Mechanical Analysis and Modeling for Tricopter

Yang Yanga, Li-ling Xiangb, Zhi Liuc, Jin-feng Cuid

Changchun University of Science and Technology, Changchun 130022 China

ae-mail:cloneyang@126.com, be-mail:dymanic@163.com,

ce-mail: liuzhi@cust.edu.cn, de-mail: 359555426@qq.com

Keywords: Tricopter; Flight attitude; Torque analysis; PID control

Abstract. Distinguished quadrotor , the research of tricopter is still in undeveloped state. This paper

focuses on torque analysis and mathematical modeling for attitude control of the tricopter aircraft.

The simulation of controller shows that the PID method can not achieve the desired states, so the

research of building the mathematical model provides some theoretical foundation for controlling the

tricopter.

Introduction

It's currently more popular to do the research of rotorcraft about miniature unmanned quadrotor all

over the world. So far, the basic theory and experimental research of micro quadrotor have made

great progress, but it is also facing many key technical challenges to maturity and utility. The main

jobs are the overall design optimizing, power and energy, establishment of mathematical model and

flight attitude control.

According to the trends of rotorcraft, the research of tricopter is still undeveloped state in the

world, so this paper puts forward the research scheme of tricopter, provides an effective way for

UAV attitude control and enriches the developments in the field of the UAV.

The Moment Analysis of tricopter UAV

1l′ 2l′

3l′

1f

2f

3f

o′

1l 2l

3l

Fig.1 Torsional forces f1, f2, f3 acting perpendicularly to respective shafts

In this system the motor produces two kinds of forces, one is the torsional forces f1, f2 and f3. The

second torque equilibrium condition is to make approximate that is vertical on l1, l2, l3 in the x-y

plane as shown in figure 1. Another forces are the lift forces F1, F2 and F3, which direction is

vertical upward the x-y plane as shown in figure 4. So when the aircraft hovering in the air, it

should meet two torque equilibrium conditions: one makes the aircraft not loop in the x-y plane, and

the other makes it not roll in the y-z plane. Under the first torque equilibrium condition the

corresponding torques of the forces f1 and f2 are vertical upward, and the torque of f3 downward.

Seen by the right hand rule, it is known that the former can make the aircraft anti-clockwise rotate,

and the latter can make it clockwise rotate. In order to make the torque balance, we need to have the

following formula:

1 1 2 2 3 3f l f l f l′ ′ ′⋅ + ⋅ = ⋅� � �

(1)

Which l1’, l2

’, l3

’are the arms of each torsional force to the center of gravity.The vehicle not roll

around y axis, the torque equilibrium as follows:

Applied Mechanics and Materials Vol. 455 (2014) pp 304-309Online available since 2013/Nov/15 at www.scientific.net© (2014) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMM.455.304

All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP,www.ttp.net. (ID: 130.207.50.37, Georgia Tech Library, Atlanta, USA-16/11/14,13:06:51)

33002211 lFlgmlFlF ⋅=⋅+⋅+⋅����

(2)

Which m0g is the counterweight and l0 is the distance from center of gravity of stuff for

counterweight to fulcrum O.

In addition, the lift forces of motors and the gravity of aircraft should be equal, so equilibrium

conditions as follows:

mggmFFF +=++ 0321 (3)

Which mg is the weight of tricopter.

Let F1, F2 and F3 be the lift forces provided by three motors of tricopter as shown in Figure 4.

Assuming that the x axis positive direction is the forward direction, the forces along x, y, z axis [3-4]

as follows:

++=

321

0

0

FFF

Ftotal (4)

Which the lift force of each motor as follows: 2

1, 2,3

i i iF K W

i

= •=

(5)

Which the constant ki represents the lift coefficient of the ith rotor wing, and wi is the angular

velocity of the ith motor.

According to the figure 2, the moment analysis of the x, y, z axis as follows [5]

:

+=

332211

44321

2211

--

-)(

-

lflflf

LFLFF

LFLF

τ

(6) Which f1, f2 and f3 are the torsional forces of tangential direction mentioned in the Part I of this

paper.

Fig.2 The torque diagram of the tricopter

Coordinate Transformation Matrix Analysis of Tricopter

The rolling unit matrix along x axis direction is as follows:

−=φφφφφ

cossin0

sincos0

001

),(xR

Which φ is the rolling angle along the direction of x axis.

φ

θ

ψ

Fig.3 (a)Rolling angle φ (b)Pitch angle θ (c) Yaw angle ψ

Applied Mechanics and Materials Vol. 455 305

The rolling matrix equation of unit points a, b, c along y axis direction as follows:

−=

θθ

θθθ

cos0sin

010

sin0cos

),( yR

Which θ is the rolling angle along the direction of y axis.

−=

100

0cossin

0sincos

),( ψψψψ

ψzR

Which ψ is the rolling angle along the direction of z axis .

By means of the above matrixes, the coordinate transformation matrix can be got:

),(),(),(),,( φθψψθφ xRyRzRR = (7)

−+−+

++=

θφθφθψθφψφφψθψφψθ

ψθφψφψθφψφθψψθφ

coscoscossinsin

sinsincoscossinsinsinsincoscossincos

cossincossinsincossinsinsincoscos

),,(

s

R

(8)

The Process of Modeling of Tricopter

The forces of tricopter along the direction of x, y, z axes are:

3

1

sin sin cos sin cos

( , , ) ( ) -sin sin cos sin cos

cos cos

x

y i

i

z

F

F F R F

F

ϕ ψ ϕ θ ψϕ θ ψ ϕ ψ ϕ θ ψ

ϕ θ=

+ = • = +

∑总 (9)

According to the third theorem of Newton, the linear equations along the x, y, z directions base

on Ground Coordinate are as follows:

⋅−=⋅−=⋅−=

mzKFz

myKFy

mxKFx

z

y

x

/)(

/)(

/)(

���

���

���

(10) Which K is the coefficient of air resistance.

According to the Euler equation, we can get the equation of angle motion for tricopter:

⋅⋅⋅=⋅⋅+⋅=

⋅⋅=

Izlflflf

IyLFLFLF

IxLFLF

/)--(

/)-(

/)-(

332211

443231

2211

ψθ

φ

��

��

��

(11) From the above formulas, we can reach an equation of modeling the dynamics for tricopter.

( (1)(sin sin cos sin cos )) /

( (1)(sin sin cos cos sin )) /

( (1) cos cos ) /

(2) 1/

(3) 3 /

(4) /

x u m

y u m

z u m g

u L Ix

u L Iy

u l Iz

ψ ϕ ψ θ ϕψ θ ϕ ψ ϕϕ θ

ϕθψ

= + = − = − = =

=

��

��

��

��

��

��

(12)

Which l is the length from center of gravity of the UAV to the arm each propeller, lk is the

moment of the inertia of the corresponding shaft.

Because the output matrix of state equation of tricopter is TzY ),,,( ψθφ�= , the input matrix of

control should be equal to the output matrix，for U = (u(1), u(2), u(3), u(4))T，let the formula of the

controlled object of the input as follows:

306 Mechanical Materials and Manufacturing Engineering III

1 2 3

1 2

1 2 3

1 2 3

(1)

(2)

(3) 2

(4)

u F F F

u F F

u F F F

u f f f

= + + = − = + − = − −

(13)

The Establishment of State Equation and the Simulation of PID Control of Tricoptor

[6-7]

Through the non-linear model of tricopter, we can get the state equation as follows:

X AX BU

Y C X D U

•

•

= +

= +

(14)

TzY ),,,( ψθφ�= TuuuuU ))4(),3(),2(),1((= .

The Coefficient matrixes are as follows:

−

=

0000010000

0000100000

0001000000

0000000000

0000000000

0000000000

0000000000

1000000000

0000000000

0000000000

A

=

0000

0000

0000

0000

/000

0/200

00/10

000/1

000/1

000/1

Izl

IyL

IxL

m

m

m

B

=

0000010000

0000000010

0000000100

0010000000

C

By the following formula, we can obtain the transfer function of tricopter with Matlab as

follows:

DBASICSG +−= −1)()( (15)

According to the design parameters of Reference [3]，assuming m = 5.4, l = 1, Ix = Iy = 0.92 and

Iz = 0.78, the transfer function is as follows:

=

2

2

2

/78.0000

0/92.000

00/92.00

000/4.5

)(

s

s

s

s

SG

According to the transfer function of the system of tricopter, we use the method of

Ziegler–Nichols to design PID controller of the control system and the unit step response, unit

impulse response are simulated in the state space with Matlab. The diagram is as shown below[4-7]

:

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Step Response

Time (sec)

Am

plit

ud

e

0 0.5 1 1.5 2 2.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (sec)

Am

plit

ude

Fig.4 (a)The simulation plot of gain P proportional control for channel Z

(b)PID control step response plot for channel Z at gain k = 1 and the time T = [0, 2.5]

Applied Mechanics and Materials Vol. 455 307

0 2 4 6 8 10 12 14 16 180

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (sec)

Am

plit

ude

Fig.5 The step response plot of P(red), PI(green), PID(blue) controls, at T = [0, 18] for channel φ

and θ

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Step Response

Time (sec)

Am

plit

ud

e

Fig.6 PID control step response plot at gain k = 1, the time T = [0, 100] for channel φ and θ

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Step Response

Time (sec)

Am

plit

ude

Fig.7 PID control step response plot at gain k = 1, the time T = [0, 2.5] for channel Z

Conclusoion

In this paper, at first the mechanical structure of unmanned tricopter is also briefly introduced, and

we analyze the overall physical torque, at the same time theoretically solve the problem of torques

offset. Secondly, we discuss the attitude of the aircraft take-off, pitch, roll, yaw, etc. Finally, through

the above steps we get the kinematics equation of flight attitude control system to describe the

unmanned tricopter, and simulate it with Matlab. The experimental results show that oscillation

time tends to about 80-100 seconds which is a long time in the control system of tricopter using

PID method at gain k = 1, time T = 0-100s. So the design of the control system of tricopter using

PID method can not reach the ideal state and need other method to improve.

308 Mechanical Materials and Manufacturing Engineering III

References

[1] LIU Lili .Research on the Modeling and Control to a Quadrotor Helicopter Simulator [D]. Zhong

Nan University. (in Chinese)

[2] LIU Huanye. Study and design of flight control systems for small scale quadrotor[D] .Shanghai

Jiaotong University. (in Chinese)

[3] WANG Shugang. Research of Quadrotor Control [D]. Harbin Institute of Technology. (in

Chinese)

[4] Sergio Salazar-Cruz, Real-Time Control of a Small-Scale Helicopter Having Three Rotors

Compiègne, France{ssalazar, fkendoul, rlozano, ifantoni}[M]

[5] S. Salazar-Cruz and R. Lozano. Stabilization and nonlinear control fora novel tri-rotor

mini-aircraft, in proceedings of the IEEE ICRA’05, Barcelone, Spain, April 2005[C].

[6] A.Tayebi, S.McGilvray, “Attitude stabilization of a four-rotor aerial robot,” IEEE Conference on

Decision and Control, 2004[C]

[7] N.Guenard, T.Hamel, V.Moreau, Dynamic modeling and intuitive control strategy for an

“X4-flyer, International Conference on Control and Automation, 2005[C].

[8] S. Salazar-cruz, F. Kendoul, R. Lozano, I. Fantoni, “Real-Time Stabilization of a Small

Three-Rotor Aircraft,” Aerospace and Electronic Systems Vol.44, No.2, 2008[C].

[9] J.Escareno, A.Sanchez, O.Garcia, R.Lozano, “Triple Tilting Rotor mini-UAV: Modeling and

Embedded Control of the Attitude,” American Control Conference, 2008[C].

Applied Mechanics and Materials Vol. 455 309

Mechanical Materials and Manufacturing Engineering III 10.4028/www.scientific.net/AMM.455 Mechanical Analysis and Modeling for Tricopter 10.4028/www.scientific.net/AMM.455.304