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Calibration and Linearization of Nonlinear Quadrotor Helicopter Model
Engr. M Yasir Amir
Department of Electronic and Power Engineering, National University of Sciences and Technology (PNEC) Karachi, Pakistan [email protected]

Dr. Valiuddin Abbass
Department of Electronic and Power Engineering, National University of Sciences and Technology (PNEC) Karachi, Pakistan [email protected]

Abstract— Quadrotor Helicopter is a rotorcraft that has four propellers. Two of the propellers spin clock wise and the other two counter-clockwise generating a vertical lifting force. Control of the machine can be achieved by varying relative speed of the propellers. Quadrotor concept is a rather old concept, however the modern quadrotor helicopters are mostly unmanned machines. Advancement in miniaturized Inertial Measurement Units and availability of high speed brushless motors and high power to weight ratio Li-Polymer battery technology, unmanned quadrotors can now be successfully designed and fabricated. This paper proposes linearized and calibrated mathematical model of quadrotor helicopter dynamics based on a nonlinear model proposed earlier [1]. The proposed model is linearized about the operating point which is the hover state. Keywords-quadrotor; dynamics; claibration; linearization; roll; pitch mathematical model;

These research efforts produced mathematical models mainly from two different approaches namely Euler’s method and Newton’s Method. These models are useful from mechanical design point of view. However a simplified model was proposed in the research work on Quadrotor Helicopter at National University of Sciences and Technology [7]. This model is very important from controller designing point of view. II. CALIBRATION OF QUADROTOR MODEL

The quadrotor model developed earlier [1] is given in Equation (1.1) to (1.5). This plant model is highly nonlinear and un-calibrated. The parameters were not assigned any realistic or practical values. Linearization about an operating point requires calibration. Calibration implies simply assigning some realistic values to various plant parameters that may be taken from any successful quadrotor project.
JZ Ω ZDΩ + KeΩ + =V Kq Kq
2

I.

INTRODUCTION

(2.1)

A Quadrotor Helicopter is a helicopter with four liftgenerating propellers mounted on motors. Two of the motors spin the propellers clockwise and other two counter-clockwise. Control of the machine can be achieved by varying the relative speed of the propellers. Quadrotor concept is an old concept. In Breguet Brothers built the first quadrotor named Gyroplane No.1 in 1907 [2]. In 1922, Georges de Bothezat contructed a truss structure of crossing beams, where the propeller were located the ends of the + shaped structure [9]. The modern quadrotors helicopters are mostly small unmanned machines. Due to the availability of high-speed brush-less dc motors, inertial measurement units based on Micro-Electro-Mechanical Systems technology and high power to weight ratio (>160W/Kg) Li-polymer batteries, unmanned quadrotors can now be successfully designed and fabricated [3]. In the past several years quite a lot of research effort has been directed towards the development of Quadrotors. Mesicopter [4] an ambitious project, was developed to explore new ways to fabricate centimeter-sized vehicles. Such vehicles can be used for gathering planetary atmospheric and meteorological data. The OS4 project [5] started in March 2003, was aimed to develop devices for searching and monitoring hostile indoor environments.

2l ρ a A ⎛ f η K t Φ= ⎜ I yy ⎜ K q ⎝
2l ρ a A ⎛ f η K t Θ= ⎜ I xx ⎜ K q ⎝
2

⎞ 2 2 ⎟ ⎡V3 − V1 ⎤ ⎦ ⎟ ⎣ ⎠
⎞ 2 2 ⎟ ⎡V2 − V4 ⎤ ⎦ ⎟ ⎣ ⎠
2

2

(2.2)

(2.3)

2ρ A ⎛ f ηKt ⎞ 2 2 2 2 az = a ⎜ ⎟ (V1 +V2 +V3 +V4 ) cos Θcos Φ− g M ⎜ Kq ⎟ ⎝ ⎠

(2.4)

2 2 I zz Ψ = J (Ω1 + Ω3 − Ω 2 − Ω 4 ) + D (Ω1 + Ω 3 − Ω 2 − Ω 2 ) 2 4

(2.5)

Table 2.1 gives the list of the various quantities that are used for calibration. These quantities have been taken form OS4 Project [6]. By using the data given in Table 2.1. these various variables present in the equations are assigned constant values which results in Equations (2.6).

Table 2.1 Calibration data for Quadrotor Helicopter.
Parameter Name 1 Rotational Inertia alon x-axis 2 Rotational Inertia alon y-axis 3 Rotational Inertia alon z-axis 4 Total Mass 5 Thrust Constant 6 Drag Constant 7 Prop Radius 8 Prop Disc Area 9 Prop Cord 10 Pitch of Incidence 11 Twist Pitch 12 Rotor Inertia 13 Arm Length 14 Motor wieght 15 Motor dc Resistance 16 Motor Torque Constant 17 Motor Speed Constant 18 Motor efficiency at Hover 19 Air Density 20 Propeller Figure of Merit 21 Prop Thrust Torq constant Numerical Symbol Value Ixx Iyy Izz M b D Rp A Cp 0.0075 0.0075 0.0013 0.65 3.13E-05 7.50E-07 0.15 0.07069 0.04 Unit Kg.m^2 Kg.m^2 Kg.m^2 Kg N.s^2 N.s^2 m m^2 m rad rad Kg.m^2 m g Ohms Nm/A V.s/rad % Kg/m^3 NIL m

A. Operating Point Caluculations The operating point for the out puts as specified above is given in (6.2) below.

ΘQ = Θ = Θ = zero Φ Q = Φ = Φ = zero Ψ Q = Ψ = Ψ = zero Ω = ΩQ (cons) Ω = Ω = zero
Substituting these values in relation between voltage input of motors and vertical acceleration in (2.6) gives:
2 0 = 0.00237 (V1Q + V22Q + V32 + V42Q ) (1) − g Q

(3.1)

Theta o 0.26 Theta-tw 0.045 Jr l m Z Kq Ke eta pa f Kt 6.04E-05 0.23 51 0.6 0.0018 0.0015 64 1.1 0.5 0.00056

As a basic assumption all the motors are assumed to be identical and design is perfectly symmetric, therefore the voltage inputs of all motors at operating point is assumed to be equal.

0.00237 ( 4VQ2 ) = g
VQ = g = 32.24volts 4X

(3.2) Here X = 0.00237. Equation (3.2) gives the operating input voltage to the quadrotor plant. The first equation in (2.6) describes the relationship of motor speed and voltage input. It’s a nonlinear differential equation. At the operating point the speed of motor is ΩQ . If P, Q and S are the constants in this equation then linearization proceeds as follows.

0.0201Ω + 0.0015Ω + 0.00025Ω2 = V

Θ = 0.04721⎡V22 − V42 ⎤ ⎣ ⎦

Φ = 0.04721⎡V32 − V12 ⎤ ⎣ ⎦

2 2 2 Ψ = 0.04646(Ω1 + Ω3 − Ω2 − Ω4 ) + 0.000577(Ω1 + Ω3 − Ω2 − Ω2 ) 4

az = 0.00237 (V12 + V22 + V32 + V42 ) cos Θ cos Φ − g

(2.6)

Ω=−

Q S V Ω − Ω 2 + = f (Ω, V ) P P P
(3.3)

All the results of quadrotor dynamics and control will be based on (2.6). The value “g” is taken to be 9.86 m/s2 [10]. III. LINEARIZATION OF QUADROTOR DYNAMICS speed
ΩQ = −Q ± Q 2 + 4SVQ

Using (3.1) and equating f(.) to zero, the operating point In the process of linearization the most important and initial step is specification of operating point. In our case we need to operate the machine in the hover state. Hover state is defined to be one where the machine stays at a fixed point in the air. It neither climbs nor stalls, does not roll pitch or yaw and does not cruise in any of the four horizontal directions. In terms of the system variables roll, pitch, yaw angles, height and their first and second time derivatives are all zero. The operating input voltage that corresponds to achieving the above, is termed as operating input. All operating points are represented with subscript “Q”.
= 356.2r / s is obtained, 2S discarding the negative result. Now it is needed to find the coefficients of the first order Taylor Series, therefore:

A= B=

−Q − S Ω Q ∂f = ∂Ω Q P ∂f ∂V =
Q

1 P

Using the first order Taylor Series following linearized differential equation is obtained.

⎛ −Q − S Ω Q ⎞ 1 d (∆Ω) = ⎜ ⎟ ∆Ω + ∆V dt P P ⎝ ⎠
where : ∆Ω = Ω ( t ) − Ω Q , ∆V = V ( t ) − VQ . In the second equation in (2.6), let

(3.4)

If ∆vz = vz − vzQ then the linearized equation can be written as,

az = 2 XVQ [ ∆V1 + ∆V2 + ∆V3 + ∆V4 ]
(3.5)

(3.7)

Finally the equation describing the yaw dynamics is linearized. Let,
2 2 W = Ψ = F (Ω1 + Ω3 − Ω 2 − Ω 4 ) + G (Ω1 + Ω3 − Ω 2 − Ω 2 ) 2 4

W = Θ = M (V2 2 − V4 2 ) = f (V2 , V4 )
At operating point Θ = 0 .
∂f A= ∂W B= C= ∂f ∂V2 ∂f ∂V4 =0
Q

= 2 MVQ
Q

At operating point Ω = Ω = 0, Ω = ΩQ . Therefore

= −2 MVQ
Q

Ψ Q = f (ΩiQ ) = 0 . It should be noted that at operating point
the speeds of all the motors are equal in their magnitude. The Taylor series coefficients are found as given below.

Using above equations in (2.6) the result is:

d ∆W = 2MVQ (V2 − V4 ) dt
Therefore using (3.5) following is obtained:

A=

∂f ∂W

= 0, B =
Q

∂f = 2GΩQ ∂Ω Q

B = C, B = −D = − E
F=
Therefore, (3.6)
Ψ = 2GΩQ ( ∆Ω1 + ∆Ω3 − ∆Ω2 − ∆Ω 4 ) + F ( ∆Ω1 + ∆Ω3 − ∆Ω 2 − ∆Ω 4 )

d (∆Θ) = 2MVQ ( ∆V2 − ∆V4 ) dt
Similarly,

∂f = F , G = G, H = − F , F = − I ∂Ω Q

d (∆Φ ) = 2MVQ ( ∆V3 − ∆V1 ) dt

Finally in (2.6) the equation for vertical acceleration is linearized. It is assumed that about the operating point the roll, pitch and yaw angles are small, hence there cosine functions are close to one. Therefore the cosines of roll and pitch angles can be neglected. At hover the vertical velocity and acceleration are zero and height is constant. Therefore,

(3.8) IV. CONCLUSION

So, finally (3.4) and (3.6) to (3.8) constitutes the linearized model of the quadrotor plant. In the linearized plant model the

vz = X (V + V
2 1

2 2

+V

2 3

+ V 4 ) − g = f (V1 , V2 , V3 , V4 )
2

variables are ∆V , ∆Ω etc. as opposed to V , Ω where the former signify the change from operating point and the later the actual variable. This model can used to develop a controller using any appropriate linear control methodology. REFERENCES
[1]

At operating point f(.) = 0. And the Taylor Series coefficients are [8],
Y. Amir, V. Abbas, Modelling of Quadrotor Helicopter Dynamics, In Proc, of International Conference on Smart Manufacturing Application, pg 100-105, Goyang-si, South Korea, 2008. Leishman, J.G, Principles of helicopter Aerodynamics, Cambridge University Press, NewYork NY, 2000. P. Ponds, R. Mahony, J. Gresham, P. Corke and J. Roberts, Towards Dynamically Favourable Quad-Rotor Aerial Robots, In Proc, of

A=

∂f ∂vz

= 0, B =
Q

∂f ∂Vi

= C = D = E = 2 XVQ
Q

[2] [3]

[4] [5] [6]

Australasian Conference on Robotics and Automation, Canberra, Australia 2004. I. Prinz, The Mesicopter: A Meso-Scale Flight vehicle, http://aero.stanford.edu/mesicopter/. S. BouabdAllah, P. Murrieri, R. Siegwart, Towards Autonomous Indoor Micro VTOL, Autonomous Robots 18,171-183, Springer Inc, 2005. S. BouadAllah, A. Noth and R. Siegwart, PID versus LQ Control Techniques Applied to an Indoor Micro Quadrotor, In Proc, of the IEEE International Conference on Intelligent Robots and Systems, Sendai Japan, 2004.

[7]

Y. Amir, Algorithm Development for In-flight Control of Quadrotor UAV, Masters Thesis, NUST, Pakistan, 2008. [8] C. Chen, Linear System Theory and Design, Third Edition, pg 16 to 32, Published by Oxford University Press. [9] Arda Özgur Kivrak, Design of Control Systems for a Quadrotor Flight Vehicle equipped with inertial Sensors, pg 4 to 29, Masters Thesis Atilim University Turkey, 2006. [10] R. Resnick, D. Halliday, K. Krane, Physics Volume 1 Fourth Edition, pg 231 to 240, Published by John Wiley and Sons, Inc.

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