Load distribution strategy for multi-lift system with helicopters based on power consumption and robust adaptive game control

Dengyan DUAN, Gen LENG, Jie GAO, Xinming FENG, Jianbo LI

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

KEYWORDSDifferential game;Helicopter;Load distribution;Multi-lift system;Power consumption;Tracking control

AbstractIt is of great significance to reasonably distribute the slung load to each helicopter while considering difference in power consumption, relative position and interaction comprehensively.Therefore, the load distribution strategy based on power consumption and robust adaptive game control is proposed in this paper.The study is on a ‘‘2-lead”multi-lift system of four tandem helicopters carrying a load cooperatively.First,based on the hierarchical control,the load distribution problem is divided into two parts:the calculation of expected cable force and the calculation of the anti-disturbance cable force.Then, aimed at minimizing the maximum equivalent power of helicopter,an optimization problem is set up to calculate the expected cable force.Specially,the agent power model is trained by BP neural network,the safe distance constraint between helicopters is set to 2.5 rotor diameters to reduce aerodynamic interference,and the helicopters with different performance can be considered by introducing the equivalent power factor into the objective function.Next, considering the difference and interaction between helicopters, the robust adaptive differential game control is proposed to calculate the anti-disturbance cable force.Particularly,to solve the coupled Hamiltonian equations,an adaptive solving method for value function is proposed,and its stability is proved in the sense of Lyapunov.The simulation results indicate that the proposed load distribution method based on power consumption is applicable to the entire flight trajectory even there are differences between helicopters.The game control can consider interaction between helicopters, can deal with different objective functions, and has strong robustness and small steadystate error.Based on the entire strategy, the cable force can be reasonably allocated so as to resist disturbance and improve the flight performance of the whole system.

1.Introduction

The helicopter’s ability to take off vertically and hover at fixed point makes the helicopter/load system not limited by cargo shape and geographical terrain.So, it has been widely used in both military and civil aspects.1–4Considering the high complexity and low economics of heavy transport helicopter,multi-lift as an alternative has attracted more and more attention.5–7According to ‘‘cask effect”, the capability of a system depends on its weakest component.For the multi-lift system,its load capacity and endurance depend on the helicopter with the largest power consumption or the helicopter with the smallest endurance under the same power consumption.Therefore,it is necessary to distribute load reasonably to avoid premature failure of one or more helicopters during the flight process.

Load distribution strategy to equalize cable forces has been proposed in many studies.For a twin-lift system with two Yamaha RMAX helicopters, Bernard8discussed a load distribution control concept to equalize the cable force and the related flight tests verified the effectiveness of the method.Berrios et al.9designed a feedforward controller to calculate the position of the following helicopter relative to the leading helicopter, and proposed a feedback control strategy to make the cable forces of the front and rear helicopters equal.Based on the load distribution controller mentioned above,the difference between the cable forces will be less than 2 % of the load weight.Geng and Langelaan10,11studied a multi-lift system with four quadrotors based on hierarchical approach.The load distribution problem is converted into a convex optimization problem and flight tests are carried out.The social spider algorithm is introduced by Duan et al.12to solve the above convex optimization problem, which has high robustness and is not easy to fall into local optimal solution.However, the above convex optimization problem is aimed at minimizing the sum of cable forces, which cannot achieve equal distribution of the load.So, in the subsequent research by Geng et al.,13,14a trajectory planning and tracking control method that seeks to evenly distribute cable tensions is proposed.But the introduction of load trajectory will double the design variables, result in long calculation time and make it difficult to converge.

However,equal cable tension cannot guarantee equal power consumption if the helicopters are different in performance or at forward flight.The flight performance optimization of a multi-lift rotorcraft formation with four UH-60 Black Hawk helicopters was carried out by Enciu and Horn15And it was pointed out that the performance of the whole system can be optimized by being aimed at minimizing the maximum power of the helicopter and considering the dynamics constraints,formation constraint and safe distance constraint meanwhile.The calculation results show that for the multi-lift system with ‘‘2-lead”formation at forward flight speed of 100 ft/s (1 ft/s = 0.3048 m/s), the maximum power consumption by equal power-based load distribution can be reduced by 16 % compared to that by equal force-based load distribution.However,the above calculation method is only applicable to the steady flight state and the approach for high maneuver is not given.Similar result is obtained by Song et al.16Trimming and stability analyses constrained by maintaining 100 ft(1 ft=0.3048 m)distance between helicopters are carried out at different forward flight speed and the results show that with the increase of flight speed, the collective pitch demand of the front helicopter is much greater than that of the rear helicopter.

Moreover,for the tracking control of the load,it is usually assumed that there is no difference in control requirements between helicopters, or the weighted objective functions are simply added together as the overall control goal of the multi-lift system.As a result,the differences among helicopters cannot be considered and the interaction influence cannot be introduced.Aimed at minimizing the sum of state error and related manipulations, a load tracking control based on Bryson’s backward sweep method was proposed by Geng et al.13,14For cooperative transport of a bar-shaped payload with rotorcrafts,Gimenez et al.17introduced the trajectory following target, the collision avoidance target and the weight distribution target, and then designed the control law based on null-space theory.Arab et al.18proposed a leaderless distributed control algorithm to ensure safe transport of the load.But the influences from load to helicopters are regarded as disturbances,so the trajectory and the attitude of load cannot be controlled explicitly and the interaction between helicopters cannot be considered.Chopra and Ghose19equipped the load with IMU and an optical sensor,and then the outputs of the sensors were compared to the reference trajectory to generate control inputs.However, the proposed controller acts equally on all UAVs in the formation, and the interaction between helicopters can still not be considered.

Game theory has been widely used in the field of multi-agent control.Jimenez-Lizarraga et al.20designed a novel control strategy based on differential game for the formation flight of n quadrotors, in which the lead vehicle follows the predesigned trajectory and other vehicles follow the leader.Jiang et al.21proposed a formation control for multiple UAVs based on cooperative game theory.Each agent interacts with each other and reaches a consensus by reducing the weighted team cost.The simulation results show the superiority of cooperative differential game over non-cooperative game and original optimal control in UAV formation flight.Chai et al.22studied an onorbit assembly strategy based on robust event-triggered game theory.The disturbance is taken as a special player to the game and the coupled Ricatti equations are solved by Lyapunov iteration method.It can be seen that based on game control,each agent can design individual objective according to its own performance and task independently and can consider interactions with each other meanwhile.This brings inspiration to the proposal of robust adaptive game-based tracking control.

In light of the preceding discussion, the load distribution strategy based on power consumption and robust adaptive game control is proposed.First, for the baseline configuration of four tandem helicopters carrying a load cooperatively,based on hierarchical control, Section 2 divides the load distribution problem into two parts:the calculation of expected cable force and the calculation of the anti-disturbance cable force.Modelling of the multi-lift system is carried out in Section 3.Then,Section 4 proposes the load distribution strategy based on power consumption for the calculation of expected cable force.Section 5 proposes the robust adaptive game-based tracking control for the calculation of the anti-disturbance cable force.In Section 6, some simulations are carried out to verify the effectiveness of the whole proposed load distribution strategy.Finally, this article is concluded in Section 7.

2.Problem formulation

2.1.Hierarchical control for multi-lift system

It is well known that if one of the helicopters runs out of energy or loses effectiveness during flight, cooperative transportation will break down.Therefore, it is necessary to distribute load reasonably to make the endurances of helicopters equal during the flight process.

Suppose path is given, and the load-leading hierarchical control for the multi-lift system shown in Fig.1 is mainly comprised of three layers as follows:

(A)The load layer: calculates expected cable force and moment based on expected trajectory and state of the load,and meanwhile calculates the anti-disturbance one by tracking control.Then the total required cable force and moment can be obtained by adding the expected cable force and the antidisturbance one together.

(B)The cable layer: calculates force of each cable based on the total required cable force and moment.The cable force and cable angle are related to the power consumption of helicopter in a specific flight state.

(C)The helicopter layer: calculates the desired position of each helicopter according to the cable force.Then the control strategy is designed to follow the expected position.

By combining the load and cable layers, it can be seen that the load distribution problem can be divided into the calculations of expected and anti-disturbance forces for each cable.Therefore, considering the performance, position, state and constraint differences of each helicopter, load distribution strategy based on power consumption and robust adaptive game control is proposed.

2.2.Baseline configuration

This study is on a multi-lift system with‘‘2-lead”formation as shown in Fig.2.The baseline configuration describes four tandem helicopters carrying a heavy load cooperatively through four 7.2-meters-long cables.The length, width and height of the cuboid slung load are 1 m, 0.4 m and 0.4 m respectively.Each tandem helicopter has two two-blade rotors with horizontal offset of 1.165 m.The rotational speed is 113 rad/s and the rotor diameter is 1.8 m.The weight of the load is 10 kg which is larger than the load capacity(3.5 kg)of a single helicopter and smaller than the whole capacity (14 kg) of the multi-lift system.And in Fig.2, xeyeze, xibyibzib, xLyLzLrepresent the north-east-down earth coordinate, the body coordinate of helicopter i, and the body coordinate of the load respectively.

3.Modelling

3.1.Load model

Suppose the origin of coordinate xLyLzLis located at the mass center of the load,and then the dynamics of the slung load can be described by Newton-Euler approach3as.

Fig.1 Load leading hierarchical control for multi-lift system.

Fig.2 Baseline configuration of multi-lift system with four tandem helicopters.

Specially, the 0.4 m × 0.4 m × 1 m slung load is similar in shape to the standard 8 ft × 8 ft × 20 ft MILVAN container(Military-Owned Demountable Container).With a length scale factor sload= 6.1, we can obtain the aerodynamic force Maerousing the wind tunnel test data23–25of the standard MILVAN container.Here, a steady load model is introduced by

Where Fx;ref, Fy;ref, Fz;refare the reference aerodynamic force along xL, yL, zLaxis respectively; Mx;ref, My;ref, Mz;refare the reference aerodynamic moment along xL, yL, zLaxis respectively.And these six variables are derived from wind tunnel data23–25related to the attack angle and the sideslip angle.Besides, Qloadis the dynamic pressure of the load.

It should be noted that the only unknown variable in Eq.(1) is Mc.In other words, if the desired trajectory and state are given,Mccan be calculated by Eq.(1)conveniently.In this paper, Mcis written as

where Fc;sumis a 3×1 vector representing the sum of cable force, and Mc;sumis a 3×1 vector representing the sum of cable moment.

3.2.Cable model

In real applications of the multi-lift system,compared to cable forces, as well as the gravity and aerodynamic forces of the helicopters and the load,the cables have much smaller weights and windward areas.So,in this paper,the four 7.2-meters-long cables are modelled as spring-damper systems ignoring the effects of their cable gravity and aerodynamic drag force.Besides, the cable forces are assumed to be positive and the cables are not loose.

Based on the cable forces,we can obtain the positionePiin xeyezeof helicopter i by

where liis the length of cable i,K is the spring stiffness,C is the damping coefficient, ˙lc;iis a 3×1 vector representing the change rates of cable length in xLyLzLcoordinate, and αi, βiandLfcable,i= [Lfcable,ixLfcable,iyLfcable,iz] are the cone angle,regional angle and force vector as defined in Fig.3,respectively.

3.3.Helicopter model

The tandem helicopter is modelled as in Refs.26,27For helicopter i, its flight dynamics can be described as

Besides, Mextrais a vector representing the forces and moments from main rotors, fuselage and cables.

Based on the Pitt-Peters dynamic inflow model,28the rotor inflow can be modeled as

Therefore, for the i th helicopter, there are 30 state parameters in total including 12 rigid body state parameters, 9 front rotor state parameters and 9 rear rotor state parameters, as follows:

Fig.3 Force, cone angle and regional angle of the ith cable.

4.Load distribution based on power consumption

4.1.Agent model of power consumption

The power consumption of helicopter should be calculated under trimming condition.If the power consumption is calculated by trimming online in each optimization iteration, it will be time-consuming.Therefore, a power agent model based on Back Propagation (BP) neural network is proposed.

4.1.1.Power consumption calculation

The slung point of the tandem helicopter is assumed to be on the gravity center.Then the sling forces and moments for helicopter i in xibyibzibcan be calculated by

Newton’s method is introduced to trim helicopters loaded by cable forces.This is a process to set parameters in vector ˙xito given values or zeros by changing the manipulation variables in Eqs.(9),(11)and(12).After trimming,the power consumption of the helicopter can be calculated by

where QFand QRare the torque generated by the front and rear rotors while trimming respectively,and Ω is the rotational speed of the rotor.

Fig.4 show the calculated power consumptions in hovering and at advance ratio of 0.1.As we can see, while hovering, the power consumption is more related to the magnitude of cable force and angle α.This is the reason why most studies proposed the even-force-based load distribution strategy with fixed angle α.However, at relatively high flight speed as shown in Fig.4(b), the power consumption is related to the cable force, the angle α and the angle β.For the multi-lift system with ‘‘2-lead”formation, the values of angle β of front helicopters are between -90?and 90°, and the cable force corresponds to backward cable force acting on the helicopter.Meanwhile, the values of angle β of rear helicopters are between -180?and -90?or 90° and 180°, and the cable force corresponds to forward cable force.Obviously, at forward flight, the two rear helicopters are in favorable positions to reduce power consumption while the two front helicopters are in disadvantageous positions.Therefore, it is essential and necessary to study the load distribution method based on even power consumption to achieve long endurance.

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Besides, it is worth mentioning that multi-lift system as a close formation flight, there exists not only complex dynamics coupling but also serious aerodynamic interference between helicopters especially at forward flight.However, according to previous work,30–32aerodynamic interference induced power addition is complex and difficult to model.And a separation of 2.5 rotor diameters between helicopters is enough to avoid interferences for the baseline configuration.Therefore,instead of introducing aerodynamic interference into the agent model, a safe distance constraint of 2.5 rotor diameters is utilized in the following optimization.

4.1.2.Neural-network-based agent model

The agent model is established based on BP neural network with structure shown in Fig.5.The input layer has four variables:advance ratio,cone angle,regional angle and the magnitude of cable force.Forty neurons are utilized in the hidden layer considering the complexity of the problem to be solved.The final output is the power consumption.The active function f in the hidden layer is hyperbolic tangent function, and the one in the output layer is identity function.

Besides,in Fig.5,m′is the number of the input variables,l′is the number of neurons in hidden layer, n′is the number of neurons in output layer and meanwhile the number of the output variables, and p and a represent the vector of the input variables and the vector of the output variables of neurons respectively.Specially, the superscripts of p and a represent the indexes of layers, IW1;1and LW2;1represent the weights of neurons in hidden and output layers respectively, and b1and b2represent the bias of neurons in hidden and output layers respectively.

For each small tandem helicopter, the database is used to train agent model spans advance ratios from 0 to 0.11, cone angles from 0 to π/2, regional angles from -π to π, and cable forces from 0 to 100 N (weight of the load).As a result, there are 4680 points in the database.4000 of these points are used to train the network,and the rest for validation.As we can see in Fig.6,the agent model is effective and can be utilized in the optimization problem.

Fig.4 Calculated power consumption.

Fig.5 BP neural network.

4.2.Mathematical formulation of cable layer optimization

4.2.1.Design variables

For the baseline configuration of four tandem helicopters transporting a load cooperatively,there are 12 design variables as follows:

where （hi）*is the skew symmetric matrix of vector hi.

In order to avoid helicopter collision and reduce aerodynamic interference, a safe separation constraint is introduced as

Fig.6 Fitting results of power consumption.

where dminequals 4.5 m (2.5 rotor diameters).Indeed, the farther the distance, the larger the cone angle, and then the greater the cable force required to carry the same load.To avoid this, longer cables are generally required and 7.2-meters-long ropes of the baseline configuration are enough.Some bound constraints in Eq.(19) are introduced to increase the convergence speed.The constraints for lateral cable force components are used to reduce the possibility of cable crossing.

where αi;minand αi;maxare the minimum and maximum allowed values of angle α respectively,and βi;minand βi;maxare the minimum and maximum allowed values of angle β respectively.

4.2.3.Objective function

To distribute the load to each helicopter based on power consumption evenly, meanwhile considering the performance differences between helicopters, the objective function is defined as

where Pirepresents the power consumption of helicopter i,and kiis the scaling factor related to performance differences.For instance, the endurance of helicopters 1 and 2 is 60 minutes and 30 minutes respectively under the same power consumption of 500 W.Then k2is set to 2 to avoid helicopter 2 running out of energy prematurely.

4.2.4.Numerical solution

The MATLAB/fmincon function with the sqp algorithm is adopted to solve the nonlinear numerical optimization problem, which is of good robustness and computational efficiency.The tolerance on design variables, objective function and constraints are set to 1 × 10-6, 1 × 10-6and 1 × 10-4respectively.

5.Load distribution based on tracking control

There may be different control requirements between helicopters participating in cooperative transportation.If each helicopter adopts the same objective function or takes the weighted sum of the objective function of each helicopter as the system objective function, the influence of the interaction between helicopters cannot be fully considered and system potential cannot be brought into full play.Therefore, for the calculation of anti-disturbance cable force,the robust adaptive game control is proposed.

5.1.Differential game formulation

According to Section 4, the optimal trajectory and manipulation achieving even power consumption, defined as xrand urrespectively, can be obtained.On this basis, the linear model at each optimal point is obtained by linearizing Eq.(1) with Jacobi linearization method as follows:

where vFis a 3×1 vector representing the disturbances related to force,and vMis a 3×1 vector representing the disturbances related to moment.Besides,

5.2.Robust Nash equilibrium

Based on given u～j and v, the value function of helicopter i can be expressed as

5.3.Adaptive learning

Under the above analysis, ^Wican be updated according to Eq.(42) first, and then the robust equilibrium solution can be calculated by Eq.(39).

5.4.Stability analysis

Theorem 1.Consider a linear system in the form Eq.(25).Using control law Eq.(39), the stability of the closed-loop system will be guaranteed with the adaptation law Eq.(42).

Proof Consider a Lyapunov function as follows:

Substitute Eq.(46)into Eq.(45),and ˙Li（t ）can be rewritten as

It should be noticed that once the inequality(54)is satisfied,˙L（t ）<0.The system is asymptotically stable in the sense of Lyapunov.In addition, it can be seen that ‖Z‖ is uniformly ultimately bounded.Moreover, ‖W～i‖ is uniformly ultimately bounded.

6.Results and discussion

In this section,numerical simulation results are presented.The baseline configuration is scheduled flying from waypoint A to B.In general,choose A as the origin of coordinate xeyezewith zero velocity and acceleration.B is a waypoint withePL= [80 10 10]Tand zero velocity and acceleration.

6.1.Load trajectory

During flight,Euler angles of the load are set as 0°in order to have minimal aerodynamic forces.Load trajectory is developed based on the third order polynomial as shown in Fig.7, where the legend x, y and z represent the related component along xe, yeand zeaxis respectively.It can be noted that at 10 s the forward flight speed gets its maximum value 7.5 m/s corresponds to the advance ratio of 0.07.Besides,there exists a maximum forward acceleration of 1.2 m/s2at 4 s and a minimum one of -1.2 m/s2at 16 s.The curves of position,velocity and acceleration along yeand zeaxis are the same because the position targets along yeand zeaxis are 10 m.

6.2.Power-based load distribution

6.2.1.Comparison of power-based and force-based load distribution strategies

Simulation results by power-based and force-based strategies are shown in Fig.8 and Fig.9 respectively.To facilitate comparative analysis,it is assumed that there is no difference in the performance between the four tandem helicopters in this subsection.Therefore, kiin Eq.(22) equals 1.Different from Ref.,14cable angle α instead of load trajectory is introduced to equalize cable tension and reduce design variables in the force-based strategy.

As we can see,in Fig.8(b),the power consumptions of four helicopters are equal.As shown in Fig.9(a),at 10 s,there exists the maximum mean square deviation 7.7 × 10-3among the four cable tensions, but it is not an order of magnitude compared with the real value of the cable tension.These are consistent with the optimization objectives.

The difference between cable forces in Fig.8(a) exists to make the power consumption of each helicopter equal.And four cable forces all change smoothly without large fluctuation, which indicates the feasibility of the power-based load distribution strategy.

As shown in Fig.9(b), from 4 to 16 s, the power difference between the front helicopters (Helicopters 3, 4) and the rear helicopters (Helicopters 1, 2) is large.Referring to Fig.7, this corresponds to the time period when the forward flight speed is larger than 3.5 m/s.Specially, at 10 s when the forward flight speed reaches its maximum value of 7.5 m/s, the power consumptions of Helicopters 1,2 are 487.5 W, and the ones of Helicopters 3,4 are 623.0 W.The difference reaches 21.8 %.By power-based strategy, at 10 s, the power consumptions of four helicopters are 525.5 W.Compared with those by forcebased strategy, the power consumptions of Helicopters 3, 4,i.e., the maximum power consumptions of the multi-lift system, are reduced by 15.7 %.Although the powers of Helicopters 1,2 are increased by 7.8 %, the system power consumption is reduced by 5.4 %.

Fig.7 Designed load trajectory.

Fig.8 Simulation results by power-based strategy.

Fig.9 Simulation results by force-based strategy.

There is also big difference in cable angles between powerbased and force-based strategy, as shown in Fig.8(c), (d) and Fig.9(c), (d) respectively.Taking 10 s as an example, for power-based strategy, we have

It can be seen that in Eq.(55) the absolute values of β for Cables 1, 2 (3, 4) are smaller (larger) than those in Eq.(56).Referring to Fig.4(b),this means that the power consumption of Helicopters 1, 2 (3,4) are increased (reduced), which is beneficial to equalize the power consumption.The change of angle α is to meet the dynamic constraints of the load.

In addition,no matter by power-based or force-based strategy,the distances between helicopters are less than the safe distance constraint 4.5 m, i.e., 2.5 rotor diameters.Fig.8(f) and Fig.9(f) depict the three-dimensional trajectories of the multi-lift system.As we can see, the trajectories of the load and four helicopters change smoothly,which verifies the effectiveness of the two load strategies.

In summary, when flying at low speeds with small maneuver, the power-based and force-based load distribution strategies are equivalent.However, with the increase of forward flight speed, the difference of power consumption between the front and rear helicopters increases.Considering that the power consumption directly affects the endurance and flight range of each helicopter, the power-based strategy is better.

6.2.2.Simulation results when k3, k4equal 0.85

This subsection discusses the feasibility of power-based strategy when the performance of the four helicopters are different.It is assumed that the endurances of Helicopters 1,2 are 85%of those of Helicopters 3,4 under the condition of equal power consumption.Therefore,in Eq.(22),k1,k2are set to 1 and k3,k4are set to 0.85.The simulation results are depicted in Fig.10.The power consumptions of helicopters 1, 2 are basically 85 % of those of Helicopters 3, 4 during the simulation process.

Taking 0 s as an example, in Fig.10, we have

Referring to Fig.4(a), the power consumption in hovering is mainly related to the cable force and angle α.To realize that powers of Helicopters 1,2 are 85%of those of Helicopters 3,4, the cable forces and angles α of Helicopters 3, 4 all need to be increased.Moreover, to satisfy the load dynamics constraint in Eq.(17),the cable forces and angles α of Helicopters 1, 2 are increased too.Besides, the change of β is utilized to meet the load dynamics and safe distance constraints.

In addition,it can be seen from Fig.10(e), (f)that the distances between helicopters are safe and the cables are not crossed, which indicates the validity of power-based strategy when there are performance differences between helicopters.

6.3.Game-based tracking control

The purpose of game-based tracking control is to calculate u～to make x～equal 0 no matter under disturbances or when there is difference in control requirement between helicopters.To facilitate comparative analysis, in the following simulations, the initial condition is that u～Lequals 1 and other items of x～equal 0.

6.3.1.Results with different value function

The following three cases are designed to validate the ability of game-based control to deal with the situation when there is difference in control requirement between helicopters.

(A) Case 1: the value functions of four helicopters are the same.And the weight matrices are

(B)Case 2:the value functions of Helicopters 1,2,3 are the same as those in Case 1.For the value function of Helicopter 4,Q4is set to 2 I12×12and other items related to Helicopter 4 in Eq.(59) remain unchanged.

(C)Case 3:the value functions of Helicopters 1,2,3 are the same as those in Case 1.For the value function of Helicopter 4,R44is set to 0.02 I3×3and other items related to Helicopter 4 in Eq.(59) remain unchanged.

Considering that there is no difference among the value functions of Helicopters 1, 2, 3, Helicopter 1 is taken as an example for analysis below.As can be seen in Fig.12,in three cases, the components of f～

cable;1are almost equal.

Fig.10 Simulation results when k3, k4equal 0.8.

Fig.11 Simulation results of items of x～by game-based control.

The simulation results of value function are shown in Fig.13.As we can see, the value functions of Helicopter 1 in three cases are almost equal.For Helicopter 4,the value function in Case 2 is the largest because Q4is set to 2 I12×12.In Case 3,though R44is twice those in Cases 1 and 2,the value of R44is 0.02 I3×3.It is not of the same order of magnitude as the value of Q4,so there is almost no difference between the value functions in Cases 1 and 3.

Fig.12 Simulation results of u～by game-based control.

6.3.2.Results with disturbances

This subsection shows the robustness of game-based tracking control in the presence of disturbances.The disturbance v in Eq.(25) is set to

It can be seen that the amplitude of Eq.(60) is 0.2 and the period is 2 s.The simulation results by game-based control with disturbances are shown in Fig.14.

It can be seen that the trends of items of x～in Fig.14 are not different from those in Fig.11.Meanwhile, there exist fluctuations with periods of about 2 s for items of x～resulting from the additional disturbances.But the fluctuation amplitude is small.After 5 s, the fluctuation amplitude of u～L is less than 0.02 m/s, and the fluctuation amplitudes of q～Land θ～Lare less than 1×10-3.This indicates that the game-based tracking control has strong robustness.In addition,as shown in Fig.14(b)-(e),after 2.3 s there also exists fluctuation with period of 2 s for the cable forces, which is the compensation for the additional disturbances.

Fig.13 Simulation results of value function by game-based control.

6.4.Simulation results based on whole load distribution strategy

The whole load distribution strategy is obtained by combining the power-based strategy and the game-based tracking control.During the simulation process, disturbance v is set as that in Eq.(60) too.As can be seen in Fig.15, the value of fcable;iis the sum of the cable force calculated by power-based strategy and the one by game-based control.Similarly, the fluctuation with a period of about 2 s is introduced to compensate for the disturbance.The legends xtra, ytra, ztrarepresent the results by power-based strategy only.

Fig.14 Simulation results by game-based control with disturbances.

Fig.15 Simulation results of u～based on whole load distribution strategy.

Fig.16 Simulation results of x～only by power-based strategy under disturbances.

Fig.17 Simulation results of x～based on whole load distribution strategy.

Fig.16 and Fig.17 show the results of x～by the power-based strategy only and by the whole load distribution strategy respectively.In the presence of disturbances, by the powerbased strategy only,the calculated expected cable force cannot carry the load to the specified position,but will cause the oscillation and divergence of the whole multi-lift system.At 20 s,xL, yL, zL, φL, θLand ψLconverge to -494.6 m, 200.2 m,980.6 m,130.0°,34.4°and 114.3°respectively.And in this status, the frontal area of the load is large and there exists unsteady aerodynamic force.

Referring to Fig.17, four helicopters can carry the load to the desired position cooperatively by the whole load distribution strategy.At 20 s, xL, yLand zLare 80.00 m, 10.00 m and 9.99 m respectively.Although there exist fluctuations with periods of about 2 s for the velocity, Euler angle and angular velocity, the amplitude is small.And it is acceptable for baseline configuration.

7.Conclusions

To improve the flight performance of the baseline configuration, the load distribution strategy based on power consumption and robust adaptive game control is proposed considering the differences in power consumption and relative position between helicopters as well as the interactions.First,based on the load-leading hierarchical control,the load allocation problem is divided into two parts: the calculation of expected cable force and the calculation of the antidisturbance cable force.The effects of forward flight speeds,cable forces and cable angles on the power consumption of helicopter are considered in detail.Then, the power agent model is trained based on BP neural network and the safe distance constraint is set as 2.5 rotor diameters to reduce aerodynamic interference.Subsequently, aimed at minimizing the maximum equivalent power of helicopters, the expected cable force is calculated by optimization method based on MATLAB/fmincon algorithm.For the calculation of the anti-disturbance cable force, considering the difference and interaction between helicopters, the robust adaptive differential game control is proposed.The simulation results indicate that the power-based strategy is applicable to the whole flight trajectory and the situation when there are performance differences among helicopters.The game-based tracking control can consider interaction between helicopters, can deal with different objective functions, and has strong robustness and small steady-state error.Based on the entire strategy,the cable force can be reasonably allocated to resist disturbance and further improve the flight performance of the whole system.

Finally,it should be pointed out that the proposed load distribution strategy not only is applicable to the baseline configuration, but also can be applied to the multi-lift system with two or more aircraft by substituting the power agent model,changing the number of aircraft,increasing or decreasing numbers of constraints and so on.The Appendix describes how to apply the proposed load distribution strategy to other multilift system in detail.

In the future, we will concentrate on speeding up the optimization solving algorithm for real-world experiments,getting load distribution strategy considering obstacle avoidance, and designing helicopter control strategy to transport the load to target position precisely.

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.

Appendix.The appendix describes how to apply the proposed load distribution strategy to other multi-lift systems.Particularly, the establishment of the power agent model, the effects of the number of aircraft and the constrains are discussed.

(1) Power agent model

First,to obtain the original data of power consumption,we should trim the helicopter loaded by cable forces.For other aircraft, though their manipulation methods or aerodynamic components may be different from the tandem helicopter,the trim equations all include the key flight dynamic equations and the trimming methods are essentially consistent no matter by Newton’s approach or other advanced optimization methods.Besides, the input variables of the power agent model based on BP neural network include: advance ratio, cone angle, regional angle and the magnitude of cable force.And the output variable is the power consumption.It is obvious that the above five variables are convenient to get after trimming.In this way, the power agent model of other aircraft can be obtained.

(2) Effects of the number of aircraft

In the condition where there are only two aircraft in the multi-lift system, the two aircraft are generally arranged side by side or in front and back.In other words, the cable points on the load are on the axis of coordinates xLyLzL.Therefore in this case, the six-freedom-movement of the load cannot be achieved and some other constraints must be added to solve Eq.(17).

If there are three or more aircraft in the multi-lift system,Eq.(17)is generally solvable as long as the rank of the matrix in the left side of Eq.(17)is not less than 6.And if there are too many aircraft, the biggest challenge becomes collision avoidance.

(3) Constraints

If there are only two aircraft and if the two aircraft are in the side-by-side arrangement, to solve Eq.(17), the pitch movement must be ignored; if the two aircraft are in the front-by-rear arrangement, the roll movement must be ignored.And if there are too many aircraft,we should reasonably and carefully arrange the positions of aircraft and determine the safe distance constraints among these aircraft.