A general method for closed-loop inverse simulation of helicopter maneuver flight

Wei WU

National Key Laboratory of Rotorcraft Aeromechanics,College of Aerospace Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China

A general method for closed-loop inverse simulation of helicopter maneuver flight

Wei WU*

National Key Laboratory of Rotorcraft Aeromechanics,College of Aerospace Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China

Closed-loop; Flying quality; Helicopters; Inverse simulation; Maneuver flight

Maneuverability is a key factor to determine whether a helicopter could finish certain flight missions successfully or not.Inverse simulation is commonly used to calculate the pilot controls of a helicopter to complete a certain kind of maneuver flight and to assess its maneuverability.A general method for inverse simulation of maneuver flight for helicopters with the flight control system online is developed in this paper.A general mathematical describing function is established to provide mathematical descriptions of different kinds of maneuvers.A comprehensive control solver based on the optimal linear quadratic regulator theory is developed to calculate the pilot controls of different maneuvers.The coupling problem between pilot controls and flight control system outputs is well solved by taking the flight control system model into the control solver.Inverse simulation of three different kinds of maneuvers with different agility requirements de fined in the ADS-33E-PRF is implemented based on the developed method for a UH-60 helicopter.The results show that the method developed in this paper can solve the closed-loop inverse simulation problem of helicopter maneuver flight with high reliability as well as efficiency.

1.Introduction

A helicopter is a special aircraft which can perform hover,vertical takeoff and landing as well as low-speed maneuver flight.However,it is also an aircraft that is difficult to fly due to itsunstable and heavy coupling characteristics,and this problem will be more severe during maneuver flight.Therefore,relevant topics on helicopter maneuver flight such as how to assess the maneuverability of a helicopter,how to find a best control strategy for certain maneuver,etc.need to be studied.At the early stage,maneuverability was not considered in helicopter design,and the only standard for helicopter design is performance.This situation lasted for decades until the first flying quality specification for rotorcraft appeared in 1961.1In the first flying quality specification,the MIL-H-8501A,various flying quality criteria such as control stick force,acceleration to stick input,etc.were proposed,and some of these criteria have obvious in fluences on helicopter maneuverability.Currently,the state-of-art flying quality specification for rotorcraft is the ADS-33E-PRF,2in which maneuverability is described more explicitly.There are totally 23 mission task elements(MTEs)defined in the ADS-33E-PRF with different agility requirements,and the flying quality as well as the maneuverability of a certain type of helicopter can be assessed by performing these maneuvers.

A flight test is a direct way and the most accurate method to determine flying quality as well as maneuverability for helicopters.However,a flight test can only be performed very limited times,so it is usually used to obtain an assigned level of flying quality only.In order to get more information about maneuverability of helicopters,some researchers use ground flight simulators to do simulation flight experiments instead of real flighttests.3,4One of the key techniques of a flight simulator is the mathematical model of helicopters.A helicopter is a very complex system,so flight dynamics modeling is also complicated,and it will be more difficult to obtain an accurate flight dynamics model of a helicopter duringmaneuver flight.Inorder to solve this problem,there are a lot of research works carried out in this domain.5–7The advantage of using a flight simulator to study maneuverability is that a pilot can perform different kinds of maneuvers without considering the safety problem,which is quite useful to help finding the maximum maneuverability of a certain helicopter.On the other hand,the cost of a simulation flight test is much lower than that of a real flight test,so it can be performed much more times than the latter one.The deficiency of a simulation flight test is that it cannot tell a pilot how to control a helicopter to finish each maneuver,and it cannot be used to optimize the flight trajectory as well as the control strategy for different kinds of maneuvers.Inverse simulation was proposed then to deal with these problems.

In verse simulation uses some mathematical tools to calculate the pilot control time history for certain maneuvers,and it does not need a real pilot during the simulation procedure.Therefore,the cost of inverse simulation is very low,and it can provide very useful information to a pilot when he/she conducts a simulated or real flight test for the same maneuver.One commonly used inverse simulation technique is an optimization based method,8–13which gives a prescribed flight path for a certain maneuver at first,then establishes a cost function related to the error between the calculated flight path and the prescribed one,and finally,pilot controls for the maneuver are obtained based on some optimization algorithms.In order to increase the efficiency and practicality of inverse simulation,some improved methods such as sensitivity analysis12,and trajectory optimiaztion,13were proposed.Although the optimization based method is quite useful and has been used in inverse simulation for decades,its deficiencies are also obvious.Firstly,the calculation efficiency is poor,because it requires several iterations at each time step during the whole maneuver,and typically there are about 100 time steps in 1 s simulation.Secondly,the numeric stability of the optimization procedure is also poor since the dynamic characteristics of a helicopter are very complex,and sometimes in verse simulation may fail due to the divergence of numerical optimization.Finally,the optimization based method requires a prescribed trajectory of the maneuver to implement the optimization calculation.However,a lot of maneuvers do not have explicit trajectories.Therefore,this kind of method cannot be used to solve the inverse simulation problems of all kinds of maneuvers.In order to consider the pilot behavior during the maneuver flight,there are some research works focused on pilot modeling;14–16however,pilot modeling is also a complicated problem,and the introduction of a pilot model makes it more difficult to obtain inverse simulation results.Therefore,only simple pilot models and simple maneuvers are implemented currently.In recent years,another inverse simulation technique based on automatic control theory17,18is developed to conquer the difficulties encountered in the optimization based method.In this kind of method,no optimization calculation is required,so there is no numeric stability problem,and the inverse simulation efficiency is increased considerably.On the other hand,this kind of method does not depend on the flight trajectory,which indicates that this kind of method can be used to inversely simulate a wider range of maneuvers.Although it has been proven effective,the automatic control based method still has many problems to be solved.Firstly,in the current technique,only several typical maneuvers are inversely simulated by using this kind of method,and there is no general inverse simulation scheme based on this kind of method for all kinds of maneuvers.Secondly,the flight control system is not considered in current approaches,and neglecting the influence of the flight control system will make inverse simulation results a bit unreasonable.

In order to solve the above difficulties,a general method based on the optimal control theory for the helicopter closed-loop inverse simulation problem is developed in this paper.The influence of the flight control system is considered in the developed method.Three different maneuvers with different agility requirements are implemented for a UH-60 helicopter with a flight control system on line.The differences between the inverse simulation results with and without considering the flight control system’s influence are also studied.

2.Flight dynamics model for inverse simulation

The helicopter is a nonlinear,unsteady,high-order system,which is extremely true in maneuver flight.In order to increase the confidence of inverse simulation results,a nonlinear flight dynamics model as shown in Eq.(1)is used.

where u is the control input vector,t is the time variable,f(·)is a nonlinear function,and y is the state vector of the helicopter which can be expressed in a more detailed form as Eq.(2).

Since the unsteady aerodynamic phenomenon,dynamic stall,and dynamic wake distortion are considered in this flight dynamics model,it can be used to simulate different kinds of helicopter maneuver flight with different agility.More detailed information about this flight dynamics model can be found in Ref.7.

3.General method for helicopter closed-loop inverse simulation

In all the current inverse simulation methods,the influences of a flight control system on inverse simulation results are neglected.In this paper, flight control system models with different levels,the flight dynamics model,and the pilot control solver are combined to overcome this deficiency.Therefore,the general method developed in this paper can be used to solve the closed-loop inverse problem which is not possible for existing methods.

The general closed-loop inverse simulation method can be divided into three parts: flight control system modeling,mathematical description of maneuvers,and pilot control calculation.

3.1.Flight control system modeling

Closed-loop inverse simulation requires a flight control system model with two different levels.A simulation model which is composed of a helicopter flight dynamics model and a flight control system model requires a high-level model.At this level,the flight control system model should be close to the real flight control system as much as possible.Pilot control calculation requires a lower level of the flight control system model,at which hardware characteristics such as filter,sensor,and actuator dynamics can be neglected,and only a simplified control law is remained.This is because at this level,the flight control system model is only used to calculate the closed-loop stability matrix as well as the pilot control solver’s coefficients.

In this paper,the real engineering flight control system model of a UH-60 helicopter which can be found in Ref.19is used as the high-level model for the inverse simulation purpose.The model is implemented in Matlab Simulink environment,and then compiled to a dynamic link library(dll) file for further use.Based on the real flight control system model,a simplified flight control law is obtained by neglecting all the filter,sensor,and actuator transfer functions.Then the control law is transformed into a multi-input-multi-output(MIMO)feedback control form as Eq.(3)which has a very compact format.

where ufis the flight control system output,x is the helicopter responses including airspeed,Euler angles,and angular rates,and Kfis the feedback coefficient matrix.

3.2.Mathematical description of maneuvers

The mathematical description of maneuvers is a key factor for implementing inverse simulation successfully.For most of the current methods,the mathematical description depends on detailed flight trajectory,so it is not possible to describe all kinds of maneuvers for these methods.In this paper,a general form of the mathematical description function is established as Eq.(4).

where the description vector Des consists of 8 description variables,Hdis the altitude,is the changing rate of altitude,pd,qd,andrdare the roll rate,pitch rate,and yaw rate,respectively,φd,θd,and ψdare the roll angle,pitch angle,and yaw angle,respectively,Kdis the description coefficients vector,xdis the selected state vector of the helicopter for a certain maneuver,and g(·)is the mathematical description function.

Eq.(4)is a general form for describing helicopter maneuver flight.It is applicable not only to maneuvers that have explicit flight trajectory,but also to all kinds of maneuvers.Therefore,no matter what the maneuver is,it can be described by the 8 description variables.The functional structure and elements of g(·)for all kinds of maneuvers are also the same,and the only difference between each maneuver is the expression of this function.More detailed information of the description function will be discussed in the next section with specific maneuvers.

3.3.Pilot control calculation

Pilot control calculation is the final step for inverse simulation,and in order to avoid numeric optimization which may cause numeric problems,a direct computation based on the automatic control theory is established in this paper.On the other hand,the flight control system model is used in the control solver design procedure in order to separate pilot control from the control system output during maneuver flight.

The basic solution of pilot control for any maneuver can be obtained by using Eq.(5).

Since the integration and compensation coefficients are relatively small,and it is not difficult to determine these coefficients based on engineering experiences,KIand Kcare set to be constant matrices manually for each maneuver.However,the Kpmatrix is not so easy to obtain,and the values are changed for different maneuvers that have different initial states and agility requirements.On the other hand,the determination of the Kpmatrix should consider the influence of the flight control system.Therefore,a comprehensive algorithm based on the optimal quadratic regulator theory to calculate the Kpmatrix for each maneuver is developed as follows.

Firstly,trim the helicopter at the initial steady flight state for each maneuver,then linearize the helicopter flight dynamics model in a trim condition,and a standard state space model as shown in Eq.(9)is obtained.In order to consider the influence of the flight control system in determining Kp,a closed loop state space model is then established as shown in Eq.(10)by combing Eq.(9)and Eq.(3).

where A is the stability matrix,B is the control matrix,X is the linearized state vector and U is the linearized control vector.

Secondly,define a cost function based on the optimal quadratic regulator theory as shown in Eq.(11),where Q and R are non-negative de finite and positive definite symmetric matrices,respectively.Then minimize the cost function to find a best feedback controller that has high control precision as well as minimum control power.In order to solve this optimization problem,assume the original closed-loop stability matrix and the control matrix in Eq.(10)are constant for one maneuver,and then the solution of the optimization problem can be found as Eq.(12),in which P is the solution of an algebraic Riccati equation as shown in Eq.(13).There are many comprehensive tools to solve the Riccati equation,so it is easy to obtain P.

Finally,based on Eq.(12),the optimal feedback coefficient matrix can be obtained by using Eq.(14).Then eliminate all the cross coupling control coefficients in K*,and a solution of Kpfor a certain maneuver is obtained.

4.Applications to typical helicopter maneuvers

The detailed inverse simulation procedure by using the developed method will be addressed in this section.In order to show that the developed method is capable of solving any maneuver flight problems,three typical maneuvers defined in the ADS-33E-PRF with large differences in course patterns as well as agility requirements are calculated in this paper.These three maneuvers are pirouette,vertical remask,and high yo-yo.

4.1.Pirouette

The pirouette maneuver is a high-precision flight mission with moderate agility requirements.This maneuver starts from a steady hover condition,and then accomplishes a lateral translation around a circle while keeping the nose of the helicopter pointing at the center of the circle.The maneuver will be terminated at a hover condition over the starting point.The main performance standards of the desired level at good visual environment(GVE)are concluded in Table 1.

The first procedure of inverse simulation of this maneuver is to determine the mathematical description function g(Kd，xd).The pirouette maneuver can be divided into 2 steps:the first step is lateral translation around the circle,and the second step is hover when finishing the maneuver.Therefore,there are also 2 different mathematical description functions for this maneuver,as shown in Eqs.(15)and(16).

In Eq.(15),Hcomis the constant altitude command which can be set according to the ADS-33E-PRF,H˙,p,and q are the real-time helicopter altitude changing rate,pitch rate,and roll rate,respectively.The reason of using real-time values of these responses as description variables is that during the first step of this maneuver,these states should be kept at 0 in an ideal case,and this is the stability augmentation function which can be done by the flight control system.Therefore,the pilot does not need to make compensation control for undesired responses of these 3 state variables.According to the first term of Eq.(5),the difference between these 3 rows will be 0 because xdis the helicopter’s real-time response vector too,and the result is that no matter what the values of these 3 states are,pilot compensation control for these states willalways be 0.Rcomand R are the required radius of pirouette and the helicopter’s real-time radius around the circle,respectively,u and v are helicopter forward and lateral velocities in body axis,respectively,vcomis the required lateral velocity in body axis,Xhand Yhindicate the current position of the helicopter,Xcand Ycare the coordinates of pirouette center,while θtrimand φtrimare the pitch and roll angles at the trim condition,respectively.The heading of the helicopter is changing all the time during the pirouette maneuver,so the yaw rate should not be zero,which is why the yaw rate dose not set to be the helicopter’s real-time value.The pilot should control the yaw rate based on current lateral translational speed and the distance to the circle center.Finally,the head of the helicopter should always point at the center of the circle,and the required yaw angle can be calculated based on the X,Y coordinates of the helicopter and the center point.

Table 1 Desired performance of the pirouette maneuver at GVE.

In Eq.(16),˙u and˙v are time derivatives of u and v,while Xhovand Yhovare coordinates of hover point.In this step,the helicopter enters a hover condition,and the yaw rate should be 0 at this time.Therefore,the required yaw rate is set to be the real-time value for the same reason as above.The yaw angle is set to be constant,ψconstant,the same as the value when entering the hover condition.

In Eq.(15)and Eq.(16),Ku,Kv,KR,Kx,Ky,Kudot,and Kvdotare description coefficients in Kdvector.

When the mathematical description function is determined,the next step is to calculate the coefficient matrix in Eq.(5)at the hover condition based on the preceding developed method.Finally,combining Eqs.(1),(4),and(5),the pilot control time history of pirouette can be obtained.

Fig.1 shows the pilot control solution for pirouette.At the beginning of this maneuver,the pilot moves the lateral stick to the right a bit to induce a lateral speed,and at the same time,steps the left pedal to make the helicopter turn left in order to keep the nose pointing at the center of the circle.The longitudinal stick control is used to prevent pitching up due to a lateral sideslip velocity.Since the roll and pitch angles are small during the whole maneuver,the collective stick compensation is not very obvious.

Taking the calculated pilot controls into Eq.(1),the flight states time histories can be solved as shown in Figs.2–4.It is obvious that the simulation results satisfy the entire performance standards in Table 1,which means that the flying quality of the UH-60 helicopter can reach the desired level in this maneuver.

An open-loop solution for pilot control is also calculated as shown in Fig.5 to check the differences between closed-and open-loop inverse simulations.It can be found that in an open-loop situation where the flight control system is offline,the pilot will make compensation controls to eliminate all the undesired responses,and the workload will be increased considerably.

4.2.Vertical remask

Vertical remask is a vertical and lateral maneuver with aggressive agility requirements.The main performance standards for this maneuver are concluded in Table 2.

The vertical remask maneuver can be divided into 2 steps:the first step is rapid vertical descent,and the second step is rapid lateral displacement.Therefore,there are also 2 different mathematical description functions for this maneuver,as shown in Eqs.(17)and(18).

In the first step,the helicopter performs rapid vertical descent to a prescribed altitude relative to a constant point.Therefore,in Eq.(17),the altitude is set to a constant value,and the changing rate of altitude is also set to a constant value at first few seconds.When the altitude is close to the required one,the changing rate of altitude will set to be 0.Since the helicopter is required to hold longitudinal and lateral positions in the vertical descent phase,the remaining description variables are calculated similar to the hovering case in Eq.(16).

In the second step,the main flight course is lateral displacement,so the roll angle description variable is set according to the sideslip speed requirement.The determinations of other description variables are similar to those of the hovering case.

Table 2 Performance standards of vertical remask maneuver.

When the mathematical description of the vertical remask maneuver is done,the same procedure as in the pirouette maneuver simulation is performed to obtain inverse simulation results.Fig.6 shows the pilot control time history for implementing vertical remask,while Figs.7 and 8 provide the simulated state responses time histories of this maneuver.

At the beginning of this maneuver,the pilot decreases the collective to make the helicopter descend,while adjusting the pedal to keep the heading of the helicopter.Then at about 7 s,the pilot pulls the cyclic stick to the right to make the helicopter sideslip to the right rapidly.At the same time,the longitudinal stick control is used to prevent pitching up caused by the longitudinal and lateral rotor flap coupling phenomenon.

It can be found in Figs.7 and 8 that the UH-60 helicopter can perform the vertical remask maneuver with sufficient precision.However,it cannot finish this maneuver in 15 s,but it can be done within 25 s.Therefore,the UH-60 helicopter cannot reach the desired level of this maneuver,but it meets the adequate requirements.Since the UH-60 helicopter is a utility but not attack helicopter,it lacks aggressive maneuverability.Therefore,the inverse simulation result is reasonable.

Inverse simulation of this maneuver with the flight control system offline is also implemented as shown in Fig.9.The same conclusion can be made as in the pirouette maneuver.

4.3.High Yo-Yo

High yo-yo is a target acquisition and tracking maneuver with aggressive agility requirements.This maneuver requires two aircraft to perform at the same time,and the test course is not static.Therefore,it is impossible to use a conventional optimization based method to implement the inverse simulation of this maneuver.The performance standard for this maneuver in the ADS-33E-PRF is qualitative.

The high yo-yo maneuver can be divided into three steps.The first step is following,the second step is deceleration by means of climbing,and the last step is pursuit.The relevant mathematical description functions of these three steps can be found in Eqs.(19)–(21).Where,the subscript ‘target” in all variables indicates the target aircraft’s states,TDcomis the trace distance between the two aircrafts,Khu,Krψand Ktdare also the description coefficients in Kdvector.yaw rate is also set based on the current and required heading in order to make a rapid turn.

In the last step,the test helicopter will dive to the target helicopter and keep pursuing it.The mathematical description of this step is similar to that of the second step.The main differences are that the altitude description variable is set to be the height of the target helicopter,and the pitch angle description variable is set based on both the airspeed difference between the two helicopters as well as the required pursuit distance between them.

When the mathematical description of high yo-yo is done,the same procedure can be performed to implement the inverse simulation of this maneuver.Fig.10 gives the solution of the pilot control time history,while Figs.11–13 show the simulated states time histories of this maneuver.

In the first step,the test helicopter only needs to track the target helicopter at a constant altitude and heading.It is quite close to a steady forward flight condition,and since the flight control system is online which can stabilize the helicopter in a steady flight state,the pilot only needs to perform very small compensation control in this step.At about 8 s,the test heli-

In the first step,the test helicopter chases the target helicopter straightforward,so the altitude description variable is set to be the target helicopter’s altitude,while the stabilization of the changing rate of altitude is left to the flight control system.The main control in this step is using the cyclic stick to keep the test helicopter at a required forward flight speed,so the pitch angle description variable is set according to the speed command.In the lateral channel,the roll angle description variable is set based on the sideslip speed and lateral displacement to ensure the test helicopter behind the target helicopter all the time.The stability augmentation of angular rates is left to the flight control system.

In the second step,the test helicopter reduces the airspeed since the target helicopter does the same thing.On the other hand,the target helicopter makes a transient turn to try getting rid of the test helicopter,so the test helicopter should change its heading and keep pointing at the target helicopter.In order to implement these works,the pitch angle description variable is set according to the airspeed difference between the two helicopters.The altitude description variable is also set based on the airspeed difference because the collective determines the rotor thrust which will influence the acceleration and deceleration of the helicopter.The yaw angle description variable is set according to the relative coordinate of the two aircraft,and the copter begins pitching up to reduce its airspeed.The longitudinal cyclic stick is pulled back a bit then,and the collective is also increased to provide a large rotor thrust which is helpful of reducing the airspeed.At the meantime,the pedal and lateral controls are applied to make the test helicopter turn toward the target helicopter.After a rapid deceleration,the test helicopter begins to pursue the target helicopter,and the pilot applies all of the four controls to implement this.

The flight trajectory in Fig.11 shows the good performance of inverse simulation.The test helicopter has a delay in initiating a turn to the right which is just required in the ADS-33EPRF.The reason is to keep the test helicopter behind the target helicopter all the time for the missile launch requirement.

Fig.14 shows the inverse simulation result for the high yoyo maneuver with the flight control system offline.

5.Conclusions

(1)A general method for closed-loop inverse simulation of helicopter maneuver flight is developed which includes flight control system modeling,general mathematical description of maneuvers,and pilot control calculation.The influence problem of a flight control system on inverse simulation is well solved by taking flight control system models with different levels into each inverse simulation step.

(2)Closed-loop inverse simulation of three typical helicopter maneuvers de fined in the ADS-33E_PRF with large differences in course pattern and agility requirements is implemented.

(3)The results show that the method developed in this paper is capable of solving the inverse simulation problem for different kinds of maneuvers with high reliability as well as efficiency.

(4)The developed method can separate pure pilot controls of each maneuver from flight control system outputs,and this improvement is very useful for engineering applications such as pilot training and simulation flight test.

Acknowledgements

This study was co-supported by the National Natural Science Foundation of China(No.61503183)and the Aeronautical Science Foundation of China(No.2015ZA52002).

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25 July 2016;revised 6 January 2017;accepted 13 March 2017

Available online 30 August 2017

?2017 Production and hosting by Elsevier Ltd.on behalf of Chinese Society of Aeronautics and Astronautics.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.

E-mail address:scorpio_nuaa@nuaa.edu.cn.

Peer review under responsibility of Editorial Committee of CJA.

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