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Influence of Aerodynamics on Quadrotor Dynamics
MARCO C. DE SIMONEUniversity of Salerno
Department of Industrial EngineeringVia Giovanni Paolo II, 84084 Fisciano
SERENA RUSSOUniversity of Naples, Federico II
Department of Industrial EngineeringCorso Umberto I 40 - 80138 Napoli
ALESSANDRO RUGGIEROUniversity of Salerno
Department of Industrial EngineeringVia Giovanni Paolo II, 84084 Fisciano
Abstract: The aim of this work is to study the aerodynamics effect of the propellers on the dynamic behaviourof a quadrotor helicopter system. Here an inverse dynamics analysis and a standard PID linear controller forfeedback control are considered for a simplified model of UAV represented by a linear under-actuated system.Aerodynamic forces are modeled with the blade element theory to consider the variation of the thrust coefficientswith the different flight conditions, in order to calculate the aerodynamic effect of drag and thrust on the rotation ofthe propellers. This aerodynamic analysis is very useful to predict the dynamic behaviour of the system in presenceof wind fields and gives us indications about the type of propellers and motor characteristics to be installed on theUAV. The proposed dynamical model and its control strategies can be source for future works.
Key–Words: UAV, Quadcopter, PID Control, Under-Actuated System.
1 Introduction
A quadrotor UAV is a unmanned mini aerial vehicles(UMAVs) whose lift is generated by four independentrotors. They were initially designed for military ap-plication such as surveillance, border patrolling, minedetection, aerial delivery of payload, while, at themoment, they are widely used in several applicationof life such as civilian purposes like disaster man-agement during floods, earthquakes, fire, commercialmissions like aerial photography, television and cin-ema shootings, and research and development pro-gram which need flying vehicles to perform variousexperiments. They are characterized by vertical take-off and landing (VTOL) capabilities which gives ithigher maneuverability and hovering capabilities.
The first real concept of a Quadrotor is due toBreguet and Richet [1]: it was a heavy vehicle ca-pable of flying only over a small height and for shortduration. However, over the last years, there has beena great development in the area of quadrotor control,e. g. [2], [3], where the quadrotor dynamics is ap-proximated with a linear system and a standard linearcontroller is designed (in [4] a PI and PID control hasbeen considered).
The main difficulties in the tracking control of aquadrotor are represented by the parametrization andmodelization of external disturbances, such as windgusts, that could deteriorate the quadrotor abilitiesof fly and, then, cause instabilities ([5] and [6]). Inthe flight altitudes typical of these vehicles, quadrotor
aerodynamic performances are more sensible to windfields; so the design of the flight control systems rep-resents one of the most critical aspects to be consid-ered in order to track the desired trajectory in presenceof wind external disturbances. One novel aspect thatis considered in this article and will be deepened inthe successive sperimental activity is represented bythe study of the aerodynamics of the propellers that,in general, affects the dynamic response and perfor-mance of the UAVs. It depends on the different flightconditions because it is important to include relativewind velocity and gusts (see [6] and [7]). Unless whatis already done in several applications, here a differ-ent approach will be considered in which the bladeelement theory will be taken into account in the dif-ferent flight conditions because there is a change inthe trajectory of the drone and also of its velocity, assuggested by the classical propellers theory [8].
In this study, we present a preliminary study fora simplified model of quadrotor in which an inversedynamic model and a standard PDI linear controlleris considered. Inverse dynamics control has been al-ready applied in the past (see [9]) and it representsan approach to nonlinear control design of which themain scope is to construct an inner loop control basedon the motion base dynamic model which, in the idealcase, exactly linearizes the nonlinear system and anouter loop control to drive tracking errors to zero[15, 16, 17, 18]. Nonetheless, this technique is basedon the assumption of exact cancellation of nonlinearterms, therefore, parametric uncertainty, unmodeled
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dynamics and external disturbances may deterioratethe controller performance. The standard PID linearcontroller is considered in the second part of this pa-per, where the effect of wind fields will be taken intoaccount.
The paper is organized as follows. Quadrotor Dy-namic Model section discusses the quadrotor dynamicmodel: it focuses the attention on the set of equationsused to model the quadrotor dynamics with particularattention to the treatment of the thrust forces gener-ated by each propeller whose expressions are derivedthrough blade element theory. Numerical Simulationscontains some details about the prescribed trajectorygiven to the quadrotor and is divided into two parts:the first part relates with the inverse dynamic analysisof the model, while the second part is dedicated to thefeedback control law evaluation in presence of winddisturbances. Final section presents the conclusions.
1.1 Quadrotor dynamic modelIn figure 1 the classical configuration of quadrotor ispresented. The four rotors are labelled from 1 to 4 andare mounted at the end of each cross arm. One pair ofopposite propellers rotates clockwise, while the otherpair rotates anticlockwise: in this way it is possibleto avoid the yaw drift due to reactive torques and thelateral motion without changing the pitch of the pro-peller blades. Fixed pitch simplifies rotor mechanicsand reduces the gyroscopic effects.
Figure 1: Quadrotor axis system
By imposing different velocities to different pro-pellers, so differential aerodynamic forces and mo-ments, it is possible to control the UAV. For hover-ing, all four propellers need to rotate at same speed.The vertical motion is allowed because the speed of allfour propellers is increased or decreased by the sameamount, simultaneously. In order to pitch and movelaterally in the x direction, speed of propellers 1 and2 is changed conversely. Similarly, for roll and corre-sponding lateral motion in the z, speed of propellers 3
and 4 is changed conversely. Yaw along the y axis ispossible when the speed of one pair of two oppositelyplaced propellers is increased while the speed of theother pair is decreased by the same amount. This way,overall thrust produced is same, but differential dragmoment creates yawing motion. In spite of four actu-ators, the quadrotor is still an under-actuated system.In the hypothesis that the center of mass coincideswith the origin of body frame B, the propellers arerigid in plane and the structure is rigid and symmet-rical, the translational dynamics of the drone in thecoordinate system I can be written as
Mv = −
0mg0
+ R{I−B}
0T0
− FD + FW (1)
where I is the inertial frame, v is the velocity vectorof the center of gravity of the aerial vehicle in the in-ertial frame, M represents the mass matrix, R{I−B}is the rotation matrix from the body to the inertialframe, g is the gravitational acceleration, m is the to-tal mass of the system, T =
∑Ti is the total upward
thrust, FD = [FDx, FDy, FDz]T is drag force acting
opposite to the relative motion of the quadrotor ob-ject moving with respect to the surrounding air andFW = [FWx, FWy, FWz]
T represents the wind dis-turbance.gives the rotational acceleration of the body frame isgiven by Euler’s equation of motion
Iω = −ω × ωI + Γ (2)
where I is the 3 × 3 inertia matrix of the system, ωis the angular velocity vector and Γ = [τx, τy, τz]
T
is the torque applied to the frame. As already said,differences in rotor thrust produce rotation along theaxis. Rolling is caused by the torque along the x-axis,τx = d(T4 − T3) where d is the distance of the pro-peller from the center of gravity of the system. Pitch-ing torque, τz = d(T2 − T1), is responsible of the ro-tation of the frame along the z-axis. Yawing rotation,instead, is due to the total reaction torque of each pro-peller which acts at rotating the airframe in the oppo-site direction of its rotation τy = Q1 +Q2−Q3−Q4.In this work, a simplified 2D model is consideredwhere the rolling, yawing, traslation along the z axisbalance equations are satisfied, so that the set of equa-tions simplify to{
−mx+ Tx − FDx + FWx = 0
−my −mg + Ty − FDy + FWy = 0(3)
The pitch angle cannot be arbitrarily set, but it is ob-tained from the following condition constraint
TxTy
= tan(θ) ' θ, (4)
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The moment balance equation (2) becomes ascalar equation because the only pitching torque τz isdifferent from zero.
2 Aerodynamic forces and torques
The resulting thrust force Ti produced from each pro-peller is proportional to the square of its angular speedrotation using momentum theory [8] according to thefollowing relation
Ti = bω2i , i = 1, 2, 3, 4; (5)
where parameter b needs to be determined for eachpropeller. Since the quadrotor is characterized by atime constant of the motor dynamics less than 1msthat is faster than the time scale typical of rigid bodydynamics, it is possible to assume that Ti changes is-tantaneously. The main error of this model is repre-sented by the simplistic assumption underlying for-mula 5 that the thrust forces are directly proportionalto the square of the motor speed. In reality they arecomplex functions of the motor speed and environ-mental conditions.The drag force is imposed to have the following rela-tion (see [11]), so it is proportional to the translationalvelocity of drone
FD =1
2ρSCDV2, (6)
where S is the cross-sectional area of the quadrotor,CD = diag(CDx, CDy, CDz) is the traslational dragcoefficient.
2.1 Aerodynamics of vertical and forwardflight
A rotor generates thrust by inducing a velocity on theair that passes through it. In hovering (that is the situa-tion in which the quadrotor maintains a constant posi-tion over the ground), the thrust (Ti,h) generated fromeach propeller is linked to the (vertical) induced veloc-ity (vh), that is the velocity induced on the air withoutrelative velocity between the rotor and the air. It isderived from the momentum analysis as
vh =
√Ti,h2ρA
. (7)
The relative velocity between the rotor plane and thesurrounding air (V) and the pitch angle, also, influ-ence the thrust produced by each rotor (see figure 2).
Figure 2: Diagram of rotor.
According to the blade element theory, it is possibleto find an expression for the thrust like this
Ti = ctρπR4pω
2i . (8)
Here ρ is the air density and Rp represents the radiusof the propeller. When θ = π/2, the quadrotor isin vertical (climb or descendent) flight and the thrustcoefficients (ct) can be derived with the following ex-pression [14]
ct =nca
4πRp
(βt −
vh ± VωRp
), (9)
where βt is the pitch angle at the blade tip that is func-tion of rotor geometry alone as we are consideringfixed pitch rotors [14], a is the lift slope curve, n isthe number of blades on each propeller and c is theblade chord. Formula 9 is derived in the hypothesis ofconstant chord and linear twist and assuming uniforminflow. Given the vertical velocity y and the geometricparameters, it is possible to find the ωi which producesa given thrust at the current vertical velocity.In forward flight, it is important to consider the an-gle of attack (θ) to the model, so the equation for thethrust coefficient becomes
ct =nca
2πRp
(βt3
+V 2βcos2θ
2ω2iR
2p
+V sinθ + vh
2ωRp
). (10)
In order to test the reliability of the above expressionof ct (formula 9), the results are compared with an ex-perimental database ([19]). Figure 3 depicts the com-parison between the values of the thrust coefficientsobtained from the database (the blue circles) and thecurve obtained from formula 9: the thrust coefficientsare plotted against J
J =V
NDp, (11)
where N is the angular velocity in rpm and Dp repre-sents the diameter of the propeller.Fig. 4 displays the thrust coefficient trends for the in-teresting range of N of our quadrotor. One of the novel
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0.25 0.3 0.35 0.40.02
0.025
0.03
0.035
0.04
0.045
0.05
0.055
0.06
J
c t
N=1000
Formula [Leishman]Database
Figure 3: Thrust coefficients.
0.25 0.3 0.35 0.40.02
0.025
0.03
0.035
0.04
0.045
0.05
0.055
0.06
J
c t
N=1000N=2000N=3000
Figure 4: Thrust coefficients.
contribution of this paper is the introduction of a vari-ation of the ct coefficients for every flight conditions,instead of using a constant value of it and then of b asit is generally done in these applications. These coef-ficients cannot be directly derived from Fig. 4 becauseJ is function of N that is the parametrization constantof the curve, for this reason the corrected values havebeen extrapolated by trial.
3 Numerical Simulations
The dynamic model, that includes rotor dynamics andexternal perturbation like drag force and wind distur-bances, is developed using a multibody approach con-sisting in interconnected rigid components by differ-ent kinds of joints. Mass and inertia matrices are ob-tained by using a CAD model of the quadcopter andare represented by the following expressions (12 and13), i.e.
M =
0.37958 0 00 0.37958 00 0 0.37958
kg (12)
Segment xin[m] xfin[m] yin[m] yfin[m] tifin[s] Tfin[s]
1 0 0 0 5 10 102 0 100 5 5 40 503 100 100 5 0 10 60
Table 1: Waypoint Coordinates used for desired tra-jectory
J =
0.0053 −2.492e− 05 1.249e− 05−2.492e− 06 0.0092 7.245e− 051.249e− 05 7.245e− 05 0.0043
kgm/s2.(13)
The mathematical model used for the evaluation offeedforward control is reported in eq. 14, where T =∑4
i=1 Ti and τz = d(T2 − T1), i.e.
{T = Mv + F (v)
τz = Izz θ +MD(θ).(14)
In the general case, F contains the gravity effects, theaerodynamic drag terms and the wind disturbancescontributions, while MD is function of the aerody-namic drag moment.For the given trajectory, reported in figure 5(a), thelaw of motion for the two degrees of freedom systemis evaluated by using a quintic polynomial function.Smoothness represents an important characteristic ofthe law of motion, so that position and orientation varyslowly and their temporal derivatives are continuouswith time. The respective boundary conditions for ev-ery segment of the trajectory are reported in tabular 1,while velocity and acceleration boundary conditionsare all set equal to zero. By solving system 14, itis possible to derive the laws of motion for the threedegrees of freedom (x-axis, y-axis and pitch angle θ)and the feedforward control laws of the four propellerT1, T2, T3 and T4 necessary to obtain the desired tra-jectory.From the thrust forces, one can evaluate the angularvelocity that each motor must deliver in order to en-sure the tracing of the desired trajectory by using eq.5. Here we have verified if it is valid the hypothesis,that is generally made in such applications [3] [11], ofa constant parameter b and so of a constant thrust co-efficient ct. For this reason, we have extrapolated ct,that it is function not only of N , but also of the rel-ative velocity v, from fig.4 for each integration timeand we have verified that the results obtained here aredifferent from the case in which a constant value isconsidered. In this way it is possible to have the cor-rected angular velocities for every time step.
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0 20 40 60 80 100
1
2
3
4
5
6
7
Desired xy-plane trajectory
x[m]
y[m
]
trajectorystart pointtarget point
(a)
0 20 40 600
50
100
150x, x, x motion profile
t[s]
x[m
]
0 20 40 600
2
4
6
t[s]
x[m
/s]
0 20 40 60−0.4
−0.2
0
0.2
0.4
t[s]
x[m
/s2]
0 20 40 600
2
4
6y, y, y motion profile
t[s]
y[m
]
0 20 40 60−1
−0.5
0
0.5
1
t[s]
y[m
/s]
0 20 40 60−0.4
−0.2
0
0.2
0.4
t[s]
y[m
/s2]
(b)
Figure 5: (a) Target Trajectory, (b) Law of Motion forx-axis and y-axis
.
0 10 20 30 40 50 60−40
−20
0
20θ, θ, θ yaw motion profile
t[s]
θ[deg]
0 10 20 30 40 50 60−4
−2
0
2
4
t[s]
θ[deg/s]
0 10 20 30 40 50 60−0.5
0
0.5
1
t[s]
θ[deg/s2]
Figure 6: Derived Law of Motion for Pitch Angle
3.1 Inverse Dynamics Analysis in absence ofwind disturbances
In this section, the effect of wind disturbances is notconsidered in 14, so F only contains the gravity termand aerodynamic drag contributions, i.e.
F (v) = FD (v)−mg. (15)
By solving system 14 with the above expression for F,it is possible to derive the laws of motion for the twodegrees of freedom, x-axis, y-axis and the boundedpitch angle θ, that are depicted in figures 5(a), 5(b)and 6). Figures 5(a), 5(b) and 6 demonstrate that theyare in agreement with the given trajectory and lawsof motion. The thrust laws are reported in fig. 7,in which the curves that refer to propellers 4 and 3,can be overlapped because of the simplified 2D con-figuration. The angular velocities of each propeller
0 10 20 30 40 50 600.9
0.95
1
1.05
1.1
1.15Feedforward Control Law Evaluated
t[s]
T[N
]
T
1
T2
T3
T4
Figure 7: Thrust Forces Evaluated by Inverse Dynam-ics Analysis
are calculated from the thrust forces and the interpo-lated thrust coefficients (using eq. 5 and fig. 4) andare plotted in fig. 8: the different sign is due to thedifferent direction of rotation of each propeller, nec-essary to lateral displacement and the balance of mo-ments. In these cases, thrust coefficients are variablefor each time step: by comparing these results withthose obtained in a first approximation in which a con-stant value of b has been considered, we have observeddifferences of the order of 100rpm in the calculatedangular velocities.
Forward Dynamics Analysis in presence ofwind disturbances
In this section, a feedback control has been consideredin order to take into account the aerodynamic distur-
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0 20 40 6030.5
31
31.5
32
32.5
33
33.5
34
t[s]
w[rad/s]
0 20 40 6030.5
31
31.5
32
32.5
33
33.5
34
t[s]
w[rad/s]
0 20 40 60−33.5
−33
−32.5
−32
−31.5
−31
−30.5
t[s]
w[rad/s]
0 20 40 60−33.5
−33
−32.5
−32
−31.5
−31
−30.5
t[s]
w[rad/s]
w1
w2
w3
w4
Figure 8: Angular Velocity of each Propeller
bances represented by wind gusts. To counteract theseeffects a PID controller is developed, i.e.
Ffb (t) = kpe(t) + kde(t) + ki
∫ t
0e(t)dt, (16)
where kp, kd and ki respectively represent the propor-tional, derivative and integral gain values vector. Theerror signal vector has the following expression
e(t) =
[ex(t)ey(t)
]=
[xref − x(t)yref − y(t)
], (17)
where [xref , yref ]T is the desired solution. As sug-gested by Chen et al. ([11]), the wind field is modeledas an external force term, so that, in the balance equa-tions 14, F is the sum of three contributions, i.e.
F (v) = FD (v)−mg + FW (v) . (18)
To test the robustness of the controller, the vehicle issubjected to two wind fields W1 =
[2, 0]T m/s and
W2 =[2, 1]T m/s.
Figures 9(a) and 9(b) depict the comparison betweenthe open–loop real trajectories covered by the UAVunder feed–forward control law in presence of windgusts (continuous line) and the desired trajectory(dashed line): in both cases of wind velocities, it ispossible to observe a discrepancy between the two tra-jectories.Figures 10(a) and 10(b) display the effect of theclosed–loop control in the comparison between thereal trajectory (continuous line) and the desired one(dashed line). Unlike the previous case, curves per-fectly overlap: this evidence demonstrates the effec-tiveness of the projected closed–loop control.Finally the two control laws have been traced back,
0 50 100 150 200
1
2
3
4
5
6
7
Comparison of trajectories with wind disturbance
x[m]y[m
]
feedforward control trajectorytarget trajectory
(a)
0 50 100 150 200
20
40
60
80
100
120
Comparison of trajectories with wind disturbance
x[m]
y[m
]
feedforward control trajectorytarget trajectory
(b)
Figure 9: Comparison between the Open Loop Trajec-tories Subjected to Wind Fields and the Desired Tra-jectories: (a) refers to W1, (b) refers to W2
.
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0 10 20 30 40 50 60 70 80 90 100
1
2
3
4
5
6
7
x[m]
y[m
]
F
fw + F
b control trajectory
target trajectory
(a)
0 10 20 30 40 50 60 70 80 90 100
1
2
3
4
5
6
7
x[m]
y[m
]
F
fw + F
b control trajectory
target trajectory
(b)
Figure 10: Comparison between the Closed Loop Tra-jectories Subjected to Wind Fields and the DesiredTrajectories: (a) refers to W1, (b) refers to W2.
through the use of eq 5, to the values of the angularvelocities required by the four engines and have beenreported in figure 11. The first oscillations that char-acterize each curve of the angular velocity relative toW1 for the first time step are due to the sudden ar-rangement that the system must have in order to adaptto the presence of the wind field.
4 ConclusionIn this paper a 2D dynamical model of an under-actuated UAV has been presented. Firstly, an inversedynamics analysis has been performed on the modelfor feed-forward thrust evaluation from a given trajec-tory. In this phase only aerodynamics effects of draghave been considered and pitch angle motion law hasbeen evaluated together with the pitching torque re-quired.Thereafter, a linear PID controller has been designedfor feedback control in presence of wind fields. Thecomparison between the real and the desired pathshave demonstrated the effectiveness of the projectedclosed–loop control.For both cases of open–loop and closed–loop thrustlaws, we have verified the reliability of the hypothesisof constant thrust coefficient that is generally made in
0 20 40 601600
1650
1700
1750
t[s]
N[rpm]
Propeller 1
N
ffw
Nffw+fb
0 20 40 601600
1650
1700
1750
t[s]
N[rpm]
Propeller 2
N
ffw
Nffw+fb
0 20 40 60−1750
−1700
−1650
−1600
t[s]
N[rpm]
Propeller 3
N
ffw
Nffw+fb
0 20 40 60−1750
−1700
−1650
−1600
t[s]
N[rpm]
Propeller 4
N
ffw
Nffw+fb
(a)
0 20 40 601500
1550
1600
1650
1700
t[s]
N[rpm]
Propeller 1
N
ffw
Nffw+fb
0 20 40 601500
1550
1600
1650
1700
t[s]N[rpm]
Propeller 2
N
ffw
Nffw+fb
0 20 40 60−1700
−1650
−1600
−1550
−1500
t[s]
N[rpm]
Propeller 3
N
ffw
Nffw+fb
0 20 40 60−1700
−1650
−1600
−1550
−1500
t[s]
N[rpm]
Propeller 4
N
ffw
Nffw+fb
(b)
Figure 11: Comparison between the Open Loop (con-tinuous line) and the Closed Loop (dashed line) Tra-jectories Subjected to Wind Fields in term of the an-gular velocity of each propeller: (a) refers to W1, (b)refers to W2.
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these applications. Since it is generally function of theengine rpm, of the thrust forces required and of rela-tive velocity of the vehicle and its variation with theseparameters is not neglibible, all the values are extrap-olated from different curves parameterized as a func-tion of the rpm for every time steps. By comparingthese results with those obtained by using a constantcoefficient, there is an error in estimating the correctrpm of about ±100.The considerations developed in this work about asimplified 2D will be the starting point for a subse-quent analysis of a 3D model in which a variation ofthe thrust coefficient will be taken into account.
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