Modeling and Calibration of a Structured-light Optical - Michigan ...

Chenggang Che 

Graduate Research Assistant. 

Jun Ni 

Associate Professor. 

Department of Mechanical Engineering 

and Applied Mechanics, 

The University of Michigan, 

Ann Arbor, Ml 48109 

1 Introduction 

The accurate measurement, or digitization of free-form surfaces 

and parts with complex features, such as dies and molds, 

ship hulls, airplane fuselages, wings and turbine blades, etc., 

has always been a tedious and time consuming task. Although 

designers often specify a tight control tolerance over the entire 

three dimensional contour surface, manufacturers and/or inspectors 

have difficulty employing a rigorous method to measure 

and check the entire surface. The most popular measurement 

tools used to perform sampled measurement of these surfaces 

are Coordinate Measurement Machines (CMMs) using 

contact probes. These devices, while having more than sufficient 

measurement accuracy for the defined task, suffer from the 

relatively low measurement throughput which makes the measurement 

task time consuming. 

To address this problem, a number of vendors and researchers 

have developed non-contact laser probes for digitizing surfaces 

by employing a single-spot beam to sweep the entire surface 

to be measured (Aromat, 1992; Keyence, 1993; Okada, 1992; 

Selcom, 1993; Venture, 1993; CyberOptics, 1992; Hymarc, 

1994; Chesapeake Laser, 1994; Digibotics, 1994; Goh, et al., 

1985; Saito and Miyoshi, 1991; Bradley and Vickers, 1992, 

etc.). These devices, while faster than their contact probe equivalents, 

suffer similar throughput limitations and often have difficulty 

locating certain features accurately, particularly edges & 

holes. Thus, the development of high-throughput dimensional 

measurement/digitization techniques for large and complex 

free-form surfaces becomes necessary. 

With the emergence of large field of view structured-light 

laser stripe sensors in recent years (Perceptron, 1994ab; Sami, 

1994; Geometric Research, 1994; 3D Technology, 1994, etc.), 

integration of such 2D contour sensors to multiple-axis machines 

such as CNC, CMM or robot systems has generated some 

interests (Chen, 1991; Champ, 1992). Chen (1991) studied the 

case of single axis scanning. 

Some earlier work was done by Will and Pennington (1971, 

1972), Popplestone et al. (1975), Agin and Binford (1973), 

Agin and Highnam (1982), Agin (1985) and Mansbach 

(1986). Their modeling methodologies were all based on a pinhole 

camera model. Their calibration approaches required the 

measurement of both camera/projector parameters and robot 

joint parameters, thus making on-line use impossible. 

Contributed by the Manufacturing Engineering Division for publication in die 

JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received 

Jan. 1995; revised Aug. 1995. Technical Editor: S. K. Kapoor. 

Modeling and Calibration of a 

Structured-light Optical CMM via 

Skewed Frame Representation 

A new tetrahedron-target-based approach is presented for the extrinsic calibration 

of a non-contact "light-striping" (structured light) optical coordinate measuring 

machine (CMM). The procedure makes automated on-line calibration possible. The 

system modeling is based on a unique skewed frame representation without the use 

of pin-hole camera model assumption. It is demonstrated that the extrinsic calibration 

matrix can be decomposed into two classes of transformations, one homogeneous 

and the other nonhomogeneous. The nonhomogeneous transformation between a 

Cartesian world frame and the non-Cartesian skewed sensor frame is studied. The 

sensitivity of the dimensional deformation on the two skew angles is simulated. 

Experimental studies show that a micron level calibration accuracy can be achieved. 

The mechanism of a multiple-axis, arbitrary sensor orientation 

laser stripe scanning system was studied recently by Chen 

and Kak (1987), and Theodoracatos and Calkins (1993). They 

mounted a CCD camera and a structured line laser source on a 

pan-tilt device. Their intrinsic and extrinsic calibration methods 

were based on a simple pin-hole camera model. Pin-hole camera 

model is only an approximation of the real imaging system. 

It is based on the following assumptions (Theodoracatos and 

Calkins, 1993): (1) lens system behaves as an ideal, thin lens; 

(2) reflecting surface is perfectly diffusive; (3) pixel raster is 

ideal; and (4) image is ideal. These assumptions allow the use 

of an optical center of the lens to determine spatial location of 

the image. Consequently intrinsic and extrinsic parameters can 

be derived. If these assumptions were valid, the power of the 

laser source would have to be extremely high so that the CCD 

camera can have enough exposure because the laser beam has 

to get through an infinitely small "pin-hole." As noted by 

Dewar (1994), the power of the laser beam has to be so high 

that practically eliminates its use for safety concerns. In addition, 

if assumptions (2) through (4) were valid, laser speckle 

would not have been a problem perplexing optical scientists 

and engineers for decades. The pin-hole camera model may be 

accurate enough for pattern or object recognition purposes, it 

is not adequate for high precision dimensional measurement. 

The purpose of this work is to provide a fast on-line extrinsic 

calibration scheme for the integrated system. The organization 

of the paper is as follows: first, the system configuration is 

introduced; secondly, the system modeling approach is discussed; 

thirdly, system calibration technique is proposed; in the 

end, experimental studies and sensitivity analysis are performed. 

2 System Configuration 

A commercially available structured light sensor (Perceptron, 

1994a) is mounted on the shaft of a mechanical coordinate 

measurement machine (CMM), as shown in Fig. 1. This offthe-shelf 

sensor had already been intrinsically calibrated by the 

manufacturer via a "blackbox" approach (Dewar, 1994; Wang, 

W. W., 1991). Using Scheimpflug condition to guarantee focus 

throughout the sensor field of view, image distortion inevitably 

occurs. The "blackbox" approach gets around the inaccuracy 

problems inherent in any analytical camera geometric models. 

By finding a mapping from a 2-D sensor field of view to a 

2-D CCD array, image distortion can be corrected. Thus, the 

sensor can directly output 2-D world coordinate information. 

This enables us to focus on the extrinsic calibration of the 

integrated system. 

Journal of Manufacturing Science and Engineering NOVEMBER 1996, Vol. 118/595 

Copyright © 1996 by ASME 

Downloaded 26 Sep 2011 to 141.212.97.74. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

Fig. 1 Integration of a laser stripe sensor to a CMM 

The multiple-axis integrated system allows the sensor to be 

rotated to any angle for optimal view of the part surface to be 

measured. The same sensor can also be attached to the endeffector 

of a robot. The robot itself cannot be used for scanning 

due to its poor positioning accuracy. However, when combined 

with a precision X-Y motion stage, the robot integrated system 

has its advantages: the robot arm can easily position the sensor 

to reach and measure large parts from many positions and orientations. 

Since the laser scanner is only a two dimensional sensor, a 

third dimension needs to be integrated to the system to realize 

3-D measurements. When scanning a part, the CMM shaft 

moves the sensor along a certain axis, e.g., the X axis, in small 

increments at a fixed speed. The sensor provides Ys and Zs data 

while the optical (or magnetic) scale of the CMM X axis provides 

the Xs information (Fig. 2). If the part instead of the 

sensor moves, the scanning mechanism is the same. 

If the laser plane, on which the sensor coordinate frame Ys- 

Zs lies, is always perpendicular to the surface to be measured 

(i.e., the laser plane is placed perpendicular to the CMM Xaxis), 

the coordinate system thus formed is Cartesian. However, 

this severely limits the application of the system, practically 

precluding the measurement of complex parts requiring multiple-axis 

scanner motions and multiple orientations of the laser 

sensor to achieve an optimal projecting angle. 

During scanning, the laser scanning coordinate system usually 

is non-orthogonal, i.e., skewed. If data obtained from this 

mechanical driving 

direction(Xs) 

Structured laser 

line lighting 

/ Sensing optics 

>• Xs: added axis 

Fig. 2 2D sensor used for 3D measurement 

596 / Vol. 118, NOVEMBER 1996 

imaginary Cartesian sensor Zs(Zs') 

coordinate system {S'} 

world coordinate system {W} 

Xw 

YsCYs 1 ) 

real skewed sensor 

coordinate system (S} 

Fig. 3 Characterization of the laser stripe scanning system 

skewed coordinate system are to be used for dimensional measurement, 

they have to be transformed to a Cartesian frame. 

Otherwise, the dimensions on the part will be distorted, e.g., 

after non-Cartesian scanning, a sphere in real space will look 

like an ellipsoid and a cube will look like an inclined parallelepiped. 

3 System Modeling 

3.1 Skewed Frame Representation. Three frames, the 

world coordinate frame {W}, an imaginary sensor coordinate 

system {S'} and the sensor's body frame {S}, as shown in 

Figs. 3 and 4, are used in the following discussions. Both {W} 

and {S'} are Cartesian frames, but {S} frame is not. {S} and 

{S'} frames share the same origin. Let b aTbe the 4 X 4 matrix 

representation of coordinate transformation from {a} to {b}, 

a, b e {W,S',S). Then 

T is a homogeneous transform of which various equivalent 

representations exist (e.g., Craig, 1992; Wang, 1992). This paper 

adopts the matrix representation using the roll-pitch-yaw 

angles (a, /3, y): 

"T = 

cacfi 

sac (3 

-sp 

0 

cas0sy — sacy 

sasfisy + cacy 

cfisy 

0 

caspcy + sasy 

sas/3cy — easy 

c0cy 

0 

: where sin a sa, cos a •• = COL and (qx,qy,qz, 1 ) 

vector Neither fT nor 

form. Their representations are explored below 

T is a translation 

W 

T constitutes a homogeneous trans- 

Ys(Ys 

A x^ 

Fig. 4 Spatial relationship between the real skewed sensor coordinate 

system {S} and the imaginary Cartesian coordinate system {S } 

(1) 

(2) 

Transactions of the ASME 


In Figs. 3 and 4, the orientation of the real sensor Xs axis in 

the imaginary Cartesian sensor coordinate system {S'} can be 

defined by two angles, 9 and ip. 9 is defined as the angle between 

axes Xs and Xs'. ip is defined as the angle between Xprj (the 

orthogonal projection of Xs axis onto the Ys-ZslYs'-Zs' plane) 

and Ys. 

Point P is used to find the transformation from {S} to {S'}. 

P has two sets of coordinates, {x, y, z) in {S} and (x', y', z') 

m{S'}. 

Since the laser plane [Ys - Zs] moves along the Xs axis 

during scanning, the (x, y, z) coordinates of P in sensor frame 

are not all perpendicular projections onto the three axes of {S}. 

P' is the intersection of line PP' and the Ys-Zs plane, and PP' 

the second or the third column of the corresponding transformation 

matrices of Eq. (4) and Eq. (6). 

From Eq. (6), when 9 = 0°, fTbecomes an identical matrix. 

This implies that no skew transformations are involved, which 

is intuitively correct. When the Xs' axis coincides with the Xs 

axis, i.e., when the mechanical driving system is perpendicular 

to the 2D sensor plane, the coordinate system will become 

orthogonal. Here we limit the parameters to 0 deg s 9 < 90 

deg and 0 deg s \p =s 360 deg. We have no interest in 180 deg 

a 6 a 90 deg because the coordinate frame will be left-handed. 

Combining the nonhomogeneous coordinate transformation 

matrix f'Tand the homogeneous coordinate transformation matrix 

y T, the composition gives 

cached + (casfisy — sacy)s9cip + (casficy + sasy)s9stp caspsy — sacy casficy + sasy qx 

sac0c9 + (sasfisy + cacy)s9cip + (sas/3cy — casy)s9sip sasPsy + cacy sasficy — easy qy 

-sficO + c{3sys9cip + cficys8sip cfisy cficy qz 

is parallel to axis Xs. The y and z coordinates of P, which 

are the lengths of OB and OA, are formed by the perpendicular 

projections of P' onto the Ys (Ys') and Zs (Zs') axes, 

respectively. Coordinate x of P is formed by a parallel projection 

of P onto Xs axis. The line of parallel projection PC is parallel 

to OP'. Thus, the x coordinate of P is formed by a parallel 

projection while the y and z coordinates are formed by perpendicular 

projections. This characteristic is very different from 

that of a normal Cartesian coordinate system, where all the 

point coordinates are formed by perpendicular projections. 

To derive the transformation between the two frames, the 

algebraic relationship between (x, y, z) and (x', y', z') needs 

to be established. We have, 

or in matrix form, 

s T - 

X = x' • sec 9 

• y = y' — x' -tan 9 -cos ip 

. z = z' - x' -tan #-sin ip 

|~ sec 9 0 0 0 1 

-tan 9 cos tp 1 0 0 

-tan 9 sin ip 0 1 0 

0 0 0 1 J 

Note that S S>T\s non-orthonormal. Besides, it is undefined at 

9 = 90 deg which disables the data collection. 

From Eq. (3), the inverse transformation | Teem be obtained 

as 

or in matrix form 

x = x • cos 9 

y' = y + x • sin 9 cos \p 

z' = z + x• sin 9 sin ip 

• cos 9 0 0 0' 

sin 8 cos ip 1 0 0 

sin 9 sin ip 0 1 0 

0 0 0 1 

Note that both f. T and f' T are non-orthonormal. If the Ys or 

Zs axis were skewed, non-orthogonal terms would appear in 

0 0 0 1 

(3) 

(4) 

(5) 

(6) 

which describes the entire laser stripe scanning process. 

3.2 Properties of the Composition Matrix and Identification 

of Extrinsic Parameters. When the angular encoder 

of the pan-tilt device is accurate enough, external calibration 

needs to be done only once when the system is initially setup. 

All the rest of the external re-calibrations could be avoided by 

knowing all the parameters of the position and orientation of 

the sensor. However, if the accuracy of the angular encoder is 

not good enough, the calibration matrix of (7) has to be identified 

whenever a change in sensor position or orientation occurs. 

In this section, the properties of the calibration matrix are explored 

and extrinsic parameters are derived to facilitate onetime 

calibration when system angular encoder is accurate. 

Since the three projections of the skewed sensor coordinate 

system are not all perpendicular projections, six parameters 

which are usually required of any homogeneous coordinate 

transformation can not fully describe the skewed frame transformation 

here. Two extra parameters, 9 and ip, are needed. Therefore, 

the number of extrinsic parameters extends to eight: five 

angular parameters {a, (3, y, 9, tp] and three translational parameters 

{qx, qy, qz]. 

We can also write the overall transformation as 

tn fl2 fl3 Pi 

til hi ?23 Pi 

hx hi ?33 P3 

0 0 0 1 

" R p~ 

0 1 

where R is a composition of rotation, shearing and scaling transformations 

and P = (pi, p2, PT, ) T is the translational vector. The 

orthonormality of a matrix governs that the norms of its three 

column vectors are one and that the three cross products of the 

vectors are zero, which gives six constraints to twelve variables. 

For the multiple-axis laser scanning process, due to the introduction 

of a skewed axis, two of the three orthogonality constraints 

are relaxed and it can be shown that the following four constraints 

exist for the overall transformation fT, 

Journal of Manufacturing Science and Engineering NOVEMBER 1996, Vol. 118/597 

Downloaded 26 Sep 2011 to 141.212.97.74. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm 

(7) 

(8) 

(9)

tu + ?2i + hi 

t\2 + ?22 + ^32 

'l3 1" J 23 T" '33 

?12'?13 + ^22'?23 + ?32 ''33 = 0 

(10) 

of which the top three are normality constraints and the fourth 

one is an orthogonality constraint (governs the orthogonality 

of the sensor Ys and Zs axis). We can also write Eq. (10) in 

matrix form, 

where 

R T -R = 

1 K\ K2 

K, 1 0 

K2 0 1 

" 1 — ?llf 12 + ^21^22 "t~ ^31^32 

K2 = fll^l3 + ^21^23 + ?31^33 

(ID 

(12) 

^ is the inner product of the first and second column vectors 

of the transformation matrix fT, thus represents the non-orthogonality 

of the sensor Xs axis and Ys axis. K2 is the inner product 

of the first and third column vectors of JT, thus represents the 

non-orthogonality of the sensor Xs axis and Zs axis. It can be 

shown that the two skew angles are uniquely determined by Ki 

and K2, 

ip = tan \K2IKX) 

= sin ' (VK? + /c|) 

(13) 

Were the coordinate systems orthogonal, the transformation 

matrix f T would have been orthonormal, which means that we 

should have had R T -R = /. When T is orthonormal, «i and 

K2 approach zero and 9 becomes zero. This makes sense because 

6 is the skew angle of Xs axis. In the meantime, ip turns undefined, 

which is also true when the definition of ip is referred in 

Figure 4. 

To solve for the initial extrinsic parameters of the system, 

equate the corresponding terms of matrix fT in Eqs. (7) and 

(8), twelve equations are found. Solving the 12 simultaneous 

equations, we have 

aa = tan" 1 [(tl3t32 - t^h^lih^ - t23tn)] 

(30 = cos" 1 Mi + '33 

, y„ = tair'tez/fss) 

(14) 

Subsequent angular motions of the pan-tilt device can be measured 

by the angular encoder as Aa, A/?, Ay. Then the current 

roll-pitch-yaw angles become a = a0 + Aa, p = /3a + A/3, y 

= yD + Ay. Knowing these angles, a new transformation matrix 

can be readily calculated from (7), thus saving subsequent 

calibrations. 

4 Sensitivity Analysis of Dimensional Deformation 

The nonhomogeneous coordinate transformation of the laser 

scanning process is not a shape and dimension preserving transformation. 

In this section, sensitivity analysis of deformation 

due to the two skew angles are performed by computer simulation. 

4.1 Euclidean Length Deformation. Given two arbitrary 

points P,(x[, y\, z[) and P2(^2, yk, Z2) in a 3D Cartesian 

space, e.g., the imaginary Cartesian sensor frame {S'}, it is of 

interest to know how the distance between the two changes 

after the nonhomogeneous coordinate transformation. The coordinates 

of the same two points in the true skewed sensor frame 

598 / Vol. 118, NOVEMBER 1996 

{S} can be calculated from Eq. (3) as Pi (xt, yx, Z\) and P2 (x2, 

y2,z2). If the distance between the two points in Cartesian space 

is d\2, after the transformation of Eq. (4), the length deformation 

is 

where 

Adn(6, i},) = U\l + Dl2(d, i//) - d[2 

Dn(0,

Fig. 6 Angular deformation as a function of the skew angles 

Aa(9, i/0 = cos" 1 , 

U\l + D12 Ull + D 

where 

-cos"' (19) 

d[2-d[3 

AC = -tan #-(cos t/j-(dx[2 • dy'n + dx'n • dy[2) 

D,2 = 2 tan 0- dx[2 (tan 9- dx',2 

+ sin tp-{dx\2 • dz[3 + dx'n • dz[2)) 

\ — cos t/j- dy'l2 — sin if/ • dz \2) 

Dn = 2 tan 0-dx[3 (tan 9- dx'n 

— cos (//• dy[3 — sin iff- dz'n) 

. C = dx[2 • dx'n + dy[2 • dy'n + dz[2' dz'n (20) 

The percentage error of angular deformation Ea as a function 

of the two skew angles is shown in Fig. 6. When 9 increases, 

the angular deformation does not always increase. Indeed it 

looks like a "saddle." The "saddle" of angular deformation 

in Fig. 6 is greater than that of length deformation in Fig. 5. 

Aside from the "saddle" effect, angular deformation increases 

almost linearly with the increase of 9. Although the 

deformations do depend on different selections of coordinate 

systems, the overall shape, i.e., the characteristics of the deformation 

surface does not change. 

4.3 Computer Simulation of Shape Deformation. Unlike 

length and angular deformation analysis, shape deformation 

analysis cannot proceed in an explicit way due to difficulties 

of shape representation. Instead, an example of sphere deformation 

is presented by computer simulation. Take a computer generated 

spherical surface and map all the points on the surface 

from a 3D Cartesian space to a skewed non-Cartesian space, 

we want to know how the shape of the sphere changes by 

observing the changes in radius and center locations. For a better 

observation of center location changes, two sphere surfaces are 

generated so that the distance between the centers of the two 

spheres reflects the changes in center locations. 

The mapping of data from a Cartesian space to a skewed 

non-Cartesian space is done by multiplying matrix f. T to all 

the spherical surface points. The deformations of radius and 

center distance are again represented in a percentage error form. 

The angles 9 and ip range from 5 deg to 80 deg. As can be seen 

Fig. 7 Sensitivity of shape deformation reflected by radius change 

Fig. 8 Sensitivity of shape deformation reflected by distance change 

between centroids of two spheres 

from Figs. 7 and 8, both the radius and distance deformations 

are smoother than that of length and angle deformations of Figs. 

5 and 6. Also, the maximum changes of radius (up to 140 times 

larger than original radius) and distance (up to 25 times larger 

than original distance) are much larger than that of length (only 

5 times larger than original length) and angular (only 12 times 

larger than original angle) deformations. This is due to the fact 

that sec 9 and tan 9 terms in S s-T of Eq. (4) go to infinity when 

9 approaches 90 deg. The radius deformation surface in Fig. 7 

has the same pattern as the distance deformation surface in Fig. 

8. When 9 is fixed, both deformations almost linearly increase 

with the increase of tp. When ip is fixed, both deformations 

increase nearly exponentially with increase of 9. This agrees 

with our previous conclusion that 9 has more influence than

Given a commercial laser stripe sensor, it is not only unnecessary 

but also impossible to perform intrinsic calibration. Therefore 

it is impossible to combine the intrinsic and extrinsic calibration. 

The sensor frame is regarded as stationary instead of moving 

when skewed frame representation is employed. The advantage 

of this modeling is that scan data can be pooled as they are 

collected, i.e., no postprocessing of the data is necessary before 

performing extrinsic calibration. This makes it possible to scan 

a target in the sensor field of view and extract desirable feature 

points directly from the scanned data for extrinsic calibration. 

The use of moving frame model prevented Chen and Kak 

(1987) from knowing where exactly the object points are located 

so that they had to scan at least six non-parallel lines on a 

polyhedron target for calibration. We present a new tetrahedrontarget-based 

extrinsic calibration approach without the use of 

pin-hole assumption. Two scan lines are all that are needed to 

calculate the four conjugate pairs, and the two lines can be 

parallel. The new calibration procedure becomes so simple that 

frequent on-line calibration becomes feasible whenever the system 

configuration is altered. 

5.1 Calibration Algorithm Formulation. The same 

point Pt in a 3D Euclidean space can be represented by two 

sets of coordinate vectors, one in the non-Cartesian skewed 

sensor frame, Psi = (xsi, ysi, zst) T and the other in the Cartesian 

world frame, Pwi = (xni,ywi, zwd T • These two coordinate vectors 

are often called "conjugate pairs" because they describe the 

same point in two different coordinate systems. For extrinsic 

calibration, we need to find the matrix JT from n conjugate 

pairs. Then the absolute orientation solution becomes a constrained 

least squares optimization problem (Che, 1995). That 

is, we want to minimize the augmented objective function 

F = I HP* - JT-Psi || 2 + \,[r?, + t 2 2l + t 2 M - 1] 

i=l 

+ X2-+[f?2 + t\2 + t\2 " 1] + \3[t\3 + th + tj3 - 1] 

t22' in + h2' hi ] (21) 

where the \,'s are Lagrange multipliers. Since there are four 

constraints among the twelve unknowns of T, only eight independent 

parameters exist. Therefore a minimum of three conjugate 

pairs are needed as they provide nine coordinates. 

However, this makes the algorithm nonlinear. For on-line 

use, a linear algorithm has to be found to provide an initial 

guess to the nonlinear algorithm. This is achieved by ignoring 

the four constraints in Eq. (21). The objective function then 

becomes 

F = y ii p 

*• A^ II * w 

sT-Psi 

(22) 

and a minimum of four conjugate pairs are needed. 

With four pairs of calibration points, an initial guess of $T 

can be determined by 

scan lines I and n 

(a) schematic of the target (b) actual target 

Fig. 9(a)(6) Tetrahedron target 

where 

and 

Fig. 10 Scanned image of a tetrahedron target 

W 

S = 

P. i 

1 

w-s~ 

Pwl Pw3 

1 1 

Psl Psi 

1 1 

Pw4 

1 

PsA 

1 

(23) 

(24) 

(25) 

If more than four conjugate pairs are used to achieve higher 

calibration accuracy, a least squares linear regression will give 

the initial guess for fT (Horn, 1986) 

where 

and 

W = 

S = 

fT= W-S T -(S-S T y 

PS2 

1 

and n is the number of conjugate pairs. 

(26) 

(27) 

(28) 

5.2 Tetrahedron-target-based Calibration. From the 

calibration algorithm formulation' we know that at least four 

conjugate pairs are needed for a linear initial guess. So the 

calibration target should have at least four vertices to be used 

as the four conjugate pairs. A theodolite-based calibration approach 

was proposed (Dewar, 1988; Greer, 1988; Dewar and 

Greer, 1989). They use a four-wire optical fiber target to create 

the conjugate pairs. However, the accuracy of the theodolite is 

not satisfactory and the procedure is time consuming and involves 

human operator, which makes the automation of on-line 

calibration impossible. We propose to use a tetrahedron as one 

of the possible calibration targets. 

As can be seen from Fig. 9(a), there are four planes on the 

target. Four points A, B, C, and D can be uniquely defined as 

the intersection points of the four planes. The procedure for the 

identification of the four conjugate pairs is as follows: 

(1) Use a mechanical CMM (accuracy should be at least 

one order higher than the laser scanner) to sample several 

points (at least three) on each face of the target. 

(2) Apply a least squares plane fitting algorithm to the 

600 / Vol. 118, NOVEMBER 1996 Transactions of the ASME 


Table 1 Comparative analysis of errors of the angles among the four 

planes of the tetrahedron target before and after calibration (unit: degree) 

True angles measured by 

CMM 

Angle error before 

calibration 

Angle error after 

calibration 

angle_l_2 

angle_l_3 

angle_l_4 

angle_2_3 

angle_2_4 

50.9588 

50.9382 

50.9396 

29.7757 

29.7862 

8.8664 

9.0837 

21.6584 

-3.0446 

13.0731 

0.0025 

0.0016 

0.0084 

-0.0028 

0.0057 

angle_3_4 29.7672 13.3489 0.0045 

sampled data points to obtain the equation for the four 

planes. 

(3) Combine the four plane equations and solve for the 

coordinates of the four vertex points A, B, C and D of 

the target. 

(4) Use a multiple-axis structured-light laser stripe scanning 

system to scan the entire top surface of the target. 

A minimum of two scan lines are needed to cover all 

four surfaces, as shown in Fig. 9(b) by lines I and 

II. However, more scan lines are obtained to reduce 

sampling error. 

(5) Use an image segmentation algorithm to extract data 

clouds for each plane from the entire image into separate 

data files. 

(6) Find the coordinates of points A, B, C and D in the 

skewed sensor frame following procedures (2) and (3). 

For plane fitting on the CMM measurement data, the equation 

of the plane is of the form z = ax + by + c. For skewed 

coordinates of the laser scanner measurement, it can be proved 

that the plane equation is still of the form z = ax + by + c, as 

the nonhomogeneous transformation of a laser scanning process 

is still linear. 

6 Experimental Studies 

A tetrahedron target in Fig. 9(b) is used to validate the 

proposed laser stripe scanning calibration algorithm. The material 

is a high density foam with white paint, which approximates 

a Lambertian surface. The target is first measured on a Sheffield 

RS-30 bridge-type CMM by sampling 14 points on each plane 

surface of the target. After measurement on the CMM, the (x, 

y,z) coordinates of both the vertices and the angles among the 

four planes on the target are known relative to a Cartesian world 

coordinate system (here we use the CMM coordinate system). 

Then, the target is scanned on the integrated multiple-axis 

laser stripe optical CMM when the sensor is rotated around the 

Ys axis about 37 deg. Under this rotation, the scanned image 

of the target is obtained in the non-Cartesian laser sensor coordinate 

system, as shown in Fig. 10. 

From these two sets of data, the coordinate transformation 

matrix ^T can be calculated as 

1.0045 -0.0091 0.6073 595.0213 

0.0050 -1.0160 -0.0148 26.6586 

0.0014 -0.0040 0.7952 -139.7755 

0.0000 0.0000 0.0000 0.0000 

(29) 

Table 2 Comparative analysis of errors of the distances between vertices 

of the tetrahedron before and after calibration (unit: mm) 

True distance measured 

by CMM 

Distance error before 

calibration 

Distance error after 

calibration 

distance_l_2 76.9113 9.6711 0.0073 

distance 1 3 

distance_l_4 

distance 2 3 

distance_2_4 

distance 3 4 

76.9198 

76.8763 

128.0921 

128.0579 

128.0352 

9.3278 

-13.5267 

-2.0328 

-0.9545 

-1.0657 

0.0091 

-0.0040 

0.0067 

0.0056 

0.0057 

Note: distanced j refers to the distance between the i" and the /* vertex of the target 

Journal of Manufacturing Science and Engineering 

Fig. 11 Top portion of the panel 

The nonhomogeneity can be clearly identified by looking at 

the non-orthogonality of the column vectors of JT, or more 

clearly, at the off-diagonal terms of R T -R, 

R T -R 

1.0034 -0.0048 

-0.0048 0.9990 

-0.3419 -6.917x10" 

-0.3419 

-6.917 X 10" 

1.0001 

(30) 

The nonzero inner products of the column vectors in (30) 

demonstrate that the coordinate transformation from a non- 

Cartesian laser sensor frame to a Cartesian frame (the CMM 

coordinate system in this case) is nonhomogeneous and the 

mapping of the whole image to a Cartesian world coordinate 

system is required. 

The nonlinear optimization search program we use is the 

"constr.m" in the Matlab Optimization Toolbox (Math Works, 

1992). The implementation of the Sequential Quadratic Programming 

algorithm used in the constrained optimization routine 

"constr.m" is by updating the Hessian matrix. Usually it 

takes several hundreds function evaluation to converge. The 

iteration termination tolerances for independent variables and 

objective function have been set as 0.001 mm, and the termination 

criterion on constraint violation has also been set at 0.001 

mm. 

Using the obtained matrix, all the data on the target are transformed 

from the skewed coordinate system to the CMM 

Cartesian coordinate system. Now, the real shape of the tetrahedron 

target is recovered, as seen from the comparison of the 

errors from Tables 1 and 2. 

From Table 1, we can see that the angle errors of the four 

planes before the nonhomogeneous coordinate transformation 

is much greater than that obtained after the transformation. 

Fig. 12 Bottom portion of the panel scanned at a different angle 

NOVEMBER 1996, Vol. 118/601 


Fig. 13 Combined image of the panel after the new sensor position and 

orientation is calibrated 

Through calibration, the maximum angular distortion of 

21.6584 degree has been reduced to a maximum distortion of 

0.0084 degrees (or, 30.24 arc second). Also, a comparison of 

errors of the six distances between the four vertices of the target 

before and after the transformation shows another significant 

improvement, as seen from Table 2. Through calibration, dimensional 

distortion of 13.5267 mm has been reduced to 9.1 

microns. So, the significant improvements in the shape and 

relative position representations of the target after the calibration 

shows that the multiple-axis laser-stripe scanning process 

is indeed nonhomogeneous and the calibration is successful. 

We have included a scanned panel in Figs. 11 through 13. 

Figures 11 and 12 show the two scanning passes on the panel, 

each covering one side of it. This is a very typical application 

of the multi-axis system. When the part is large and curvatured, 

it is not possible or optimal to scan the whole part in one pass. 

In this case, we scanned the top portion of the panel first. Then 

we rotated the sensor around the scanning axis by approximately 

30 degrees and scanned the lower portion of the panel to have 

a better view of the steep area. After the rotation, we scanned 

the tetrahedron target and calculated the new sensor position 

and orientation. The scan obtained after the rotation is then 

transformed into the same coordinate system as the first pass. 

Figure 13 shows the merged panel image. The increment between 

lines is 4 mm, and there are altogether 100 lines. Since 

the body panel is not perfectly flat, it is hard to compare any 

dimensions to show the calibration accuracy. However, this has 

been done in Tables 1 and 2 on the more prismatic tetrahedron 

target. 

7 Conclusions 

This paper presents a new modeling scheme that treats sensor 

body frame as stationary so that fast on-line calibration becomes 

possible. It has been demonstrated that a general laser stripe 

scanning process can be modeled as a composition of two transformations, 

one homogeneous and the other nonhomogeneous. 

An analysis of the nonhomogeneous coordinate transformation 

between a Cartesian frame and a skewed non-Cartesian frame 

has been performed. It is concluded that in order for the skewedaxis 

sensor frame to be fully described relative to a Cartesian 

common world coordinate system, eight parameters are needed. 

A unique tetrahedron-target-based calibration method is proposed 

and calibration algorithms discussed. Length, angular and 

shape deformations are studied to see how sensitive they are to 

skewness of the axes. Experimental results have shown that the 

scanning process is indeed nonhomogeneous and the calibration 

is accurate to micron level. 

Acknowledgments 

This research is supported by the National Science Foundation 

Industrial/University Cooperative Research Center for Di 

mensional Measurement and Control in Manufacturing at the 

University of Michigan, Ann Arbor. Fruitful discussions with 

Dr. Yi Zhang and Mr. Dale R. Greer are gratefully acknowledged. 

The authors would like to thank Mr. Dale Simon and 

Dr. Yudong Chen of Perceptron, Inc. for their help in providing 

the sensor, and Mr. Grant Gildner of Modern Engineering for 

constructing the tetrahedron target used in the experiments. 

References 

Agin, G. J., 1985, "Calibration and Use of a Light Stripe Range Sensor 

Mounted on the Hand of a Robot," Proceedings of the IEEE International Conference 

on Robotics and Automation, St. Louis. 

Agin, G. J., and Binford, T. O., 1973, "Computer Description of Curved 

Objects," Proceedings of the 3rd International Joint Conference on Artificial 

Intelligence, Stanford, CA., pp. 629-640. 

Agin, G. J., and Highnam, P. T., 1982, "Movable Light-stripe Sensor for 

Obtaining Three-dimensional Coordinate Measurement," Proceedings of the SPIE 

International Tech. Sym., San Diego, CA., Aug. 21-27, pp. 326-333. 

Aromat Corporation, 1992, "Technical Manual: MQ Laser Analog Sensor." 

Bradley, C, and Vickers, G. W., 1992, "Automated Rapid Prototyping Utilizing 

Laser Scanning and Free-Form Machining," Annals of CIRP, pp. 437-440. 

Champ, P., 1992, "Scanning the Third Dimension," Image Processing, December. 

Chesapeake Laser, 1994, "Chesapeake Laser Systems—SpatialMetrix Newsletter," 

CLSMX, Kennett Square, PA. 

Che, C, 1995, "Multi-axis, Three-dimensional, Structured-light Laser Scanning 

System: Modeling, Calibration, and Measurement Uncertainty Assessment," 

Ph.D. Dissertation, the University of Michigan, Ann Arbor. 

Chen, C. H., and Kak, A. C, 1987, "Modeling and Calibration of a Structured 

Light Scanner for 3-D Robot Vision," Proceedings of the 1987 IEEE International 

Conference on Robotics and Automation, CH2413-3, pp. 807-815, 1987. 

Chen, Y., 1991, "Free Form Curve and Surface Measurement, Modeling and 

Machining," Ph.D. dissertation, the University of Michigan, Ann Arbor. 

Craig, J. J., 1992, Introduction to Robotics: Mechanics and Control, 2nd edition, 

Addison-Wesley. 

CyberOptics, 1992, "CyberScan, Measurement Systems: Non-contact Data Acquisition 

and Measurement," CyberOptics Corporation. 

Dewar, R., 1994, Personal Communication. 

Dewar, R., 1988, "Self-Generated Targets for Spatial Calibration of Structured- 

Light Optical Sectioning Sensors with Respect to an External Coordinate System," 

SME Vision '88 Conference Proceedings, Detroit, MI, June, 1988. 

Dewar, R„ and Greer, D. R., 1989, "Method and Apparatus for Calibrating a 

Non-Contact Gauging Sensor with Respect to an External Coordinate System,'' 

United States Patent, Number 4,841,460. 

Digibotics, 1994, "Automated 4-axis 3D Laser Digitizing," product catalogue, 

Digibotics, Inc., Austin, TX. 

Geometric Research, 1994, preliminary data sheet, Geometric Research Inc., 

Bob Thoreson, (206) 820-0574. 

Goh, K. H., Phillips, N„ and Bell, R„ 1985, "The Applicability of the Laser 

Triangulation Probe to Non-contacting Inspection," International Journal of Production, 

Vol. 24, No. 6, pp. 1331-1348. 

Greer, D. R., 1988, "On-line Machine Vision Sensor Measurements in a Coordinate 

System," SME Vision '88 Conference Proceedings, Detroit, MI, June, 

1988. 

Horn, B. K. P., 1986, Robot Vision, the MIT Press, Cambidge, Massachusetts, 

p. 462. 

Hymarc, 1994, Production Information, Ottawa, Ontario, Canada. 

Keyence, 1993, "LB Series Laser Displacement Sensors," Catalog No. MLB- 

KJ-02, Keyence Corporation, Japan. 

Mansbach, P., 1986, "Calibration of a Camera and Light Source by Fitting to 

a Physical Model," Computer Vision, Graphics, Image Processing, Vol. 35, pp. 

200-219. 

Math Works, 1992, Matlab Optimization Toolbox User's Guide, the Math 

Works, Inc., Natick, Massachusetts, December, 1992. 

Okada Co., Ltd., 1992, "Vigitizer: Non Contact Laser Scanning Digitizing and 

3D Modeling CAD/CAM System." 

Perceptron, 1994a, "Contour Sensor Family," Product catalogue, Perceptron, 

Inc., Farmington Hills, MI. 

Perceptron, 1994b, "TriCam: Non-contact Measurement Solutions," Product 

catalogue, Perceptron, Inc., Farmington Hills, MI. 

Popplestone, R. J., Brown, C. M., Ambler, A. P., and Crawford, G. F., 1975, 

"Forming Models of Plane-and-cylinder Faceted Bodies from Light Stripes," 

Proc. 4th International Joint Conference on Artificial Intelligence, pp. 664-668. 

Sami, 1994, "Smartprox Sensors," Product Catalogue, Sensor Adaptive Machines 

Inc., Windsor, Ont., Canada. 

Saito, K., and Miyoshi, T., 1991, "Non-contact 3-D Digitizing and Machining 

System for Free-form Surfaces," Annals of the CIRP, Vol. 40/1, pp. 483-486. 

Selcom, 1993, "OPTOCATOR, the Rapid, Accurate, Non-contact Measurement 

System," Selective Electronic, Inc. product catalog. 

Theodoracatos, V. E„ and Calkins, D. E., 1993, "A 3-D Vision System Model 

for Automatic Object Surface Sensing," International Journal of Computer Vision, 

11:1, pp. 75-99. 

3D Technology, 1994, "Optica: Non-contact Laser Based Scanners for CNC/ 

CMM Machines," 3D Technology, Inc., Trumbull, CT. 

602 / Vol. 118, NOVEMBER 1996 Transactions of the ASME 


Venture Laser Technologies, Inc., 1993, "3D Laser Digitizing Systems, Software, 

and Services." 

Wang, C. C, 1992, "Extrinsic Calibration of a Vision Sensor Mounted on a 

Robot," IEEE Journal of Robotics and Automation, Vol. 8, No. 2, April, pp. 

161-175, 1992. 

Wang, W. W., 1991, "Accuracy Analysis of an In-line Optical Coordinate 

Journal of Manufacturing Science and Engineering 

Measuring Machine (IOCMM)," Ph.D. dissertation, the University of Michigan. 

Will, P. M., and Pennington, K. S., 1971, "Grid Coding: Preprocessing Technique 

for Robot and Machine Vision," Artificial Intelligence, Vol. 2, No. 3, pp. 

319-329. 

Will, P. M., and Pennington, K. S., 1972, "Grid Coding: Novel Technique for 

Image Processing," Proceedings of IEEE, Vol. 60, No. 6, pp. 669-680.

Modeling and Calibration of a Structured-light Optical - Michigan ...

Create successful ePaper yourself

Delete template?

Save as template?