1 Modeling and Control of Quadrotor MAV Using Vision-based Measurement

7/21/2019 1 Modeling and Control of Quadrotor MAV Using Vision-based Measurement

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1

Moling an Conol o aoo MAV

Using Visionbas Masmn

L Dat inh Deparment of Aerospace Engineering

University of UlsaUlsan, Korea

[email protected]

bstct In this paper we review the mathematics model ofquadrotor using Lagrange's equation. We propose a vision-basedmeasurement to stabilize this model. A dual camera method isused for estimating the pose of the quadrotor; positions andattitudes of the quadrotor MAV One of these cameras is locatedon-board the quadrotor MAV, and the orther is located on theground. The control system is developed in Matlab/Simulink. In

this paper, we consider a linear controller for our purpose.Linear Quadratic tracking controller with integral action, andOptimal Linear Quadratic Gaussian (LQG) control with integralaction are designed for stabilization of the attitude of thequadrotor MAV Moreover, some measurement noises will beconsidered in the controller design, too. Finally, this paper willdemonstrate how well this control works for a certain ightmission : quadrotor M V on the ground, and then startinghover, sideway move and keeping the position with a pointing toa certain object xed in space.

Kd d MA V p n vndhvn nl

. NTRODUTION

The ppose of this study is to explore conol methodologies and vision algorithms to develop a autonomous miniature aerial vehicle (MAV). MA Vs areimportant when it comes to performing desired tasks in

dagerous or inaccessible environments. autonomous MA V brings enormous benets and is suitable for applications likesech ad rescue, surveillace, remote inspection, milit applications; therefore saving time, reducing costs ad keeping huma pilots away om dgerous ight conditions.

Many resech oups are interesting in MA V. There ae two types of MAV: ula-lit xed-wing plaes and rot wing aerial vehicle such as helicopter, quad-rotor ad etc.Ultra-light xed-wing planes cannot hover so they requirelage space to navigate. Rot ing, hoever, can te olad in limited space ad easily hover above ay target.Furthermore, rot wing aerial vehicles ae hily maeuverable. A quad-rotor MAV is a four-rotor helicopter. A quad-rotor MA V has two pairs of propellers ing in opposite directions in order to cacel out the resultant moment found insingle-rotor helicopter. This eliminates the need for a tail rotor

to stabilize the yaw motions of the quad-rotor. n this paper, the quad-rotor MA V has xed pitch gle rotors ad the rotorspeeds e conolled to produce the desired li forces.

Chelkeun Ha Deptment of Aerospace Engineering

University of UlsanUlsan, Korea

[email protected]

The Dragayer, a commercial aerial vehicle shown in Fig.1, is a radio-conolled fo-rotor helicopter available om RCToys. t is a uni for studying the problems of realizing autonomous ight of MAV. n this paper, we model d aalyze the dynamics of the Dragyer in order to design the conoller to stabilize the attitde in hover. We present some

conol methods such as optimal control techniques of LQ(Line Quadratic) acking controller with integral action tostabilize and conol the quad-rotor model.

n order to create an autonomous MA V, precise knowledge of the helicopter position and orientation is needed. n the previous work involving autonomous ying vehicles, thisinformation was obtained om nerial Navigation Systems(NS), Global Positioning Systems (GPS) or other sensors like ultrasonic sensor. Typically, mltiple sensors e used to overcome limitations of individual sensors, thereby increasing the reliability and decreasing the eors. Vision sensors e primaily used for estimating the relative positions of some target. Limited payload capacity may not permit the use of

heavy navigation systems or GPS. Moreover, GPS does not work in indoor. Vision information can also be used to stabilize and hover the helicopter, and also to ack a moving object. n this study, we inoduce vision algorithms of pose ad motion estimation using two-caera system [1].

The pose estimation problem has been a subject of may resech projects for may yeas. Most of the pose estimation techniques e image-based and they fall into these two categories: point-based methods ad model-based methods.Point-based methods use the feate points identied on a 2Dimage while model-based methods use the geometric models(e.g. lines, curves) and its image to estimate the motion. We aeinterested in point-based techniques in this paper.

. ATHEATIAL ODEL

Unlike regular helicopters that have viable pitch angles, a quadrotor MA V has xed pitch agle rotors ad its rotorspeeds are conolled to produce the desired li forces. Basic

motion of a quad-rtor MA V ca be described in Fige 1.

A quaotor MAV has two pairs of rotor (1,3) and (2,4), ing in opposite directions. By vying the rotor speed, one ca chage the li force ad create motion. Vertical motion ca be achieved by simultaneous chage in rotor speed. Moving

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leright motion ca be obtained by converse change of rotorspeeds 2 and 4. Similaly, forwarbackwad motion ca be obtained by converse change of rotor speeds 1 ad 3. The yaw motion is obtained by rotor speed imbalance between the two pairs. A good conoller should be able to reach a desired yaw agle while keeping the tilt agles and height constat. There is no chage in the direction of rotation of the rotors.

o

R rh

�

Gg up M

Figure 1. Basic motion of a quadrotor; the arrow width is proportional to

prpeller rotational speed

The following are some assumptions that are used in developing the mathematical model of the quaotor V.

• The center of gravity ad the body ame origin are assumed to coincide.

• nteraction with ground or other surfaces is neglected.

• The blade apping is not modeled.

• Friction is only considered in yaw motion.• The cbon-ber sucte is supposed rigid.

• The sucte is supposed symmetric (i.e. diagonalinertia matrix).

• Trust ad ag e supposed proportional to thesque of the propellers speed.

Let ={ } denote inertial ame, ad

B {x,y,} denote a body xed ame. The axes of the rotors

are pallel to the axis in E. The coordinate ames of inertial ad body aes e shown in Fig. 2.

E

Z g�YX

Figure 2. Quadrotor coordinates system

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n hover, conol of the quaotor V can be thought of as achieving force ad torque balace. Using Euler agles paeterization, e airame orientation in space is given by a

rotation matrix R B2E om B to E, where R B2E E S03 is the

rotation matrix.

TLE!. PHYSIAL PATS DFINTION

Paraeter Description

g Gravity [m / S 2]

m Mass [kg]

I length [m]

¢ Roll angle [rad]

( Pitch angle [rad]

aw angle [rad]

°i Rotor speed (i 1,2,3,4) [rad / s]

F Force (i 1,2,3,4) [N] Thrust force (i 1,2,3,4) [N]

D i Drag moment (i 1,2,3,4) [Nm]

b Thrust constant

dDrag constant

R B2E Rotaion matrix om ody to Eath

R Rotor inertia [kg . m2]

Or Or O +0 2 -03 +04 [rad / s]

The Lagrange-Euler approach is based on the concept of kinetic ad potential energy :

!( OLJ- o L r dt , oq, 'LTV

qi : generalized coordinates

ri : generalized forces given by non-conservative forces

T: total kinetic energy

V: total potential enerThe total kinetic energy is dened as

IET=- v dm

(1)

.. e e · • • e (2)=-1( -lsm ) +-1 ( co

s +lsm

cos

) +... w

e · · • e+1 ( sm +lcos cos ) z

And the potential energy can be expressed by

V JF . dgJE zd (3)

Jxd gsinB JydgcosBsin JzdgcosBcos

2



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Main forces ad moments come om propellers. n this paper the quadrotor MAV is in hover at trim, thus thrust ad drag ae proportional to the squaed propellers rotationalspeed, shown in Eq. 4:T=hQ; Thrust force

D i=

dQ; Drag moment

The non-conservative moments come fm the propeller

thrusts, propeller ag, ad the gyroscopic eects due to thesimultaneous rotation propeller ad the quaotor. Aer

calculating, the following ll quadrotor dynaic model is obtained as shown in Eq. 5 ad 6:

4=(cossincos +sin sin LT.

;=4

mY=(cos sin sin sin cosLT (5);=

4mi=m g (coscos)LT

;=

Note that Eq. 5 means the slational motion and Eq. 6 e attitude motion.

. ONTROLLER DESIGN

n this section we present a control technique for attitudestabilization of the quaotor MAV in hover. We use the ll

dynamic model in Eq. 3 and 4 to mae the quad-rotor nonlinear model in Matlab/Simulink, which will be used to aalyze e

conoller to be desied. this paper a linear conoller design is considered so that the nonline model has to belineized about a im in hover. Note that this im is usually unstable. The lineized state equation is expressed as

X= A +B U} with x(O) = o

y=Cx+Du(7)

where x E Rn, u E R, y E RI ae the state vector, e conol

input vector ad the ouut vector, respectively. The state vector is deed as

x = [ X y Y z Z x x z J (8)

The associated matrices A ,B,C,D have proper dimension toEq 7 Linearization of the nonline state equations at im is done by using Jacobis of the nonline state equations with respect to the states ad the inuts, evaluated at hovering condition selected as

x = [00000.100 00 0 00] (9)

The Jacobian matrix A of the nonline state equation with respect to all states evaluated at the given initial conditions inEq. 9 is shown in Eq. 10:

A OF _x '

o o o o

o o o o o

o o

o o o o

(10)

The inut matrix, B, of the state-space representation is formed by taing the Jacobian of the state equation with respect to theinput vector in Eq. 11.

(11)

where the input u is deed as the speed of the rotor

U i hver =�

(12)

Also the matrix B for the state equation in Eq. 7 is given

(13)

B= a F I = 24 24 24 24au u

3242 3242

3242 3242

27 27 27 27The ouut matrix C consists of four state viables, the traslation {X,Y,Z} in inertial ae E ad the yaw angle .

n this paper these vaiables e used as conolled viables by commads. Also the conoller design is based on these variables. The direct trasition matrix D is taken to be zero( D = 0 ) because there is no direct coupling between the input and the ouut of the state equation in Eq. 7.

n this study, we choose Linea Quaatic Gaussia (LQG) conoller for o problem. order to use LQG control, the nonline system in Eq. 5 d 6 must be lineaized ound a certain operating point, where that is hover. t is well-known that LQG conoller is desied om combination of Line Quaratic Regulator (LQR) ad Line Quaratic Estimator(LQE), called as Kalman Filter, with preserving optimality of the control. This idea is behind the sepation principle, meaing that optimal controller ad optimal estimator ca be designed sepately.

A . Liear Quadratic Regulator (LQ R ) Given a line system dened in Eq. 7, LQR conoller ies

to minimize the performance index is given as

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FOST 2010 PrcdgsJ(u)= (xTQx+uTRu)dt

The state feedback for a line system is given as

u=-

where the feedback gain is given as

K=IBTp

(14)

(15)

(16)Here the positive denite ad symmetric matrix P satises the following Riccati equation:

(17)

The positive deite matrix Q weighs the size of the state responses. Usually we choose it diagonal. f the enies of Q aelage numbers, then small deviation of the state om zero will

cause the cost to increase siicatly, thus the conol will have high gain. The symmetric matrix R, which should be positive matrix, weighs the conol. may also be chosen diagonal. f the entries of this matrix e lge, then the conol action will be small, opposite to what happens with matrix Q in

Eq. 14. f we wt to increase the size of the conol action, we c reduce the entries of the control weighting matrix R.

LQ trackig cooler with itegral actioA state feedback controller in Eq. 15 may not allow to

reject the eects of distbances (paiculaly of input distbaces or references). A ve usel method of rejecting the disturbace is to add a integral term to the feedback conol to ensure a it static closed-loop gain, i.e . gain between the reference command and the ouut associated with the reference. ntegral action is used to eliminate steady state eors when tracking constant signals. The objective is to keep

the ouut y close to a reference signal r .Let us dene the acking eor such that its time derivative obeys the following dierential equation:

i=rt - y e t )=rt - Cx(t) (18)

Dee a augmented state vector is also deed as

=[x eThen the augmented state equation is realized as

A 0 � B 0 x=

- C° x+ ° u+

1r

y= [ C 0XThe feedback control to be designed is now structured as

u=-+Kr=-K K1 [; ]+Kr

Optimal LQG cotrol with itegral actio

(19)

(20)

(21)

Consider the line system model with exteal distbaces as shown in Eq. 22:

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xt = A x(t) +R u(t) +Gw ( t )yt = Cx(t) +vt

(22)

where wt and vt e a process disturbace and a

measurement noise, respectively, ad uncorrelated radom processes with coviace matrices Q f ad R f, respectively.

The feedback gain K r , related with the reference command

r ), ca be yielded as

K r =IBTpeA- BKT Q (23)

where K satises Eq. 16 ad P is associated with in Eq. 17.

Next, let us dee the optimal estimator, called Kalma Filter, with the estimator gain L :

t =A( t )+R u(t) +L(y(t) yt (24)

jt = C(t)

where the estimator gain is obtained as

L=SCTg/ (25)

Note the positive deite symmetric matrix S satises the Riccati equation dual to Eq. 17.

So f we designed the optimal LQG controller to stabilize the position and atitude motion of the quad-rotor V ne hover. n this design approach we assume that the ouut y in

Eq. 7 is available as measurement to the feedback controller.However, in our problem e ouut shall be estimated by vision caera. So in next section we will discuss the pose d motion estimation algorithm[] estimating the output in Eq. 7.

V. POSE ESTIMATION

n this paper, we will describe the two-caera pose estimation method. This method is based on the point-based techniques. n general, point-based method uses the feature points identied on a 2D image. A pair of cameras is being used to ack the image features. These cameras ack multi color blobs located under the quad-rotor V and ond.These blobs are located on a known geometric shape. A blob tracking algorithm installed in the ground caera is used to obtain the positions and eas of the blobs on the image plaes.

Two ceras e set to see each other. ne of the camerasis located at the ound (CAM2) and the other is a on-bod

caera (CAM) looking downwards. Colored blobs ae

attached to the bottom of the quad-rotor V and to e ground caera. Tracking two blobs on the quad-rotor image plane and one blob on the ground image ame is found to be enough for accate pose estimation. The blob tracking algorithm tracks e blobs d res image values (ui,V i) The cameras have matrices of ininsic paetersAl for CAM

2 and for CAM. Let W i be the vector om each camera to

the blobs.



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W i= nv I.[ui V i 1] for iI,3,4,5,6

W z= nv .[u V 1] (26)

= WiI norm(wi )

Let �3 be the vector pointing om blob 1 to blob 3, ad

A i e the unknown scalars. Also the vector ad W3 e

related by

W3= W I+\3 (27)

To simpli, let us tae cross product of Eq. 27 by W3 :

0= (W XW3)+(3 x w31 (28

Figure 3. Dual cera method

n order to solve Eq. 28, let the rotation matrix R assume to be

composed of two rotations: the rotation ound the vector formed the cross product of W and , the rotation ound

. This can be expressed in this form shown in Eq. 29:

R= Rot( l x w z, B)· Rot(wl,a (29)

where Rot(a,b means the rotation of b degrees aound the

unit vector a. Note the value of B ca be obtained om

B= I l X W zI cos(W .W z}

The only unknown le in (29) is the angle a .

Rewriting (28) gives

(W3x WI) X (W3 X (Rot(wlx wz,B)·Rot(wl,a»)\3)= 0

Let us assume M= (w3 x wl)x {W3 x (Rot(wl x w 2,B).

Using Roigues' formula given as

(30)

(31)

Rot(wp)= I+I sina+ l z(1- cosa ) (32)

where (33)

5

Eq. 31 becomes- - 2-M ·L\3 +saM ·I�3 +1-csaM .I L\3= 0 (34)

Moreover, Eq. 34 has a set of ree equations in form of

Acs+BsinC

A solution of Eq. 35 is yielded as

_ • -I B

C

±

B

Z

C

2

_ ( A z +B z ) ( C z - A

Z

]

a - s 2 2 A +B

(35)

(36)

Finally, the estimated rotation matrix R can be found om Eq. 29, meaing that Euler agles (f,B,) are obtained.

n order to nd the relative position of the quad-rotor V with respect to the inertial e located at the ground caeraame, we need to d scals A i . First, d out om Eq. 28.

Then the rest of A i e obtained om

(37)

where i is the position vector of the i-th blob in the body

ae. Hence the center of the quad-rotor V will be

X y Z]'=W3 +A4W4 +AsWs +A6Wd/ (38)

V. IMULATION

n this section we present e simulation result of the conoller design which is presented in section . We mae the quaotor nonline model ad line state space model inMatlab/Simulink.

We consider ll-state feedback in LQR with integral

action. The simulation perfos the motion of the quaotorom position (O.2m, O.m, 1m) to the position (0, 0, 3m), while reducing the yaw angle om -20 to zero degrees. So theinitial state is selected as

= [0.200.101 00 00 -90] (39)

The simulation result using the linear state space model isshown in Figure 5.

K "

J �'- f :re i

! : [ x

.

. >:

.. .

N '

'

time (secnds) me (secnds) tme (secnds)

I

f

.

,

i

tme sens) me (secnds) tme (secnds)

Figure 4. Simulation result of linear state-space model in LQ

tracking with integral

Also the simulation result using the nonline state model isshown in Figure 5.

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I

�

£

-

:

J

me seconds) me seconds)ph \S tme ea vs tme

lV time15

m.

f 10 ·

• 5 . . " � . . .N .

00 5 10me seconds)

ps s tme

� �

':r · ..

. �

E � 0 { 0 ! .10 ... .. l .. a 2

·

·

· . � . � [-40 5 10 -50 5 0

-2 00 0tme (seconds) lme seconds) me seconds)

Figure 5. Simulion result of nonlinear state model

Figure 6 and Figure 7 show the simulation of LQG withintegral action controller design using the line model d

rotor speed, respectively.

2

02

A

c

]

;i

3

u

"

lime (econd) I,me (scon) tme (cond

ph

Vlm ha v tm m

�

o

n

�

:

F

.

.

t: E 2 � 1 -2 1 2 1lm cond lm cond lm (ond)

Figure 6. Simulation of linear state model in LQG wih integral action

l

g

�l

o 5 s o 5 10

loa

'

i i 50

C 0 C 0 50

o 10 0 5 10

Figure 7. Speed response of 4 rotors

Figure 8 ad Figure 9 show the simulation of LQG withiegral action design ad of the rotor speed in hover at 3[m]

height for the nonline quadrotor MV, respectively.

Y

,

;

m

e

m

.

!

-

:K

r

t

me seconds) time scond) tme sends)

n

�

D

O

�

:

-2 00 0 2 0 50 10 202 0

0 0 2 0lme seconds) lme seconds lme seconds)

Figure 8 Simulation of nonlinear state model in LQG wih integral action

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! � ! �o 2 1

! � ! Eo 5 1 5 2 5 0 5 2 0

Figure 9. Speed response of 4 rotors of nonlinear state model in LQG

wih integral action

V. CONLUSION

n this paper we have reviewed the pose estimation algorithm[] ad designed the conollers to keep the quaotorMV at a specic position in hover. The tee dimensional

quadrotor MV model has been developed. LQ ad LQG withintegral action controller have been used to stabilize d performed the ouut-acking objective. Simulations performed on Matlab/Simulink showed the ability of the conollers to perfo ouut acking control objective in hover.

ACKNWLEDGEMENT

This resech was supported by Basic Science ResechPro through the National Reseach Foundation ofKorea(NF) nded by the Minist of Education, Science dTechnology(No.2010-0016929)

EFERENES

[] Altug, E.; Ostrowski, 1. P. & Taylor, . 1. (2005), "ontrol of a Quadrotor Helicopter Using Dual amera Visual Feedback, TheInteational Joual of Robotics Research, Vol. 24, No. 5, May 2005,pp. 329-341.

[2] P.McKerrow, "Modelling the Drager four-rotor helicopter,Proceedings of the IEEE Inteational onference on Robotics d Automation, 26April -May 2004, 4, 3596-360.

[3] Arda Ozgur Kivrak, "Design of control systems for a Quadrotor light vehicle equipped with inertial sensors , Master Thesis, in MechatronicsEngineering , Atilim University, December 2006.

[4] S. Boubdallah, A.Noth and R.Siegwart, "PID vs LQ ontrol Techniques Applied to Indoor Micro Quadrotor, IROS 2004, Sendai (Japan),

October 2004.[5] S. Boubdallah, P.Murrieri d R.Siegwart, "Design d ontrol of Indoor Micro Quadrotor, IA 2004, New Orleas (USA), April 2004.

[6] Amidi O. (1996) " Autonomous Vision Guided Helicopter, Ph.D.Thesis, egie Mellon University, USA.

[7] hristopher Kemp "Visual ontrl of a Miniature QuadrotorHelicopter, PhD thesis, University of ambridge, 2006.

1 Modeling and Control of Quadrotor MAV Using Vision-based Measurement

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