Architectural stability analysis of the rotary-laser scanning technique

Optics and Lasers in Engineering 78 (2016) 26–34

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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng

Architectural stability analysis of the rotary-laser scanning technique Bin Xue a,1, Xiaoxia Yang b,n,1, Jigui Zhu c a b c

School of Marine Science and Technology, Tianjin University, Tianjin 300072, China Tianjin Key Laboratory of Information Sensing & Intelligent Control, Tianjin University of Technology and Education, Tianjin 300222, China State Key Laboratory of Precision Measurement Technology and Instruments, Tianjin University, Tianjin 300072, China

art ic l e i nf o

a b s t r a c t

Article history: Received 29 March 2015 Received in revised form 9 September 2015 Accepted 9 September 2015

The rotary-laser scanning technique is an important method in scale measurements due to its high accuracy and large measurement range. This paper first introduces a newly designed measurement station which is able to provide two-dimensional measurement information including the azimuth and elevation by using the rotary-laser scanning technique, then presents the architectural stability analysis of this technique by detailed theoretical derivations. Based on the designed station, a validation using both experiment and simulation is presented in order to verify the analytic conclusion. The results show that the architectural stability of the rotary-laser scanning technique is only affected by the two scanning angles' difference. And the difference which brings the best architectural stability can be calculated by using pre-calibrated parameters of the two laser planes. This research gives us an insight into the rotarylaser scanning technique. Moreover, the measurement accuracy of the rotary-laser scanning technique can be further improved based on the results of the study. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Rotary-laser scanning Azimuth Elevation Measurement stability

1. Introduction For geometric and positioning information monitoring and measurement in large-scale aircraft docking, shipbuilding and manufacturing, the rotary-laser scanning technology is a useful and an indispensable measurement technique for the good performance of lasers including the wavelength stabilization, strong coherence and well directional property [1–4]. Lots of experts develop such scanners using similar principles. Lopez et al. have developed scanners based on the novel method of precise measurement of plane spatial angles applying to geometrical monitoring to predict their structural health during their lifetime [5]. Básaca-Preciado et al. designed an optical 3D laser measurement system for navigation of autonomous mobile robot, which is capable of performing obstacle avoiding task on an unknown environment [6]. Rodríguez-Quiñonez et al. presented a 3D medical laser scanner based on the novel principle of dynamic triangulation, and they analyzed the method of operation, medical applications, orthopedically diseases as Scoliosis and the most common types of skin to employ the system the most proper way [7]. Guan et al. use mobile laser scanning data for automated extraction of road markings and develop relevant algorithms [8]. Lots of other

n

Corresponding author. Tel.: þ 86 13820085223. E-mail address: [email protected] (X. Yang). 1 These authors contributed equally to this work.

http://dx.doi.org/10.1016/j.optlaseng.2015.09.005 0143-8166/& 2015 Elsevier Ltd. All rights reserved.

experts also make achievements in this area, which are unable to detail here [9–17]. The rotary-laser scanning technique of this kind can provide very good performance like high accuracy, large measurement range, high usability and high efficiency, which extremely meet the demand of measurement applications at the scale of tens of meters [18–22]. A single station based on the rotary-laser scanning technique is designed and presented in this paper. As an important component of a large-scale 3D measurement network, the single station’s characteristic determines the performance of the whole network. So, precisely mastering the characteristic of this rotary-laser scanning technique is an important work both in theory and practice. In recent years, some researchers have done a lot of work on studying the characteristic of a single station based on the rotarylaser scanning technique. For instance: Muelaner demonstrated how the angular uncertainties can be determined for a rotary-laser automatic theodolite of the type used in (indoor-GPS) iGPS networks [23]; Xiong used the similar method as Muelaner to find the angle-measuring ability of a single transmitter of the (workspace Measuring and Positioning System) wMPS [24,25], and so on. However, the above work just focused on the accuracy of the observed angles by testing procedures, not profoundly analyzed the construction feature of this technique using a pure and rigorous mathematical derivation, nor did they give precise instructions on how to utilize the feature to get the best performance in practical applications. In this paper, a rigorous perturbation

B. Xue et al. / Optics and Lasers in Engineering 78 (2016) 26–34

analysis on the matrix equation abstracted from the rotary-laser scanning technique is first presented which gives a geometrically concise result, then, Monte Carlo simulations combined with some experimental results are illustrated which are used to verify the theoretical conclusion. As an instructive result, we find that the architectural stability remains the same in the circle direction around the single station, whereas it changes a lot with the change of the elevation angle. Moreover, the elevation is directly determined by the two scanning angles’ difference, which is gotten from the two laser fans’ continuously strikes on the receiver (the photoelectric sensor used to receive the laser signal). So, we have that the volume with the optimum architectural stability around the station can be obtained by detecting the difference of the two scanning angles. If it meets the value which is calculated using the pre-calibrated parameters of the laser planes, we get the optimum measurement volume. Obviously, using the volume with the optimum stability to implement some monitoring or measurement work is the best solution we should choose. Furthermore, our analysis reveals the internal architecture of the rotary-laser scanning technique. A good understanding on the internal architecture will be helpful in tapping the accuracy potential of this technique in the large-scale measurement networks. We divide this paper into the following sections: Section 2 describes the measurement model of the rotary-laser scanning station and demonstrates the characteristic of the station from a rigorous mathematical point of view. Section 3 designs the validation procedures whose results noticeably meet our derived result. In Section 4, we present concluding remarks and a brief overview of future improvements.

2. Measurement model of the station based on rotary-laser scanning technique

27

As shown in Fig. 2(a), the initial time is defined as the time when the head of station rotates to a predefined position (initial position) and the pulsed lasers emit the synchronous laser strobe. At the initial time, the receiver captures the synchronous laser strobe and records the time as t 0 . Rotating with the head, the two laser planes scan the measurement space around the station. As shown in Fig. 2(b), when laser plane 1 sweeps past the receiver, the time is recorded as t 1 . Assuming that the angular velocity of the rotating head is ω, then the scanning angle θ1 of laser plane 1 from the initial position to the position where it passes through the receiver can be obtained:

θ1 ¼ ωðt 1  t 0 Þ

ð1Þ

In a similar way, to laser plane 2:

θ2 ¼ ωðt 2  t 0 Þ

ð2Þ

For the problem that the laser plane's thickness becomes wider as it travels longer, like a cone which may deteriorate the measurement accuracy, we give explanations as follows: This is a phenomenon that exists objectively. It can deteriorate the measurement accuracy especially at a distance around 10 m. At present, we resolve this problem through two methods. One is to select a laser with good parameters which has less than 3 mm width at 10 m distance, the other is to detect the energy center of the spot at the receiver using some method like in references [26,27]. However, what we want to say is that the topic we talk about in this paper is the architectural stability of the laser scanning technique we introduced above. The slight cone-like property of lasers affects the measurement accuracy, but not significantly affects the architectural stability as the 3 mm width at 10 m has not much impact on our topic. So, in this paper, we assume that the laser performs ideally as a plane.

2.1. General introduction of the station 2.2. The construction of the line equation The station can be treated as a spatial angle measurement system. The components of the system and construction of the station are shown in Fig. 1. As illustrated in Fig. 1, the rotary-laser station consists of a rotating head and a stationary base. Two line laser modules are fixed on the rotating head, which can emit planar laser beams with different tilt angles. Several pulsed lasers are mounted around the stationary base. With the rotating head spinning at a predefined speed in anti-clockwise direction, the station generates three optical signals: two nonparallel planar laser fans rotating with the head of the station and a cyclical omnidirectional laser strobe emitted by the pulsed lasers synchronously every time the head rotates to the predefined position of every cycle. The pulsed laser works at the frequency around 2000 r/min. This is a working frequency not requiring much on the rotor structure. The photoelectric receiver captures the three optical signals and then converts them into electrical signals through a photoelectrical sensor. The signal processor is used to binarize the electrical signals into logic pulses, differentiate the logic pulses and record the time information of the laser planes.

Before going any further, we define the coordinate frame of the station. The z-axis is the rotation axis. Origin is the intersection of the rotation axis and laser plane 1 at the initial position. The x-axis through the origin is on plane 1 and perpendicular to z-axis. The yaxis is determined according to the right-hand rule, see Fig. 3. Therefore, the rotary-laser station can be treated as two nonparallel half-planes rotating around axis Z and an omnidirectional pointolite emitting laser pulse train with a fixed-frequency at the origin O. As described in Section 2.1, we can obtain the scanning angles of the rotary-laser planes using a single station and a receiver ðP i Þ. Benefiting from the tilted structure of the two line laser models, we can obtain the elevation information as illustrated in Fig. 3(b). To better explain the elevation measurement principle from the slope of the two laser planes, we mathematically describe the two

ω

ω

Laser plane 1

Laser plane 1

Laser plane2 Receiver

Receiver

Rotating shaft Rotating head Pulsed lasers

Synchronous laser strobe

Receiver

Laser plane1 Station

Terminal Signal computer processor

Fig. 1. Spatial angle measurement system.

Initial position

Station t = t0

θ1

Initial position

Station t = t1

Fig. 2. Schematic of the scanning angle measurement: (a) the initial time and (b) the time when laser plane 1 sweeps past the receiver.

28

planes as follows, 8 > < a11 x þa12 y þ a13 z ¼ 0 > : a x þa y þ a z ¼  Δd 21 22 23

B. Xue et al. / Optics and Lasers in Engineering 78 (2016) 26–34

ð3Þ

As we only take the intersecting line direction or the azimuth and elevation into consideration, omit the Δd and transform this expression of line into the follow one, x y z ¼ ¼ ð4Þ a22 a13  a12 a23 a21 a13  a11 a23 a11 a22  a12 a21  T  T In Eq. (3), the vectors a11 a12 a13 and a21 a22 a23 represent the normal directions of the two planes, respectively, further, the a13 and a23 represent  the tilting levels of those two planes. In Eq. (4), the vectors a22 a13  a12 a23 a21 a13  a11 a23 a11 a22  a12 a21 ÞT represent the intersecting line direction of the two planes including the azimuth and elevation. Then we know that if the a13 and a23 equal zero, which means the two planes have no  T tilt, the vector becomes 0 0 a11 a22 a12 a21 , a constant vertical vector with no tilt, too, which means the intersection of the two planes cannot reflect the elevation information. So, as a result, we must keep the two planes tilting in the measurement process. As the robustness with respect to the difference of inclination of the planes, we qualitatively illustrate it in Fig. 4. In Fig. 4, n1 and n2 are the normal vectors of the two planes, respectively, we only use the normal vectors to represent the planes for simplification. From Fig. 4(a), we see that if the inclinations of the two planes are large, the angle between their normal vectors are small, see angle α. Then, no matter how to rotate the two vectors around axis Z, the angle α always keep small, and make the intersection of the two planes less robust. From Fig. 4(b), we see the opposite situation. The angle α is a large angle when the two planes' inclinations are relatively small compared to the former situation. In this situation, we can change the intersection angle from large to small through rotating the two vectors around axis Z to proper positions. So, different inclinations give different

robustness. In practical applications, we should keep the inclinations relatively small in order to give more robustness by selecting the proper rotating angles, which will be depicted in the rest of paper. Looking back to Fig. 3(b), we see that the two laser fans do not always intersect at the origin of the local frame on the station for the reason of assembly error, see Δd. To rigorously express the measurement information obtained from the single station, we use the system of equations as follows: 0 10 1 8 x cos θ1  sin θ1 0 > >   > CB C > > n11 n12 n13 B cos θ1 0 A@ y A ¼ 0 @ sin θ1 > > > > < z 0 0 1 0 10 1 ð5Þ x cos θ2  sin θ2 0 > > > B >n C B C > n22 n23 @ sin θ2 > cos θ2 0 A@ y A ¼  Δ d 21 > > > : z 0 0 1   where ni1 ni2 ni3 ¼ ni ; i ¼ 1; 2 denote the unit normal vectors of the two laser planes at the initial time, respectively, Δd is the deviation between the two laser planes along axis Z, these parameters are pre-calibrated. The two 3  3 matrices denote the rotations of each plane around axis Z. To facilitate the discussion, we change Eq. (5) to the following matrix form: Ax ¼ d ( 3

a11 x þ a12 yþ a13 z ¼ 0 a21 x þ a22 yþ a23 z ¼  Δd

where 8 a ¼ n11 cos θ1  n12 > > > 11 > > a12 ¼ n11 sin θ1 þ n12 > > > > < a13 ¼ n13 > a ¼ n cos θ2  n22 21 > 21 > > > > a ¼ n sin θ 21 2 þ n22 > 22 > > : a ¼n 23

ð6Þ

sin θ1

cos θ1 sin θ2

ð7Þ

cos θ2

23

 T construct the matrix A; vector x denotes x y z , and vector d  T denotes 0  Δd . Actually, Eq. (6) constructs a line. And that is the accurate description. 2.3. The analysis of the line equation Based on the solution structure of linear equation group, we know that the general solution of an underdetermined system of non-homogeneous linear equations consists of a system of basic solutions and a particular solution [28]. Fig. 3. Line-determining model: (a) the initial position of the two laser planes and (b) the position of the two laser planes at the time when they pass through the receiver point P i successively.

2.3.1. The stability of the system of basic solutions In order to analyze the architectural stability, we first analyze the stability of the system of basic solutions of Eq. (6), i.e., the solutions of the following system of homogeneous linear equations Ax ¼ 0

Fig. 4. Different inclinations of the planes: (a) inclination is large (the normal vector inclination is small) and (b) inclination is small (the normal vector inclination is large).

ð8Þ

where A is a 2  3 matrix with the rank equaling two, which is  T is the smaller than the number of columns three; x ¼ x y z solution we are searching for. Obviously, there are infinitely many solutions for Eq. (8), and in geometry, they construct a line. Now, we try to find the appropriate θ1 and θ2 which lie in the  T matrix A to make the solution x ¼ x y z , or the line, has the best stability. Let us decompose matrix A using the (singular value decomposition) SVD principle [29]: A ¼ U ΛVT

B. Xue et al. / Optics and Lasers in Engineering 78 (2016) 26–34



¼ u1

u2

 σ1 0

0

σ2

! 0  v1 0

v2

v3

T

¼ σ 1 u1 vT1 þ σ 2 u2 vT2 þ0vT3

ð9Þ

where U is a 2  2 orthogonal matrix, Λ is a 2  3 matrix, and V is a 3  3 orthogonal matrix; ui ; i ¼ 1; 2 are the column vectors of U, vi ; i ¼ 1; 2; 3 are the column vectors of V, and σ i ; i ¼ 1; 2 are the singular values of matrix A. Substituting the right side of Eq. (9) for the matrix A of Eq. (8) yields: ðσ 1 u1 vT1 þ σ 2 u2 vT2 þ 0vT3 Þx ¼ 0

ð10Þ

As vi ; i ¼ 1; 2; 3 are orthogonal to one another, one can easily get that x ¼ tv3 is the solution space of Eq. (8), where t A ℝ is a variable and v3 is the rightest singular value vector of A. To verify this, we substitute tv3 for x in Eq. (10): ðσ 1 u1 vT1 þ σ 2 u2 vT2 þ 0vT3 Þtv3

The perturbation level is mainly decided by the stability of the rotating head of the measurement station, which is guaranteed by a good mechanical structure that is beyond our discussion scope. Here, one obtains that the larger σ 22 is, the more stabilized v3 will be. Also, one easily verifies from Eqs. (6) and (5) that the two rows of the 2  3 matrix A are both unit vectors that satisfy 0 1 ai1  B ai1 ai2 ai3 @ ai2 C A ai3    T ¼ ni1 ni2 ni3 R θi R Tθi ni1 ni2 ni3 0 1 ni1  B C ¼ ni1 ni2 ni3 @ ni2 A ni3 ¼ 1; i ¼ 1; 2 where 0

¼ σ 1 tu1 vT1 v3 þ σ 2 tu2 vT2 v3 þ 0tvT3 v3 ¼0

ð11Þ

where ðvi ; vj Þ ¼ vTj vi ¼ δij . Here the Kronecker's symbol δij is used for a concise representation. Now, we know that the system of basic solutions of Eq. (8) is the rightest singular value vector of A. To facilitate the further discussion, we use the symmetric matrix AT A, which can also give the vector v3 for the following reason:

ð15Þ

cos θi

 sin θi

0

0

0

1

B Rθi ¼ @ sin θi

cos θi

i ¼ 1;2

¼

T

v2

0

0

0

1

0

X

0 1 ai1 B ai3 @ ai2 C A ai3

ai1

ai2

i ¼ 1;2



σ 22 0 C A v1 v2 v3

T

¼2

ð12Þ

0

where we utilized A ¼ U ΛV in Eq. (9). Hence, we know that the system of basic solutions can be obtained by eigenvalue decomposing the symmetric matrix AT A and finding the eigenvector corresponding to the eigenvalue zero. To find the condition which brings the best stability to the eigenvector v3 , we should introduce a definition and a theorem in advance. Definition. Let the eigenvalues of a symmetric matrix A A ℝnn be λ1 Z λ2 Z U U U Z λn , then the gap between λi and the rest eigenvalues is defined by   gapðλi ; AÞ ¼ minλi  λj  ð13Þ T

iaj

^ Q^ T be the eigenvalue Theorem 1. Let A ¼ Q ΛQ T and A þ E ¼ Q^ Λ decompositions of symmetric matrices A A ℝnn and A þE A ℝnn ,   and Q^ ¼ respectively, where Q ¼ q1 q2 U U U qn   q^ 1 q^ 2 U U U q^ n are both orthogonal matrices, and the eigenvector q^ i is the perturbed eigenvector qi . Use θi to denote the acute angle between qi and q^ i , then the following inequation 1 ‖E‖2 sin ð2θi Þ r ð14Þ 2 gapðλi ; AÞ holds if gapðλi ; AÞ a0[30]. In Theorem 1, the acute angle θi between a vector and a perturbed vector is used to characterize the stability of that vector. Certainly, the smaller the angle θi is, the more stabilized the vector will be. From Theorem 1 and Eq. (12), we know that the stability of v3 is mainly determined by gap ð0; AT AÞ, i.e., the σ 22 , under a relatively fixed perturbation level ‖E‖2 compared to the element gap ðλi ; AÞ.

ð16Þ

¼ TraceðAAT Þ X ða2i1 þ a2i2 þ a2i3 Þ ¼

T

 ¼ v1

C 0 A; i ¼ 1; 2

TraceðAT AÞ

¼ V Λ UT U Λ V T 0 2 σ B 1 v3 @ 0

1

So, we have

AT A ¼ ðU ΛVT ÞT U ΛVT ¼ VΛ ΛVT

29

ð17Þ

and TraceðAT AÞ ¼ TraceðVΛ ΛVT Þ T

¼ TraceðVT VΛ ΛÞ T

¼ TraceðΛ ΛÞ 00 2 T

σ1

BB ¼ Trace@@ 0 0 ¼σ

2 1þ

σ

0

0

0

0

11

C σ 22 0 C AA

2 2

ð18Þ

where Eq. (12) is recalled, and the cyclic reordering property of trace operator is used in both Eqs. (17) and (18). Hence, from the results of Eqs. (17) and (18), we obtain

σ 21 þ σ 22 ¼ 2

ð19Þ

Based on this derived result, our problem to find the largest σ 22 becomes a constraint optimization problem which can be depicted as seeking max ½ minðσ 21 ; σ 22 Þ

ð20Þ

subject to

σ 21 þ σ 22 ¼ 2

ð190 Þ

Obviously, the only solution is

σ ¼ σ 22 ¼ 1 2 1

ð21Þ T

Next, we derive the expression of AA using Eq. (9) as follows: AA ¼ U ΛVT ðU ΛVT ÞT T

¼ U Λ V T V Λ UT T

¼ U ΛΛ U T

T

30

B. Xue et al. / Optics and Lasers in Engineering 78 (2016) 26–34



¼ u1

u2

 σ 21 0

0

σ 22

! 

u2

u1

T

ð22Þ

Plugging σ 21 ¼ σ 22 ¼ 1 into Eq. (22), we have !   σ 21 0   u1 u2 T AAT ¼ u1 u2 0 σ 22     1 0   u1 u2 T ¼ u1 u2 0 1   T ¼ u1 u2 u1 u2   1 0 ¼ 0 1

One easily finds that the solution may be the following two expressions:

π θ2  θ1 ¼ þ 2kπ  φ

B ¼B @

i ¼ 1;2;3j ¼ 1;2

ji

i ¼ 1;2;3

P

1

ð23Þ

P

2i

∏ aji

i ¼ 1;2;3j ¼ 1;2

∏ aji

1

1 C C A

ð24Þ

i ¼ 1;2;3j ¼ 1;2

Comparing Eq. (24) with (23), we have the central result of this paper X ∏ aji ¼ 0 ð25Þ i ¼ 1;2;3j ¼ 1;2



Then, from Eqs. (6) and (5), we know it is equivalent to    n21 n22 n23 Rθ2 R Tθ n11 n12 n13 T 1    T ¼ n21 n22 n23 R θ2  θ1 n11 n12 n13 ¼0

ð26Þ

where 0

cos θi B R θi ¼ @ sin θi 0 and

0

 sin θi

0

0

1

1

C 0 A; i ¼ 1; 2

cos θi

ð16Þ

 sin ðθ2  θ1 Þ

0

0

0

1

cos ðθ2  θ1 Þ

C 0 A; i ¼ 1; 2

ð17Þ

where N 1 ¼ n12 n21  n11 n22 N 2 ¼ n11 n21 þ n12 n22

ð19Þ

N 3 ¼  n13 n23 2 tan φ ¼ N N1

Then, the optimum result can be expressed as 0 1 C 3 θ2  θ1 ¼ arcsinB @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA  φ N

N 21 þ N 22

0

θ2  θ 1 ¼

3π þ 2kπ  φ 2

ð24Þ

where k A Z. The value of k is determined by the actual geometric meaning. Now, we obtain the sectional conclusion that the system of basic solutions of Eq. (6) has the best stability if the two rows of the coefficient matrix A are perpendicular to each other. Next, for the other part of the architectural stability, this condition will be checked that whether it also brings the best stability to the particular solution of Eq. (6). 2.3.2. The stability of the particular solution Theorem 2. The unique minimal norm solution of a system of linear equations Ax ¼ d can be written as x ¼ A þ d where A þ is the MoorePenrose generalized inverse matrix of A [29]. Accordingly, we use x ¼ A þ d as the particular solution of Eq. (6). The geometric meaning of this particular solution refers to the line's slight deviation off the origin of the coordinate frame, which is caused by the assembly error of the two line lasers. From x ¼ A þ d, we know that in addition to d, which is controlled by the mechanical structure, the stability of x is dependent on the stability of A þ . Moreover, the stability of A þ can be measured by the condition number of A, which is known as

ð20Þ

A ¼ U ΛVT  S ¼U 0

0 0

ð21Þ

þ

can be calculated by the SVD of A

 VT

ð26Þ

where S is a diagonal matrix with the singular values arranged on the diagonal line, then, ! S1 0 ð27Þ UT Aþ ¼ V 0 0 Hence, using Eq. (9), we get 0 1 1 σ1 0 B C 1 C T A þ ¼ VB @ 0 σ 2 AU 0 0

ð28Þ

Then, by the definition that the maximum of all the singular values of a matrix is defined as the 2-norm of that matrix [29], we obtain   1 1 1 ; ð29Þ ¼ ‖A þ ‖2 ¼ max

σ1 σ2

1

C θ2  θ1 ¼ arcsinB @N 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA  φ þ 2π 2 2 N 1 þN 2

ð25Þ

As an existing knowledge, A [29], that is, if

1

In Eq. (26), θ2  θ1 is the only unknown, and can be expressed as follows after a series of trigonometric manipulations: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N 21 þ N22 sin ðθ2  θ1 þ φÞ  N 3 ¼ 0 ð18Þ

or

or

condðAÞ ¼ ‖A þ ‖2 ‖A‖2

cos ðθ2  θ1 Þ

B R θ2  θ1 ¼ @ sin ðθ2  θ1 Þ

ð23Þ

2

However, the actual expression of AAT should be as follows: 0 1 P 2 P a1i ∏ aji i ¼ 1;2;3j ¼ 1;2 B i ¼ 1;2;3 C P 2 C AAT ¼ B @ P ∏ a A a 0

However, in some cases,ffi Eq. (28) is insoluble under the conqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dition that jN 3 j 4 N 21 þ N22 . Hence, the closest solution to the optimum one can be found by seeking for the minimal value of the following equation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð22Þ j N 21 þ N22 sin ðθ2  θ1 þ φÞ  N3 j

σ2

Similarly, ‖A‖2 can be written as ‖A‖2 ¼ maxðσ 1 ; σ 2 Þ ¼ σ 1

ð30Þ

B. Xue et al. / Optics and Lasers in Engineering 78 (2016) 26–34

31

created as follows:

Hence,

σ1 condðAÞ ¼ ‖A þ ‖2 ‖A‖2 ¼ σ2

ð31Þ

As what we have known that the smaller the condition number is, the more stability we will get, then, the problem can be expressed as finding   σ1 ð32Þ minðcondðAÞÞ ¼ min

σ2

subject to

σ 21 þ σ 22 ¼ 2 and σ 1 Z σ 2

n1 Rθ1i ðR c_s ½ xi

: n2 R θ2i ðR c_s ½ xi

ð340 Þ

is the solution, which meets our previous result. Then, benefiting from the same derivation from Eq. (19)–(34), we conclude that the two rows of the coefficient matrix A being perpendicular to each other also gives the best stability to the particular solution of matrix Eq. (6). Furthermore, the stability is only affected by θ2  θ1 , which has a concise geometric explanation, that is, the scanning angles' difference between the two laser fans striking the receiver. In addition, we get an obvious corollary that the more θ2  θ1 is off the best angle, the less stabilized it will be. That will be checked by the validation procedures in Section 3.

3. Validation

yi yi

zi T þ Tc_s Þ ¼ 0 zi T þTc_s Þ ¼  Δd

; i ¼ 1; 2:::

ð35Þ

where ½ xi yi zi T is the ith coordinate of the control points; R θ1i is the rotational matrix constructed by the plane 1's observed angle θ1i , which represents the rotation around the Z-axis of the frame of the station, and can be denoted as 0

ð33Þ

One easily verifies that

σ 21 ¼ σ 22 ¼ 1

8 <

cos θ1i B Rθ1i ¼ @ sin θ1i 0

 sin θ1i

0

0

1

1

C 0 A; i ¼ 1; 2:::

cos θ1i

ð36Þ

Rθ2i has the similar meaning, which is constructed by the plane 2's observed angle θ2i ; Rc_s and Tc_s are the transformation from the frame of the control points to that of the station, and n1 and n2 are the unit normal vectors of laser plane 1 and plane 2, respectively. Here, Rc_s ,Tc_s , n1 , n2 and Δd are the unknowns to be solved. Given the nonlinear problem, the bundle adjustment algorithm [31] is often used to solve this kind of calibration problem. By several consecutive calibrations, a set of the values of n1 , n2 and Δd can be obtained, which have slight perturbations from one another. Through estimating the variance of them, we can approximately get the perturbation level of these parameters. Moreover, for a relatively more precise validation, we can get the perturbation level of the observed angles by continuingly measuring a fixed receiver and estimating the variances of them.

3.1. The determination of perturbation level To validate the analysis in Section 2, we should first determine the perturbation level of the matrix A by experiment. As foreshadows, in a nutshell, we introduce the calibration method by which the parameters of the two laser planes of a station can be obtained. It is illustrated in Fig. 5. As shown, the control points are fixed on the wall and the ceiling of a room, the station is placed at a fixed spot where all the control points can be swept over. The coordinates of those control points are known in a coordinate frame. By collecting an amount of observed angles which are gained by the laser planes sweeping over the control points, a system of nonlinear equations can be

3.2. The validation for the system of basic solutions 3.2.1. The evaluation of the stability of a line The range of the acute angles caused by perturbations on a vector is used to characterize the stability of this vector here. For a clearer demonstration, we use the distance deviation l illustrated in Fig. 6 to represent the angle. In Fig. 6, the acute angle θa is a measure of the actual vector off the nominal vector caused by perturbation. Here, using 10,000 mm to multiply the angle as the evaluation is based on the reason that the station is often used to work at the scale of meters. This helps to produce an intuitionistic observation which more approaches the actual error scale.

om en Th

100

0 0m

in a lv

m

ec

tor

The distance deviation l

The acute angle θa d ≈ θa×10000mm Fig. 5. The calibration method for the parameters of the two laser planes.

Fig. 6. The evaluation of the stability of a line.

32

B. Xue et al. / Optics and Lasers in Engineering 78 (2016) 26–34

3.2.2. Validation by simulation For clarity, we rewrite Eq. (8) in the form of Eq. (46) 0 Bn B 11 B B B B B B B n21 @

n12

n22

0 cos θ1 B n13 @ sin θ1 0 0 cos θ2  n23 B @ sin θ2

 sin θ1 cos θ1 0  sin θ2 cos θ2

0

0

0

11

C 0AC C0 1 C x 1 C B C 1C C@ y A ¼ 0 0 C CC z 0AC A 1

ð37Þ

The perturbation levels of the parameters including n11 , n12 , n13 , n21 , n22 , n23 , Δd, θ1 and θ2 are tabulated in Table 1, which are acquired using the method in Section 3.1. In Table 1, the parameter n11 is always defined to be zero for the rule we used to construct the local frame of station which is mentioned in Section 2.2. Using Eq. (33) and the nominal parameters in Table 1, we obtain that θ2  θ1 equals 89:7 3 , which gives the best stability. As derived in Section 2, we know the stability of the system of basic solutions is not affected by the azimuth, so we randomly fix θ1 at 60°, and let θ2  θ1 change from  180° to 0° with the step length  1°. The results are shown in Fig. 7. The vertical axis of Fig. 7 represents the stability, for instance, 0.36 denotes a limit deviation of 0.36 mm off the nominal value. The horizontal axis indicates the varying of θ2  θ1 . As indicated, the best stability appears around  90°, which meets our calculation 89:7 3 . Obviously, the angle 2π þ θ2  θ1 ¼ 360 3  89:7 3 ¼ 270:3 3 also brings the best stability. Which one to choose depends on the actual geometric meaning. But it is not the area of our concern in this paper. Table 1 The parameters of laser plane 1 and plane 2, and their perturbation levels. Parameters

Nominal values

Standard deviation

n11 n12 n13 n21 n22 n23 Δd (mm) θ1 (″) θ2 (″)

0.00000 0.744553 0.667563  0.580521 0.002964  0.814240 0.3647 N/A N/A

0 1.03E-5 1.02E-5 1.19E-5 0.376E-5 1.08E-5 0.0448 1.07 1.13

Discretely setting θ1 to be 60 3 i; i ¼ 1; 2; :::; 6, and using the Monte Carlo method as what we used to get Fig. 7, we obtain the three dimensional image as shown in Fig. 8. Comparing with Fig. 7, Fig. 8 adds another axis which describes the varying of azimuth θ1 or θ2 . As shown in Fig. 8, the stability little changes with the varying of azimuth, but noticeably changes with θ2  θ1 which reflects the elevation information. This result is also completely consistent with our derivation. 3.3. The validation for the particular solution For the system of basic solutions, i.e., the line, a slight perturbation on the laser planes' parameters will be noticeably amplified as the increase of the measurement scale. But the particular solution expressed by x ¼ A þ d, whose geometric meaning is the distance between the origin and the line, does not have the above amplification effect. So, we can directly use the variance of ‖x‖2 to evaluate the stability. For clarify, we rewrite the expression x ¼ A þ d as the form of Eq. (47). 0 1 1þ 0 cos θ1  sin θ1 0 B C Bn cos θ1 0AC B 11 n12 n13 @ sin θ1 C B C   B 0 0 1 C 0 B 0 1C x¼B C cos θ2  sin θ2 0 C B  Δd B  CC B n21 n22 n23 B cos θ2 0AC @ sin θ2 @ A 0 0 1 ð38Þ Plugging the parameters and the perturbation levels in Table 1 into Eq. (47), and using Monte Carlo method, we get the results shown in Fig. 9. The same to Fig. 7, the best stability also appears around  90°, which fairly meets the calculation. For a three dimensional demonstration with a varying θ1 or θ2 added, we get the results shown in Fig. 10.

Fig. 8. Three dimensional results of the stability of the system of basic solutions with a varying azimuth included.

Fig. 7. The stability of the system of basic solutions v.s. the scanning angles’ difference.

Fig. 9. The stability of the particular solution v.s. the scanning angles’ difference.

B. Xue et al. / Optics and Lasers in Engineering 78 (2016) 26–34

Fig. 10. Three dimensional results of the stability of the particular solution with a varying azimuth included.

The results shown in Fig. 10 indicate a same conclusion to the one of the system of basic solutions that the azimuth has no effect on the stability, but the value θ2  θ1 markedly affects the stability of the lines determined by the station. Moreover, the θ2  θ1 can be pre-calculated using Eqs. (28)–(34). Now, from both derivation and validation, we conclude that the equation abstracted from the rotary-laser scanning technique can give the best stability on both the system of basic solutions and the particular solution by finding the value θ2  θ1 which is given by the pre-calibrated parameters of the two laser planes. In other words, the found θ2  θ1 brings the best stability to the system of basic solutions and the particular solution at the same time. Combining with the geometric meaning of this rotary-laser scanning technique, we figure out that around the calculated θ2  θ1 , the lines we get are the most stabilized ones, while off the value θ2  θ1 , the stability gets bad. This characteristic is an inherent feature of the rotary-laser scanning technique. In addition to the rotating stability of the rotating head of the station, this feature is another very important point we should pay attention to. The improvement of measurement accuracy not only depends on the mechanical structure of the station or the good performance of the line lasers, but also the internal architectural stability of this rotary-laser scanning technique itself. Like the famous steel structure of the Bird's Nest National Olympic Stadium in Beijing, the stabilized structure guarantees a stabilized result though the error is unavoidable. Better mastering this architectural feature will guide us to better utilize this technique. This is the central conclusion we want to point out.

4. Conclusion The rotary-laser scanning is a more and more popular measurement technique, especially in the area of large-scale measurement. Although a lot of work have been conducted to evaluate the measurement ability, little utilizes a pure perturbation analysis method, nor did they give the solution for choosing the best working volume from a theoretical point of view. This paper is not confined to the pure azimuth and elevation measurement which comes from the traditional idea of theodolites. It tries to analyze the stability of the lines determined by the single station based on the mathematical model of the station itself, which reflects the architectural stability of this rotary-laser scanning technique. Utilizing the mathematical manipulations, such as the solution structure of linear equation group, the perturbation theorem and the theory of Moore-Penrose generalized inverse matrices, we pointed out that the stability determined by a single station is only affected by the value θ2  θ1 , which is the scanning angles’ difference. The value θ1 or θ2 has no effect on the stability. Furthermore, inspired by this theoretical result, we can consciously find the best working volume by inspecting the value θ2  θ1 , which can be calculated using the pre-calibrated parameters of the two laser planes. Moreover, in practical applications where only a

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single measurement station is needed, such as using a single station to correct the absolute positioning error of an industrial robot, and compensating the tiny drift of a work-piece in manufacture, our study provides the working principle in theory. Besides, using several stations to construct a distributed network which can provide the 3D coordinates in a workspace will also benefit from our study. For instance, as a distributed measurement network, the build-in problem of weighting for each measuring station may find solutions from this paper’s result, also, the problem of the layout for the measurement network may become another beneficiary.

Acknowledgments This research was supported by China National Funds for Distinguished Young Scientists (51225505), the National Hightechnology & Development Program of China (863 Program, 2012AA041205), the National Natural Science Foundation of China (61505140) and the R & D start-up project from Tianjin University of Technology and Education (No.KYQD14049). The authors would like to express their sincere thanks to them and comments from the reviewers and the editor would be very much appreciated.

References [1] Xie ZX, Wang JT, Jin M. Study on a full field of view laser scanning system. Int J Mach Tool Manuf 2007;47:33–43. [2] Qiu H, Nisitani H, Kubo A, Yue Y. Autonomous form measurement on machining centers for free-form surfaces. Int J Mach Tool Manuf 2004;44:961–9. [3] Zhao WQ, Sun RD, Qiu LR, Sha DG. Laser differential confocal radius measurement. Opt Express 2010;18:2345–60. [4] Lu L, Yang JY, Zhai LH, Wang R, Cao ZG, Yu BL. Self-mixing interference measurement system of a fiber ring laser with ultra-narrow linewidth. Opt Express 2012;20:8598–607. [5] Lopez MR, Sergiyenko OY, Tyrsa VV, Perdomo WH, Cruz LFD, Balbuena DH, et al. Optoelectronic method for structural health monitoring. Struct Health Monit 2010;9:105–20. [6] Basaca-Preciado LC, Sergiyenko OY, Rodriguez-Quinonez JC, Garcia X, Tyrsa VV, Rivas-Lopez M, et al. Optical 3D laser measurement system for navigation of autonomous mobile robot. Opt Laser Eng 2014;54:159–69. [7] Rodriguez-Quinonez JC, Sergiyenko OY, Preciado LCB, Tyrsa VV, Gurko AG, Podrygalo MA, et al. Optical monitoring of scoliosis by 3D medical laser scanner. Opt Laser Eng 2014;54:175–86. [8] Guan HY, Li J, Yu YT, Wang C, Chapman M, Yang BS. Using mobile laser scanning data for automated extraction of road markings. ISPRS J Photogramm 2014;87:93–107. [9] Malowany K, Magda K, Rutkiewicz J, Malesa M, Kantor J, Michonski J, et al. Measurements of geometry of a boiler drum by time-of-flight laser scanning. Measurement 2015;72:88–95. [10] Gong XL, Bansmer S. Laser scanning applied for ice shape measurements. Cold Reg Sci Technol 2015;115:64–76. [11] Park HS, Kim JM, Choi SW, Kim Y. A wireless laser displacement sensor node for structural health monitoring. Sensors 2013;13:13204–16. [12] Fritzsche M, Schell J, Ochs W, Neumann MN, Lammering R. 3D scanning laser doppler vibrometry for structural health monitoring using ultrasonic surface waves. Porto: INEGI-Inst Engenharia Mecanica E Gestao Industrial; 2012. [13] Rodriguez-Quinonez JC, Sergiyenko O, Gonzalez-Navarro FF, Basaca-Preciado L, Tyrsa V. Surface recognition improvement in 3D medical laser scanner using Levenberg-Marquardt method. Signal Process 2013;93:378–86. [14] Rivas M, Sergiyenko O, Aguirre M, Devia L, Tyrsa V, Rendon I. Spatial data acquisition by laser scanning for robot or SHM task. In: Proceedings of IEEE conference on Industrial Electronics. New York; 2008. [15] Surmann H, Nuchter A, Hertzberg J. An autonomous mobile robot with a 3D laser range finder for 3D exploration and digitalization of indoor environments. Robot Auton Syst 2003;45:181–98. [16] Shi JL, Sun ZX, Bai SQ. Large-scale three-dimensional measurement via combining 3D scanner and laser rangefinder. Appl Opt 2015;54:2814–23. [17] Zhao ZY, Zhu JG, Xue B, Yang LH. Optimization for calibration of large-scale optical measurement positioning system by using spherical constraint. J Opt Soc Am A 2014;31:1427–35. [18] Liu Z, Xu Y, Liu Z, Wu J. A large scale 3D positioning method based on a network of rotating laser automatic theodolites. In: Proceedings of 2010 IEEE international conference on information and automation (ICIA). 2010. pp. 513–18. [19] Xue B, Zhu JG, Zhao ZY, Wu J, Liu ZX, Wang Q. Validation and mathematical model of workspace Measuring and Positioning System as an integrated metrology system for improving industrial robot positioning. Proceedings of

34

[20] [21]

[22] [23]

[24]

[25]

B. Xue et al. / Optics and Lasers in Engineering 78 (2016) 26–34

the institution of mechanical engineers part b-journal of engineering manufacture p i mech eng b-j eng. The abbreviation is : p i mech eng b-j eng the issn is: 0954-4054, 2014;228:422–40. Hada Y, Hemeldan E, Takase K, Gakuhari H. Trajectory tracking control of a non-holonomic mobile robot using iGPS and odometry. IEEE: New York; 2003. Karelin AV, Adriani O, Barbarino GC, Bazilevskaya GA, Bellotti R, Boezio M, et al. Measurement of the large-scale anisotropy of cosmic rays in the PAMELA experiment. JETP Lett. 2015;101:295–8. Xue B, Zhu JG, Ren YJ, Yang LH, Wu J, Liu ZX. Study on reducing the precision loss of a portable probe in large-scale measurements. Opt Eng 2014;53:10. Muelaner JE, Wang Z, Martin O, Jamshidi J, Maropoulos PG. Verification of the indoor GPS system, by comparison with calibrated coordinates and by angular reference. J Intell Manuf 2012;23:2323–31. Xiong Z, Zhu JG, Geng L, Yang XY, Ye SH. Verification of horizontal angular survey performance for workspace measuring and positioning system. J Optoelectron Laser 2012;23:291–6. Xiong Z, Zhu JG, Zhao ZY, Yang XY, Ye SH. Workspace measuring and positioning system based on rotating laser planes. Mechanika 2012:94–8.

[26] Flores-Fuentes W, Rivas-Lopez M, Sergiyenko O, Gonzalez-Navarro FF, RiveraCastillo J, Hernandez-Balbuena D, et al. Combined application of power spectrum centroid and support vector machines for measurement improvement in optical scanning systems. Signal Process 2014;98:37–51. [27] Flores-Fuentes W, Rivas-Lopez M, Sergiyenko O, Rodriguez-Quinonez JC, Hernandez-Balbuena D, Rivera-Castillo J. Energy center detection in light scanning sensors for structural health monitoring accuracy enhancement. IEEE Sens J 2014;14:2355–61. [28] Williams G. Linear algebra with applications. Massachusetts: Jones & Bartlett Learning; 2011. [29] Xiong H, Zeng S, Mao Y. Fundamentals of applied mathematics. 4th ed. TianJin: Tianjin University Press; 2004. [30] Davis C. The rotation of eigenvectors by a perturbation. J Math Anal Appl 1963;6:159–73. [31] Lourakis M, Argyros A. SBA: a software package for generic sparse bundle adjustment. ACM Trans Math Softw 2009;36:2.