Bosché, Frédéric

Haas, Carl T.

Kwon, Soon-Wook

Liapi, Katherine A.

English

Current methods for construction site modeling employ large, expensive laser range scanners that produce dense range point clouds of a scene from different perspectives. While useful for many purposes, this approach is not feasible for real-time applications, which would enable automated obstacle avoidance and semiautomated equipment control, and could improve both safety and productivity significantly. This paper presents human-assisted rapid environmental modeling algorithms for construction, and focuses on cylindrical object fitting algorithms. The presented algorithms address construction site material of cylindrical shape. Experiments were conducted to determine: (1) the effect of the ratio of length to diameter of the cylinder to the accuracy of the results, (2) the effect of the angle of view to the accuracy of the results, (3) the minimum number of scanned points required to give adequate modeling accuracy for cylinders of various length to diameter ratios. The results indicate that the proposed algorithms can model geometric primitives used in a construction site rapidly and with sufficient accuracy for automated obstacle avoidance and equipment control functions

Heriot Watt Pure

    Heriot-Watt University Research Gateway                                                          Heriot-Watt UniversityAlgorithms for fitting cylindrical objects to sparse range point clouds for rapidworkspace modelingKwon, Soon-Wook; Liapi, Katherine; Haas, Carl T.; Bosche, Frederic NicolasPublication date:2003Link to publication in Heriot-Watt Research GatewayCitation for published version (APA):Kwon, S-W., Liapi, K., Haas, C. T., & Bosche, F. N. (2003). Algorithms for fitting cylindrical objects to sparserange point clouds for rapid workspace modeling. 20th Annual International Symposium for Automation andRobotics in Construction, Eindhoven, Netherlands.173Algorithms for fitting cylindrical objects to sparse range point clouds for rapid workspace modeling Soon-Wook Kwon GRA/Field Systems and Construction Automation Lab., The University of Texas at Austin, Austin, TX 78712-0276 USA (e-mail: swkwon@mail.utexas.edu) Katharine A. Liapi Assistant Professor, The University of Texas at Austin, Austin, TX 78712-0276 USA (e-mail: kliap@mail.utexas.edu) Carl T. Haas Professor, The University of Texas at Austin, Austin, TX 78712-0276 USA (e-mail: haas@mail.utexas.edu) Frederic Bosche  GRA/Field Systems and Construction Automation Lab., The University of Texas at Austin, Austin, TX 78712-0276 USA (e-mail: bosche.frederic@mail.utexas.edu) ABSTRACT: Current methods for construction site modeling employ large, expensive laser range scanners that produce dense range point clouds of a scene from different perspectives. While useful for many purposes, this approach is not feasible for real-time applications, which would enable automated obstacle avoidance and semi-automated equipment control, and could improve both safety and productivity significantly. This paper presents human-assisted rapid environmental modeling algorithms for construction, and focuses on cylindrical object fitting algorithms. The presented algorithms address construction site material of cylindrical shape. Experiments were conducted to determine: (1) the effect of the ratio of length to diameter of the cylinder to the accuracy of the results, (2) the effect of the angle of view to the accuracy of the results, (3) the minimum number of scanned points required to give adequate modeling accuracy for cylinders of various length to diameter ratios. The results indicate that the proposed algorithms can model geometric primitives used in a construction site rapidly and with sufficient accuracy for automated obstacle avoidance and equipment control functions.  KEYWORDS:  CONSTRUCTION  AUTOMATION;  LASER  RANGE  FINDER;  OBJECT FITTING; OBJECT MATCHING; WORKSPACE MODELING 1. INTRODUCTIONConsiderable  effort  has  been  devoted  to  the development of methods of extracting geometrical information  from  a  scene,  which  is  still  a  major concern for both computer vision and robot vision (Lebegue 1993, Tsukiyama 1996). Determining the dimensions, distance and orientations of planar and curved  object  surfaces  such  as  wall,  doors,  pipes and other man-made objects is still a key issue and therefore modeling of the geometric features of a workspace  typically  demands  a  large  amount  of computation due to very large data sets (Tsukiyama 1996).  A  new  method  to  extract  geometric information  from  a  scene  by  taking  advantage  of human cognitive skills is under development at the University  of  Texas  at  Austin  (Cho  et  al  2002, Kwon et al 2002). A significant advantage of this approach  is  the  ability  to  extract  models  of  real world  objects  in  a  construction  workspace  from only a limited number of scanned points (less than 50  pts.  per  object),which  are  termed  sparse  point clouds here. Current  methods  for  construction  site  modeling employ large, expensive laser range scanners that produce dense range point clouds of a scene from different  perspectives.  While  useful  for  many purposes, this approach is not feasible for real-time  applications,  which  would  enable automated  obstacle  avoidance  and  semi-automated  equipment  control,  and  could improve  both  safety  and  productivity significantly  (McLaughlin  2002). The  dynamic nature of the construction environment requires that a real-time local area modeling system be capable of handling a rapidly changing and uncertain work environment.  However,  in  practice,  simple,  and reasonably accurate geometric primitives can give a sufficient  feedback  to  an  operator,  who  is controlling an equipment to place objects in an unstructured  construction  site.  For  real-time obstacle  avoidance,  such  volumes  also  facilitate computational tractability.  With  regard  to  the  geometric  objects  most frequently encountered in a workspace, it appears 174that a few types of primitives can be used to model a  wide  range  of  construction  scenes.  These  are planar  objects,  cuboids,  and  cylindrical  objects. Particularly, cylindrical objects can be used in plant construction  to  fit  and  match  chemical  pipes, ventilation pipes, and plumbing pipes. This paper presents  algorithms  that  accurately  fit  and  match objects of cylindrical shape, with regard to location and orientation, to sparse point clouds. Experiments were conducted to determine modeling accuracy of the algorithm at the Field Systems and Automation Laboratory of The University of Texas at Austin. 2. THE FITTING METHOD FOR CYLINDRICAL OBJECTS  A  single-axis  laser  range  finder,  a  pan/tilt  unit (PTU), and a personal computer were used for the experimental  set  up.  The  single-axis  laser  range finder (DistoMemo) that is mounted on the PTU is designed not only for hand-held operation, but also for  computer  use  through  an  interface.  The measurements  can  be  remotely  taken  and transferred directly into the computer.  Recent  research  results  indicate  that  graphical workspace  modeling  can  improve  construction equipment  control  and  operations.  Equipment operators can use graphical workspace models as an interactive  visual  feedback  tool  while  controlling equipment  (Kim  and  Haas  2000).  For  the  rapid modeling of construction site objects from sparse point  clouds  three  basic  algorithms  have  been developed  that  address  construction  site  objects. These are: (1) cuboid fitting algorithm, (2) cylinder fitting  algorithm,  (3)  sphere  algorithm,  and  (4) planar  algorithm.  This  paper  focuses  on  the cylinder fitting algorithm. Algorithm  development  and  revisions were  based on lab experiments. By using these algorithms we achieve: (1) accurate and reliable methods to save computational cost and time, (2) improved fitting algorithms  to  attain  real-time  execution,  and  (3) increased  modeling  accuracy  with  operator’s assistance. Figure 1 shows the entire fitting process.  2.1 Solid cylinder fitting algorithm Four parameters are necessary to define a bounded cylinder:  a  scalar  radius  r;  an  axis  vector,  a;  a center point to place the axis vector, c = (Xc, Yc, Zc);  and  a  length  of  cylinder  that  defines  the boundary of the cylinder. This algorithm uses the nearest  neighbor  algorithm  to  define  the  normal vector. Four scanned points are used to compute the planar  surface  of  the  cylinder.  By  projecting  the points  on  the  curved  surface  onto  the  computed planar surface, parameters r and c can be estimated. The radius of the circle is defined as the distance from the center of the circle to any point on the optimized  curve.  A  primary  estimation  of  the radius, Ĝ, is found by Ĝ = mean (|ƙ - k'|) (k' = {the projected points on the optimized curve of planar surface}). Consequently the final values of a, c, rare found by the least squares method using data d.2.2 Hollow cylinder fitting algorithm In the hollow cylinder fitting algorithm, Principal Components Analysis (PCA) was used to determine the primary axis of cylinder. Excluding the steps for computing the primary axis, the other steps of the algorithm  follow  the  same  sequence  as  the  solid cylinder algorithm.   PCA  is  a  distribution-based  ordination method  in which the distance between sites in an ordination diagram  is  correlated  with  multi  dimensional distribution (Duda et al, 2001). PCA assumes that all vectors in a set of n dimensional samples a1 … an  can  be  explained  by  a  single  vector  a0.  The vector a0 is derived using the least squares method, in which the sum of the squared distances between a0 and the various ak are minimized. We define the square-error criterion function F0(a0) by 210 0 0 ) ( ¦   nbk a a a F                                (1) ¦  nbb anp11                                       (2) ¦     nbb p a p a F120 0 ) ( ) (                       (3) Projecting the sample data onto a line through the sample  mean,  one-dimensional  representation  can be computed. If we let e be a unit vector of the line direction, the line equation is  de p a                                                   (4) Scalar d  is  the  distance  between  the  sample  data and  the  sample  mean  p.    We  can  find  the coefficients dk by minimizing the squared criterion function.211 0 ) ( ) , ,..., ( ¦    nbk b n a e d p e d d F               (5) ) ( p a e d btb                                            (6) 175The best direction e of the line can  be found by solving scatter matrix U, which is defined by tbnbb p a p a U ) )( (1   ¦                          (7) ¦     nbbt p a Ue e e F121 ) (                         (8) LaGrange multipliers can be used to maximize the etUe, which is subject to the constraint ||e||=1. Let ĳ  be  an  undetermined  multiplier.  We  can  do  the differentiation of) 1 (     e e Ue e vt t I                                   (9) with regard to e gettinge UeeuI 2 2   ww.                                       (10) By setting the gradient vector equal to zero, we see that e  should  be  an  eigenvector  of  the  scatter matrix. The eigenvector will be the primary axis of the cylinder that can be obtained by reducing the dimensionality  of  the  feature  space  and  by restricting  attention  to  the  directions  along  the scatter of the cloud (Vemuri et al.1986, Vemuri and Aggarwal  1987,  Schweikert  1966).  It  will  be  the primary axis of the cylinder. e Ue I                                                     (11) After  finding  the  primary  axis  of  a  cylinder,  the estimated planar surfaces can be generated on the top and bottom of a hollow cylinder.  By projecting the  points  of  the  curved  surface  onto  the  planar surfaces, the radius and center point of the hollow cylinder can be estimated. The radius of the circle is found using the same method used in the solid cylinder algorithm. Consequently the final values of the radius, length, center point, and primary axis are found by this fitting algorithm using scanned data. Figure  2  shows  the  scanned  points  and  the computed  primary  axis  of  the  cylinder.  Figure  3 illustrates scanned points and projected points onto planar surface of the cylinder. Figure 4 shows the points projected from the curved surface onto the planar  surface  of  the  cylinder.  Figure  5  shows modeled cylinders computed by the scanned points from actual objects.  3. EXPERIMENTAL RESULTS  Experiments were conducted to determine: (1) the effect of the ratio of radius to length (L/D) of the cylinder to the accuracy of the results, (2) the effect of the angle of view to the accuracy of the results, (3)  the  minimum  number  of  scanned  points required  to  give  adequate  modeling  accuracy  for cylinders of various length to diameter ratios.  Experiments were performed for cylinders with 3-inch through 5-inch radii. For each size of cylinder, several  measurements  have  been  conducted. Specifically, measurements based on 10, 20, 30 and 40 scanned points, were conducted for each tested cylindrical object. To increase the accuracy of the experiments, each test has been repeated 30 times. Thus, 120 tests per factor were conducted for the evaluation of the performance of the algorithm in respect to each one of factors (radius, length, and axis). The following results were obtained (Table 1, Figures 6 and 7):  x The relationship between radius and length of the visible section of a cylinder affects the  accuracy  of  the  cylinder  fitting algorithm x The most accurate results in radius, length, and  axis  were  obtained  for  a  90  degree angle of view. Results  of  tests  using  fixed  radius  (3  to  5-inch), various  lengths  (range  from  10  to  25–inch),  and various angles (30, 60, 90 degrees) show that: x Ten  scanned  data  points  give  adequately accurate  estimates  of  radius,  length,  and axis for all tested cylinders.  x Minimal  improvement  in  accuracy  is achieved by scanning more than 10 points. With  respect  to  the  accuracy  of  the  estimates  of length, both the number of data points and the way scanned data points are distributed on the surface of the  cylindrical  object  play  a  significant  role.  In other words, it is of prime importance to select well distributed points on the visible surface of cylinder. Restriction  of  points  only  on  a  small  area  of  the cylinder’s surface results in lower correlations and poorer  parameter  estimates.  In  addition,  the distance from the object to the laser scanner affects modeling  accuracy,  precision  error,  and  accuracy error.1764. CONCLUSION The  proposed  algorithms  for  fitting  cylindrical objects  are  computationally  efficient  and  suitable for use in equipment control and obstacle avoidance for safety applications. They are also acceptable for generating construction as-builts, however for long pipe sections they would have to be corrected with pipe  end  points.  These  algorithms  should  be broadly applicable. 5.  ACKNOWLEDGEMENTS This paper is based on the research funded by the National  Science  Foundation  (Grant#:  CMS-0000137) and the National Institute Standard andTechnology  (Project  NBR:  NA1341-02-W-0742).The authors gratefully acknowledge their financial support and encouragement throughout this study. The  authors  also  appreciate  the  assistance  from Sang-Wook  Han,  J.K.  Aggarwal,  and  Mohan Sridharan.6. REFERENCES McLaughlin, J., “Rapid Human-assisted Creation of Bounding  Models  for  Obstacle  Avoidance  in Construction”  Master's  thesis,  Department  of Mechanical  Engineering,  University  of  Texas  at Austin, Austin, Texas, 2002 Kim, Y., and Haas, C., “A Model for Automation of  Infrastructure  Maintenance  Using Representational  Forms,”  J.of  Automation  in Construction, Elsevier, pp.57-68, October 2000. Lebegue,  X.,  Aggarwal,  J.  K.,  “Extraction  and interpretation  of  semantically  significant  line segments  for  a  mobile  robot,”  Proceedings  of Robotics  and  Automation,  pp  1778  -1785,  May 1992. Tsukiyama,  T.,  “Understanding  man-made environments  using  nonstructured  lighting-3D world  modeling  for  indoor  mobile  robots,”Intelligent  Robots  and  Systems,  IROS  '97, Proceedings  of  the  1997  IEEE/RSJ  International Conference,  Volume:  3,  pp  1250  -12577, September 1997.  Cho, Y., Haas, C. T., Liapi, K., and Sreenivasan, S. V., “A Framework for Rapid Local Area Modeling for  Construction  Automation.”  Journal  of Automation  in  Construction,  11(6),  pp.  629-641, 2002. Kwon, S., “Human-assisted Object Fitting to Sparse Cloud  Points  for  Rapid  Workspace  Modeling  in Construction Automation”, Proceedings of the 19th International  Symposium  on  Automation  and Robotics  in  Construction  (ISARC),  pp  357-362, September 2002 Duda,  R.,  Hart,  P.,  Stork,  D.,  “Pattern Classification,”  Wiley-Interscience,  2nd  edition, October 2000. Vermuri, B.C, Aggarwal, J. K., “Representaion and Recognition of Objects from Dense Range Maps,” IEEE Trans. on Circuits and Systems, Vol. CAS-34, No. 11, November 1987. Vermuri, B.C., Mitiche, A., and Aggarwal, J. K., “Curvature-based  Representation  of  Objects  from Range Data,” Image and Vision Computing, Vol. 4, no. 2, pp. 107-114, 1986  Schweikert, D. G., “An Interpolation Curve Using Spline under Tension,” Journal. of Math. Phys., Vol 45, pp312-317, 1966 177Figure 1 .Object Fitting Method Figure 2. Scanned Points Computed Primary Axis          Figure 3. Result of Fitting (1) Figure 4. Result of Fitting (2)                       Figure 5. Modeled Cylinders178Table 1. Summary of Experimental Results of Analysis 10 pts.  20 pts  30 pts  40 pts Percentage of Error (%) Percentage of Error (%) Percentage of Error (%) Percentage of Error (%) Radius Length Devi.of Axis  Radius Length Devi.of Axis  Radius Length Devi.of Axis  Radius Length Devi.of Axis L/D §1.011.50  11.00  14.19  14.50  14.40  15.68  15.50  12.20  20.89  21.50  16.80  24.43 L/D §1.54.75  8.67  2.66  3.75  5.33  2.58  3.50  5.13  2.51  3.50  4.87  2.41 L/D §2.04.50  5.35  2.58  4.00  5.30  2.54  3.50  4.40  2.52  3.50  3.85  2.52 L/D §2.53.00  2.88  1.94  3.00  2.84  1.80  2.75  2.84  1.78  2.00  2.24  1.49 Percentage of error for radius of cylinder012345678910L/D §1.5 L/D §2 L/D §2.5Ratio of Length to diameterPercentage of error of radius (%)10 pts20 pts30 pts40 ptsFigure 6. Percentage of Error for Radius of CylinderPercentage of error for axis deviation of cylinder012345678910L/D §1.5 L/D §2 L/D §2.5Ratio of Length to diameterPercentage of error of axis deviation (%10 pts20 pts30 pts40 ptsFigure 7. Percentage of Error for Axis Deviation of Cylinder   

A Framework for Rapid Local Area Modeling for Construction Automation.”

Curvature-based Representation of Objects from Range Data,”

Extraction and interpretation of semantically significant line segments for a mobile robot,”

Human-assisted Object Fitting to Sparse Cloud Points for Rapid Workspace Modeling in Construction Automation”,

Rapid Human-assisted Creation of Bounding Models for Obstacle Avoidance in Construction”

Representaion and Recognition of Objects from Dense Range Maps,”

Understanding man-made environments using nonstructured lighting-3D world modeling for indoor mobile robots,” Intelligent Robots and Systems,

Algorithms for fitting cylindrical objects to sparse range point clouds for rapid workspace modeling

Soon-Wook Kwon

Katharine A. Liapi

Carl T. Haas

Frederic Bosche

Crossref

Algorithms for Fitting Cylindrical Objects to Sparse Range Point Clouds for Rapid Workspace Modeling

173Algorithms for fitting cylindrical objects to sparse range point clouds for rapid workspace modeling Soon-Wook Kwon GRA/Field Systems and Construction Automation Lab., The University of Texas at Austin, Austin, TX 78712-0276 USA (e-mail: swkwon@mail.utexas.edu) Katharine A. Liapi Assistant Professor, The University of Texas at Austin, Austin, TX 78712-0276 USA (e-mail: kliap@mail.utexas.edu) Carl T. Haas Professor, The University of Texas at Austin, Austin, TX 78712-0276 USA (e-mail: haas@mail.utexas.edu) Frederic Bosche  GRA/Field Systems and Construction Automation Lab., The University of Texas at Austin, Austin, TX 78712-0276 USA (e-mail: bosche.frederic@mail.utexas.edu) ABSTRACT: Current methods for construction site modeling employ large, expensive laser range scanners that produce dense range point clouds of a scene from different perspectives. While useful for many purposes, this approach is not feasible for real-time applications, which would enable automated obstacle avoidance and semi-automated equipment control, and could improve both safety and productivity significantly. This paper presents human-assisted rapid environmental modeling algorithms for construction, and focuses on cylindrical object fitting algorithms. The presented algorithms address construction site material of cylindrical shape. Experiments were conducted to determine: (1) the effect of the ratio of length to diameter of the cylinder to the accuracy of the results, (2) the effect of the angle of view to the accuracy of the results, (3) the minimum number of scanned points required to give adequate modeling accuracy for cylinders of various length to diameter ratios. The results indicate that the proposed algorithms can model geometric primitives used in a construction site rapidly and with sufficient accuracy for automated obstacle avoidance and equipment control functions.  KEYWORDS: CONSTRUCTION AUTOMATION; LASER RANGE FINDER; OBJECT FITTING; OBJECT MATCHING; WORKSPACE MODELING 1. INTRODUCTIONConsiderable effort has been devoted to the development of methods of extracting geometrical information from a scene, which is still a major concern for both computer vision and robot vision (Lebegue 1993, Tsukiyama 1996). Determining the dimensions, distance and orientations of planar and curved object surfaces such as wall, doors, pipes and other man-made objects is still a key issue and therefore modeling of the geometric features of a workspace typically demands a large amount of computation due to very large data sets (Tsukiyama 1996). A new method to extract geometric information from a scene by taking advantage of human cognitive skills is under development at the University of Texas at Austin (Cho et al 2002, Kwon et al 2002). A significant advantage of this approach is the ability to extract models of real world objects in a construction workspace from only a limited number of scanned points (less than 50 pts. per object),which are termed sparse point clouds here. Current methods for construction site modeling employ large, expensive laser range scanners that produce dense range point clouds of a scene from different perspectives. While useful for many purposes, this approach is not feasible for real-time applications, which would enable automated obstacle avoidance and semi-automated equipment control, and could improve both safety and productivity significantly (McLaughlin 2002). The dynamic nature of the construction environment requires that a real-time local area modeling system be capable of handling a rapidly changing and uncertain work environment. However, in practice, simple, and reasonably accurate geometric primitives can give a sufficient feedback to an operator, who is controlling an equipment to place objects in an unstructured construction site. For real-time obstacle avoidance, such volumes also facilitate computational tractability.  With regard to the geometric objects most frequently encountered in a workspace, it appears 174that a few types of primitives can be used to model a wide range of construction scenes. These are planar objects, cuboids, and cylindrical objects. Particularly, cylindrical objects can be used in plant construction to fit and match chemical pipes, ventilation pipes, and plumbing pipes. This paper presents algorithms that accurately fit and match objects of cylindrical shape, with regard to location and orientation, to sparse point clouds. Experiments were conducted to determine modeling accuracy of the algorithm at the Field Systems and Automation Laboratory of The University of Texas at Austin. 2. THE FITTING METHOD FOR CYLINDRICAL OBJECTS  A single-axis laser range finder, a pan/tilt unit (PTU), and a personal computer were used for the experimental set up. The single-axis laser range finder (DistoMemo) that is mounted on the PTU is designed not only for hand-held operation, but also for computer use through an interface. The measurements can be remotely taken and transferred directly into the computer.  Recent research results indicate that graphical workspace modeling can improve construction equipment control and operations. Equipment operators can use graphical workspace models as an interactive visual feedback tool while controlling equipment (Kim and Haas 2000). For the rapid modeling of construction site objects from sparse point clouds three basic algorithms have been developed that address construction site objects. These are: (1) cuboid fitting algorithm, (2) cylinder fitting algorithm, (3) sphere algorithm, and (4) planar algorithm. This paper focuses on the cylinder fitting algorithm. Algorithm development and revisions were based on lab experiments. By using these algorithms we achieve: (1) accurate and reliable methods to save computational cost and time, (2) improved fitting algorithms to attain real-time execution, and (3) increased modeling accuracy with operator’s assistance. Figure 1 shows the entire fitting process.  2.1 Solid cylinder fitting algorithm Four parameters are necessary to define a bounded cylinder: a scalar radius r; an axis vector, a; a center point to place the axis vector, c = (Xc, Yc, Zc); and a length of cylinder that defines the boundary of the cylinder. This algorithm uses the nearest neighbor algorithm to define the normal vector. Four scanned points are used to compute the planar surface of the cylinder. By projecting the points on the curved surface onto the computed planar surface, parameters r and c can be estimated. The radius of the circle is defined as the distance from the center of the circle to any point on the optimized curve. A primary estimation of the radius, ?, is found by ? = mean (|? - k'|) (k' = {the projected points on the optimized curve of planar surface}). Consequently the final values of a, c, rare found by the least squares method using data d.2.2 Hollow cylinder fitting algorithm In the hollow cylinder fitting algorithm, Principal Components Analysis (PCA) was used to determine the primary axis of cylinder. Excluding the steps for computing the primary axis, the other steps of the algorithm follow the same sequence as the solid cylinder algorithm.   PCA is a distribution-based ordination method in which the distance between sites in an ordination diagram is correlated with multi dimensional distribution (Duda et al, 2001). PCA assumes that all vectors in a set of n dimensional samples a1 … an can be explained by a single vector a0. The vector a0 is derived using the least squares method, in which the sum of the squared distances between a0 and the various ak are minimized. We define the square-error criterion function F0(a0) by 21000 )( ????nbkaaaF                                (1) ???nbbanp11                                       (2) ??????nbb papaF1200 )()(                       (3) Projecting the sample data onto a line through the sample mean, one-dimensional representation can be computed. If we let e be a unit vector of the line direction, the line equation is  depa ??                                                (4) Scalar d is the distance between the sample data and the sample mean p.  We can find the coefficients dk by minimizing the squared criterion function.2110 )(),,...,( ?????nbkbn aedpeddF               (5) )( paed btb ??                                         (6) 175The best direction e of the line can be found by solving scatter matrix U, which is defined by tbnbb papaU ))((1?????                         (7) ??????nbbt paUeeeF121 )(                         (8) LaGrange multipliers can be used to maximize the etUe, which is subject to the constraint ||e||=1. Let ? be an undetermined multiplier. We can do the differentiation of)1( ??? eeUeev tt ?                                   (9) with regard to e gettingeUeeu ?22 ????.                                       (10) By setting the gradient vector equal to zero, we see that e should be an eigenvector of the scatter matrix. The eigenvector will be the primary axis of the cylinder that can be obtained by reducing the dimensionality of the feature space and by restricting attention to the directions along the scatter of the cloud (Vemuri et al.1986, Vemuri and Aggarwal 1987, Schweikert 1966). It will be the primary axis of the cylinder. eUe ??                                                   (11) After finding the primary axis of a cylinder, the estimated planar surfaces can be generated on the top and bottom of a hollow cylinder.  By projecting the points of the curved surface onto the planar surfaces, the radius and center point of the hollow cylinder can be estimated. The radius of the circle is found using the same method used in the solid cylinder algorithm. Consequently the final values of the radius, length, center point, and primary axis are found by this fitting algorithm using scanned data. Figure 2 shows the scanned points and the computed primary axis of the cylinder. Figure 3 illustrates scanned points and projected points onto planar surface of the cylinder. Figure 4 shows the points projected from the curved surface onto the planar surface of the cylinder. Figure 5 shows modeled cylinders computed by the scanned points from actual objects.  3. EXPERIMENTAL RESULTS  Experiments were conducted to determine: (1) the effect of the ratio of radius to length (L/D) of the cylinder to the accuracy of the results, (2) the effect of the angle of view to the accuracy of the results, (3) the minimum number of scanned points required to give adequate modeling accuracy for cylinders of various length to diameter ratios.  Experiments were performed for cylinders with 3-inch through 5-inch radii. For each size of cylinder, several measurements have been conducted. Specifically, measurements based on 10, 20, 30 and 40 scanned points, were conducted for each tested cylindrical object. To increase the accuracy of the experiments, each test has been repeated 30 times. Thus, 120 tests per factor were conducted for the evaluation of the performance of the algorithm in respect to each one of factors (radius, length, and axis). The following results were obtained (Table 1, Figures 6 and 7):  ? The relationship between radius and length of the visible section of a cylinder affects the accuracy of the cylinder fitting algorithm ? The most accurate results in radius, length, and axis were obtained for a 90 degree angle of view. Results of tests using fixed radius (3 to 5-inch), various lengths (range from 10 to 25–inch), and various angles (30, 60, 90 degrees) show that: ? Ten scanned data points give adequately accurate estimates of radius, length, and axis for all tested cylinders.  ? Minimal improvement in accuracy is achieved by scanning more than 10 points. With respect to the accuracy of the estimates of length, both the number of data points and the way scanned data points are distributed on the surface of the cylindrical object play a significant role. In other words, it is of prime importance to select well distributed points on the visible surface of cylinder. Restriction of points only on a small area of the cylinder’s surface results in lower correlations and poorer parameter estimates. In addition, the distance from the object to the laser scanner affects modeling accuracy, precision error, and accuracy error.1764. CONCLUSION The proposed algorithms for fitting cylindrical objects are computationally efficient and suitable for use in equipment control and obstacle avoidance for safety applications. They are also acceptable for generating construction as-builts, however for long pipe sections they would have to be corrected with pipe end points. These algorithms should be broadly applicable. 5.  ACKNOWLEDGEMENTS This paper is based on the research funded by the National Science Foundation (Grant#: CMS-0000137) and the National Institute Standard andTechnology (Project NBR: NA1341-02-W-0742).The authors gratefully acknowledge their financial support and encouragement throughout this study. The authors also appreciate the assistance from Sang-Wook Han, J.K. Aggarwal, and Mohan Sridharan.6. REFERENCES McLaughlin, J., “Rapid Human-assisted Creation of Bounding Models for Obstacle Avoidance in Construction” Master's thesis, Department of Mechanical Engineering, University of Texas at Austin, Austin, Texas, 2002 Kim, Y., and Haas, C., “A Model for Automation of Infrastructure Maintenance Using Representational Forms,” J.of Automation in Construction, Elsevier, pp.57-68, October 2000. Lebegue, X., Aggarwal, J. K., “Extraction and interpretation of semantically significant line segments for a mobile robot,” Proceedings of Robotics and Automation, pp 1778 -1785, May 1992. Tsukiyama, T., “Understanding man-made environments using nonstructured lighting-3D world modeling for indoor mobile robots,”Intelligent Robots and Systems, IROS '97, Proceedings of the 1997 IEEE/RSJ International Conference, Volume: 3, pp 1250 -12577, September 1997.  Cho, Y., Haas, C. T., Liapi, K., and Sreenivasan, S. V., “A Framework for Rapid Local Area Modeling for Construction Automation.” Journal of Automation in Construction, 11(6), pp. 629-641, 2002. Kwon, S., “Human-assisted Object Fitting to Sparse Cloud Points for Rapid Workspace Modeling in Construction Automation”, Proceedings of the 19th International Symposium on Automation and Robotics in Construction (ISARC), pp 357-362, September 2002 Duda, R., Hart, P., Stork, D., “Pattern Classification,” Wiley-Interscience, 2nd edition, October 2000. Vermuri, B.C, Aggarwal, J. K., “Representaion and Recognition of Objects from Dense Range Maps,” IEEE Trans. on Circuits and Systems, Vol. CAS-34, No. 11, November 1987. Vermuri, B.C., Mitiche, A., and Aggarwal, J. K., “Curvature-based Representation of Objects from Range Data,” Image and Vision Computing, Vol. 4, no. 2, pp. 107-114, 1986  Schweikert, D. G., “An Interpolation Curve Using Spline under Tension,” Journal. of Math. Phys., Vol 45, pp312-317, 1966 177Figure 1 .Object Fitting Method Figure 2. Scanned Points Computed Primary Axis          Figure 3. Result of Fitting (1) Figure 4. Result of Fitting (2)                       Figure 5. Modeled Cylinders178Table 1. Summary of Experimental Results of Analysis 10 pts. 20 pts 30 pts 40 pts Percentage of Error (%) Percentage of Error (%) Percentage of Error (%) Percentage of Error (%) Radius LengthDevi.of Axis Radius LengthDevi.of Axis Radius LengthDevi.of Axis Radius LengthDevi.of Axis L/D ?1.011.50 11.00 14.19 14.50 14.40 15.68 15.50 12.20 20.89 21.50 16.80 24.43 L/D ?1.54.75 8.67 2.66 3.75 5.33 2.58 3.50 5.13 2.51 3.50 4.87 2.41 L/D ?2.04.50 5.35 2.58 4.00 5.30 2.54 3.50 4.40 2.52 3.50 3.85 2.52 L/D ?2.53.00 2.88 1.94 3.00 2.84 1.80 2.75 2.84 1.78 2.00 2.24 1.49 Percentage of error for radius of cylinder012345678910L/D ?1.5 L/D ?2 L/D ?2.5Ratio of Length to diameterPercentage of error of radius (%)10 pts20 pts30 pts40 ptsFigure 6. Percentage of Error for Radius of CylinderPercentage of error for axis deviation of cylinder012345678910L/D ?1.5 L/D ?2 L/D ?2.5Ratio of Length to diameterPercentage of error of axis deviation (%10 pts20 pts30 pts40 ptsFigure 7. Percentage of Error for Axis Deviation of Cylinder   

Algorithms for fitting cylindrical objects to sparse range point clouds for rapid workspace modeling

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