Optimal Camera Placement of Large Scale Volume

Localization System for Mobile Robot

Yingfeng WU1,a, Gangyan LI1,b, Huan Yan1,c

1(School of Mechanical and Electric Engineering,

Wuhan University of Technology, Wuhan 430070,China) awuyf1997@whut.edu.cn, bganyanl@ whut.edu.cn, cahyanhuan@163.com

Keywords: large scale volume localization system, camera network, relative position algorithm

Abstract: Large Scale Volume Localization System (LSVLS) is applied widely in industry. Large

Scale Volume Localization System with camera network has appropriate precise and cost, which is

a promising system in metrology and localization in industry and lives. Optimal camera placement

is significant to lower cost and facilitate target’s auto-control for mobile robot in the large

workspace. The author optimized cameras placement with their relative position algorithm (RPA).

The result of optimal camera placement enhances greatly the efficiency of camera placement in

LSVLS and is verified with a model of field-winding mobile vehicle.

Introduction

Mobile robots are used in our lives and industry, they can sweep the floor, welding workpiece,

transport goods automatically or not. Some mobile robots should working in a large workspace

automatically. In order to track the mobile robots, Large Scale Volume Localization System can be

used to tracking the mobile robots in the large workspace.

Large Scale Volume Localization System (LSVLS) is applied widely in industry, including

aircraft and ship manufacturing, robot guidance, motion analysis, which is used for 3D coordinate

metrology accurately and tracking of moving object[1]

. LSVLS is constituted with several tech-

nologies, including laser tracker, theodolite, iGPS and high density CCD cameras, etc. Camera net-

work can enlarge the field of tracking with high measurement accuracy (millimetre-sized). And

CCD camera is cheaper than other technologies. A mobile spatial coordinate measuring system-II

(MScMS-II) [2]

is representative of using multiple CCD cameras, which is a promising system in

metrology and localization in industry. We use multiple CCD cameras (camera network) to build a

LSVLS to tracking the mobile robots in the large workspace. Wherever the mobile robots are, the

location of mobile robots can be detected precisely. Anyplace in the large scale volume must been

“seen” with three or more cameras, so dozens even hundreds cameras may used to “seen” precisely

the mobile robot in the such workspace. But how to arrange so many cameras to reduce overlap

between adjacent cameras and avoid any point missing by cameras. So the optimized camera arran-

gement is significant to lower cost and assure mobile robot’s auto-control, and automated camera

placement is essential to set the cameras to the right places to reduce the time spent to assemble the

LSVLS. Some LSVLS[3,4]

have encountered the same problem, or even not take it into account[2]

.

Visual coverage is an important quantifiable property of camera networks. Some algorithms, such

as genetic algorithms [5]

, quality metric [6]

, greedy selection heuristic [7]

, particle swarm optimization [8]

, etc., are used to optimize the placement of cameras. But their researches are mainly focused on

camera placement in 2D[5,9]

, or camera placement of subset in 3D[6]

, or only taking some single

Advanced Materials Research Vols. 945-949 (2014) pp 1390-1395Online available since 2014/Jun/06 at www.scientific.net© (2014) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMR.945-949.1390

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factor into account [6,7]

. We will explore automated camera placement of dozens even hundreds

cameras of LSVLS, based on integer linear programming (ILP) [9]

.

Problem Definition

Modeling a camera’s field-of-view in 3-D Space

The field-of-view of a camera can be described as a rectangular pyramid, as Fig.1(a). The

parameters of this pyramid can be easily calculated according to the intrinsic camera parameters.

Fig.1 Model of a camera’s field-of-view

A camera can be modeled as a matrix FOV . In which, row is the coordinate of the apex of

rectangular pyramid. Camera in Fig.1(a), whose projection centre is at O(0,0,0) and focus axis is

pointed to –Z, can be modeled as))0,0,0(,0,0,0(FOV .

−⋅−⋅−

−⋅⋅−

−⋅⋅

−⋅−⋅

=

ddd

ddd

ddd

ddd

FOV

yx

yx

yx

yx

)2

tan()2

tan(

)2

tan()2

tan(

)2

tan()2

tan(

)2

tan()2

tan(

000

))0,0,0(,0,0,0(

αα

αα

αα

αα

A random camera in Fig.1(b) can be regarded as a camera which is translated to point S and

rotated ryrxrz ,, around the axis Z,X and Y respectively from the camera in Fig.1(a). The random

camera can be modeled as),,,( SryrxrzFOV .

SRyRxRzFOV

z

z

z

z

z

y

y

y

y

y

x

x

x

x

x

FOV

D

C

B

A

s

D

C

B

A

s

D

C

B

A

s

Sryrxrz +⋅⋅⋅=

= ))0,0,0,(0,0,0(),,,(

In which,

−

=

100

0)cos()sin(

0)sin()cos(

zz

zz

rr

rr

Rz,

−=

)cos()sin(0

)sin()cos(0

001

1

xx

xx

rr

rrRx,

−

=

)cos(0)sin(

010

)sin(0)cos(

1

yy

yy

rr

rr

Ry,

=

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

z

z

z

z

z

y

y

y

y

y

x

x

x

x

x

S

(1)

(2)

(a)Green circle means a camera, rectangular

pyramid means the field-of-view of camera, d is

depth of field-of-view, xα are the angle

between the plane ODAandOBC , and yα are the angle between the plane OAB and OCD .

(b)A random camera translatedand rotated

from the camera in Fig.2 (a) to S .

Advanced Materials Research Vols. 945-949 1391

Modeling Workspace

In the ideal case cameras can be placed continuously in the space, but the continuous case

can’t be solved. So the workspace is divided into grids, the distance ∆ between two grids→0, the

approximated solution converges to the continuous-case solution. The workspace is divided into

grids respectively in observation area (a plane near the ground, ∆= obser∆ ) and camera placement

area (a plane near the ceiling, ∆= piace∆ ) . The grid points in camera placement area are the points

where the cameras can be placed and the grid points in observation area are the points where the

target can stay as Fig.2. Assume that the number of grid points in observation area and camera

placement area is ConPn _ and CamPn _ respectively. The 3D coordinates of these grid points

can be expressed with the matrix 3_ConP ×ConPn and 3_CamP ×CamPn . It is obvious that smaller the

obser∆ and piace∆ are, more grid points and more accurate visual coverage will be, and more time

and computer resource are needed.

Fig.2 The workspace is divided into grids respectively in observation area

and camera placement area, a camera is placed on grid point S , and

camera’s optical axis is pointed to observation area.

Solution of Optimal Camera Placement

Optimal camera placement has two sides: any camera which is placed on a grid point in

camera placement area must see the most grid points in observation area, and any grid point in

observation area must be seen by no less than 3 cameras. Firstly, we assume that a camera is placed

on grid point CP in camera placement area, as Fig.2, and camera’s optical axis is pointed to

observation area. The camera on CP can be expressed with ),,,( CPryrxrzFOV , according to Eq.1. In

which, yxz rrr ,, range from π− to π . We discretize yxz rrr ,, from π− to π step by 0.1. All of

the discretized yxz rrr ,, can be listed with a matrix 3_ ×PosMnPosM , which means all postures of the

camera on CP . PosMn _ is a variable of the number of the postures. We proposed relative position

algorithm (RPA) to find optimal camera placement on a grid point in camera placement area, and

solved based ILP Algorithm[9]

.

A program in matlab is designed to get optimal camera placement.

Grid the workspace and input the matrix 3_ConP ×ConPn and 3_CamP ×CamPn automatically.

Input the posture matrix 3_PosM ×PosMn , which contains all the postures of the camera.

Define a matrix ConPnCamPn __CAM × .

for 1=i to n_CamP

Define a matrix i__CovP ConPnPosMn ×

for 1=j to n_PosM

for 1=k to n_ConP

if ( ) 1:),(,:),(:),,( =kConPiCamPjPosMf , whether any camera placed on a grid point in

camera placement area can see the most grid points in observation area can be

solved with RPA.

1392 Advances in Manufacturing Science and Engineering V

1=×

ikjCovP ; which means that the camera on :),(iCamP can see the grid point

kn_ConP , the camera placed on i.NO placement point with the j.NO posture

can cover the k.NO grid point.

else

0=×

ikjCovP ; otherwise

End

End

End

In matrix iCovP , rows means postures of camera. Now the row which contains maximum

number of 1 is founded and assign the number of this row to iBPn _ . iBPn _ means that theiBPn _.NO posture is the best posture of camera. :),_(CovP:),(CAM ii BPni =

End

In matrix CAM , the minimum number of rows can be founded, which ensure that the sum of

every column is no less than 3. So any grid point in observation area can be seen by no less than 3

cameras. The minimum number of rows is defined as minn_ , Define a linear array SerN which

contains the serial number of these rows. Define a matrix 6min_OPTI_CAM ×n . m.NO row of

OPTI_CAM means the 3D coordination and the best posture yxz rrr ,, of the NO.

SerN(m) camera.

Simulation and Experimental Results

A large workpiece, whose diameter is 25m, is manufactured by field-winding composite mate-

rial. A winding mobile vehicle can be used to wind composite material onto the surface of the work-

piece. The winding mobile vehicle is constituted with two parts: mobile vehicle and winding equip-

ment on the platform of mobile vehicle. The mobile vehicle moves around the workpiece at a cons-

tant speed in a trajectory of 30m. Large Scale Volume Localization System with camera network is

used to navigate the mobile vehicle. The workspace is devided into grids, as Fig.3. Cameras place-

ment in the LSVLS is achieved with our algorithm. 26 cameras are needed to ensure that the mobile

vehicle circling in its trajectory can been seen. And the 3D coordinates and the postures of the

cameras are calculated, listed in Table 1.

Fig.3 The workspace of field-winding a large workpiece is divided into grids respectively in

observation area (blue area) and camera placement area (yellow area), green circles in camera

placement area are the coordinate of cameras.

y

z

x

x

y

(a)3D drawing of optimal camera placement (b)Projection drawing of optimal camera placement

Advanced Materials Research Vols. 945-949 1393

Table 1: The 3D coordinates and the postures of optimal camera placement

camera x y z rz rx ry camer x y z rz rx ry

1 1.3 -4.3 5.5 1.8 0.1 -0.1 14 -3.6 5.4 5.5 1.3 0.2 0.1

2 5.4 1 5.5 0.1 -0.1 -0.1 15 -4.7 4.5 5.5 1.2 0.2 0.2

3 3 4.6 5.5 2.5 0 -0.1 16 -6.1 2.4 5.5 0.7 0.1 0.3

4 -1.3 5.3 5.5 1.6 0.1 -0.1 17 -6.4 0.9 5.5 0.4 0.1 0.2

5 -3.9 3.9 5.5 1.3 0 0.1 18 -6.5 -0.1 5.5 2.7 0 0.3

6 -3.6 -4.2 5.5 1.9 -0.1 0 19 -6.4 -1.1 5.5 2.5 0 0.3

7 2 -5.1 5.5 1.5 -0.1 0.1 20 -5 -4.2 5.5 2.1 -0.2 0.2

8 6.3 1.5 5.5 2.5 0.1 -0.2 21 -3 -5.8 5.5 2.4 -0.2 0.1

9 5.6 3.3 5.5 3.1 0.2 -0.2 22 -1.6 -6.3 5.5 2.2 -0.2 0.1

10 4.3 4.9 5.5 1.9 0.2 -0.2 23 1.8 -6.2 5.5 1.8 -0.2 -0.1

11 2.6 5.9 5.5 2.4 0.2 -0.1 24 5.4 -3.6 5.5 1 -0.1 -0.2

12 0.7 6.5 5.5 2.1 0.3 0 25 5.7 -3.2 5.5 3.1 -0.1 -0.2

13 -1.3 6.4 5.5 0.9 0.3 0 26 6.2 -1.8 5.5 0.6 -0.1 -0.2

Conclusion and Futurework

Optimal camera placement is significant to lower cost and facilitate target’s auto-control in

Large Scale Volume Localization System (LSVLS). The author proposed a relative position

algorithm (RPA) to find optimal camera placement. The result of optimal camera placement

enhances greatly the efficiency of camera placement in LSVLS. The relative position algorithm can

be used to solve camera placement. But as the grids of workspace increase, the computing time of

RPA is increases largely too.

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Advanced Materials Research Vols. 945-949 1395

Advances in Manufacturing Science and Engineering V 10.4028/www.scientific.net/AMR.945-949 Optimal Camera Placement of Large Scale Volume Localization System for Mobile Robot 10.4028/www.scientific.net/AMR.945-949.1390

DOI References

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Manufacturing Technology, 2002, 51(2): 587-609.

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sensor networks[J]. Computers, IEEE Transactions on, 2002, 51(12): 1448-1453.

http://dx.doi.org/10.1109/TC.2002.1146711