Scale-Invariant Isotropy of an Optical Mouse Array for Mobile Robot Velocity Estimation

7/26/2019 Scale-Invariant Isotropy of an Optical Mouse Array for Mobile Robot Velocity Estimation

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013 13th nteational Conference on Control, Automation and ystems (CCA 013)Oct. 0-3, 013 in Kimdaejg Convention Center, Gwanu, Korea

lIv Is f Ol Ms f Ml R Vl Es

Sungbok Kim Geunhung Lim and Hunin Kim

Dept. of Digital nfoation Eng., H Universit of Foreign tudies, Korea(EL: +8-31-330-4091, E-mail: [email protected])

r his paper presents the scale-invariant isotropy analysis of an optical mouse array for the mobilerobot velocit estimation. First, with the characteristic length introduced, the velocit kinematics of a mobilerobot with optical mice is described. hen, the uncertaint ellipsoid is obtained to represent the eorcharacteristics of the optical mouse based mobile robot velocit estimation, and the isotropy index is denedas the closeness of the uncertaint ellipsoid to a sphere. Next, using the dened isotropy index, theconditions for the anisotropic, isotropic, and regular optical mouse aays are derived in terms of theinstallation positions of optical mice, along with the corresponding characteristic lengths.

K Optical mice, mobile robot, velocit estimation, scale-invariant isotropy.

ODUCTIO

optical mouse was originally invented as anadvanced computer pointing device, which is aninexpensive but high perfoance motion detectionsensor with a sophisticated image processing engineinside. Optical mice installed at the bottom of amobile robot can detect motions over a plane oor,om which the velocit of a mobile robot canestimated [1-6]. optical mouse suffers omrandom noises in velocit measurement, which willaffect the accuracy of mobile robot velocitestimation. Using the redundant number of opticalmice, the effect of noisy velocit measurements canbe reduced signicantly. With the same number of

optical mice, the accacy of mobile robot velocitestimation also depends on the installation locationsof optical mice.

o characterize the mobile robot velocitestimation om noisy optical mouse measements,the so-called unceraint ellipsoid can be deed.he uncertaint ellipsoid is an ellipsoid in themobile robot velocit space mapped om the unitsphere in the optical mouse velocit space toughthe Jacobian matrix of the velocit kinematics. Notethat the shape and orientation of the uncertaintellipsoid changes according to the installationpositlons of optical mice. For the optimal placementof optical mice, one can minimize the volume ofthe uncertaint ellipsoid, or maximize its closenessto a sphere, so-called the isotropy. he measure ofisotropy has been adopted for the optimal design ofserial and parallel manipulators [7-11], as well as,omnidirectional mobile robots [1, 13].

Tn this paper, we present the scale-invariantisotropy analysis of an optical mouse aay for themobile robot velocit estimation. his paper isorganized as follows. With the introduction of thecharacteristic length, ection describes thevelocit kinematics of a mobile robot with opticalmice. ection 3 obtains the unceraint ellipsoid torepresent the error characteristics of the mobilerobot velocit estimation, and denes the isotropy

----//$ @r 9

Y

Y

i

ri I

w

Mbe

Fig. 1. ree coordinate ames for a mobile robot

and the th optical mouse.

index as the closeness of the uncertaint ellipsoidto a sphere. Using the dened isotropy index,ections 4, 5, and 6 derive the geometricalconditions for the anisotropic, isotropic, and regularoptical mouse aays, along with the correspondingcharacteristic lengths. Finally, the conclusion ismade in ection 7.

VELOC KETICS

he velocit of a mobile robot traveling on a

plane can be estimated using the velocitmeasurements om an array of N(: optical miceinstalled at the bottom of a mobile robot. Fig. 1shows tree coordinate ames used for thedescription of a mobile robot and the th opticalmouse. ree subscripts, 'W, 'R', and 'i' are used todistinguish the origin and the x & Y axes of theworld coordinate ame, the mobile robot coordinateame, and, the optical mouse coordinate ame.From Fig. 1, the position vector,

Pi = Yilt, = 1,,N of the th optical mouse canbe expressed by

xi1 i COSi1 .Pi= = z=l,,NYi i S (1)


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where Pi and i are the polar coordinates of theinstallation position i.

Let Vi and Vi!' = 1,,N be the lateral andlongitudnal velocit measurements of the th opticalmouse. Let V and VY be two linear velocitcomponents of a mobile robot long the x axis and

the axis, respectively, and w be its angularvelocit component about the z axis. Using thevelocit relationship between a mobile robot andthe th optical mouse, the velocit kinematics of amobile robot with an array of N optical mIce canbe obtained by [5]

(2)Tn the above, = [v Vy w ER3l represents thevelocit vector of a mobile robot, v, = [ v VtyrER2XI represents the velocit vector of N opticalmice, with i=[ViXvi,=I,,N and A represents

the Jacobian matrix mapping v to v" given by1 -l 1 l1 -2

A= 1 x2 ER23 ()

1 -Y

l

1 xNote that the expression of A is a nction of thepositions of N optical mice, Pi Yilt,= 1,,N .

Tn (), the velocit vector v of a mobile robotconsists of linear and angular components, while

the velocit vector v, of optical mice consists oflinear components only. o eliminate the problemof inconsistency in physical units, the characteristiclength, denoted by L can be introduced [8]:

= s ()where = vTU L W. ER3l and

1 l-L 1 lL

1 o

L= ER2X 1 ()L

1 - L 1 xL

he characteristic length L can be regarded as aneffective radius which converts the angular velocitto the corresponding linear velocit.

3 ISOOPY DEXTn this paper, the concept of uncertaint ellipsoid

is adopted to represent the eor characteristics ofthe mobile robot velocit estimation owing to thenoisy velocit measurements of optical mice. Basedon the velocit kinematics, given by (4), theuncertaint ellipsoid can be obtained as the3-dimensional ellipsoid in the mobile robot velocity

space, which is mapped om the 2-dimensionalunit sphere in the optical mouse velocit space:A ) A: v A A v = 1 ()

he shape and orientation of the uncertaintellipsoid given by (6), ae deeined by a seof eigenvalues of a matrix ' and theireigen vectors, respectively. Let I ' 2 and 3 betree eigenvalues of NA, and let Ul u2 and Uthe coesponding eigen vectors. he directions ofprincipal axes of the 3-dimensional uncertaintellipsoid are given by l, 2 and U and theircorresponding lengths are given by 1/ 1/ and 1/ .

From (7), can be written as

= Nx

where

R=

1

1

c

CL L

1 C = Ni=l1 l

c= Ni

c

LC

L ER33

R2L2

(7)

(8)

(9)

(10)

Tn the above,C

and ' respectively, represent theaverages of the and coordinates of the positionvectors, i =[i yJ, = 1, ,N of N optical mice,and R represents the root mean square of thedistances of N optical mice om the center of amobile robot.

Using (7), the characteristic polynomial of ISgiven by

(A-N) A2- 1+ A+N2_ 1{ 1 R2 ) ( 2 C C

L2 L2 L2 J (11)=

ince the solutions of (11) are tree eigenvalues ofA A, denoted by l , 2 and ,

l = {(1 + + r1--

2

+

2

+

2

}(1)

(13)

(14)


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t can be shown that there hold the followinginequalit relationships among tee eigenvalues, A],, and A3:

A A2( N) A] (15)

Note that A] and A are the largest and smallest.

t

.eIgenvalues of A A, while IS the middleeigenvalue of having the constant value of

Based on the uncertaint ellipsoid the. . mease of the Isotropy of the matx A A can be

dened by

(1)

he isotropy index j is deed as the inverse of

the well-known condition number of takinginto account the fact that the smallest eigenvalue ca become zero. From (15) and (16), the range of

j is given by(1)

n the case of < j < 1 the uncertaint ellipsoid remains as a 3-dimensional ellipsoid. However,in the case of r = that is, A = the uncertaintellipsoid degenerates into a -dimensionalellipse(more precisely, an elliptical cylinder); and, inthe case of j = 1 that is, A =A =N, thecertai ellipsoid becomes a 3-dimensionalsphere. For better placement of N optical mice, itis desirable to make the value of "r closer to unitas much as possible.

Although the unceraint ellipsoid canrepresent the 3-dimensional error characteristics ofthe mobile robot velocit estimation caused by thenoisy velocit measements of optical mice,another simple but effective performance measurecan be deed as

(18)he performance index "rr IS dened to beinversely proporional to the volume of theuncertaint ellipsoid For better placement of Noptical mice, it is more desirable to make the valueof

jjas large as possible.

4 ISOOPIC OPTICL MOUSE Y

Using (1) and (14), (16) can be expressed as

"r= + (19)

where

= 1+ (0)2

een om (19)-(1), the isotropy index j IS a

nction of the positions of N optical mice,including T y and R2 and also thecharacteristic length . For a given number N ofoptical mice, the value of j varies, depending onthe value of the characteristic length .

Using (19), om8"r =

the condition for

maximal value of the isotropy index j is obtainedby

where

B= B

, 8 = = L8

()

(3)

B'= =4L2_R2)XL+8(C;+CXL (4)

Plugging (0), (1), (3), and (4) into (), wehave

(L2_R2 )X {R2_ C

I

2+}=

0 (5)

t can be shown that

(6)

unless IS singular. With (6), om (5), weobtain

(7)

which results in

L*= R= (28)

(8) represents the optimal value of thecharacteristic length , which is required formaximal value of the isotropy index j' Note that

the optimal characteristic length * is the rootmean square of the distances of N optical miceom the center of a mobile robot.

As mentioned earlier, the characteristic length is introduced to make all elements of the Jacobianmatrix given by (5), to be physicallydimensionless. With the specic choice of thecharacteristic length, =* all elements of thethird column of / and / = 1" Nbecome independent of the relative size of

Noptical mouse arr. he isotropy of N opticalmouse aay made subject to the condition of (8)can be said to be scale-invariant.

5 ISOOPIC OPTICL MOUSE Y

Under the condition of (7) or (8), theexpressions of the largest and smallest eigenvalues

of A and A3, redu e to.I

C

2

+

C

2

A = N 1+ V

*

I

_ _ Jc;+ ] Nl *(9)

(30)


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With (9) and (30), the isotropy index IS gIvenby

*T=L*- J+L*+

JC;+C:

(1)

Now, the condition for which the value of K;becomes it is obtained by

(3)

(3) represents that the geometric center of optical mice coincides with the center of a mobilerobot. t should be mentioned that both (8) and(3) constitute two conditions required to achieveK; =1 that is, Al = 3 =, that is, the isotropy of

t . Note that with (8) and (3), (7) becomes t = (33)

where 3 represents the identit matrix.On the other hand, with (9) and (30), anotherperformance index Krr can be expressed as

2( + )K- 1*

L(34)

Under the condition of (3), the perfoance indexKr reaches its maximum value of N2 Note thatthe minimum volume as well as the completeisotropy of the unceraint ellipsoid can beattained simultaneously under the conditions of (8)and (3):

Kr =1 & Kr = N2

L*=R & CX==OFor comparison, it can be also shown that

(35)

K=0 & K=0 L*=R & +=L*2 (36)

6. EGULR POLYGOL OPTICLMOUSE Y

Using (), (8), and (9), the condition of (3) canbe expressed as

1 c = -Lp; = Nl

(37)

which represents the rst moment of the positionvectors of optical mice, P; = y;, i=1 . . .With respect to the rst moment c= 0, the secondmoment of position vectors is given by

or

1 M= (p; c )(p; c y1= _ Lppt E R2x2Nl I I

(38)

1M=

Lx=l

Lx;y;=l

Lx;y;=lj

LY=l

(39)

For a regular polygonal array of optical mice,

(37) becomes

P;=cos{l+ (il) 1}sin {l+ il) X 1}

, i=l, . . (0)

where represents the radius of the circumscribedcircle of a regular -gon and I represents theoffset angle of the rst optical mouse. Under thecondition of (40), it can be shown that

2M = 2 (41)

where 2 represents the identit matrix. Notethat a regular polygonal aay of optical mice isthe simplest fo which leads to the isotropy ofthe second moment M given by (41).

7 COCLUSIO

n this paper, we presented the scale-invariantisotropy analysis of an optical mouse aay for themobile robot velocit estimation. he maincontribution of this paper was the derivation of thegeometrical conditions for the anisotropic, isotropic,and regular optical mouse arrays, along with the

coesponding characteristic lengths. mportantly, thechoice of the characteristic length as the root meansquare of the distances to optical mice made theisotropic analysis to be scale-invariant. he resultsof this paper can help especially for thedevelopment of omnidirectional mobile platformshaving a non-circular base.

COWLEDGETS

his research was suppored by Basic cienceResearch Program tough the National ResearchFoundation of Korea(NF) nded by the Ministr

of Education(F-01RA A00175).

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Scale-Invariant Isotropy of an Optical Mouse Array for Mobile Robot Velocity Estimation

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