quadrotor helicopter flight dynamics and control: theory and experiment · 2015-07-29 · theory...

Quadrotor Helicopter Flight Dynamics and Control:

Theory and Experiment∗

Gabriel M. Hoffmann† Haomiao Huang‡ Steven L. Waslander§ Claire J. Tomlin ¶

Quadrotors are rapidly emerging as a popular platform for unmanned aerial vehicle(UAV) research, due to the simplicity of their construction and maintenance, their abilityto hover, and their vertical take off and landing (VTOL) capability. Current designs haveoften considered only nominal operating conditions for vehicle control design. This workseeks to address issues that arise when deviating significantly from the hover flight regime.Aided by well established research for helicopter flight control, four separate aerodynamiceffects are investigated as they pertain to quadrotor flight. The effects result from eithertranslational or vertical vehicular velocity components, and cause both moments that af-fect attitude control and thrust variation that affects altitude control. Where possible, atheoretical development is first presented, and is then validated through both thrust teststand measurements and vehicle flight tests using the Stanford Testbed of AutonomousRotorcraft for Multi-Agent Control (STARMAC) quadrotor helicopter. The results haveenabled improved controller tracking throughout the flight envelope, including at higherspeeds and in gusting winds.

I. Introduction

Quadrotor helicopters are an emerging rotorcraft concept for unmanned aerial vehicle (UAV) platforms.The vehicle consists of four rotors in total, with two pairs of counter-rotating, fixed-pitch blades located atthe four corners of the aircraft, an example of which is shown in Figure 1. Due to its specific capabilities, useof autonomous quadrotor vehicles has been envisaged for a variety of applications both as individual vehiclesand in multiple vehicle teams, including surveillance, search and rescue and mobile sensor networks.1

The particular interest of the research community in the quadrotor design can be linked to two mainadvantages over comparable vertical take off and landing (VTOL) UAVs, such as helicopters. First, quadro-tors do not require complex mechanical control linkages for rotor actuation, relying instead on fixed pitchrotors and using variation in motor speed for vehicle control. This simplifies both the design and mainte-nance of the vehicle. Second, the use of four rotors ensures each individual rotor is smaller in diameter thanthe equivalent helicopter rotor, relative to the size of the airframe. The individual rotors, therefore, storesignificantly less kinetic energy during flight, mitigating the risk posed by the rotors should they entrain anyobjects. Furthermore, by enclosing the rotors within a frame, the rotors can be protected from breakingduring collisions, permitting flights indoors and in obstacle-dense environments, with low risk of damagingthe vehicle, its operators, or its surroundings. These added safety benefits greatly accelerate the design andtest flight process by allowing testing to take place indoors, by inexperienced pilots, with a short turnaroundtime for recovery from incidents.

Quadrotor vehicle dynamics are often assumed to be accurately modeled as linear, without any depen-dence on velocity of attitude or altitude control. At slow velocities around hover, this is indeed a reasonableassumption, but even at moderate velocities, the impact of the aerodynamic effects resulting from variation

∗This research was supported by ONR under the CoMotion MURI contract N00014-02-1-0720, and by NASA grantNNAO5CS67G.

†Ph.D. Candidate, Department of Aeronautics and Astronautics, Stanford University. AIAA Member. [email protected]‡M.S. Student, Department of Aeronautics and Astronautics, Stanford University, AIAA Member, [email protected]§Ph.D. Candidate, Department of Aeronautics and Astronautics, Stanford University. AIAA Member. [email protected]¶Associate Professor, Department of Aeronautics and Astronautics; Director, Hybrid Systems Laboratory, Stanford Univer-

sity. Associate Professor, Department of Electrical Engineering and Computer Sciences, University of California at Berkeley.AIAA Member. [email protected]

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American Institute of Aeronautics and Astronautics

Figure 1. STARMAC II quadrotor aircraft unmanned aerial vehicle (UAV), in flight, with autonomous attitudeand altitude control. This is a vehicle of the Stanford Testbed of Autonomous Rotorcraft for Multi-AgentControl (STARMAC) project. Applications include search and rescue, surveillance operation in clutteredenvironments, and mobile sensor networks. Operation throughout the flight envelope allows characterizationof the aerodynamic disturbance effects on the control system, caused by vehicle motion relative to the freestream.

in air speed is significant. Although many of the effects have been discussed in the helicopter literature,2–5

their influence on quadrotors has not been comprehensively explored. This work focuses on four main effects,two that impact attitude control and two that impact altitude control as they pertain to quadrotors. First,for attitude control, blade flapping results from the differing inflow velocities experienced by the advancingand retreating blades and induces both roll and pitch moments applied at the blade root as well as a changein the direction of the thrust vector away from the horizontal plane. Second, it was observed that interfer-ence caused by the vehicle body in the downstream rotor flow resulted in unsteady thrust behavior and poorattitude tracking. This effect was demonstrated to be significantly reduced by locating the rotors furtherapart. Third, for altitude control, the vertical component of the vehicle velocity directly impacts the thrustproduced, resulting in a negative effect when the vehicle is climbing and a positive effect while descending.Finally, the combination of translational velocity and vehicle angle of attack can result in increases to theexpected thrust produced at a given power. For all but motor positioning, a theoretical derivation of theeffect is developed based on previous work, and the specific impact on quadrotor dynamics are defined. Theeffects are then validated through thrust test stand experiments, and by flight tests of the Stanford Testbedof Autonomous Rotorcraft for Multi-Agent Control (STARMAC).

We proceed with a brief survey of development efforts for quadrotor vehicles in Section II. Section IIIpresents details of the test stand apparatus and the STARMAC II testbed, and the nonlinear vehicle dynamicsfor quadrotors are then summarized in Section IV. In Sections V and VI, we present analysis of each of thefour aerodynamic effects as they pertain to quadrotor vehicles, along with experimental results demonstratingtheir presence in both thrust test stand experiments and in flight test recordings. Possible control techniquesare also presented to accommodate the resulting nonlinearities, and finally, flight results for outdoor hoverare presented in Section VII.

II. Background

Although the first successful quadrotors flew in the 1920’s,4 no practical quadrotor helicopters havebeen built until recently, largely due to the difficulty of controlling 4 motors simultaneously. The onlymanned quadrotor helicopter to leave ground effect was the Curtiss-Wright X-19A in 1963, though it lackeda stability augmentation system to reduce pilot work load, rendering stationary hover near impossible,6

and development stopped at the prototype stage. Recently, advances in microprocessor capabilities and inmicro-electro-mechanical-system (MEMS) inertial sensors have spawned a series of radio-controlled (RC)quadrotor toys, such as the Roswell flyer (HMX-4),7 and Draganflyer,8 which include stability augmentationsystems to make flight more accessible for RC pilots.

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Many research groups are now working on quadrotors as UAV testbeds for control algorithms for au-tonomous control and sensing,7,9–15 consistently selecting vehicle sizes in the range of 0.3 - 4.0 kg. Severaltestbeds have achieved control with external tethers and stabilizing devices. One such system,7 based on theHMX-4, was flown with the RC gyro augmentation system active, and with X-Y motion constraints. Alti-tude and yaw control were demonstrated using feedback linearized attitude control and backstepping controlwas applied for position, while state estimation was accomplished with an offboard computer vision system.Another tethered testbed14 used an extensive outward facing sensor suite of IR and ultrasonic rangers toperform collision avoidance. Control of the vehicle was achieved using a robust internal-loop compensator,and computer vision was used for positioning. A third project16 relied on a tether to use a POLYHEMUSmagnetic positioning system. Tight position control at slow speeds was demonstrated using a non-linearcontrol technique based on nested saturation for lateral control with linearized equations of motion, thoughcompensating for attitude tilted thrust vectors in altitude control.

Other projects have relied on various nonlinear control techniques to perform indoor flights at low ve-locities without a tether. One such project,11 consisting of a modified Draganflyer quadrotor helicopter,has demonstrated successful attitude and altitude control tests using a nonlinear control scheme. The OS4quadrotor project10 features its own vehicle design and identifies dynamics of the vehicle beyond the basicnonlinear equations of motion, including gyroscopic torque, angular acceleration of blades, drag force onthe vehicle, and rotor blade flapping as being potentially significant, though the forces are not quantified oranalyzed further. A proportional-derivative (PD) control law led to adequate hovering capability, althoughthe derivative of the command rate was not included in the control law to maneuver the vehicle. A Lyapunovproof proved stability of the simplified system in hover, and successful attitude and altitude control flightswere achieved. A third project12 achieved autonomous hover with IR range positioning to walls indoors, witha stability proof under the assumed dynamics. The system was modified to incorporate ultrasonic sensors,17

and later incorporated two cameras for state estimation18 as well.Several vehicles saw success using Linear Quadratic Regulator (LQR) controllers on linearized dynamic

models. The Cornell Autonomous Flying Vehicle (AFV)13 designed and built a custom airframe with brush-less motors controlled by custom circuitry to improve resolution. Position control was accomplished usingdead-reckoning estimation, with a human input to null integration error. The MIT multi-vehicle quadrotorproject19 uses an offboard Vicon position system to achieve very accurate indoor flight of the Draganflyer VTi Pro, and demonstrated multiple vehicles flying simultaneously. The vehicles are capable of tracking slowtrajectories throughout an enclosed area that is visible to the Vicon system. It is possible to observe, inflight videos presented with the paper, the downwash from one vehicle disturbing another vehicle in flight,causing a small rocking motion, possibly due to blade flapping.

At Stanford, there has been considerable prior work on quadrotor helicopters as well. The STARMAC Iproject was a testbed of two vehicles that performed GPS waypoint tracking using an inertial measurementunit (IMU), an ultrasonic ranger for altitude, and an L1 GPS receiver.9 The testbed was derived from aDraganflyer aircraft, and weighed 0.7 kg. In order to improve attitude control, this project found that framestiffening greatly improved attitude estimation from the IMU. Also, aerodynamic disturbances were observedwith this testbed, and modeled using flight data.20

Despite the substantial interest in quadrotor design for autonomous vehicle testbeds, very little attentionhas been paid to the aerodynamic effects that result from multiple rotors, and from motion through thefree stream. Exceptions to this trend include work from a group in Velizy, France21 which investigates dragforces due to wind and presents a control law to handle such forces should they be estimated. Also, manyimportant aerodynamic phenomenon were identified in the X-4 Flyer project at the Australian NationalUniversity.22 The project considers the effects of blade flapping, roll and pitch damping due to differingrelative ascent rates of opposite rotors, as well as dynamic motor modeling. Preliminary results of theinclusion of aerodynamic phenomena in vehicle and rotor design show promise in flight tests, although aninstability currently occurs as rotor speed increases, making untethered flight of the vehicle impossible.15

In the following sections, this paper extends the investigation of quadrotor aerodynamics, as they pertainto position control and trajectory tracking flight. The effects of aerodynamics on a moving quadrotorhelicopter are analyzed, through theory, and by experiment. The next section presents the test apparatusused.

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Figure 2. Thrust test stand used to capture experimental aerodynamic data. A load cell measures the forceand torque exerted on the mounting point. Battery monitoring circuitry measures motor voltage and current.Data is captured to the computer using an Atmel processor for A/D at 400 Hz.

III. Experimental Setup

The experimental equipment consisted of two primary components: a thrust test stand and prototypequadrotor aircraft, STARMAC II. The thrust test stand permitted research into the performance of individualmotors and rotors, in varying flight conditions, while STARMAC II permitted experiments with an actualquadrotor vehicle through indoor and outdoor flight testing. This section presents the relevant details of thetwo systems.

A. Thrust Test Stand

In order to evaluate motor and rotor characteristics, a thrust test stand was developed, shown in Figure 2.It measures the forces exerted on a lever using a load cell. The mounting point on the lever is adjustableto allow load sensitivity to be varied. An Atmel microprocessor board was programmed to perform motorcontrol through its pulse width modulation (PWM) outputs, and to acquire analog inputs from the load cell,current sensor, and battery voltage.

The microprocessor board interfaces with a data acquisition program on the PC to perform automatedtests, making measurements at 400 samples per second, well over twice the frequency of typical rotor rota-tions. To perform some experiments, external wind was applied using a fan. Wind speeds were measuredusing a Kestral 1000 wind meter, with a rated accuracy of ±3%.

B. STARMAC II Quadrotor

The STARMAC II quadrotor helicopter, part of the Stanford Testbed of Autonomous Rotorcraft for Multi-Agent Control (STARMAC) project, was designed and built to carry computer and sensor payloads for UAVprojects, as shown in Figure 3. The vehicle was fabricated from carbon fiber and fiberglass honeycomb, usinglessons learned from STARMAC I. The nylon rotors, with a 0.127m radius, are driven by sensorless, brushlessmotors, controlled by commercial electronic speed controllers (ESCs). The combined thrust capacity is 4.0 kgfor the 1.3 kg vehicle. It has flown with up to 1.1 kg of additional payload, including a Pentium-M computerand environment sensors.

Vehicle state sensors include a Microstrain 3DM-GX1 inertial measurement unit (IMU), an ultrasonicranger for altitude, and a Novatel Superstar II GPS receiver. An onboard processor computes the carrierphase differential GPS solution for single-centimeter level accuracy. The sensor suite is able to accuratelycharacterize the state of the aircraft to measure the effects of aerodynamic disturbances.

The vehicle is flown in the confined space of the lab, as well as outdoors. The control capabilities currentlyallow the vehicle to be flown in two modes, both with altitude control. In the first mode, attitude-controlled,a human uses a joystick to send attitude and altitude commands to the aircraft. In doing so, the human

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Figure 3. STARMAC II quadrotor aircraft unmanned aerial vehicle (UAV), part of the Stanford Testbed ofAutonomous Rotorcraft for Multi-Agent Control project. The air frame is 0.56 m diagonally from motor tomotor. The vehicle is capable of autonomous GPS waypoint tracking and flying aggressive maneuvers followingattitude commands, indoors and outdoors.

controls the position. In the second mode, position-controlled, the aircraft controls its attitude to navigateto a commanded GPS waypoint.

IV. Inertial Dynamics

The derivation of the nonlinear dynamics is performed in North-East-Down (NED) inertial and bodyfixed coordinates. Let eN, eE, eD denote unit vectors along the respective inertial axes, and xB,yB, zBdenote unit vectors along the repsective body axes, as defined in Figure 4. Euler angles of the body axes areφ, θ, ψ with respect to the eN, eE and eD axes, respectively, and are referred to as roll, pitch and yaw.The current velocity direction unit vector is ev, in inertial coordinates, and defines coordinates relative tothe c.g. referred to as longitudinal, lateral and vertical. The rotor plane does not necessarily align with thexB, yB plane, so let xR,yR, zR denote unit vectors aligned with the plane of the rotor and oriented withrespect to the lateral, longitudinal, and vertical directions as shown in Figure 5. Let r be defined as theposition vector from the inertial origin to the vehicle center of gravity (c.g.), and let ωB be defined as theangular velocity of the aircraft in the body frame.

Figure 4. Free body diagram of a quadrotor aircraft.

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The rotors, numbered 1 − 4, are mounted outboard on the xB, yB, −xB and −yB axes, respectively,with position vectors ri with respect to the c.g. Each rotor produces a thrust, Ti, that varies with rotorangular velocity, Ωi,

Ti = CT ρA(ΩiR)2 (1)

where CT is the dimensionless coefficient for rotor thrust, A = πR2 is the area swept out by the rotor, ρ is thedensity of air and R is the radius of the rotor. The value for CT can change due to wind and vehicle motion.Note that the thrust acts perpendicularly to the rotor plane along the zR axis. Each rotor also produces anaerodynamic moment, Mi, which is a function of the motor torque as well as various aerodynamic effectswhich are described in Section V. Rotors 1 and 3 rotate in the opposite direction of rotors 2 and 4, so thatcounteracting aerodynamic torques can be used for yaw control, where the aerodynamic torques are equalto the torques applied by the motors.

Note that the thrust is proportional to the voltage, applied to the motors, squared. The derivation andsupporting experimental data will be in the final version of this paper. Also note that in the operatingregime of the aircraft, this quadratic curve is well approximated for small variations in voltage by linearizedvariation in voltage.

Figure 5. Free body diagram of the moments and forces on a single rotor.

The vehicle drag force is defined as Dv, vehicle mass is m, acceleration due to gravity is g, and the inertiamatrix is I ∈ R

3×3. A free body diagram is depicted in Figure 4, with a depiction of the rotor forces andmoments in Figure 5. The total force, F, can be summed as,

F = −Dvev +mgeD +

4∑

i=1

(−RRi,IzRTi) (2)

where RRi,I is the rotation matrix from the plane of rotor i to inertial coordinatesa. Similarly, the totalmoment, M, is,

M =4∑

i=1

(Mi + ri × (−TiRRi,BzR)) (3)

where RRi,B is the rotation matrix from the plane of rotor i to body coordinates. The full nonlinear dynamicscan be described as,

mr = F

IωB + ωB × IωB = M(4)

where the total angular momentum of the rotors is assumed to be near zero, because they are counter-rotating.

V. Attitude Control

A. Blade Flapping in Translational Flight

Initial outdoor flights with STARMAC encountered an unexpected restorative effect damping commandedattitudes. As translational velocity increased or in the presence of wind, the vehicle tended to “settle” toward

aThe notation RA,B shall refer to rotation matrices from coordinate system A to B throughout.

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0 pitch and roll, limiting the velocities which could be attained (see Figure 9a). This damping was lateridentified to be the result of rotor blade flapping. In the following, first the aerodynamics of blade flappingare presented from theoretical basis. Then, a comparison is made to experimental data, measured using thethrust test stand. This section is concluded with STARMAC II flight test results, comparing the performanceof the attitude controller under nominal, low speed conditions, to the controller under translational motion,with the center of gravity (c.g.) at varying heights. The results demonstrate the effect of speed of bladeflapping, and consequently attitude control performance, as well as the impact that the vertical position ofthe c.g. has on attitude dynamics.

1. Aerodynamics of Blade Flapping

In translational flight, the advancing blade of a rotor sees a higher effective velocity relative to the air, whilethe retreating blade sees a lower effective velocity. This results in a difference in lift between the two rotor,causing the rotor blades to flap up and down once per revolution.5 This flapping of the blades tilts the rotorplane back away from the direction of motion, which has a variety of effects on the dynamics of the vehicle,in particular affecting stability in attitude.15 The backwards tilt of the rotor plane generates a longitudinalthrust, Tlon,

Tlon = T sin a1s (5)

where a1s is the angle by which the thrust vector T is deflected (see Figure 1). If the center of gravity ofthe vehicle is not aligned with the rotor plane, this longitudinal force will generate a moment about the c.g.For stiff rotors, the tilt of the blades also generates a moment at the rotor hub

M = kβa1s (6)

where kβ is the stiffness of the rotor blade in Nm/rad.Coning (the upward flexure of the rotor blades from the lift force on each blade) causes the impinging

airflow to have another unbalanced forcing of the blades which causes a lateral tilt of the rotor plane, thedetails of which are developed in Prouty.5 This lateral tilt generates moments at right angles to the velocityvector, but because of the symmetric position of quadrotor rotors, the lateral effects cancel. For stiff 2-bladedrotors, the moments due to the coning angles are symmetric about the rotor hub and also cancel.

A distinction must be noted here in the use of the terminology flap angle β and deflection angle a1s. Theflap angle β of a rotor blade is typically defined in the helicopter literature as the total deflection of a rotorblade away from the horizontal in body coordinates at any point in the rotation, and is calculated as

β = a0s − a1s cos Ψ + b1s sin Ψ (7)

where a0s is the blade deflection due to coning, a1s and b1s are the longitudinal and lateral blade deflectionamplitudes, respectively, due to flapping. Ψ is the azimuth angle of the blade, and is defined as 0 at therear. Since coning affects both blades equally, the deflection of the thrust vector is due to both longitudinaland lateral tilts. For quadrotor vehicles, however, the moments generated by lateral deflections cancel,and generation of unbalanced moments is due entirely to the longitudinal deflection, a1s. The longitudinaldeflection gives the amplitude of the rotor tilt fore and aft (Ψ = π, 0 rad), which we will refer to from hereon as the deflection angle to avoid confusion with the flapping angle, β.

The equation for deflection angle of a flapping rotor with hinged blades is presented in Pounds as:15

a1s =1

1 +µ2

lon

2

4

3

(

CT

σ

2

3

µlonγ

a0

+ µlon

)

(8)

where a0 is the slope of the lift curve per radian (typically about 6.0 for conventional airfoils at low Machnumbers according to Prouty5), µlon is the longitudinal rotor advance ratio, defined as the ratio of thelongitudinal to blade tip speed,

µlon =vlon

vt(9)

and γ is the nondimensional Lock number, which gives the ratio aerodynamic to centrifugal forces and isdefined as

γ =ρa0cR

4

Ib(10)

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Figure 6. Diagram representing the effects of rotor blade flapping5 and the modeling of stiff rotor blades ashinged blades with an effective offset and blade stiffness kβ.3 The rotor plane becomes tilted, resulting in adeflection of the thrust vector, and a moment is generated at the blade root.

where Ib is the moment of inertial of the blade about the hinge, c is the chord of the blade, and R is therotor radius. σ is the solidity ratio of the rotor, and is defined as

σ =Ab

A(11)

where Ab is the total area of the rotor blades.Equation (8) predicts a roughly linear relationship between velocity and deflection angle in the STARMAC

II operating regime. In practice this equation over-predicts the flapping seen by rotors with unhinged blades(see Figure 7) where the stiffness of the blades must accounted for.

The flapping properties of a stiff, fixed-pitch rotor blade can be analyzed by modeling the blade as beinghinged at an effective offset ef from the center of rotation (expressed as a percentage of the rotor radius) anda torsional spring with stiffness kβ Nm/rad at the hinge.3 This approximates the first bending mode of theblade and is sufficient for the small angles we are concerned with. Both ef and and kβ can be determinedby measuring the natural frequency ωn of blade vibration and using the following relations:3,5

ωn =

√

kβ

Ib(12)

ef =1

34

bΩ2Ib

kβ

(13)

where b is the number of blades and Ib is the moment of effective moment of inertia of the blade about thehinge at ef . Substituting Equation (12) into Equation (13), we have

ef =1

34

bΩ2

ω2n

(14)

The constants kβ and Ib can be obtained by determining the force required at the tip to deflect the bladethrough some angle δ and balancing moments:

F (1 − ef )Rδ = kβδ (15)

Substituting the value for kβ determined in Equation (15) back into Equation (12) yields Ib. With theseparameters, the equilibrium flapping constants can be determined by solving the following matrix equationdeveloped in Newman:3

λ2β 0 0 0

γ6µlon (1 − λ2

β) −γ8

0

0 γ8

(1 − λ2β) 0

0 0 0 1

a0s

a1s

b1sCT

σa0

=

γ8

0 −γ6µlon −γ

6

0 −γ8

0 0γ3µlon 0 −γ

80

13

0 − 12µlon − 1

2

Θavg

A

B

µver + λi

(16)

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2 4 6 8

145

150

155

160

165

170

175

time (s)

Mea

sure

d H

oriz

onta

l For

ce (

gram

s fo

rce)

0 m/s0.6 m/s1.1 m/s1.5 m/s1.8 m/s2.5 m/s3.4 m/s

(a)

0 1 2 3 40

1

2

3

4

5

Wind Velocity (m/s)

Def

lect

ion

Ang

le a

1s (

deg)

measuredpredicted (stiff blades)predicted (hinged blades)

(b)

Figure 7. (a) Horizontal force is measured at different wind velocities in order to calculate the flapping anglesand to see the effect of flapping. (b) The measured deflection angle is compared with predicted values usingequations which assume hinged, freely flapping blades and more complex ones which model stiff, unhingedblades. The hinged equations greatly over-predict the flapping effect if used to analyze the behavior of stiffblades.

Equation (16) depends heavily on several parameters. Once again, µlon and µver are the horizontal andvertical advance ratios, respectively (µver = 0 in translational flight). Θavg, A, and B are constants whichdefine the rotor pitch control:

Θ = Θavg + A cos Ψ + B sin Ψ (17)

For fixed pitch propellers, A, B = 0 and Θavg is the average pitch angle.λβ is the ratio of the flapping frequency ωβ to the angular rate Ω of the rotor, and for stiff propellers is

defined asλβ =

ωβ

Ω(18)

and can be calculated for use in Equation (16) as

λβ =

√

(1 +3

2ef ) +

kβ

IbΩ2(19)

2. Static Measurements of Flapping Effect

The lateral force due to the deflection of the thrust vector by flapping was measured for a single rotor byblowing air at fixed velocities across a spinning rotor attached to the test stand. This data was filtered andused to calculate the average deflection angle as a function of incident wind velocity and compared to thepredictions of the flapping equations developed in Newman.3 ωβ for the flapping equations was also measuredusing the test stand, giving an effective hinge offset of 25%. kbeta was measured to be 0.23Nm/rad. Theresults are plotted in Figure 7.

Although the deflection angles noted in Figure 7 are very small, they represent a significant effect onSTARMAC II. The typical control effort for low-velocity attitude tracking as shown in Figure 8 is a differentialon the order of 3 percent of hover thrust, generating a moment of 0.06Nm. At the STARMAC II hoverthrust of 3.4N per motor, a 0.75 deflection of the thrust vector, seen at speeds of about 2.5m/s, results ina moment about the c.g. of 0.022Nm, when the c.g. was located 12cm below the rotor plane. The totalmoment from the blade stiffness of all four rotors with deflection angles of 0.75 was 0.024Nm. Therefore,at 2.5m/s, the aerodynamic moments generated on the vehicle by flapping can be as high as 75% of thecommanded torque. STARMAC II has been flown at speeds of up to 3.5m/s, where the moment caused byflapping can dominate over attitude control commands.

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62 64 66 68 70 72

−10

−5

0

5

10

15

20

25

time (s)

Rol

l (de

g) a

nd V

eloc

ity (

m/s

)

Actual RollCommanded RollVelocity

Figure 8. At low velocities, i.e. for small displacements from hover, a PID controller is sufficient for goodattitude control.

3. Attitude Control in Nominal Conditions

At low velocities and with small aerodynamic disturbances (for example in indoor flight), PID control is fullysufficient for good tracking of commanded attitude since the vehicle approximates a double-integrator witha first-order lag from the motor dynamics. For initial test flights indoors with STARMAC, good trackingwas obtained with simple PD controllers for pitch and roll (see Figure 8) giving tracking errors on the orderof 2 − 3.

4. Effect of Flapping on Attitude Control

In translational flight, the pitch and roll dynamics of a quadrotor are very sensitive to rotor blade flapping.There is a moment generated at the hub of each rotor by the aerodynamic forces creating the blade deflection,and a moment about the center of gravity from the horizontal deflection of the thrust if the plane of therotors is not in line with the c.g. This means that position and attitude control are not completely decoupled,as has been assumed in the past.20 In Figure 9, the effects of the blade flapping moments and c.g. locationcan be seen during a step input in the pitch command. Initially, the control effort commanded by the PDcontroller is sufficient to bring the vehicle toward the commanded pitch. As the speed increases, the restoringmoments caused by blade flapping increase until the commanded torque is insufficient to hold the vehicleat commanded pitch despite an increase in the pitch error. With the c.g. below the plane of the rotorsas in Figure 9a, the effect is especially strong, since the horizontal component of the thrust deflection dueto flapping adds another moment in addition to that generated directly by the blades flapping. With thec.g. relocated to be level with the rotor plane, this extra moment is eliminated. This difference can be seenin Figure 9b, where the pitch gets much closer to the commanded value, and the aircraft also settles at ahigher equilibrium velocity since the only restoring moment is through the aerodynamic moments actingdirectly on the blades. The effect of the position of the center of gravity is strong. Flight logs show thatthe commanded torque in any axis is linearly correlated to the speed in the corresponding direction. Themagnitude of the linear coefficient increases as the c.g. is shifted below the plane of the rotors. Using linearregression on filtered flight data, the coefficient was found to be 0.014Nm when the c.g. was 0.02m belowthe rotor plane, and 0.023Nm when the c.g. was 0.12m below the rotor plane. Note that typical variationin the commanded torque in stationary hover is on the order of these values.

B. Effect of Airflow Disruption on Attitude Control

1. Motor Placement and Vortex Impingement

Motor placement and vortex impingement on the aircraft body had a significant effect on attitude stability.Unfortunately, we did not have time to prepare results for inclusion by the paper deadline. Details willbe included in the final version of this paper. They will include the predicted dynamics, using the inertial

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57 58 59 60 61 62 63−10

0

10

20

Rol

l Ang

le (

deg)

ActualCommanded

57 58 59 60 61 62 63−1

0

1

2

3

4

Time(s)

Wes

twar

d V

eloc

ity (

m/s

)

(a)

17 18 19 20 21 22 23−10

0

10

20

Rol

l Ang

le (

deg)

ActualCommanded

17 18 19 20 21 22 23−1

0

1

2

3

4

Time(s)

Wes

twar

d V

eloc

ity (

m/s

)

(b)

Figure 9. The effect of rotor blade flapping is shown in the vehicle’s response to a step input command in pitchangle. The controller is a simple PD controller, without an integral component, in order to demonstrate theeffects of rotor blade flapping on the quadrotor in translational motion. a Center of gravity c.g. 0.12 m belowthe plane of the rotors. As the aircraft translates, the thrust vector is deflected, causes a restoring momentabout the c.g., adding to the moment from the flapping of the blades to force the vehicle to tilt back. b C.G.0.02 m below the plane of rotors, so horizontal force from the tilted thrust vectors generates less moment. Notethe reduced undershoot and higher speed achieved for the same commanded attitude.

dynamics and a motor model,23 and the dynamics observed experimentally, given the initial frame design.Modifications were made to the air frame to test several physical configurations for aerodynamic interferencewith the rotor wash. Upon converging on a suitable frame design, a root locus analysis was done forproportional and derivative parameters, separately factored out, to aid in online controller tuning to improveperformance.

2. Effect of Protective Shrouds on Yaw Control

Yaw control is achieved by giving differential commands to rotors spinning in opposite directions to generatea torque about the vertical axis which is used for control through a PID controller. We have noticed that thetorque of a motor at a given command voltage depends on the airflow through the rotor. The initial designof STARMAC II called for protective circular shrouds around each propeller. In this configuration, it wasvery difficult to get good or even consistent yaw tracking performance (see Figure 10a). The shrouds weresimple protective enclosures which were not designed with aerodynamic considerations in mind. With theshrouds removed, yaw tracking instantly improved from errors of roughly ±10 to less about ±3 (see Figure10b). Subsequent analysis of the data showed that during shrouded flights, measured angular accelerationsof the vehicle did not consistently match with motor commands. The mostly likely explanation is that theshrouds were disturbing or disrupting the flow of air through the rotors, causing the actual motor torque tovary for a given commanded voltage level. Replacing the shrouds with fixed guards away from the rotorseliminated this problem.

VI. Altitude Control

A. Effect of Angle-of-Attack on Thrust in Translational Flight

1. Aerodynamics of Induced Power

The induced velocity, vi, is the change in air speed across the blades, with respect to the free stream velocity,vinf , induced by the rotor blades. The induced power is the required power input to create the inducedvelocity. As a rotorcraft undergoes translational motion, the induced power requirement of a rotorcraftchanges. Note that vinf is the total free stream velocity, not just the translational component. To derivethe effect of free stream velocity on induced power, from conservation of momentum, the induced velocity,

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0 10 20 30 40 50

−10

−5

0

5

10

15

20

time (s)

Yaw

(de

g)

actual yawcommanded yaw

(a)

0 10 20 30 40 50 60

−10

−5

0

5

10

15

20

time (s)

Yaw

(de

g)

actual yawcommanded yaw

(b)

Figure 10. The effect of shrouds on yaw control. a) With shrouds, unless the shrouds are built to very exactingspecifications yaw control performance is highly variable since instability in the air flowing through the shroudsleads to highly varying torques on the rotors. b) With the shrouds removed, performance of the yaw controlloop is instantly improved.

vi for an ideal vehicle can be found by solving4

vi =v2

h√

(v∞ cosα)2 + (v∞ sinα+ vi)2(20)

for vi, α is the angle of attack, with positive corresponding to pitching forward, and vh is the induced velocityat hover conditions. From momentum theory,4

vh =

√

T

2ρA(21)

where T is the thrust produced by the rotor to remain in hover. For a quadrotor helicopter, this is equal to14Tnom, where the nominal thrust Tnom is equal to the weight of the vehicle.

The analytical solution for vi is the solution to the quartic polynomial of Equation (20), and its expressionis too large to includeb. Using the expression for vi, or a numerical solution, the ideal thrust per power inputcan be computed, using

T =P

v∞ sinα+ vi(22)

where the denominator corresponds to the air speed across the rotors. The value of the ratio of thrust tohover thrust, T/Th, is plotted for the vh of STARMAC II, in Figure 12. At low speeds, the angle of attackhas vanishingly little effect on T/Th. However, as speed increases, the ratio T/Th becomes increasinglysensitive to the angle of attack, varying by a substantial fraction of the aircraft’s capabilities, within theflight envelope. Similar to an airplane, pitching up increases the lift force. The angle of attack for whichthrust is at the hover value increases with forward speed. For level flight, the power required to retainaltitude increases with the forward speed.

2. Altitude Control in Nominal Conditions

Altitude is controlled using a modified version of the Integral Sliding Mode (ISM) controller described inprevious work.20 The modification adjusts the control input ualt to be

ualt = up + ud +Tnom

cos θ cosφ(23)

bAlthough the solution is too long to print here, it will be provided in a Matlab script online.

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0.96

0.980.98

1

1

1.02

1.02

1.04

1.04

1.06

1.06

1.08

1.08

1.1

1.1

1.12

1.12

1.14

1.14

1.16

1.18

1.2

1.22

1.24

1.26 1.28

1.3 1.32

1.34

1.36

1.38

1.4

1.42

1.44

1.46

1.48

1.5

1.54 1.6

1.66

Flight Speed (m/s)

Ang

le o

f Atta

ck (

deg)

(T/Th)P=const

for vh=6 m/s

1 2 3 4 5 6−20

−15

−10

−5

0

5

10

15

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Figure 11. The thrust generated at a constant power, Pconst, varies depending on vehicle’s speed and angle ofattack relative to the incoming flow (the angle of attack is positive when pitching down). As speed increases,the variation of thrust with the angle of attack becomes more extreme, causing strong control disturbances,particularly in attitude, as shown in Figure 12.

where up and ud are the proportional and derivative control terms commanded by the ISM controller, and θand φ are the pitch and roll angles respectively. The third term is the steady state offset required to maintainan altitude adjusted for the loss of vertical thrust due the deflection of the thrust vector from rolling andpitching.20

3. Flight Results

In general, the ISM controller has proved to be very effective in altitude control, though performance canbe improved by better filtering of the ultrasonic altitude sensor readings (see Figure 12a). It must providestrong active damping whenever descent velocity is encountered. Otherwise, attitude oscillations have beenobserved to occur, due to an apparent drop in thrust during small descent velocities, as predicted by theinduced velocity model results, shown in Figure 13. However, with strong damping, this effect has beenreduced, as shown in Figure 12a.

Apparent degradation in the altitude control capabilities as speed increases has been found to be dueto a strong aerodynamic effect. The thrust variation created by different angles of attack at varying speedsand wind conditions can have a substantial, systematic, nonlinear effect which requires wind estimation tobe compensated for, as was derived in Section 1.

These effects of forward speed are strongly felt by the altitude control loop, even with the disturbancerejection provided by an integrator term. For slow changes in motion, although disturbances occur, theirslow speed allows the controller to reject them effectively. However, the quadrotor is able to rapidly pitch androll, leading to disturbances that are difficult to reject at non-zero speeds, as shown in Figure 12. The abilityto counteract this effect in control is compounded by the difficulty of accurately measuring the ambient wind.

This is demonstrated in Figure 12b, where the vehicle was operating under human control of attitudecommands, and with the roll axis aligned with North/South. A positive roll was commanded to arrestthe vehicle’s sideways progress, but as it rotated through 0 roll while at a velocity of 3.5m/s, the thrustincreased such that the controller was unable to compensate for the upward force and caused the vehicle to“balloon” high above the commanded altitude.

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75 80 85 90 95

0.4

0.45

0.5

0.55

0.6

0.65

0.7

Time (s)

Alti

tude

(m

)

ActualCommanded

(a)

20 21 22 23 24 25

0

20

40

Rol

l (de

g)

20 21 22 23 24 250

0.5

1

Alti

tude

(m

)

20 21 22 23 24 25

−2

0

2

Time (s)

Eas

t Vel

ocity

(m

/s)

actualcommanded

actualcommanded

(b)

Figure 12. The effect of angle of attack and velocity on thrust is demonstrated during a flight with human-commanded attitude. (a) Altitude data from a typical flight near hover with sinusoidal roll commands, witha usual error of ±10cm. Note the sharp spikes which indicate noise in the ultrasonic altitude sensor. (b) Datafrom a subsequent flight. At just past 22, s, a positive roll is commanded to reduce the vehicle’s westwardvelocity (the vehicle is aligned such that the roll axis points north/south). As the vehicle rolls through 0, itstill has a large non-zero velocity, giving it zero angle of attack relative to the incoming flow. This causes thethrust to increase (see Figure 11), pushing the vehicle above the commanded altitude. Note that the altitudereference is time varying, though not shown here.

B. Ascending and Descending Rotors

Rotorcraft have three operational modes for climb velocity,4 vc,

1. Normal working state: 0 ≤ vc

vh

2. Vortex ring state (VRS): −2 ≤ vc

vh< 0

3. Windmill brake state: vc

vh< −2

In normal working state, air is flowing down through the rotor. In windmill brake state, air is flowingup through the rotor due to rapid descent. For these two states, conservation of momentum can be usedto derive the induced velocity. For the normal working state, the hover and ascent condition, the inducedvelocity is4

vi = −vc

2+

√

(vc

2

)2

+ v2h (24)

For the windmill braking state, rapid descent, it is

vi = −vc

2−

√

(vc

2

)2

− v2h (25)

In the vortex ring state, air recirculates through the blades in a periodic and somewhat random fashion.As a result, the induced velocity varies greatly, particularly over the domain −1.4 ≥ vc/vh ≥ −0.4, reducingaerodynamic damping.24 An empirical model of induced velocity in vortex ring state is4

vi = −vh

(

κ+ k1

(

vc

vh

)

+ k1

(

vc

vh

)

+ k2

(

vc

vh

)2

+ k3

(

vc

vh

)3

+ k4

(

vc

vh

)4)

(26)

where k1 = −1.125, k2 = −1.372, k3 = −1.718, k4 = −0.655. This model compares with the mean ofexperimental results in literature, though it fails to capture the periodic nature of the vortex entrapment.4

To model the dynamics during climbing, the power is the thrust times the speed it is applied at,

P = T (vc + vi) (27)

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−1 −0.5 0 0.5 10.85

0.9

0.95

1

1.05

Vc

(

ms

)

T/T

h

P(W

−1)

(a)

−3 −2 −1 0 1 20.9

1

1.1

1.2

1.3

Velocity (m/s)

T /

T h

(b)

Figure 13. a) Theoretically predicted effect of climb velocity, vc, on normalized thrust, per unit of power inputand b) thrust test stand measured effect of vc on normalized thrust. The hover velocity is for a vehicle with thesame parameters as STARMAC II. In ascent, there is a virtual damping force close to linearly proportional tospeed. However, for descent at a slow velocity, there is no damping force, rather there may be a small negativedamping force. The ascent equations are based on momentum conservation of the airstream, and the descentequations are based on published experimental curve fits.

ignoring profile power losses. Note that Tvc is the power consumed by the climbing motion, whereas Tvi isthe power transfered into the air. It is typically desirable to avoid the vortex ring state. Note that one wayto avoid this state is to maintain a substantial forward speed.3

We can compute the thrust achieved for a given input power, as a function of climb velocity, by sub-stituting Equation (24), Equation (25), and Equation (26) into Equation (27). For the flight conditionsexperienced by STARMAC II, the ratio of the thrust to hover thrust, per power input, is shown in Figure13, where plot (a) is the theoretical curve, using the solution to the above equations, and plot (b) showsdata from a thrust test stand experiment using a vertical wind disturbance. As shown in the plots, thereis certainly a damping effect associated with climbing, with thrust reducing nearly proportionally to theclimb velocity. However, in decent, for low speeds, there is essentially no damping. There is possibly nega-tive damping for small descent rates, although the model of Equation (26) is an approximation based on acompilation of experimental data in literature.

In thrust test stand experiments, the loss of thrust with an applied climb velocity was experienced. Thedescent velocity experiments were much less conclusive. The average thrust remained close to the zero climbvelocity value, though it oscillated substantially, making assessment of the change in thrust difficult. A moreformal summary of the data will be included in the final version of this paper.

VII. Flight Results

Position control is currently implemented using a PID controller which actuates the vehicle’s roll andpitch as control inputs. Tilting the vehicle in any direction causes a component of the thrust vector to pointin that direction, so commanding pitch and roll is directly analogous to commanding accelerations in theX-Y plane. The inner-loop dynamics are assumed to be fast enough not to affect position control. Figure14 shows a typical flight using this control scheme. The vehicle is able to stay inside of a 1.5m-radius circle,but the current control implementation has little ability to reject disturbances from wind and translationalvelocity effects. A key weakness of this and similar position controllers used by other groups is the assumptionthat position control and attitude control are decoupled. This is in fact only true for very small velocities.As shown in Section V, there is significant coupling between the velocity of the aircraft and the attitudedynamics. The PD and LQR-derived PD attitude controllers currently used by many groups25 would beunable to cope with this coupling, leading to loss of performance for both attitude and position control.

Ultimately, the dynamics of a quadrotor in flight in attitude, altitude, and position are not decoupled tothe extent previously assumed.9,20,25 The aerodynamics in flight cause disturbance forces to be generatedin each set of states to be controlled. If these effects are not accounted for, control performance can be

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2 3 4 5 6

1

1.5

2

2.5

3

3.5

4

4.5

East (m)

Nor

th (

m)

trajectoryError CircleDesired PointStartStop

Figure 14. Hover position control with simple PD control commanding attitude achieves a maximum errorof about 2m, with the integrator turned off (in poth the position and the attitude control loops) to keepthe dynamics simple. Thus, the quadrotor helicopter was very vulnerable to wind gusts, demonstrating thedisturbances experienced, without the dynamics introduced by the integrator. Tests such as these showedwhat the underlying aerodynamic phenomena were that must be addressed.

greatly diminished. In order to achieve good tracking of either altitude, attitude, or position, the inner-loopcontrollers must have good enough disturbance rejection to negate the coupling effects from translationaland vertical velocities as well as external disturbances like wind. Vehicle design can also reduce or eliminatesome of these effects (i.e. location of the c.g. and rotor spacing). In the final version of the paper, we willpresent results for improved autonomous hover and trajectory tracking with controllers incorporating theaerodynamic effects discussed in earlier sections.

VIII. Conclusions

Quadrotor helicopters are popular as testbeds for small UAV development, but their aerodynamics arecomplex and have not been well addressed. Although many good control results have been reported in previ-ous work, these have focused around simple trajectories at low velocities, in a controlled indoor environment.In this paper, we have addressed a number of issues observed in quadrotor aircraft operating at speed andin the presence of wind disturbances. We have explored the resulting forces and moments applied to thevehicle through these aerodynamic effects and investigated their impact on attitude and altitude control.We have uncovered the extent of their influence using data from static measurements and flight data fromthe STARMAC II quadrotor. These results have shown that existing models and control techniques are in-adequate for accurate trajectory tracking at speed and in uncontrolled environments. Careful considerationof these disturbances has allowed us to improve both the physical configuration and control design of theSTARMAC II quadrotor, improving attitude and altitude tracking performance and permitting controlled,stable flight at higher velocities and in the presence of gusting winds. This work should open the door toimproved autonomous hover and trajectory tracking in the near future, enabling most of the applicationsthat have been envisaged for the STARMAC testbed.

Acknowledgments

The authors would like to thank Jung Soon Jang, David Shoemaker, David Dostal, Dev Gorur Raj-narayan, Vijay Pradeep, Paul Yu, and Justin Hendrickson, for their many contributions to the developmentof the STARMAC testbed.

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quadrotor helicopter flight dynamics and control: theory and experiment · 2015-07-29 · theory...

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