf=11 s.The backward transition optimization problem is formulated as

The above optimization problems are solved numerically by the direct collocation method, in which the trajectory optimization is transcribed into a nonlinear optimization problem.36After the velocity trajectory is planned, we can calculate the trajectories for the pitch angle and the thrust via substituting the planned velocity trajectory into Eq.(18).The generated transition trajectories for the horizontal velocity, the pitch angle and the thrust are denoted as(t),θt(t),Tt(t), respectively, as illustrated in Fig.8.

Remark 1.In this paper, the extra constraints Eqs.(27), (31)and (34) as well as the transition time are chosen empirically for a trade-off between the flight performance and the safety issue with regard to real-time implementation.These design decisions can be arbitrary as long as they do not contradict the transition corridor and other performance requirements.

4.Flight control

In this section,we present the flight control scheme, aiming to track the generated transition trajectory.The proposed control scheme is constructed with a typical cascaded structure, in which the inner-loop is the attitude control and the outerloop serves as the trajectory tracking control(velocity control).The complete control structure is depicted in Fig.9.

4.1.Attitude control

A full-envelope attitude controller for the ducted-fan tail sitter is developed based on the Incremental Nonlinear Dynamic Inversion (INDI) technique, which is one of the most popular attitude control methods in the late decade.7,19,37INDI is very suitable for a full-envelope attitude control of a tail-sitter aircraft due to the two main properties:(A)The external aerodynamic moment features a slow manifold,mainly influenced by the lower order states (velocity, attitude, etc.) in the angular velocity channel, while the driven moment from the control vanes is a fast manifold;(B)We can have a reliable prediction on the control effect from the driven moment.Yet, a sophisticated modeling and online estimation of the external aerodynamic moment is unpreferable.

Based on the time-scale separation principle,38let z=[vI,η,ωB,T]Tdenote the lower order states that create a slow manifold and u=BMδcvthe virtual moment input creating a fast manifold.From Eq.(1),the angular velocity dynamics can then be rewritten as

where G1denotes the control effectiveness matrix from the virtual moment input to the real angular acceleration:

L represents slow varying dynamics mainly created from the external aerodynamic moment, given by Eqs.(2), (8), (11)–(13), (16).The INDI design starts from the first-order Taylor expansion of Eq.(36):

If the control loop runs at a relatively high frequency, we can assume that Δz ≈0, o(Δz2)≈0 by applying the timescale separation principle.Then, the angular dynamics Eq.(38) can be formed as the following incremental one:

The INDI attitude controller takes two steps.First, we linearize Eq.(39) by

Second, considering the third row of Eq.(1), the attitude tracking controller can be constructed as

where eη=ηd-η, eω=-ωB,and ηd,denote the reference attitude and angular velocity, respectively;Kη,Kωare diagonal feedback gain matrices.

Fig.9 Transition strategy and flight control structure.

4.2.Control allocation

To ensure that G1in Eq.(40)is an invertible square matrix,the attitude control design ends at the virtual moment input u.However, the mapping matrix from u to the real vane angles δcv, i.e.,BM, is non-square.The control allocation problem solves the vane angles δcvfrom a given u by the following optimization problem:

Eq.(42) is referred to as the direct allocation method,39which guarantees the attainability of u by transforming the control allocation problem to linear programing.

4.3.Trajectory tracking control

In this paper,we devise the outer-loop controllers for the transition mode,the hovering mode,and the level flight mode separately because the control logics in these 3 flight modes are totally different.We mainly concentrate on the outer-loop design of the transition mode.The designs for both the hovering mode and the level flight mode will be briefly introduced since they are beyond the scope of this paper.

4.3.1.Transition mode

In general, the outer-loop design in a full-envelope flight control structure calculates the desired attitude and thrust from a given reference velocity trajectory (vId(t),ψd(t)) based on the aerodynamic knowledge and dynamic characteristics of the vehicle.However, trajectory tracking error always exists.For an altitude-hold transition process, the continuous accumulation of tracking errors in the vertical channel,both for the vertical velocity and acceleration, will lead to an evident altitude change even if the transition trajectory is well planned.For an improvement, the trick is that the aircraft should track the velocity trajectory as well as the attitude and thrust trajectories simultaneously.To this end, the references for both the attitude and the thrust should be constructed as two parts:

where θ0,T0are pre-generated from the transition corridor.Since these terms only create a feedforward control to the velocity trajectory, θe,φe,Teare herein designed to build a closed-loop system which cancels the tracking errors on the velocity channel.

First, we define the side velocity as

Substituting Eq.(17)and Eq.(44)into the velocity dynamics Eq.(1), we can get rid of the yaw angle in the rotation matrix:

where

4.3.2.Hovering mode

In the hovering mode, a widely used SE(3)-based method is adopted for near-hovering motion control.

where ev=-vI,anddenotes the reference velocity in the inertial frame.

For a desired yaw angle ψd, to create the desired acceleration, the reference attitude and thrust are given as

where

The reference roll and pitch angles can then be solved from.

4.3.3.Level flight mode

The control scheme in the level flight mode takes the same structure as Eq.(43) with the (?)eterms modified as

where Kθ,KTare feedback gains.Of course, if necessary, the feedback controllers here can be replaced by PI controllers.

When a transition is finished and the aircraft steps into cruising flight, a total energy control design might be a better solution.However, this is beyond the scope of this paper and will not be discussed here.

4.3.4.Transition strategy

To complete the corridor-based transition control strategy,an important issue that we should address is how to define each flight mode.Note that, from Fig.5(c), there is a sudden fall of the equilibrium thrust at vIh≥15 m/s.Also,from Fig.6(b), the forward transition corridor for the pitch angle shrinks to be an extremely narrow one in these flight regimes, same as the backward transition corridor in Fig.7(b).These indicate that the dynamic characteristics of the aircraft are a bit different between vIh?[0,15) m/s and vIh?[15,20] m/s where the latter one matches the region of level flight.Therefore, in this paper, the level flight mode is defined as vIh?[15,20] m/s.

Choosing the hovering mode could be more flexible since from Figs.5–7, there are no obvious interfaces between the hovering mode and the transition mode.In this paper, the hovering mode is determined as the initial status before the forward transition or the final status after the backward transition.The forward transition from 0 m/s to 15 m/s and the backward transition from 15 m/s to 0 m/s are executed in the transition mode.

where the cross-track error epy, the heading angle χdand the corresponding azimuth angle αazare illustrated in Fig.10,given as

For vIh?[0,15) m/s, the aircraft is controlled in the transition mode under Eq.(51).After vIh=15 m/s, the flight mode switches to level flight mode and the controller switches to Eq.(55).The switching from the transition mode to the level flight mode executes only once.Then, the forward transition continues in the level flight mode and ends at cruising flight with the horizontal velocity stabilized around vIh=20 m/s.

For a regular backward transition,the aircraft is initialized in the level flight mode by vIh=20 m/s.The backward transition is conducted as an inverse process of the forward transition mentioned above, which ends at hovering.

5.Flight tests

In this section, the altitude-hold transition is successfully implemented on the ducted-fan tail-sitter UAV via the proposed corridor-based flight mode transition strategy.We design a particular flight course to verify the proposed method.First,the aircraft is initialized hovering at a given position and executes a forward transition.After the forward transition finishes, the aircraft keeps cruising flight for a few seconds and then executes a backward transition to hovering again.The entire flight course is conducted along a preset line path with a constant direction.The transition trajectory is consistent with that in Section 3.3 and the flight controller strictly follows that in Section 4.

Fig.10 Illustration of corss-track error.

Table 2 Controller parameters.

Onboard sensors include IMU, compass, differential GPS,and pitot-tube.Extended Karlman filter is employed for state estimation which runs at 200 Hz.The control loop updates every 0.01 s and the flight data is logged every 0.02 s.The adopted controller parameters are listed in Table 2.

Flight test results are shown in Fig.11.Fig.11(a)shows the tracking performance of horizontal velocity and Fig.11(b)shows the tracking performance of attitude and altitude.Fig.11(c)exhibits the flight path in the XI-YIplane.The time periods for different flight phases can be recognized as: initial hovering:[0,1.5]s,forward transition:[1.5,7.5]s,cruising flight:[7.5,9.5]s,backward transition:[9.5,20.5] s,and final hovering:[20.5,23]s.For clarity,grey blocks are utilized to mark the two transition phases in both Fig.11(a) and Fig.11(b).

From 1.5 s to 7.5 s, the aircraft accomplishes the forward transition through accelerating from hovering to cruising flight.From 1.5 s to 5.5 s, controlled under the transition mode, the aircraft continuously pitches down and accelerates from hovering to level flight by 15 m/s.At t ?[5?5,7?5] s,switching to the level flight mode, the aircraft keeps accelerating levelly with very little attitude change.From 9.5 s to 20.5 s,the aircraft performs the backward transition through decelerating from cruising back to hovering,during which the aircraft is controlled under the level flight mode at t ?[9?5,13?8] s and the transition mode at t ?[13?8,20?5] s,respectively.The reference trajectory of the horizontal velocity is adopted as the generated transition trajectory vIht(t),which is shown in Fig.11(a)with the dashed curves.The reference trajectory of the pitch angle, as mentioned in Section 4.3, is a composition of θt(t)and θein order to guide the vehicle tracking at both the desired horizontal velocity and the pitch trajectories.The altitude is set as zId=-17?9 m and the yaw angle (heading angle) is set as ψd=123?7?.

According to Fig.11, the overall tracking performance is satisfactory in both the horizontal velocity as well as the pitch and roll angles.The entire flight path is about 160 m with a maximum cross-track error of 2.7 m, and yaw tracking error reaches its peek around 7° at 4.5 s.

During the forward transition, the altitude ascends from the set point by 0.6 m.During the backward transition, the altitude ascends by 0.8 m and descends by 0.3 m from the set point.The overall altitude change,i.e.,maximum ascending to maximum descending, is less than 1.1 m.In practice,although the desired transition trajectory is well planned in obedience with the transition corridor of altitude-hold transition,the altitude change still occurs due to three main factors:(1)tracking errors on the pitch angle;(2)external wind disturbance;(3)model uncertainties.Anyway,compared to the existing works,the overall performance in maintaining the altitude is satisfactory.

In all, the flight test results verify that the proposed corridor-based transition strategy is effective in accomplishing the flight mode transition as well as maintaining the altitude for an altitude-hold transition course.Flight videos can be found at https://youtube.com/playlist?list=PLi0YFpJg-5Ouwzyx91ieK2WDuUWqmO8MW.

6.Conclusions

This paper focuses on a challenging transition approach: the altitude-hold transition, in which an agile aircraft conducts the vertical/horizontal flight mode transition at fixed altitude.To this end, we have proposed a corridor-based flight mode transition strategy along with practical implementations on a small ducted-fan tail-sitter UAV.The proposed methodology consists of 3 critical steps: model and compute the transition corridor in the longitudinal plane, generate the transition trajectory within the transition corridor, and track the transition trajectory via the proposed flight control scheme.Important results and conclusions are summarized as follows:

(1) The forward transition corridor is relatively wider than the backward transition corridor,especially in the region of level flight.This indicates that the acceleration capacity of the ducted-fan UAV is greater than the deceleration capacity.

(2) The transition corridor of the altitude-hold transition is a spatial surface, implying that for a given velocity trajectory, the solution for the corresponding trajectories of pitch angle and thrust is unique.Based on this characteristic, it is possible to simplify the transition trajectory generation to an optimization problem on the velocity-acceleration plane.

(3) The proposed corridor-based flight mode transition strategy is verified through practical flight tests with satisfactory performance, in which the altitude change is less than 1.1 m during the entire transition course.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was supported by Scientific Instruments Development Program of National Natural Science Foundation of China (No.61527810) and the Fundamental Research Funds for the Central Universities, China.

The authors wish to thank the Key Laboratory of Autonomous Systems and Networked Control,Ministry of Education and the Unmanned Aerial Vehicle Systems Engineering Technology Research Center of Guangdong(China)for supporting this research.