Flight dynamics modeling and analysis for a Mars helicopter

Hong ZHAO, Zhiwei DING, Gen LENG, Jinbo LI,*

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

b China Helicopter Research & Development Institute, Jingdezhen 33300, China

KEYWORDS

Abstract Flight dynamics modeling for the Mars helicopter faces great challenges.Aerodynamic modeling of coaxial rotor with high confidence and high computational efficiency is a major difficulty for the field.This paper builds an aerodynamic model of coaxial rotor in the extremely thin Martian atmosphere using the viscous vortex particle method.The aerodynamic forces and flow characteristics of rigid coaxial rotor are computed and analyzed.Meanwhile, a high fidelity aerodynamic surrogate model is built to improve the computational efficiency of the flight dynamics model.Results in this paper reveal that rigid coaxial rotor can bring the Mars helicopter sufficient controllability but result in obvious instability and control couplings in forward flight.This highlights the great differences in flight dynamics characteristics compared with conventional helicopters on Earth.

1.Introduction

The Ingenuity Mars Helicopter, which landed on Mars successfully in February 2021 with the Perseverance rover, has accomplished many autonomous flights.As an air exploration platform,the Mars helicopter can not only assist rover navigation,but can also supplement the rover to explore inaccessible areas and explore Mars more effectively1,2.

Mars has mass around 11% of the Earth with the average gravitational acceleration about 3.73 m/s2.Though the reduced gravity is beneficial to helicopter flight on Mars,the extremely thin atmosphere (average air density is 0.0166 kg/m3which is only around 1/70 of the Earth3) imposes great challenges.To further understand the flight dynamics characteristics of the Mars helicopter and support plant models for subsequent flight control simulations, extensive flight dynamics modeling and analyses for the Mars helicopter are required.

At present, only a few authors have carried out flight dynamics research on the Ingenuity Mars helicopter4,5.Flapping response characteristics of the rigid rotors under the extremely thin atmosphere are analyzed in detail.Influences deriving from the rigid rotors for controllability and stability of the Mars helicopter are discussed.Flight dynamics characteristics of the Mars helicopter are well understood.

In the flight dynamics model built by Grip et al.,a dynamic inflow model is employed to establish the aerodynamic model of the coaxial rotor.6Aerodynamic interferences between the upper and lower rotor are simulated by applying a correction to inflow velocity.As the selection of correction coefficients is semi-empirical, deviations may occur in the results of Mars helicopter rotor aerodynamic characteristics which will reduce the modeling precision of the flight dynamics for the Mars helicopter.

Studies to establish rotor aerodynamic models with high fidelity are required not only for calculating the aerodynamic forces on the coaxial rotor more accurately, but also for evaluating the flight dynamics characteristics of the Mars helicopter more precisely.At present, high-fidelity rotor aerodynamic modeling methods mainly involve Eulerianbased Computational Fluid Dynamics (CFD) methods and Lagrange-based vortex methods.

Young et al.7and Dull et al.8have carried out many studies on aerodynamic computation and analyses of coaxial rotor for the Mars helicopter using the OVERFLOW and RotCFD software packages.The near blade surface aerodynamics,such as viscous and compressible effects,are predicted well by CFD methods, but these methods are computationally expensive and inherent numerical dissipation gives unrealistic far field rotor wake.These problems mean that CFD methods are hard to employ for flight dynamics modeling of the Mars helicopter.

An aerodynamic model of the coaxial rotor type Mars helicopter was built by Koning et al9,10.using Comprehensive Analytical Model of Rotorcraft Aerodynamics and Dynamics(CAMRAD Ⅱ), which is based on coupling lift line theory,constant two-dimensional airfoil aerodynamic data, and the free rotor wake model11.CAMRAD II has undergone extensive correlation with experimental data on helicopters, including coaxial designs12.

Compared to the CFD method, the vortex method has higher efficiency and greater flexibility on aerodynamic computation of the coaxial Mars helicopter.In this paper, the aerodynamic model of the coaxial rotor type Mars helicopter is established based upon the Viscous Vortex Particle Method(VVPM)13.The VVPM, which requires fewer empirical parameters, is better able to capture rotor wake interaction and blade-vortex collision interferences between the coaxial rotor.To ensure confidence,the model is verified and updated using measurements of rotor performance and loads on simulated Martian atmospheres.The rotor aerodynamic surrogate model is built and employed to model the flight dynamics to realize higher efficiency on analyzing the Mars helicopter flight dynamics characteristics in hovering and forward flight.

2.Aerodynamic modeling of coaxial rotor for Mars helicopter

2.1.Overview of Mars helicopter

The Mars helicopter in this paper features coaxial rotor driven by two individual motors.The coaxial rotor is capable of variable speed and can perform collective pitch and cyclic pitch control.Key properties of the Mars helicopter are given in Table 1.

2.2.Control and flapping motion of rigid rotors

The upper and lower rotors of the Mars helicopter can perform collective pitch control and individual longitudinal/lateral cyclic pitch control.The expressions for the upper and lower rotor blades’ pitch angle are given as follows:

Table 1 Summary of key helicopter properties.

where θUand θLare pitch angles for the upper and lower rotor blades; δcolis the collective blade pitch control; δpedis the differential collective blade pitch control;δlatand δlonare longitudinal and lateral cyclic pitch controls, respectively; ψUand ψLare azimuthal phase angles for the upper and lower rotor blades with contrary rotational directions; Γ is the control phase angle of the rotors.

Affected by the extremely thin atmosphere, aerodynamic flapping damping for rotor blades of the Mars helicopter is extremely low, so that flap oscillations easily occur when the rotor rotational frequency is approached.For safety, the rotors of the Mars helicopter use a rigid hingeless design with a stiffened root segment, and the mass of a single blade is less than 45 g.Flap frequency nondimensionalized by the rotor frequency ω-β1is increased to around 214,15.

In hovering, the relationship between the flapping angle and the cyclic control angle for the blades is given by

where β1sand β1care the lateral and longitudinal flapping angles,respectively,and γ is the blade Lock number with value around 0.33.

Amplitude Amp and control phase angle Γ for blade flapping are given by the following:

It can be seen from the above that the flapping angle and control phase angle are approximately 0° and as such the flapping motion of the blades can be ignored.When the Mars helicopter is performing longitudinal and lateral cyclic controls, upper and lower swashplates tilt in opposition, which is quite different from conventional coaxial helicopters on Earth.

2.3.Aerodynamic modeling of coaxial rotor

Due to the low density and speed of sound of the Mars atmosphere, Mars helicopter flies in the unusual aerodynamic regime of low Reynold number with high Mach number.Also,the axial distance of the coaxial rotor is so close that the wake from the upper rotor and lower rotor interact heavily.All the above factors make rotor aerodynamic modeling of Mars helicopters challenging.

The first principle VVPM, is adopted to simulate the wake of the coaxial rotor.VVPM is a mesh-free method based on Navier-Stokes equations in a Lagrange framework which can take vortex stretching and viscosity effects into account.As VVPM does not rely on empirical parameters, it has excellent versatility, allowing it to be applied to the working environment of Mars helicopters.As a typical N-body problem,VVPM can be integrated with the Fast Multipole Method(FMM) algorithm16.Moreover, it can be applied in multicore CPU or GPU workstations to obtain higher calculation efficiency and allow the model to be balanced in accuracy and speed.

The discretion vorticity field of the VVPM model can be expressed as:

where rp(t ),αp(t )and Rpare the spatial position,vorticity density and radius of the p-th particle respectively, and ζ(r ) is a truncation function which is used to consider the vorticity distribution induced by vortex particles.

Consider the vortex dynamic equation:

where ω=?×u is the vorticity field, D()/Dt=

Combining the formulas above, allows us to derive the VVPM governing equations:

where the local velocity,u( rp(t ),t),of vortex particles is determined by the flow-stream speed and the induced velocity due to the whole vortex field, and the induced velocity term is solved using the Biot-Savart Law.is the viscous diffusion term which is to be solved using the Particle Strength Exchange (PSE) method.In PSE, the ?2operator is substituted by an integral operator to avoid direct numerical integration, which is expressed as below:

where Vpand Vjare the p-th and j-th vortex particles’volumes,respectively.The kernel function decreases rapidly with increasing distance, so only Pinearest-neighbor particles are considered.Particles outside this region are neglected.The kinematic viscosity of Mars’s atmosphere is around 6×10-4,which is an order of magnitude higher than Earth’s.This makes the Vortex Reynolds number an order of magnitude smaller than Earth’s, making the vortex structure steady.With rotation revolutions steps, lots of steady vortex particles are shed into the wake-field and influenced by the strong tip vortex, which makes these vortex particles face the problem of density misalignment.An adaptive vortex adjustment method17is adopted to improve the distribution of the particles and ensure the calculation stability.

All aerodynamic surfaces in the flow field can generate vorticity sources,giving rise to new particles at each time step.By calculating the induced effects between any two particles and superimposing them, the overall interference and evolution of the coaxial rotor wake flow field can be simulated.In this study, the second order lifting line method is used to model the rotor blade.The effects of viscosity and compressibility are considered using the pre-prepared C81 table of the rotor airfoil.The change of the bound vorticity in azimuthal and radial direction then forms the rotor wake field.The source vorticity of a blade segment due to conservation of vorticity is:

where Γbis the bound vortex vector,vbis local velocity,is the shedding vortex, and vb??Γbis the trailing vortex.

The coaxial rotor’s rotation speed is much higher than Earth’s to generate enough lift in the thin Mars atmosphere.Besides, the speed of sound in Mars’s atmosphere is much lower than the Earth.Moreover, the Mars helicopter rotor’s taper is designed for better aerodynamic efficiency, rapidly changing the spanwise Reynolds number.The spanwise Mach number changes more rapidly than in the Earth, which highly increases the change rate of the spanwise Mach number in the outside section and makes it higher than 0.62 in the tip.As a result, the blade’s spanwise lift and drag coefficient change rapidly, especially in the outside section near the tip.

The blade spanwise aerodynamic discretization should use a tip-cluster rule to calculate the blade’s air loads accurately.In classic VVPM,the newly generated particles adopt the same rule with the aerodynamic discretization.However, this may lead to new particles in the tip having smaller sizes than the minimum vortex particle field resolution.An independent interpolation surface is adopted for the vortex particle generation to solve the problem.The geometries of the aerodynamic discretization and the interpolation surface are isolated, but they both match the physical characteristics of the blade.The blade’s bound circulation of the aerodynamic discretization is then interpolated into this interpolation surface with an accurate monotonicity-preserving cubic interpolation algorithm18to ensure the information communication of these two meshes is consistent.Compared to most popular interpolation algorithm, such as the Spline interpolation, the algorithm adopted here can keep the monotonic characteristics19of the original bound circulation to ensure the total vorticity conservative.

Fig.1 Mars helicopter technology demonstration.

Fig.2 Rotor blade of Mars helicopter.

Although the method described above is based on the expansion of a single rotor,it can be applied to both the upper and lower rotors of a coaxial rotor system in the Mars helicopter.When calculating the coaxial rotor, the upper and lower rotors are initialized synchronously at the initial time step.The individual vortex particle field of the upper and lower rotors are updated according to the same time step, and all vortex particles are incorporated into the overall vortex elements in space to calculate mutual interference20.The wake interference effects of the upper and lower rotors can thus be naturally reflected in the calculation results over time.(See Fig.1 and Fig.2)

The method described above takes the aerodynamic features of Mars helicopters into account and allows accurate and efficient coaxial rotor wake simulation.It can be used to calculate the force and moment of the Mars helicopter’s coaxial rotor following the whole solution process for the method as described in Fig.3.

Aerodynamic testing for the coaxial rotor of the Mars helicopter is carried out in a vacuum chamber with dimensions 4 m × 4 m × 10 m (see Fig.4).The atmospheric pressure in the chamber is reduced to 1.5 kPa and the air density inside is around 0.017 kg/m3, corresponding to the Martian atmosphere.Rotor thrusts and torques under specific rotational speed and different collective pitch are measured so that aerodynamic models of the coaxial rotor can be verified and amended by comparison with experimental data (see Fig.5).

2.4.Aerodynamic characteristic analysis of coaxial rotor

The rotor wake of the Mars helicopter in hovering and forward flight(10 m/s)is shown in Fig.5.Obvious blade-tip vortices can be observed.The distance between the upper and lower rotors is so small that aerodynamic interference from the two rotors is significant.In hovering, influenced by the lower rotor, the contraction of the wake from the upper rotor accelerates along the radial direction.After passing through the lower rotor disk,the upper rotor wake and the lower rotor wake interfere with each other and intertwine,making a vortex wake field with more significant distortion and greater complexity.

In forward flight, affected by the forward flow, the rotor wake tilts backward entirely.The longitudinal distribution of induced velocity in the rotor disk varies significantly.For the front rotor disk,the induced velocity is relatively small,so that the vortex wake is greatly influenced by the forward flow and the wake has a larger tilting angle.For the rear rotor disk,the opposite behavior occurs.

Lift distributions of the Mars helicopter in hovering, forward flight,and climbing are analyzed with different collective pitch controls.The calculation points are given in Table 2.

In hovering and forward flight,lift distributions of both the upper and lower rotors are illustrated in Fig.6.In hovering flight, the lift of the upper and lower rotors is distributed evenly along the circumferential direction (see Fig.7(a)).In forward flight,since the induced velocity of the front rotor disk is relatively small, lifts generated by both the upper and lower rotors are more concentrated in both windward and advancing blade sides.An obvious nose-up moment is formed by mutual superposition of pitch moments from the upper and lower rotors such that roll moments from the upper and lower rotors mitigate each other, making the roll moment a small value.

Fig.3 Flowchart of coaxial rotor aerodynamic modeling.

Fig.4 Aerodynamic test for coaxial rotor in vacuum chamber.

Fig.5 Thrust coefficient vs power coefficient, including VVPM and experimental results.

It can be observed from Fig.7(b) and Fig.7(c) that in forward flight downward vertical movement is superposed on the helicopter such that larger incremental lift is generated for the rotor blades in the windward side, forming a greater nose-up moment.Fig.7(b)and Fig.7(d)indicate that,in forward flight,increasing the rotor collective pitch also exacerbates the asymmetric distribution of lift, which in turn increases the nose-up moment such that couplings between the collective control and the longitudinal and lateral cyclic control are induced.

By adding perturbation to the coaxial rotor in hovering, to make the rotors pitch down at 30 (°)/s, a small nose-up moment acts on the coaxial rotor (Fig.8) which reflects weak damping for pitch movement.

3.Flight dynamics modeling and analysis for Mars helicopter

3.1.Flight dynamics modeling

Aerodynamic components of the Mars helicopter include the coaxial rotor, airframe and landing gear.Aerodynamic coefficients for the airframe and landing gear assembly under different attack angles and sideslip angles are computed by the CFD method (see Fig.9).Combining with flight velocities allows aerodynamic forces and moments of hovering and forward flight to be calculated.

Fig.10, as an example, shows the change of the airframe’s lift, drag, and pitch moment versus angle of attack at 10 m/s airspeed.The forces and moment are small due to the Mars atmosphere’s low density.The drag and the pitch moment have little change, with the indicated angle increasing from 0° to 8 °, while the lift gets an increase of 0.025 N.

The viscous vortex particle method,which can lead to complex rotor aerodynamic models with high computational requirements, is challenging to employ directly in flight dynamics models of the Mars helicopter.Hence, a surrogatebased approach is established using the Kriging method to improve computational efficiency.Kriging agent model as a response surface model can measure model’s inaccuracy performance and shows good approximation as well as prediction ability to aerodynamic optimization problems21–23.Suppose there are n sample points ｛x1, x2,???xn｝ and corresponding response value y= [y1,y2,???yn]T, where xiis a vector whose dimension is n, the number of the input variables.

Table 2 Computation points of the coaxial rotor.

Fig.6 Wake simulations of coaxial rotor.

The kriging model can be represented as

For the stochastic process z(x ),the correlation between two points can be expressed as

where the superscript k represents the k-th input variable,θkis the weight related to the k-th input variable.

On the basis, to minimize the mean square error, the kriging model can be rewritten as

where 1 is a n×1 matrix with all 1 elements; R is a n×n matrix representing the correlation matrix, and its elements can be calculated by Eq.(14), and

At last, the problem can be converted to find θkto get the maximum likelihood estimation value, as follows:

For the kriging agent model, a more detailed derivation process can be found in Ref.24.

The Mars helicopter features axial symmetry.The longitudinal and vertical speeds (Vxand Vz), collective pitch control δcol, longitudinal cyclic control δlon, lateral cyclic control δlat,and differential collective control δpedare taken as inputs.The aerodynamic forces and moments are taken as outputs.According to the flight envelope,296 sampling points are generated.Cross validation is employed to verify the precision of the surrogate model.The results of the cross validation indicate that the square error of the surrogate model is up to 0.99 which implies high precision of the surrogate model.Thus, the surrogate model can be used to replace the vortex particle model which also has high precision.

The two rotors of the Mars helicopter rotate in contrary directions.The gyro moments of the rotors counteract each other sufficiently well that they can be neglected.Bringing expressions of the aerodynamic forces and moments for the aerodynamic components into the equilibrium equations yields:

where F and M are aerodynamic force and moment, respectively; m is the mass of the Mars helicopter; g is the gravitational acceleration at the surface of Mars; the subscript CR means the coaxial rotor, and the subscript AF represents the airframe and landing gear ensemble.

3.2.Trimming

Transitioning from hover to forward flight at 10 m/s airspeed,the pitch angle of the Mars helicopter varies from 0°to-2.5°.Affected by side forces,the roll angle increases slightly with the growth of forward speed, as seen in Fig.11.

It can be seen from the computational values in Fig.12,affected by the upper rotor wake, that the collective pitch required by the lower rotor is around 1 degree larger than for the upper rotor.As forward flight speed increases,air flow passing through the coaxial rotor increases so that the collective control required by the coaxial rotor is reduced and the longitudinal cyclic control increases.The variation tendency of the attitude angles and control inputs is similar to conventional helicopters on Earth25.

3.3.Controllability and stability

Linearizing the flight dynamics model of the Mars helicopter under equilibrium states yields the linearized equation of motion which is given by

where A is the state matrix;B is the control effectiveness matrix;xMH= [VxVyVzφ θ ψ p q r ]is the state vector,Vx,Vyand Vzare the components of the flight velocity on the x,y and z axes of the body axis system.p,q and r are the roll rate, pitch rate and yaw rate, respectively;uMH= [δcolδlonδlatδped] is the control vector.

The derivatives in matrix A reflect the influence on aerodynamic forces induced by variation of flight states.Stability analysis can be carried out using matrix A.

Fig.7 Lift distribution of coaxial rotor.

It can be seen from the analyses in Section 2.4 that the pitch moment of the Mars helicopter increases with the growth of both forward and vertical speeds in forward flight, which corresponds to the terms ΔMy/ΔVxand ΔMy/ΔVzin matrix A,as shown in Fig.13.The two terms imply velocity stability and pitch-heave instability.

Lateral movement of the Mars helicopter is similar to the longitudinal case.The term ΔMx/ΔVy<0 in matrix A implies the dihedral effect.Due to the axial symmetry configuration,damping of the azimuth direction is almost zero, leading to weak stability in the azimuth direction.

Eigenvalues of the Mars helicopter are selected under three different speeds which are listed in Table 3.

Fig.8 Changes of pitch moments of coaxial rotor.

Fig.9 Velocity flow field of airframe at 10 m/s airspeed.

Fig.10 Forces and moment vs airframe angle of attack.

In forward flight, the larger negative real root of the longitudinal eigenvalues corresponds to the longitudinal velocity,while smaller negative real root corresponds to the vertical velocity which means low convergence speed.The conjugate complex roots correspond to the pitch-heavy mode which is induced by the couplings between attack-angle instability and speed stability.As the forward flight speed increases, the pitch-heavy mode turns into pitch divergence.The stability can be improved properly if the center of gravity of the Mars helicopter is moved forward by adjusting the location of airborne equipment.

Fig.11 Attitude angles of Mars helicopter in forward flight.

Fig.12 Control inputs of Mars helicopter in forward flight.

Fig.13 Main aerodynamic derivatives of Mars helicopter.

In forward flight,the larger negative real root of the lateral eigenvalues corresponds to the roll convergence mode.This is similar to the pitch rate converging rapidly.The conjugate complex roots in the movement appear as Dutch roll mode due to the near zero of the azimuth damping.

As forward flight speed increases, movement of the eigenvalues for the Mars helicopter are illustrated in Fig.14.It can be seen that the larger the forward flight speed,the poorer the stability becomes, making the Mars helicopter diverge easily when it suffers external disturbance.(See Fig.15)

Controllability of the Mars helicopter can be analyzed through the control effectiveness matrix B.The rigid rotor hub of the Mars helicopter has high controllability in pitchand roll movements.There are certain cross-axis couplings for the longitudinal and lateral controls in forward flight.

Table 3 Eigenvalues of Mars helicopter.

Fig.14 Root loci of Mars helicopter dynamics(hover to forward flight at 10 m/s airspeed).

Fig.15 Bode diagram of longitudinal dynamics for Mars helicopter.

In forward flight, the wake angle is changed by the collective control and the additive pitch moment is induced, which appears as significant control couplings for the collective pitch control with respect to the pitch movement.

In yaw control,due to the small moment of inertia for yaw,the Mars helicopter still has high controllability even though the rotors have high speed and small torques.Similarly, there are slight control couplings for the yaw control with respect to the longitudinal and lateral controls in forward flight.

Take the longitudinal dynamics as an example.Controllability of the Mars helicopter is analyzed from the perspective of phase and bandwidth.The gain bandwidth is around 5 rad/s which means the bandwidth is sufficient.

4.Conclusions

The viscous vortex particle method has been employed to establish aerodynamic models for the coaxial rotor of the Mars helicopter.Special aerodynamic characteristics of the coaxial rotor have been analyzed through the rotor wake and the distribution of disk loads.The aerodynamic model, which was introduced as a surrogate model, has been employed in flight dynamics modeling to give reliable results over a much broader range of operational conditions compared to previous work.The flight dynamics model can be used for flight control design in future development efforts.The main conclusions are drawn as follows:

(1) In the extremely thin atmosphere of Mars,the variation tendency of the attitude and control for the Mars helicopter are similar to conventional helicopters on Earth.The aerodynamic drag of the Mars helicopter for forward flight is small.Longitudinal cyclic control is mainly used to eliminate nose-up moments caused by the coaxial rotor in forward flight.

(2) The Mars helicopter adopts rigid rotors with little flapping motion so that adjustment from the flapping to the rotor thrusts can be neglected.As forward flight speed increases, more obvious lift gradient is generated by the rigid rotors so that the pitch stability of the Mars helicopter becomes worse, making the movements diverge quickly when the helicopter is disturbed.

(3) The rigid rotors bring larger hub moments so that the controllability of the Mars helicopter is sufficient in all directions.Meanwhile, rotor control couplings become more notable in forward flight.In addition, control phase angles of the rigid rotors are extremely small so that the upper and lower swashplates are required to tilt reversely, which is significantly different to coaxial helicopters on Earth.

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

The work was supported by the Priority Academic Program Development of Jiangsu Higher Education Institutions,China.And thanks Jinting XUAN,Chaoqun ZHANG for their helping in theoretical derivation and feasibility analysis.