Document
Hexacopter F light Dynamics on Earth and Martian Surfaces
Raghuvir Singh Aerospace Engineer NASA Ames Research Center Moffett Field , CA , USA ABSTRACT Since the atmospher ic conditions and gravitational field are different for Mars and Earth, the operating flight environments are different, and necessitate Earth - based appropriate modeling, analysis, and simulation for Mar s flight vehicl e s . This paper compares a hexacopter’s flight dynamics in the two environments. The analysis is necessary to determine if a dynamically matched surrogate hexacopter can be designed to conduct reliable testing on Earth, with the goal of successfully operation on Mars. To answer the question, the compr ehensive tool FLIGHTLAB ® is used to model the hexacopter flight dynamics. F requency responses in heave, pitch, roll, and yaw rates of the hexacopter in hover are analyzed. The simulation result shows that each attitude response of the designed hexacopter r esponds very differently in the two environments. The closed - loop response of the hexacopter can be made stable on Earth but not on Mars. Therefore, the proportional feedback technique cannot be utilized to stabilize all four responses altogether. Due to the dynamics of the model being very different in each atmosphere, it is yet not feasible to create a dynamically matched surrogate helicopter that can operate in both environments.
the way for future aerial explorers at Mars which is discussed INTRODUCTION later in the paper.
Mars , known as the Red Planet, provides an ideal terrestrial F lying on Mars constitutes a set of challenges and la boratory to understand the early history of the solar system.
requirements. The challenges are imposed due to the Martian T he p lanets Mars, Venus, and Earth all formed from the same atmosphere being very different when compared to Earth’s minerals and elements; however, all three planets went atmosphere. Carbon d ioxide composes 95.32% of the through dif ferent epo chal period s . Mar s’s surface pressure is Martian atmosphere while the remainin g 4.68% is composed 1% of the surface pressure of the Earth ( R ef. 1). Moreover, of argon, oxygen, carbon monoxide, water vapor , and trace Mars has a history of dehydration and, loss of its atmosphere , gasses ( R ef. 4). Moreover, Mars experiences a temperature and surfac e. Research is being conducted to understand Mars range from - 140 C at the poles to up to 30 C on the equator transformation as a planet. Specifically, NASA’s Mars during daytime ( R ef . 5). Besides the high variance in Exploration Program is studying the formation and early temperature, Mar s’s atmosphere is also very different from evolution of Mars as a planet, the history of geological Earth’s in numerous other ways. The pressure on Mars processes, the potential for Mars to have hosted life, and the averages 6.36 millibars which is 0.6% of the Earth's future exploration of Mars by humans .
atmospheric pressure ( R ef. 4). Air density at the Mars surface is about 0.02 kg / m . Furthermore, Mars's gravitational Orbiters like MAVEN (2003) helped explore the upper acceleration is about one - third of Earth’s gravitational atmosphere of Mars whereas, s tationary landers like Insight acceleration ( R ef. 4). Lower gravitational p ull should result in (2018) made it possible to detect quakes on Mars and revealed greater lift capability , yet the reduction in lift due to different details about the depth and composition of Mar s’s crust, atmospheric conditions eliminates this advantage .
mantle, and core ( R ef. 2) . Furthermore, t he Perseverance rover helped understand the dust processes on Mars and The challenges of flying on Mars put greater reliance on using contributed to a body of knowledge that could one day help simulation tools that can replicate Martian flight predict the dust storms that Mars is famous for, which poses environment s . This paper provides a flight dynamics a threat to future robotic, and human explorers. R over s are also designed to seek evidence of life , study rocks and soil in sit u and collect soil samples to return to Earth ( R ef. 1). Even though s tationary landers and rovers have transverse great distances in search of new scientific informatio n , the a erial dimension of Mars exploration is still yet to be fully exploited ( R ef. 3). Ingenuity (2021), a tech nology demonstrator , paved th Presented at the V FS 6 Decennial Aeromechanics Specialists’ Conference , Santa Clara, CA, USA Feb.6 - 8, 20 24. This is a work of the U.S. Government and is not subject to copyright protection.
comparison of a hexacopter in a hover state using a s tate of the art comprehensive a nalysis tool - FLIGHTLAB. The comparison of flight dynamics in the two different environments help s determine if a dynamically matched surrogate helicopter can be designed, such that the resulting model can be used to conduct flight testing on Earth during the air craft deve lopment cycle on E arth.
BACKGROUND Figure 2 . GT MA RS (Ref. 10) The idea of flying on Mars has been around since the early Georgia Institute of Technology also produced the Mars UAV days of space exploration. The idea of f lying in a thin, cold, concept . Figure 3 illustrates the concept, a combination of a and C O based environment became prevalent after the ground rover and a rotary - wing UAV, designed to be used for Viking Lander Mission of the 1970s ( R ef. 2). The idea of exploration purposes . Tohoku University (Ref. 11) also flying on Mars using compressed gas was first introduced by developed a four - rotor conceptual design (JMH) shown in Savu and Trifu in the mid - 1990s ( R ef. 6) . Soon after, the Figure 4 .
Stanford University tested a small rotorcraft under Mar s’s atmospheric conditions in the Je t Propulsion Laboratory ( JPL) vacuum chamber ( R ef. 7). Even though no data was published from the above research, they certainly opened the arena of the possibility of flying on Mars. NASA Ames conducted research on rotorcraft conceptual design s for Mars exploration. Young, Chen, and Briggs discuss ed the challenges associated with developing autonomous vertical - lift planetary aerial vehicles ( R ef. 8 ) and conclude d that vertical - l ift planetary aerial vehicles could potentially be developed for planets like Mars and Venus . Following the Figure 3 . Mars UAV (Ref. 10) research, the University of Maryland (R ef. 9) and Georgia Institute of Technology ( Ref. 10) devel oped potential designs for Martian rotorcraft. The University of Maryland produced the Martian A utonomous R otary W ing V ehicle (MARV).
MARV was a coaxial helicopter designed to carry a payload of 10.8 kg with an endurance of 39 min . Separately, the Georgia Institute of Technology developed a quad - rotor design (GTMARS) with rotors of 1.84 m in diameter and endurance of 30 min . Figure 1 shows the MARV design and Figure 2 shows the GTMARS design .
Figure 4 . Japanese Mars Helicopter (JMH) (Ref . 11) F ollowing the developments described above, the technology demonstrator Ingenuity was developed as a collaboration between the JPL , NASA Ames Research Center , and NASA Langley Research Center ( R ef. 1 2 ). Ingenuity features a coaxi al rotor system with counter - rotating hingeless two - bladed rotors. The vehicle is controlled using both upper and lower swashplates which provide both collective and cyclic control .
Figure 1 . MARV (R ef. 9) Differential collective is used to achieve yaw control while airfoils relative to the body frame and the influence of the keeping the rotor speed constant. Figure 5 shows a CAD wake - induced inflow distribution on rotor s . Periodic forces model and Table 1 shows I ngenuity vehicle characteristic s. and moments are produced by the asymmetry produced by forward flight, control inputs and/ or environmental disturbances. Blade flap damping is affected by the density difference between Earth and Mars To understand blade flapping on Mars, a simpler model of a rod rotating about a hinge is considered . Figure 6 shows an illustration of the blade model being cons idered.
Figure 6 . Blade flapping model ed by a central hinge ( R ef. 1 2 ) Figure 5 . Mars Helicopter Ingenuity ( R ef. 1 2 ) The model shown in Figure 6 acts as a classical mass - spring - Table 1 . Ingenuity specifications damper system. Centrifugal force and structural stiffening create a restoring moment on the hinge. Damping is present P arameter V alue due to the aerodynamic forces. When cyclic pitch is applied Total Mass 1 .8 kg to a helicopter blade, a periodic change in lift is produced at the rotor frequency, with maximum lift on one side of the Rotor Diameter 1 .21 m rotor di sk, and minimum lift on the opposite side . Given the Rotor Spacing 0 .1 m above co nditions, a blade responds like a mass spring damper, flapping with the same frequency, but with a different phase Ground Clearance 0 .3 m than the input ( R ef. 1 2 ). Aircr aft r oll and pitch moments are (lower Rotor) generated through a combina tion of the tilting of the thrust Landing gear 0 .6 X 0.6 m vector due t o blade flapping and di rect hub moments due to footprint resistance aga inst flapping at the hu b. (Ref 1 3 ). Figure 7 shows the magnitude and phase response of a centrally hinged Thrust - Weight 1 35 to 155% blade flap angle response to 1 - degree blade pitch input in both ratio Earth ' s and Mars ’ s densit y . The blue line r epresents the Endurance ≥ 1 . 5min response to the input pitch cyclic in Earth’s atmosphere. T he green line represents the response in Mars’s atmosphere. The Rotor Speed ≤ 2800 rpm red line represents the rotor frequency. In the Earth’s Collective control - 4.5 to 17.5 deg atmosphere, the peak flap output occurs 90 ° after the peak (both rotors) cyc lic pitch input due to rotor speed coinciding with the natural frequency of the mass - spring - damper . Specifically, a Cyclic control ± 10 deg ” peaking cyclic input applied on the right - hand side of the (both rotors) vehicle will result in a nose - up moment. Change in response is noticed due to red uced aerodynamic damping with a Challenges of Flying on Mars decrease in density (damping reduced to 2% in Figure 7) . In Mar s’s atmosphere , the phase angle drops to near zero around As mentioned before, f lying on Mars involves a set of the natural frequency if the rotor is stiffened to increase the challenges and requirements. The challenges are imposed due natural frequency.
to the Martian atmosphere having large daily temperature ranges , and lower density. The difference in atmosphere highly influences both the fight dynamics and des ign of a vehicle. Mars' s atmosphere also presents some challenges in testing, verification, and validation of a model as it is difficult to fully replicate the Mar tian atmosphere on Earth . This results in a heavy reliance on modeling and an alysis tools that can run simulations in user - defined environments.
Flap Dynamics on Mars C ompared to the flight dynamics of stationary airfoils relative to their body frame (i.e., fixed - wing aircraft) , helicopter flight dynamics are much more complicate d due to having rotating In eq. ( 1 ) T is the rotor thrust, A is t he total blade area , 𝜔 is b the rotor rotational speed, 𝜌 is the density, and R is the rotor radius. Rotor s rotational speed is increased in Mars’ s atmosphere t o compensate for the reduce in density compared to that on Earth . All rotors are controlled via collective pitch control that changes the amount of thrust produced by rotors.
Differential collective produces a yaw moment, whereas a combination of change in the thrust of specific rotors produces both roll and pitch moments. The vehicle is designed to carry a payload of 2.02 kg and has a range of 2 km. The vehicle is also able to cruise at 3 0 m/s (ref. 1 3 ).
Figure 7 . Magnitude and phase response of a blade flap response to pitch input for a centrally hinged blade with no additional stiffness ; blue Earth, green Mars (Ref. 1 2 ) H EXACOPTER OVERVIEW The vehicle presented in this paper is a rotorcraft with six Figure 9 . Hexacopter model (Ref. 1 3 ) rotors, commonly known as a hexacopter. The hexacopter is designed to fulfill a Mars science flight mission ( Figure 8 ). FLIGHTLAB MODELING The hexacopter should be able to take off in 30 seconds and To understand the flight dynamics of the hexacopt er, climb 200m above its landing site. It must have a cruising FLIGHTLAB, a finite element, multi - body , selective fidelity range of 1km along with the capability to hover over the modeling , and analysis software package is used to simulate science site for 2 min. Lastly, it should be able to land and the hexacopter in both Earth and Mars environment s .
recharge ( R ef. 1 3 ).
FLIGHTLAB allows users to build a model using structural, aerodynamic, control, and solution components. Once a model is built in the FLIGHTLAB environment, the model ( vehicle ) can be simulated in any user - desired atmosphere.
The hexacopter model consists of a fuselage, the mast, and the rotor blades. T he fusela ge of the hexacopter is modeled as a rigid fuselage with six non - linear degrees of freedom with inertial and mass properties. T he mast of the hexacopter is Figure 8 . Hexacopter flight mission (R ef. 1 3 ) modeled as a point mass with no inertial or mass properties .
The design of the hexacopter is constrained by the size of the The rotor blades are modeled as rigid b lades with flapping aeroshell that will be used to transport the vehicle from Earth dynamics . The aircraft is controlled via collective pitch .
to Mars. A spreadsheet was designed by NASA Ames Research Center that sized and produced in itial estimates . The Aerodynamic Forces and Moments estimates considered sizing constraints imposed by the size of The aerodynamic forces and moments on rotor blades can be an aeroshell in which the vehicle needs to be folded and determined by splitting the rotor blade into multiple sections packaged before being transported to Mars’ s surface ( R ef.
with individual span, chord, twist, and sweep. For the 1 3 ).
hexacopter each blade is divided into five sections. Each section of the blade is associated with lift, drag, and moment The hexacopter weighs 17.8 kg and features six fou r - bladed coefficient as a function of angle of attack and Mach number.
rotors. All six rotors are hingeless and are 0.64 m in radius.
Blade Element Theory is used to calculate the entire The rotors operate at 2782 RPM in Mars’s atmosphere and performance of each of the six rotors. The theory is based on 600 RPM in Earth’s atmosphere . The difference in RPM the lifting - line assumption and neglects any stall and between each atmosphere is to achieve the same blade lift compressibility effects on rotor performance. Based on the coefficient in bo th atmospheres .
theory, eq ( 2 ) describes the air velocity seen by the b lade , C T T where, U , U are the tangential, and normal air veloc ity T P ( 1 ) = ( ) σ ρ A ω R b component respectively.
Trim and Linearization ( 2 ) U = √ U + U ≅ U p T T An equilibrium point is required to obtain the desired linear time - invariant model for flight dynamic analysis. Linear time - The two - dimensional quasi - steady aerodynamic theory is invariant mode ls allow frequency domain analysis that can be used to compute wing/blade segment airloads with respect to used to understand the flight dynamics of the designed angle of attack and Mach number. Lift eq ( 3 ) and drag e q ( 4 ) vehicle. FLIGHTLAB is used to obtain an equilibrium point are computed in terms of air velocity.
of the vehicle by using the process of Newton - Raphson ρ 2 method. The Newton - Raphson method is an it erative process ( 3 ) L = U c { a ( θ − ϕ ) + c } a 0 that finds the root of an equation by linear approximation. In ρ other words, a nonlinear function is approximated by a linear ( 4 ) D = U cc d0 function tangent to it. Figure 10 shows a geometric In the above equations, a is the lift slope, c is the lift at zero interpretation of the Newton - Raphson method .
angle of attack, c is the chord length, θ is the blade pitch angle, ϕ is the induced angle which is simplified by a small angle a Up assumpt ion so that the angle ϕ ≅ , and L and D are lift and a UT drag per unit length.
T he drag produced by the fuselage is not a function of the angle of attack as no ne of the aerodynamic properties of the fuselage are modeled.
Induced Velocity The model uses a three - state induced in flow model derived from the Peters/He finite state model. The finite state dynamic wake model uses state space formulation and can model the Figure 10 . Newton Raphson method rotor wake dyn amics. To obtain a solution, the finite state model enforces boundary conditions such that the pressure illustration function matches the blade loading on the rotor blades and is The figure illustrates the principle of linear approximation zero at infinity . The induced flow distribution (w) at the rotor point . 𝑥 is the initial root guess of nonlinear function 𝑓 ( 𝑥 ) = 𝑖 plane is represented at a desired harmonic, N, for each 0 . The Newton - Raphson method approximates an improved harmonic, a specific number of radial shape function , 𝑆 , .
𝑟 estimate of the root by fitting a tangent line to the curve at point 𝑥 . The point of intersection where the tangent intersects 𝑖 ( 5 ) the x - axis is the improved estimate of the root i.e. 𝑥 . The 𝑖 + 1 2 S + r − 1 r r r N process is iterated until the desired root is obtained . For w ( x ̂ , ψ , t ) = ∑ ∑ ϕ ( x ̂ ) [ α ( t ) cos ( r ψ ) + i j j j = r + 1 , r + 3 … r = 0 example, a generalized force is set equal to nonlinear r ( ) ( ) β t sin r ψ ] j equations x ̂ is the radial coordinate, 𝜓 is the azimuth position, and t is ( 9 ) Q = f ( x ̈ , x ̇ , x , x ̇ , x , u ) 2 2 2 1 1 time. The dynamic wake in the tip path plane is modeled as a ( ) set of first order ordinary equations which relates the inflow y = g x ̈ , x ̇ , x , x , x , ̇ u 2 2 2 1 1 states to the induced inflow forcing functions.
In eq . ( 9 ), Q is the imbalance in satisfying the differential ( 6 ) M x ̇ + Lx = τ equations. u are the inputs to the component and y represents the outputs from the component. The objective is to find a set W here M is the apparent mas s of the system and is computed of states and their derivatives that drive Q to zero such that as, Q=0 . If the e quations are couple d with more than one solution component , then the coupled/assembled equations are used to c [ M ] 0 [ ] find the states and derivatives that derive the generalized ( 7 ) M = [ ] s [ ] 0 M forces for all components to within an acc eptable tolerance close to zero. Each s tate and derivative are perturbed one by T he apparent stiffness , L, is computed as, one to calculate the change in all generalized forces. The c − 1 nonlinear state equations for a multi - component solution [ ] L 0 T ( 8 ) [ L ] = [ T ] [ ] [ T ] s − 1 group can be written as, [ ] 0 L T he coefficients of the dynamic wake in the tip path plane are ̇ ( 10 ) δ Q = C δ x ̇ + K δ x + M δ x ̈ + C δ x + 1 11 1 11 1 12 2 12 2 described in detail in the FLIGHTLAB theory manual ( Ref .
K δ x 12 2 1 4 ) .
( 11 ) δ Q = M δ 𝑥 ̈ + C δ x ̇ + K δ x + C δ x ̇ + f lapping modes at a frequency range of 420 to 600 rad/s , and 2 22 2 22 2 22 2 21 1 K δ x airframe modes at low frequency f rom 0.1 to 100 rad/s. Inflow 21 1 modes also exist at the mid - frequency range of 340 to 420 W here, M, C, a nd K are the mass, d a mping, and stiffness rad/s.
partials respectively . Newton - Raphson process is car ried out to estimate the change in the states and the derivatives that are I n Figure 12 , FLIGHTLAB shows all poles of the system in required to drive the generalized force, Q, to near zero. Due the left half plane, predicting a stable system whereas, to the equation being nonlinear, a n iterative approach is used CAMRAD II shows one set of poles in the right half plane, to obtain the change in Q . Equation ( 12 ) shows the current thus predicting the unstable behavior of the hexacopter in iteration which is assumed to be the negative value of Q at the Earth’s atmospher e.
previous step.
i i − 1 ( 12 ) δ Q = − Q F LIGHTLAB uses a method to solve the nonlinear state and output equations which compute s th e value of Q at each time step . The method determines the level of linearization and the nature of discretization. The highest derivatives terms are linearized using the Newton - Raphson method.
i i i i − 1 ( 13 ) δ Q = C δ x ̇ + M δ x ̈ = − Q 1 11 1 12 2 1 i i i i − 1 ( 14 ) δ Q = C δ x ̇ + M δ x ̈ = − Q 2 21 1 22 2 2 T he above equation can be written as, i i − 1 ( 15 ) δ Q = M δ x = − Q hd Figure 11 . Eigenvalue s of the hexacopter in Earth's T he highest order derivate x is solved and the inverted mass hd atmos phere matrix is computed by time discretization.
i i − 1 i − 1 ( 16 ) x = x − Minv ∗ Q hd hd E quation ( 16 ) is the approximate solution. The used method is called repeatedly until a suitable degree of convergence is achieved.
LINEAR MODEL EXTRACTION AND VALIDATION The linear state space matrices in the form of 𝑥 ̇ = 𝐴𝑥 + 𝐵𝑢 were extracted through FLIGHTLAB to generat e f requency B ode plots. Frequency Bode plots are generated to understand the system response by taking raw measurements of the output amplitude and phase of the system undergoing a sinusoidal input. Frequency responses are also helpful in determining the s tability of a closed - loop system.
Figure 12 . Pole z ero m ap (Earth's atmosphere) Flight Dynamic Analysis in Earth’s Atmosphere A slight difference in inflow modes is observed between the FIGHTLAB and CAMRAD II linear models ( Figure 11 ) .
To validate the linear models obtained from FLIGHTLAB, Both tools use momentum potential flow theory to conduct the non - linear dynamics of the hexacopter were also rotor wake analysis. The difference is due to CAMRAD II linearized in CAMRAD II (Ref. 1 5 ) . Both linearized models using a wake dis tortion factor ( κ ) of 1.15 . No wake distortion were input into M ATLAB and the bode function was used to compare the flight dynamics of the model. Figure 11 shows factor or states are included in the linearized model obtained from FLIGHTAB . Furthermore, the modes retrieved from the the eigenvalue comparison between CAMRAD II and FLIGHTLAB models in Earth’s atmosphere. Both sets of CAMRAD II linear model are slightly more damped than the ones retrieved from FLIGHTLAB.
eigenvalues align very closely and predict similar damping characteristics of the hexacopter in Earth’s atmosphere. The difference is due to the difference in modeling infl ow which FLIGHTLAB f requency responses of the hexacopter were is explained later in this section. T he figure also shows also extracted and compared against the CAMRAD II linear model. Figure 13 and Figure 14 show the heave frequency Figure 15 and Figure 16 shows the pitch and roll rate response response to collective input and yaw rate response to pedal of the hexacopter in Earth’s atmosphere . Both the input respectively. Both responses are plotted from 0.1 rad/s characteristics and magnitude of FLIGHTLAB and to 1000 rad/s. For both heave and yaw rate, the results CAMRAD II linear models align very closely with each other obtained from FLIGHTLAB align very close ly with the with a difference in frequency for the phugoid mode. The results obtained from CAMRAD II , thus validating the difference is due to FLIGHTLAB linear models missing response produced by FLIGHTLAB. The response s are first inflow states and not using t he wake distortion factor as order with a rotor mode present at a higher frequency. This is me ntioned previously.
the regressive mode which is generated at the frequency of 𝜐 − 1 /rev , w here, 𝜐 is the flap frequency of the coning 𝑓𝑙𝑎𝑝 𝑓𝑙𝑎𝑝 mode . For both heave and yaw rates, the regressive mode is generated at approximately 371 rad/ s . The coning mode is the oretically at the same frequency as the rotating natural flap frequency. Specifically, a coning would be seen at approximately 430 rad/ s . F urthermore, the system is also damped which is expected when operating in Earth’s atmosphere. This is confirmed by th e gradual decrease in the phase angle. For flight control purposes, the FLIGHTLAB heave response shown in Figure 13 is acceptable as no modes are present in the mid - range frequency (rad/s) and the response matches very closely to the one obtained from CAMRAD II .
Figure 15 . Pitch rate r esponse to longitudinal input Figure 13 . Heave response to collective input Figure 16 . Roll rate response to lateral input The phugoid mode shown in both pitch and roll rates is present due to the coupling between the attitude and horizontal speed states ( R ef. 1 2 ). The existence of modes can be understood by looking at the state stability matrix from a simplified model of the longitudinal dynamics.
0 − g 0 ( 17 ) A = [ 0 0 1 ] M 0 0 u M atrix A obtained from ( R ef. 1 2 ) is restricted to longitudinal dynamics and neglects longitudinal drag X and pitch u damping M . The quantity M is the pitch rate sensitivity to a q u gust hitting the helicopter from the front. As the gust becomes Figure 14 . Yaw rate response to pedal input stronger, more nose - up moment is generated. The mode is sta bilized by a nose - down moment generated either from the system M or with the help of a c o ntrol system ( with rate 10 rad/ s . As with Earth’s atmosphere, inflow modes exist in q both low and high - frequency ranges , i.e. 0 to 200 rad/ s and feedback) . Matrix A has the characteristic equation i.e. λ + 3 3 400 to 600 rad/ s . The modes presented here are much less M g = 0 with solutions λ = − M g , λ = ( 1 ± 3j ) M g √ √ √ u 1 u 23 u damped due to Mars’ s density being lower than Earth's . Like ( Ref. 1 2 ). The solutions show that the frequency of unstable the hexacopter in Earth’s atmosphere, the hexacopter in poles increases with an i ncrease in 𝑀 and g. Even though the 𝑢 Mars’ s atmosphere is also not stable as a set of poles exists in phugoid mode is in low frequency, it is still a fundamental the right - side plane ( Figure 18 ) factor in designing a control system as it can help impose limitations on the sta bility margin of the system. The phase shift occurring at different frequencies is due to each tool, i.e., FLIGHTLAB and CAMRAD II predicting different M , i.e ., u 0.2850 for FLIGHTLAB and 0.0080 for CAMRAD II . The difference in M is potentially due to the differences in the u dynamic inflow wake modeling and wake distortion effect. A coning mode is also present at a higher frequency. This is a coning mode since it exists exactly at the frequency of the rotating flap mode, i.e., 430 rad/ s . For control system design purposes , it is important to have no resonant frequencies in mid - range frequencies of 0.1 to 100 rad/s to avoid overworking the actuators. B oth longitudinal and lateral rates can be controlled with a control system as there are no resonant f requencies in mid - range frequencies .
Flight Dynamic Analysis in Mars’s Atmosphere F igure 18 . Pole z ero map of the h exacopter in Mars’ s Unlike trimming the model i n Earth’s atmosphere , artificial atmosphere flap damping is introduced in the system to obtain an equilibrium point when in Mars’ s atmospheric conditions .
Frequency responses in Mars’s atmosphere from Specifically, a 0 . 85 kg m / s flap hinge damping coefficient FLIGHTLAB are also compared against CAMRAD II . Figure is introduced. An equilibrium point is computed with this 1 9 shows the heav e response of the hexacopter to collective added damping , but the nonlinear dynamics of the hexacopter input. Both responses match very closely with each other, thus on Mars are linearized without it. In other words, th e flap validating the heave response . A coning mode is generated in hinge coefficient of all six rotors is set to zero before the high - f requency range of the response. The mode is extracting linearized models. The extracted models are also identified as the coning mode as it is generated at the rotating validated against linearized models from CAMRAD II .
flapping frequency of 448 rad/ s . The phase shift in the heave response is very sudden. Specifically, the phase shifts to near zero suddenly due to Mars’ s atmosphere being low in density making the vehicle lightly damped .
Figure 20 shows a close match between CAMRAD II and FLIGHTLAB ’s yaw rate response to yaw ( peda l ) input in Mars’ s atmosphere. A coning mode is also present in the yaw rate at the rotating natural frequency . A similar characteristic as the heave response is seen with the phase shift to near zero.
Figure 17 . Eigenvalues of Hexacopter in Mars’s atmos ph ere Figure 17 compares the eigenvalues o f the hexacopter in Mars’ s flight condition s . As seen in the figure, the eigenvalues obtained from FLIGHTLAB match very closely with the eigenvalues from CAMRAD II . Both m odels consist of higher frequency rotor modes present between the frequency range of 450 to 742 rad/ s , and lower frequency airframe modes present between the f requency range of 0.1 to Figure 19 . Heave response to collective input in Mars' s Figure 21 . Pitch rate response to longitudinal input in atmosphere Mars' s atmosphere Figure 22 . Roll rate response to lateral input in Mars' s Figure 20 . Yaw rate response to pedal input in Mars' s atmosphere atmosphere Differences In the Flight Dynamics Between Earth and Figure 21 and Figure 22 show the pitch rate and roll rate Mars E nvironment s response of the hexacopter in Mars’ s atmosphere respectively. The responses from CAMRAD II and Here, t he inner loop dynamic characteristic of the hexacopter FLIGHTLAB do match closely besides the difference in in both Earth and Mars atmosphere s are compared. Figure 23 frequency and damping of phugoid mode, as described shows the eigenvalues obtained from FLIGHTLAB’s linear previously. The phugoid mode in both CAMRAD II and model in the Earth and Mars environment s . As seen in Figure FLIGHTLAB responses is verified by the existenc e of 23 , the rotor (flapping) modes and inflow modes in Earth's unstable poles in the right half plane ( Figure 18 ). The atmosphere are more damped than the ones in Mars’ s regressive and advancing modes are generated at 𝜈 − 𝑓𝑙𝑎𝑝 environment. This is due to rotors operating at different 1 / 𝑟𝑒𝑣 and 𝜈 + 1 / 𝑟𝑒𝑣 respectively. For the hexacopter speeds in each atmosphere which causes all modes on earth to 𝑓𝑙𝑎𝑝 occur at a much lower frequency than they do in Mars’ s operating in Mars’ s condition with a 𝜈 = 1 . 54 / 𝑟𝑒𝑣 , the 𝑓𝑙𝑎𝑝 atmosphere.
regressive and advancing modes exist at approximately 157 rad/s and approximately 740 rad/s respectively. The coning mode also exists at the rotational flapping frequency of 448 rad/s.
Figure 23 . Eigenvalue comparison between Earth and Figure 25 . Yaw rate response compared between Mar s atmosphere s Earth and Mar s atmosphere s Figure 24 and Figure 25 shows the heave and yaw rate A similar dynamic behavior difference is also noticed in both response of hexacopter in Mars and Ear th atmosphere s . Both pitch and roll rate responses . The system is not as responsive responses in Mars ’ s atmosphere are damped compared to to Mars ’s atmospheric conditions as it is in Earth's responses in the Earth's atmosphere. The damped response on atmosphere . At 0.1 rad/ s , the Mars model reacts more like t he Mars is shown by the steep change in the phase angle around model in Earth's atmosphere. The Mars model becomes less the frequency of 450 rad/ s ( Figure 24 and Figure 25 ). responsive past 10 rad/ s . Moreover, higher magnitude rotor modes exist in the high - frequency domain. The modes are The system resp onse at Mars is not as sensitive to the input at very less damped than they are in the Earth’s atmosphere.
the frequency range from 1 rad/ s to 400 rad/ s . As the system Thus, the modes are hig h in magnitude on Mars than on Earth.
transitions into the mid - range frequencies, the model on Earth The slightly damped modes are confirmed by a not so gradual is more responsive as there is a lower change in magnitude.
shift in phase for Mars model, whereas the model operating At the higher frequency, the Mars model predicts higher in Earth’s atmosphere is damped as the phase shift is very magnitude flapping modes due to Mars's density being low.
gradual. The phugoid mode in Mars ’s g ravity has a reduced T he flapping modes in Mars's atmosphere are higher in frequency compared to Earth , as frequency decreases with the magnitude by 10 dB . T he differences are further explained in decrease in gravity .
the discussion section below.
Figure 26 . Pitch rate comparison between Earth and F igure 24 . Heave response of the hexacopter in both Mar s atmosphere s Earth and Mars atmosphere s critical point: one above the critical point and one above the critical frequency (phase=1 80) . The two crossovers (N= - 2) are equivalent to two counterclockwise encirclements of the critical point in the Nyquist plot. Furthermore, the Nyquist criteria also states that the number of zeros of the characteristic function Q(s)=1+L(s) in the right - ha nd plane is: Nz=N+np= - 2+2=0. Since the pitch response of the hexacopter on Mars has no zeros, the system is stable as zeros of function Q(s) equals the number of poles of T(s) in the right - hand plane . E quation 18 states the trans fer function of a closed loop system showing numbers of zeros of Q(s) equals poles of T(s).
( ) ( ) L s L s ( ) ( 19 ) T s = = ( ) 1 + 𝑄 ( 𝑠 ) Q s E ven though the system is stable, the response will still be Figure 27 . Roll rate response comparison between Earth oscillatory as the phase margin intersects the desirable and Mar s atmosphere s stability margin block ( Figure 28 ) . The Nic hols plot gives a D ISCUSSION phase margin of ~20.5 dB which is less than the desired phase margin of 45 degrees for the system to not be oscillatory.
To better understand the feasibility of designing a dynamically matched surrogate helicopter, a flight dynamics comparison of a closed - loop system between the two environments is needed. Ba sed on the analysis conducted above, it is determined that f requency Bode plots are not sufficient to check if both systems are dynamically similar due to the model being unstable with a non - minimum phase in Mars’ s atmospheric conditions .
T he linear model obtained from FLIGHTLAB predicts the pitch response of the system in Earth’s atmosphere being stabi lizable . Furthermore, the hexacopter in Earth’s atmosphere is also a minimum phase system. Thus, one can predict the stability of the system in Earth’s atmosphere by considering the gain and phase margin. T he open loop predicts both gain and phase margin of 79 dB and 79 degrees Figure 28 . Nichols c hart (q/Lon) respectively. In the case of a hexacopter operating in Mars’ s atmospheric conditions, FLIGHTLAB predicts the model to In addition to the pitch response of the system in bo th Earth be unstable with a non - minimum phase , thus making the and Mars atmosphere s , the roll response of the vehicle is also f requency Bode plots not very informative. To determine if analyzed. The roll response on Earth is unstable with a non - the system can be stabilized , Nyquist stability criteria is minimum phase. Therefore, Nyquist stability criteria must be applied. The Nyquist criteria is based on the number and applied to understand the roll stability of the vehicle. T he r oll direction of encirclements of the critical point. The response cannot be stabilized as it gives two clockwise encirclements can be determined interchangeably either from encirclements, thus adding two zeros in the right - hand plane.
a Nyquist diagram or a Nichols char t. The Nichols chart The phase of roll response can be converted into a minimum retains a closer connection to the Bode plot quantities and gain phase by inverting the sign of input. Once the input is and phase margins can easily be determined from it. Figure inverted, mirroring the roll response on Earth (Figure 29) 28 shows the pit ch response to longitudinal input for the about 180 degrees gives one crossover above the critical point hexacopter operating in both Earth and Mars atmosphere s .
and one above the critical frequency. Then the roll response Here the Nichols chart is used to understand the closed - loop of the hexacopter on Earth can be stabilized as only two stability of the hexacopter in Mars’ s atmosphere.
eigenvalues exist that dominate the roll response. The two counterclockwise encirclements of the critical point result in A MIMO system is modeled , therefore, there exist four no zeros of the characteristic function Q(s), thus the closed unstable eigenvalues. Assuming both pitch and roll can be loop transfer function has a zero number of poles in the right - stabilized independently, only two eigenvalues dominate per hand plane.
both pitch and roll response (np=2). Therefore, there are two unstable poles for both pitch and roll respons es. As seen in Figure 28 , the Mars curve makes two crossovers above the dynamic response of both models operating under different environments .
Overall, gi ven the flight dynamics differences mentioned above, it is difficult to design a dynamically matched surrogate hexacopter . Thus, further analys is such as designing an advanced control system that can produce identical responses in both atmosphere s with a given input needs to be designed . The advanced control system design can help infer control gains that produce identical responses in both atmospheres. After identical responses are obtained in a hover flight c ondition, a forward flight configuration can be explo red.
Author contact: Raghuvir Singh raghuvir.singh@nasa.gov F igure 29 . Nichols c hart (p/lat) REFERENCES T he flight be havior of the designed hexacopter differs significantly in both the Earth and Mars environment s .
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