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On-site accuracy evaluation of laser tracking attitude measurement system

Zhi Xiong, Jing He, Luke Chang, Ningtong Liu, Zhongsheng Zhai, Dabao Lao, Dengfeng Dong, and Weihu Zhou

Open Access Open Access

Abstract

In this study, an on-site attitude accuracy evaluation method based on parallel mechanism model and indirect traceability from length to angle is proposed. Firstly, the mathematical model is established. Through orthogonal experimental design, quantitative analysis shows that the ranging accuracy and control layout have a significant impact on the accuracy of the evaluation system. On this basis, the layout of control field is optimized by genetic algorithm. Finally, the practicability of the evaluation method is verified by experiments. The results show that the yaw and pitch accuracy of the method are 0.008°and 0.007°respectively in the range of −25°to 25°within the working distance of 8 m. This method can accurately and effectively evaluate the attitude angle information of the measurement system and is adapted to various on-site environments. The research provides an innovative idea which can be used to ensure the strict requirements of attitude angle measurement in fields such as intelligent manufacturing and in situ processing.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Six-degree-of-freedom (6-DOF) laser tracking measurement technology provides accurate information on the position and attitude of the object in three-dimensional space [1]. It is widely used in aerospace [2], shipbuilding [3], rail transit [4] and other modern advanced manufacturing fields. In scenarios such as automatic welding, complex surface processing, and automatic drilling, the need for high precision of the attitude of the industrial robot end is crucial as shown in Fig. 1(a)–(c). The attitude information of the industrial robot can be collected in real time by the laser tracking attitude measurement system (LTAMS), and the measurement data is transmitted to the control part of the robot or other motion mechanism in real time to realize the automatic correction of the 6-DOF path. Through real-time measurement and feedback of industrial robot attitude, the accuracy of robot end trajectory can be effectively improved [5–7]. In recent years, the attitude measurement methods based on Laser Tracker has been continuously researched and some progress and breakthroughs in static attitude angle measurement have been made [8–10]. Currently, attitude angle measurement can be achieved within a distance range of 3-15 meters. In order to ensure the accuracy and reliability of LTAMS attitude information and the popularization and application of field measurement, it is very important to evaluate the measurement performance of the attitude measurement system [11] before use. Auxiliary equipment such as the precision turntable and articulated arm have been used to evaluate the measurement performance of LTAMS which had significant limitations in on-site applications. The precision turntable requires a flat installation surface, and suitable for laboratory environments, and the articulated arm will incur additional costs. Therefore, it is particularly important to find a low-cost and efficient evaluation method of LTAMS that can be suitable for on-site environment.

 figure: Fig. 1.

Fig. 1. LTAMS application scenarios and their structural composition (a) LTAMS used for automatic(b) LTAMS used for complex surface (c) LTAMS used for aerospace drilling (d) Real time pose error compensation for LTAMS and industrial robots (e) Composition and Coordinate System Definition of LTAMS

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The existing angle evaluation methods include direct evaluation method and indirect evaluation method. The direct evaluation method is usually based on the angle reference and the transfer chain is clear. For instance, in Ref. [12] Pan Minghua et al. used a high precision rotary table to evaluate the attitude measurement system, which combined with photoelectric targets. The standard deviations of the pitch angle and yaw angle are 0.0297° and 0.0332°respectively. Gao Yang et al. calibrated the attitude measurement system using a precision rotary table [13] and a total station. Lauryna et al. evaluated the accuracy of the angle measurement of the total station with angular indexing table [14]. However, the direct evaluation method often used in the laboratory environment, and the inconsistency between the evaluation environment and the on-site environment brings errors. Indirect evaluation method always involves length constraints. For instance, in Ref. [8] Kun Yan et al. chose to use the articulated arm as a benchmark to verify the attitude measurement results, where the cost was increased for the case of additional length measuring equipment. To meet the needs of portable, high precision and low cost for the attitude measurement evaluation in complex environments, we innovatively propose a detachable on-site evaluation method based on parallel mechanism model and indirect traceability from length to angle. Compared with the existing methods, this method has the following advantages: Firstly, due to its on-site use, our method overcomes the influence of the difference between the evaluation environment and the use environment. Secondly, since laser tracker has high precision length of measurement, there is no need to use additional length measurement equipment, which can reduce the evaluation cost. Thirdly, for different field application environments, the evaluation accuracy can be improved by optimizing the design of the control field. Finally, the installation requirements of the evaluation mechanism are guaranteed by the mechanical structure. The detachable structure facilitates on-site assembly and improves evaluation efficiency.

This study uses a homogeneous coordinate transformation method based on spatial distance constraints. Inspired by the parallel structure [15–17], the distance constraint model between the internal points of the control field and the target measurement points is established. Thus, the measurement results of the attitude angle can be traced back to the length measurement benchmark. Based on establishing the mathematical model of the method, the main factors affecting the accuracy of the evaluation model are discussed, and the optimization strategy of the control field is proposed. A series of valuable conclusions are obtained through simulations and experiments. And the main contributions in this article are summarized as follows.

  • • An on-site evaluation method based on parallel mechanism model and indirect traceability from length to angle is innovatively proposed, which can realize the field accuracy evaluation of LTAMS.
  • • In this study, orthogonal experimental design is used to analyze the main influencing factors of the evaluation model, and a control field layout optimization strategy is developed. This strategy can be applied to a wider range of application scenarios for flexible layout to ensure evaluation accuracy.
  • • The feasibility of proposed method is verified by theoretical modeling, simulation analysis and experiments. Compared with the existing evaluation methods, it has good environmental adaptability, portability and low cost.
  • • The proposed evaluation method in this article has the potential for applications in the manufacturing of large-scale equipment such as aerospace and large scientific installations.
In the rest of this article, Section 2 describes LTAMS and the measurement model based on the distance constraint accuracy evaluation method in detail. Section 3 analyses the significant factors affecting the evaluation model and optimizes the accuracy evaluation model. Section 4 presents the experimental results and discussion. Finally, Section 6 concludes the article.

2. Methods for the accuracy evaluation of an evaluation measurement system

2.1 System model

LTAMS relies on a laser tracking device as the base station and works together with a camera and a cooperative target, to conduct attitude measurements, as shown in Fig. 1. The target is equipped with a corner cube prism and a position-sensitive detector (PSD), as well as additional characteristic target points.

Four different coordinate systems are defined in LTAMS: the laser tracker frame (OLXLYLZL-coordinate, frame L), the camera frame (OCXCYCZC-coordinate, frame C), the target frame (OTXTYTZT-coordinate, frame T) and the PSD frame (ODXDYDZD-coordinate, frame D). It is noteworthy that the origin OT of frame T is set as the optical center of prism and the XT and YT axis of frame T are parallel to the corresponding axis of the PSD frame P. The ZT -axis is determined by Ampere’s Rule. The distance between the origin of the PSD coordinate system and the cooperative target coordinate system is h.

The corner cube prism in the target receives a laser beam emitted by the laser tracking device and reflects back most of the beam. From this, the corner coordinates of the prism and the tracking beam vector in frame L can be obtained. In addition, the PSD photosensitive surface receives a small portion of the laser, thereby determining the center coordinates of the light spot. By combining the installation distance of PSD and corner cube prism, the tracking beam vector is obtained in frame T.

Establish a relationship based on the uniqueness constraint of the beam as Eq. (1).

$$R_T^C\cdot \frac{{{{\boldsymbol l}_{{O_T}}}}}{{||{{{\boldsymbol l}_{{O_T}}}} ||}} = R_L^C\cdot \frac{{{{\boldsymbol l}_{{O_L}}}}}{{||{{{\boldsymbol l}_{{O_L}}}} ||}}$$
where${\; }{{\boldsymbol l}_{{O_T}}}$ and ${{\boldsymbol l}_{{O_L}}}$ are the beam vector in frame T and frame L. $R_L^C{\; }$and $R_T^C$ are the rotation and translation matrix relate frame C to frame L and to frame T.

Frame T rotates around its own Z-axis by an angle of $\psi $, followed by a rotation around its own X-axis by an angle of $\theta $, and finally a rotation around its own Y-axis by an angle of $\phi $. This series of rotations aligns Frame T with Frame C. $R_T^C $ can be parameterized by Euler angles ($\theta $, $\phi $, $\psi $.) as defined in Eq. (2).

$$\begin{array}{l} R_T^C = \left( {\begin{array}{ccc} {\cos \phi }&0&{\sin \phi }\\ 0&1&0\\ { - \sin \phi }&0&{\cos \phi } \end{array}} \right)\left( {\begin{array}{ccc} 1&0&0\\ 0&{\cos \theta }&{ - \sin \theta }\\ 0&{\sin \theta }&{\cos \theta } \end{array}} \right)\left( {\begin{array}{ccc} {\cos \psi }&{ - \sin \psi }&0\\ {\sin \psi }&{\cos \psi }&0\\ 0&0&1 \end{array}} \right)\\ = \left( {\begin{array}{ccc} {\cos \phi \cdot \cos \psi + \sin \theta \cdot \sin \phi \cdot \sin \psi }&{\cos \psi \cdot \sin \theta \cdot \sin \phi - \cos \phi \cdot \sin \psi }&{\cos \theta \cdot \sin \phi }\\ {\cos \theta \cdot \sin \psi }&{\cos \theta \cdot \cos \psi }&{ - \sin \theta }\\ {\cos \phi \cdot \sin \theta \cdot \sin \psi - \cos \psi \cdot \sin \phi }&{\sin \phi \cdot \sin \psi + \cos \phi \cdot \cos \psi \cdot \sin \theta }&{\cos \theta \cdot \cos \phi } \end{array}} \right) \end{array}$$
Using the weighted accelerated orthogonal iteration algorithm [18], we calibrate the rotation matrix $R_L^C{\; }$between frame C and frame L. Next, we use the monocular vision method (MVM) to measure the yaw angle ${\theta _1}$, pitch angle ${\phi _1}$ and roll angle ${\psi _1}$. Then, we merge the known amount of the roll angle ${\psi _1}$ obtained by MVM into the Eqs. (1) and (2) and combined with the uniqueness of the beam vector in different coordinate systems, the yaw angle ${\theta _2}$ and the pitch angle ${\phi _2}$ can be calculated, where the roll angle ${\psi _2}$ is equal to the roll angle ${\psi _1}$. The weighted least squares method is used to fuse the data of (${\theta _1}$, ${\phi _1}$, ${\psi _1}$) and (${\theta _2}$, ${\phi _2}$, ${\psi _2}$) to obtain ($\theta $, $\phi $, $\psi $). Finally, the 3 × 3 rotation matrix $R_T^C$ is obtained according to the attitude angle ($\theta $, $\phi $, $\psi $), as shown in Fig. 2.
$$R_T^L = {({R_L^C} )^{ - 1}}\cdot R_T^C = \left( {\begin{array}{ccc} {{r_{11}}}&{{r_{12}}}&{{r_{13}}}\\ {{r_{21}}}&{{r_{22}}}&{{r_{23}}}\\ {{r_{31}}}&{{r_{32}}}&{{r_{33}}} \end{array}} \right)$$

 figure: Fig. 2.

Fig. 2. Solution of attitude angle for LTAMS

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The attitude measurement information of the measured object of LTAMS is expressed by the yaw angle α around the X-axis, the pitch angle β around the Y-axis and the roll angle γ around the Z-axis of the frame T relative to the frame L. The $({\alpha ,\beta ,\gamma } )$ of the target in frame L can be from the relationship between rotation matrices and Euler angles.

$$\begin{array}{c} {\alpha ={-} \arcsin ({{r_{23}}} )}\\ {\beta = \arctan \left( {\frac{{{r_{13}}}}{{{r_{33}}}}} \right)}\\ {\gamma = \arctan \left( {\frac{{{r_{21}}}}{{{r_{22}}}}} \right)} \end{array}$$
Our study aims to develop an evaluation method for the measurement accuracy of LTAMS attitude angle $({\alpha ,\beta ,\gamma } )$ in the field working environment.

2.2 Accuracy evaluation method

The 6-6 Stewart Platform is a highly precise six-degree-of-freedom motion platform that offers strong controllability and flexibility. It is widely used in motion simulators, parallel machine tools, aerospace and other fields. The structure diagram is shown in Fig. 3(a).

 figure: Fig. 3.

Fig. 3. (a) schematic of Stewart platform (b) Geometric parameters of the Stewart platform

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The attitude change of the Stewart platform is determined through a forward kinematics numerical solution, which considers known changes in the six bar lengths of the Stewart structure. A distance constraint between the static and moving platform layout is applied to achieve the target attitude, as illustrated in Fig. 3(b). This article proposes an approach without physical objects to eliminate mechanical interference. To evaluate attitude angle, we adopt the 6-6 Stewart structure, which overcomes limitations in existing accuracy evaluation methods.

The dynamic frame (OMXMYMZM-coordinate, frame M) and the static frame (OCXCYCZC-coordinate, frame C) are established on the moving platform and the static platform respectively. The coordinate vectors of the hinge point ${M_k}({k = 1,2 \ldots ,6} )$ of the moving platform in the dynamic frame and the hinge point ${C_k}({k = 1,2, \ldots ,6} )$ in the static frame are expressed as:

$${M_k} = {\left( {\begin{array}{cccc} {{M_{kx}}}&{{M_{ky}}}&{{M_{kz}}}&1 \end{array}} \right)^T}$$
$${C_k} = {\left( {\begin{array}{cccc} {{C_{kx}}}&{{C_{ky}}}&{{C_{kz}}}&1 \end{array}} \right)^T}$$
When the dynamic platform moves, the coordinate vector of any hinge point$\; {M_k}({k = 1,2 \ldots ,6} )$ is transformed into frame C through the spatial homogeneous transformation matrix. This forms a geometric relationship with the hinge point ${C_k}({k = 1,2, \ldots ,6} )$ of the static platform.
$${l_k} = ||{T({\alpha ,\beta ,\gamma ,{t_x},{t_y},{t_z}} )} ||\times {M_k} - {C_k}$$
where $T({\alpha ,\beta ,\gamma ,{t_x},{t_y},{t_z}} )$ is a space homogeneous transformation matrix, which is expressed in Z-X-Y rotation order as formula (8).

Integrating the Kinematics model of parallel structure, the mobile platform is fixed on the target to establish a measurement field. This facilitates the acquisition of the coordinate vector of measuring point ${M_k}({k = 1,2 \ldots ,6} )$ in frame T. Afterward, the static platform is fixed around the target to establish a control field, enabling the acquisition of the coordinate vector of control point ${C_k}({k = 1,2, \ldots ,6} )$ in frame L, as shown in Fig. 4.

$$\begin{array}{l} {{\mathbf T}_{ZXY}}({\alpha ,\beta ,\gamma ,{t_x},{t_y},{t_z}} )= \\ \left( {\begin{array}{cccc} {\cos \beta \cos \gamma + \sin \alpha \sin \beta \sin \gamma }&{\cos \gamma \sin \alpha \sin \beta - \cos \beta \sin \gamma }&{\cos \alpha \sin \beta }&{{t_x}}\\ {\cos \alpha \sin \gamma }&{\cos \alpha \cos \gamma }&{ - \sin \alpha }&{{t_y}}\\ {\cos \beta \sin \alpha \sin \gamma - \cos \gamma \sin \beta }&{\sin \beta \sin \gamma + \cos \beta \cos \gamma \sin \alpha }&{\cos \alpha \cos \beta }&{{t_z}}\\ 0&0&0&1 \end{array}} \right) \end{array}$$
This laser tracking attitude angle accuracy evaluation method includes the following steps: (1) Obtain the internal point’s space vector of the measurement field and control field of the evaluation system. (2) Determine the distance constraint (the length of the rod). (3) Establish homogeneous coordinate transformation matrix equations. (4) Give the initial value of these equations. (5) Solving the attitude angle of the evaluation system. This method avoids the strict coordinate system registration requirements in the current angle-based evaluation method. The control field and measurement field are established on the site. It comprehensively reflects the field use status of the measurement system. Therefore, the mathematical model of nonlinear equations with pose $({\alpha ,\beta ,\gamma ,{t_x},{t_y},{t_z}} )$ as unknown quantity is established by spatial distance constraint condition.
$${G_1}({\alpha ,\beta ,\gamma } )_k^2 + {G_2}({\alpha ,\beta ,\gamma } )_k^2 + {G_3}({\alpha ,\beta ,\gamma } )_k^2 - {l_k} = 0$$
where:
$$\left\{ {\begin{array}{c} {{G_1}{{({\alpha ,\beta ,\gamma } )}_k} = {{\mathbf T}_{11}}{{\mathbf M}_{kx}} + {{\mathbf T}_{12}}{{\mathbf M}_{ky}} + {{\mathbf T}_{13}}{{\mathbf M}_{kz}} + {{\mathbf T}_{14}} - {{\mathbf C}_{kx}}}\\ {{G_2}{{({\alpha ,\beta ,\gamma } )}_k} = {{\mathbf T}_{21}}{{\mathbf M}_{kx}} + {{\mathbf T}_{22}}{{\mathbf M}_{ky}} + {{\mathbf T}_{23}}{{\mathbf M}_{kz}} + {{\mathbf T}_{24}} - {{\mathbf C}_{ky}}}\\ {{G_3}{{({\alpha ,\beta ,\gamma } )}_k} = {{\mathbf T}_{31}}{{\mathbf M}_{kx}} + {{\mathbf T}_{32}}{{\mathbf M}_{ky}} + {{\mathbf T}_{33}}{{\mathbf M}_{kz}} + {{\mathbf T}_{34}} - {{\mathbf C}_{kz}}} \end{array}} \right.$$
where ${T_{ij}}$ is the ith row and jth column element of space homogeneous transformation matrix T. Substitute into $X = ({\alpha ,\beta ,\gamma ,{t_x},{t_y},{t_z}} )$, rewrite Eq. (9) as:
$$\begin{array}{c} {{F_k}({X,{l_k}} )= 0}\quad{({k = 1,2, \ldots , 6} )} \end{array}$$
We know Newton iterative method for the nonlinear system F(x) = 0, where F is defined by Eq. (12).
$${X^{(N + 1)}} = {X^{(N)}} - \frac{{F({x_n})}}{{{F^{\prime}}({x_n})}}$$
where $F^{\prime}({{x_n}} )$ is the Jacobian matrix in point xn.We rewrite Eq. (12) to solve the nonlinear system F(x) = 0, this produces the following iteration scheme:
$${X^{(N + 1)}} = {X^{(N)}} - {F^{\prime}}{({x_n})^{ - 1}}({F({x_n}) + F({x^\ast }_{n + 1})} )$$
where
$${x^\ast }_{n + 1} = {x_n} - {F^{\prime}}{({x_n})^{ - 1}}F({x_n})$$

 figure: Fig. 4.

Fig. 4. Schematic diagram of accuracy evaluation model

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The iteration scheme Eq. (14) is referred to as the modified Newton method for solving systems of nonlinear equations [19]. Therefore, the third-order iterative expression based on Adomian decomposition of nonlinear Eq. (11) is as:

$${X^{({\textrm{N} + 1} )}} = {X^{(\textrm{N} )}} - {J^{(\textrm{N} )}}^{ - 1}({F({{X^{(\textrm{N} )}},{l_k}} )+ F({{X^{({\textrm{N} + 1} )}}^ \ast ,{l_k}} )} )$$
$${X^{({\textrm{N} + 1} )}}^ \ast{=} {X^{(\textrm{N} )}} - {J^{(\textrm{N} )}}^{ - 1}F({{X^{(\textrm{N} )}},{l_k}} )$$
In the above equations, N is the number of iterations, J is the Jacobian matrix and is computed as:
$${J_{ij}} = \frac{{\partial {F_i}}}{{\partial {X_j}}}$$
where ${J_{ij}}$ is the ith row and jth column element of the Jacobian matrix. ${F_i}$ is the ith line of nonlinear equations. ${X_j}\; $is the jth element of the unknown X. To obtain the pose of the Stewart platform, the initial values of the pose $({{\alpha^0},{\beta^0},{\gamma^0},{t_x}^0,{t_y}^0,{t_y}^0} )$ are iterated into Eq. (8). This results in the corresponding spatial homogeneous transformation matrix $\; {T^0}$, which is further iterated through Eqs. (15) and (16). The iterative process of obtaining the attitude angles of the target in frame L stops when the condition of Eq. (18) is met, resulting in the optimal numerical solution.
$$Max|{\triangle X} |\le \varepsilon $$
where $\varepsilon \; $is the maximum allowable error. $\Delta X$ is the difference between ${X^{({N + 1} )}}$ and ${X^{(N )}}$.

Based on the Stewart platform position solution method, the pose of the corresponding platform is determined by measuring the length of the rod. $Max|{\Delta X} |\le \varepsilon \; $is used as the iterative termination judgment condition. The procedure flow chart is shown in Fig. 5.

 figure: Fig. 5.

Fig. 5. Procedure flow chart

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The problem of attitude solving is essentially the optimization solution of nonlinear equations. The selection of initial values and iterative algorithms are two key points. Newton Raphson iteration method and Adomian decomposition method are utilized respectively on the MATLAB platform to solve the model. As shown in in Table 1, compared with the modified Newton method, the third-order method of Adomian decomposition can achieve the desired accuracy in fewer iterations and shorter time, which improves the evaluation efficiency.

Tables Icon

Table 1. Comparison between Newton Raphson iteration and Adomian decomposition method

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3. Analysis of influencing factors and optimization strategy

3.1 Analysis of influencing factors

The model's accuracy is affected by the measurement accuracy of the distance constraint and the layout of the control. The accuracy of the distance constraint measurement is determined by the accuracy of ranging the spatial distance length between the control point C and the target measurement point M. The key factors in the control layout are the size of the control field (RB) and the distance between the measurement field and the control field. Because the measurement field needs to be installed on the target, the space is limited, and the change of the measurement field size (RA) is small, which is not considered as an influencing factor. To ensure platform stability and avoid singular configurations, the control field circle’s radius RB cannot be too small [20], which needs to meet the Eq. (19).

$$0.9{R_B} > {R_A}$$
The orthogonal test design theory is used to analyze the sensitivity of the influencing factors. Control field radius, control field distance and ranging accuracy were selected as the main influencing factors for analysis. Combined with the mechanical size of the target and the Eq. (19), the control field radius is selected between [200 mm, 1800 mm] with five levels at 400 mm equidistant. The control field distance can meet the accuracy requirements in the range of [1600 mm, 3200 mm], so five levels are selected with an equal spacing of 400 mm. Ranging accuracy is affected by working distance and equipment, which is selected between [0.005 mm, 0.085 mm] taking the working distance into consideration, and the five levels of ranging accuracy are selected at an equal interval of 0.02 mm.The level of factors selected for orthogonal analysis is shown in Table 2.

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Table 2. Multiple levels of factors

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Based on the Table 2, Monte Carlo [21–23] simulation experiment is carried out on the accuracy evaluation model with reference to the orthogonal table${L_{25}}({5^3})$, and the attitude angle accuracy results at different levels are obtained. The simulation conditions are as follows: The attitude angles change in the range of [-30°,30°], the step size is 10°, and each position is repeatedly measured 1000 times for simulation purposes. Taking the yaw angle as an example, the sth measurement value$\; {\alpha _{rs}}$ of the yaw angle at the rth position is obtained. The systematic error and random error calculation formulas of each position are as:

$$\left\{ \begin{array}{l} {\sigma_{\alpha r}} = \sqrt {\frac{{\sum\nolimits_{s = 1}^M {{{({{\alpha_{rs}} - {\alpha_r}} )}^2}} }}{M}} \\ {\Delta _{\alpha r}} = \frac{{\sum\nolimits_{s = 1}^M {{\alpha_{rs}}} }}{M} - {\alpha_r} \end{array} \right.$$
where ${\alpha _r}$ is the theoretical value of the yaw of the rth position. M is the number of repeated measurements.

The standard deviation ${\sigma _\alpha }\; $of the random error of all position yaws is used to characterize the random error of the model yaw. The standard deviation ${e_\alpha }$ of the systematic error of all position yaws is used to characterize the systematic error of the model yaw. The calculation formula is as follows.

$$\left\{ \begin{array}{l} {\sigma_\alpha } = \sqrt {\frac{{\sum\nolimits_{r = 1}^P {{{({{\sigma_{\alpha r}} - {{\bar{\sigma }}_\alpha }} )}^2}} }}{{P - 1}}} \\ {e_\alpha } = \sqrt {\frac{{\sum\nolimits_{r = 1}^P {{{({{\Delta _{\alpha r}} - {{\bar{\Delta }}_\alpha }} )}^2}} }}{{P - 1}}} \end{array} \right.$$
where ${\bar{\sigma }_\alpha }$ is the average value of the root mean square of yaw at each position. ${\mathrm{\bar{\Delta }}_\alpha }$ is the average value of the deviation between the average yaw of each position and the true value. P is the number of measurement positions.

According to the 3σ criterion, the triple of the RMSE is used as the accuracy index of the evaluation model. The systematic and random errors of the accuracy evaluation model are combined to obtain the comprehensive error ${\delta _{lim\alpha }}$, which serves as an indicator of yaw accuracy for evaluating the model. When the confidence coefficient t = 3,the confidence probability is 99.73%. The calculation formula is as:

$${\delta _{\lim \alpha }} ={\pm} t\sqrt {e_\alpha ^2 + \sigma _\alpha ^2} $$
Similarly, the combined errors ${\delta _{lim\beta }}$ and$\; {\delta _{lim\gamma }}$ of pitch angle and roll angle is obtained.
$$\begin{array}{c} {{\delta _{\lim \beta }} ={\pm} t\sqrt {e_\beta ^2 + \sigma _\beta ^2} }\\ {{\delta _{\lim \gamma }} ={\pm} t\sqrt {e_\gamma ^2 + \sigma _\gamma ^2} } \end{array}$$
The variance analysis of the attitude angle accuracy results at different levels is performed using Minitab software. The results are shown in Fig. 6, with a confidence level of 95%. Based on the P value, it was determined whether the impact term had a significant effect on the dependent variable. If P > 0.05, it indicates that the effect term has no significant effect on the dependent variable. If P < 0.05, it indicates that the effect term has a significant effect on the dependent variable.

 figure: Fig. 6.

Fig. 6. The P value of each factor on α, β and γ

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As shown in Fig. 6, the P values of control field radius and ranging accuracy are less than 0.05, indicating that these factors have a significant impact on the accuracy of the evaluation method. Therefore, next, we try to improve the accuracy of the evaluation model by optimizing the control field layout and improving the ranging accuracy so that it can have a wider application.

3.2 Optimization strategy

3.2.1 Using high precision length constraints of laser tracker

As a large-scale space geometric precision measuring instrument, the laser tracker integrates advanced measurement technology theories such as laser interference ranging technology, photoelectric detection technology, and computer control technology. It shows extremely high measurement accuracy and efficiency in equipment calibration, component detection, tooling manufacturing and debugging, integrated assembly and reverse engineering [24]. For example, the Leica AT960 laser tracker has an interference length measurement accuracy of 0.5µm/m, an angle measurement accuracy of 15µm + 6µm/m, the maximum diameter of the measurement space can reach 160 m. The Leica AT901-MR laser tracker has the same accuracy level as AT960, but the maximum diameter of the measurement space is just reaching 50 meters. Assuming that the ranging range of LTAMS is 15 m, based on the interference length measurement accuracy of Leica laser tracker, the ranging accuracy can reach 0.0075 mm. This accuracy is consistent with the range of ranging accuracy given in Table 2 [0.005 mm, 0.085 mm], so ranging measurements of the laser tracker can be used as a thigh precision length constraint.

3.2.2 Optimizing the control field layout

Determining the control field optimal layout in the evaluation system is difficult due to the coupling and interdependence of multiple parameters. Genetic algorithm was used to optimize the control layout. The optimization process is based on the evaluation system’s mathematical model, using accuracy as a quantitative index to measure the effectiveness of layout design. The flow chart of the algorithm is shown in Fig. 7.

 figure: Fig. 7.

Fig. 7. The algorithm flow chart

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Each individual in the group represents a control layout. The following five parameters are selected to characterize the individuals in the population: the control field radius, the distribution angle of the control points and the rotation angles around the XL, YL and ZL axes. The fitness function is designed as:

$$f({{G_i}} )= 3\sqrt {{{({{\delta_{\alpha i}}} )}^2} + {{({{\delta_{\beta i}}} )}^2} + {{({{\delta_{\gamma i}}} )}^2}}$$
$$F({{G_i}} )= \frac{1}{{f({{G_i}} )}}$$
where Gi is the ith, F(Gi) is the fitness. ${\delta _{\alpha i}},{\delta _{\beta i}}\; $and ${\delta _{\gamma i}}\; $are the evaluation model yaw accuracy, pitch angle accuracy and roll angle accuracy under the control layout corresponding to the ith individual in the population. Take the yaw accuracy ${\delta _{\alpha i}}$ as an example.
$${\delta _{\alpha i}} = \sqrt {\frac{{\sum\nolimits_{r = 1}^P {({\alpha_{_{ir}}^\ast{-} {\alpha_{ir}}} )} }}{P}}$$
where $\alpha _{ir}^\ast $ and ${\alpha _{ir}}\; $are the solution value and theoretical value of the yaw angle of the rth position under the control layout of the ith individual in the population, respectively. P is the number of measurement positions, which is determined by the rotation range and step length of the attitude angle.

The control layout optimization algorithm is tested and analyzed. The same attitude angle variation range and Monte Carlo method as Section 3.1 are used. Three working distances of 3 m, 8 m and 15 m are selected. The simulation conditions are as follows: the laser tracker interference ranging accuracy is ${\varepsilon _d} = 0.5\mu m$.The angle measurement accuracy is ${\varepsilon _\theta } = 15\mu m + 6\mu m/m$. The genetic algorithm population size M is set to 50. The maximum number of evolution N is set to 100. The hybridization probability is set to ${p_c} = 0.01$, and the mutation probability is set to ${p_m} = 0.01$.

The simulation results are shown in Fig. 8 In the range of [-30°,30°], 15 m working distance. The original layout yielded a maximum error of 0.008°, whereas the optimized layout demonstrated a maximum error of 0.002°, signifying a remarkable improvement in accuracy. Compared with the experimental layout, this method is easy to obtain the control field that meets the measurement requirements. It greatly reduces the requirement of operator experience.

 figure: Fig. 8.

Fig. 8. Error comparison before and after layout optimization (a)layout error at 3 m (b)layout error at 8 m (c)layout error at 15m

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Referring to the distribution of hinge points on the Stewart platform, the layout of the measurement points and control points of the evaluation system has been designed, as shown in Fig. 9. Among them, the control field size is a circle with a radius of 990 mm, the angle ${\theta _a}$ between ${O_T}{A_1}$ and Y negative axis is 8°, and the Y-axis is symmetrically distributed. The six measurement points are distributed on a circle with a radius of 220 mm on the target. The angle ${\theta _b}$ between ${O_L}{B_1}$ and Y positive axis is 6 °, which is Y-axis symmetrical distribution. The distance between the measurement field and the control field is 2500 mm.

 figure: Fig. 9.

Fig. 9. The layout of the measurement field and control field (a) Position distribution of measuring point (b) Position distribution of control points

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4. Experiments

4.1 Experimental platform

Using Leica AT901-MR laser tracker as on-site tracking equipment, the attitude angle measurement accuracy is traced by the ranging accuracy of the laser tracker. Suzhou Jiehui 0.5-inch target ball and target ball seat are selected to measure with laser tracker.

In order to verify the proposed accuracy evaluation method, the relative rotation of the precision two-dimensional rotary table is used as the angle reference. The measurement accuracy of the rotary table is 2 seconds, and it can rotate automatically. The yaw rotation range is [50°, 330°], and the pitch angle rotation range is [0°,360°]. Since the registration displacement error between the turntable frame and the frame T does not affect the attitude angle evaluation error, it can be used for experimental verification.

The simulation results and accuracy requirements guide the design of the evaluation structure. The evaluation structure adapts to the target size and range, while ensuring ensure the rigidity of the evaluation framework and enhance the alignment between the measurement points. The cooperative target is shown in Fig. 10.

 figure: Fig. 10.

Fig. 10. Cooperative target and evaluation structure

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The experimental platform is shown in Fig. 11. The cooperative target and evaluation structure are placed on a precision two-dimensional rotary table. Because the precision two-dimensional turntable only rotates in two directions of yaw and pitch angle, and the simulation analysis shows that the accuracy of roll angle is higher than that of yaw and pitch angle, this study only verifies the yaw and pitch angle experimentally. In order to flexibly change the layout of the control points in the application, the virtual control points layout in the laser tracker coordinate system is directly used. Each time the cooperative target rotates, the laser tracker re-measures the coordinate of the measurement points of the cooperative target to obtain the distance value ${h_k}({k = 1,2 \ldots ,6} )$ between the control points and the measurement points.

 figure: Fig. 11.

Fig. 11. Experimental platform

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4.2 Experiment validation and results discussion

Due to the limitation of site space, the working distance of 3 m and 8 m are selected for verification experiments. Using the above experimental platform. In the range of [-25°,25°],11 groups of targets working positions are selected at an equal interval of 5°, and the yaw and pitch angles are compared. The proposed accuracy evaluation algorithm calculates the attitude information of the target in each rotation state, compared with the relative rotation of the rotary table. The root mean square errors of yaw and pitch angle are obtained, and summarized in Fig. 12 and Fig. 13.

 figure: Fig. 12.

Fig. 12. Experimental results of yaw results (a) Relative rotation of 3 m and 8 m (b) The root mean square error of yaw angle

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 figure: Fig. 13.

Fig. 13. Experimental results of pitch angle (a) Relative rotation of 3 m and 8 m (b) The root mean square error of pitch angle

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As shown in the above figures, the accuracy evaluation model indicates a maximum absolute error of 0.0031° for the yaw angle and 0.0026° for the pitch angle at the working distance of 3 m. At a working distance of 8 m, the maximum absolute error is 0.0053° for the yaw angle and 0.0049° for the pitch angle. According to Eq. (27), the RMSE in the yaw and pitch angle directions is calculated, as shown in Table 3.

$$\begin{array}{c} {RMS{E_\alpha } = \sqrt {\frac{{\sum\nolimits_{i = 1}^\textrm{M} {{{({{\nu_{\alpha i}}} )}^2}} }}{\textrm{M}}} }\\ {RMS{E_\beta } = \sqrt {\frac{{\sum\nolimits_{i = 1}^\textrm{M} {{{({{\nu_{\beta i}}} )}^2}} }}{\textrm{M}}} } \end{array}$$
where M is the number of measurements. ${\nu _{\alpha i}}$ and ${\nu _{\beta i}}$ are the ith measurement errors of yaw and pitch angles, respectively, where ${\nu _{\alpha i}} = {\alpha _{i + 1}} - {\alpha _i} - \varphi $, ${\nu _{\beta i}} = {\beta _{i + 1}} - {\beta _i} - \varphi $. ${\alpha _i}$ and ${\beta _i}$ are the ith solution values of yaw angle and pitch angle respectively. $\varphi $ is the relative rotation of the rotary Table 5°.

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Table 3. Attitude angle RMSE (unit:°)

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Based on the 3σ criterion, the three times of the RMSE is used as the accuracy index of the evaluation model. In the range of [−25°, 25°], the yaw accuracy of the evaluation system at 3 m is 0.006°, and the pitch angle accuracy is 0.004°. The yaw accuracy at 8 m is 0.008°, and the pitch angle accuracy is 0.007°. Based on the simulation rule in Fig. 7, the accuracy of roll angle is better than that of yaw angle and pitch angle. With the increase of distance, the length measurement accuracy of the laser tracker decreases, and the RMSE increases slightly. Therefore, in practical use, the working distance of the laser tracker needs to be considered.

Table 4 summarizes and compares attitude evaluation methods of LTAMS researched in recent years. Compared with the existing evaluation methods using auxiliary equipment such as rotary table and the articulated arm, the proposed method has outstanding advantages in field application, cost and optimization strategy, although its accuracy is not as high as that of precision turntables. According to the theory of small errors, the accuracy of the evaluation system can only be higher than 1/3-1/10 of the accuracy of the tested system, so this method has good application prospects.

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Table 4. Comparisons between the proposed method and the existing methods

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5. Conclusions

The subject of this article is to evaluate the attitude results of LTAMS, a detachable on-site evaluation method based on parallel mechanism model and indirect traceability from length to angle is proposed. The influence of ranging accuracy and control layout on model accuracy is the most significant through quantitative analysis of mathematical model. Based on the results of orthogonal experiment analysis, the control layout is optimized by genetic algorithm. The accuracy of the attitude angle field accuracy evaluation method is verified by experiments. The acquisition of experimental measurement data and the solution of attitude angle are realized by building an experimental platform. The relative rotation angle of the two-dimensional precision turntable is used as the angle reference, and the RMSE of yaw angle and pitch angle is calculated. In the range of [−25°, 25°], the yaw accuracy of the evaluation system at 3 m is 0.006°, and the pitch angle accuracy is 0.004°, while the yaw accuracy at 8 m is 0.008°, and the pitch angle accuracy is 0.007°. Based on the experimental results and the 3σ criterion, the evaluation method can at least evaluate the LTAMS with an attitude accuracy of 0. 024°.In addition, it is worth mentioning that this is a relatively small economic cost assessment method. It should also be noted that the system can adapt to different application environments, is simple and convenient, and has high work efficiency. Our research results provide valuable insights for the on-site accuracy evaluation of the attitude angle of LTAMS and provide a new method to ensure the strict requirements of attitude angle measurement in intelligent manufacturing, in-situ processing and other research.

Funding

The Key Technology of Six-degree-of-freedom Laser Automatic Precise Tracking and Measurement (2019YFB2006100).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (13)
Fig. 1.
Fig. 1. LTAMS application scenarios and their structural composition (a) LTAMS used for automatic(b) LTAMS used for complex surface (c) LTAMS used for aerospace drilling (d) Real time pose error compensation for LTAMS and industrial robots (e) Composition and Coordinate System Definition of LTAMS

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Fig. 2.
Fig. 2. Solution of attitude angle for LTAMS

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Fig. 3.
Fig. 3. (a) schematic of Stewart platform (b) Geometric parameters of the Stewart platform

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Fig. 4.
Fig. 4. Schematic diagram of accuracy evaluation model

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Fig. 5.
Fig. 5. Procedure flow chart

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Fig. 6.
Fig. 6. The P value of each factor on α, β and γ

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Fig. 7.
Fig. 7. The algorithm flow chart

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Fig. 8.
Fig. 8. Error comparison before and after layout optimization (a)layout error at 3 m (b)layout error at 8 m (c)layout error at 15m

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Fig. 9.
Fig. 9. The layout of the measurement field and control field (a) Position distribution of measuring point (b) Position distribution of control points

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Fig. 10.
Fig. 10. Cooperative target and evaluation structure

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Fig. 11.
Fig. 11. Experimental platform

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Fig. 12.
Fig. 12. Experimental results of yaw results (a) Relative rotation of 3 m and 8 m (b) The root mean square error of yaw angle

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Fig. 13.
Fig. 13. Experimental results of pitch angle (a) Relative rotation of 3 m and 8 m (b) The root mean square error of pitch angle

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Tables (4)

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Table 1. Comparison between Newton Raphson iteration and Adomian decomposition method

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Table 2. Multiple levels of factors

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Table 3. Attitude angle RMSE (unit:°)

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Table 4. Comparisons between the proposed method and the existing methods

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Equations (27)

Equations on this page are rendered with MathJax. Learn more.

R T C ⋅ l O T | | l O T | | = R L C ⋅ l O L | | l O L | |
R T C = ( cos ⁡ ϕ 0 sin ⁡ ϕ 0 1 0 − sin ⁡ ϕ 0 cos ⁡ ϕ ) ( 1 0 0 0 cos ⁡ θ − sin ⁡ θ 0 sin ⁡ θ cos ⁡ θ ) ( cos ⁡ ψ − sin ⁡ ψ 0 sin ⁡ ψ cos ⁡ ψ 0 0 0 1 ) = ( cos ⁡ ϕ ⋅ cos ⁡ ψ + sin ⁡ θ ⋅ sin ⁡ ϕ ⋅ sin ⁡ ψ cos ⁡ ψ ⋅ sin ⁡ θ ⋅ sin ⁡ ϕ − cos ⁡ ϕ ⋅ sin ⁡ ψ cos ⁡ θ ⋅ sin ⁡ ϕ cos ⁡ θ ⋅ sin ⁡ ψ cos ⁡ θ ⋅ cos ⁡ ψ − sin ⁡ θ cos ⁡ ϕ ⋅ sin ⁡ θ ⋅ sin ⁡ ψ − cos ⁡ ψ ⋅ sin ⁡ ϕ sin ⁡ ϕ ⋅ sin ⁡ ψ + cos ⁡ ϕ ⋅ cos ⁡ ψ ⋅ sin ⁡ θ cos ⁡ θ ⋅ cos ⁡ ϕ )
R T L = ( R L C ) − 1 ⋅ R T C = ( r 11 r 12 r 13 r 21 r 22 r 23 r 31 r 32 r 33 )
α = − arcsin ⁡ ( r 23 ) β = arctan ⁡ ( r 13 r 33 ) γ = arctan ⁡ ( r 21 r 22 )
M k = ( M k x M k y M k z 1 ) T
C k = ( C k x C k y C k z 1 ) T
l k = | | T ( α , β , γ , t x , t y , t z ) | | × M k − C k
T Z X Y ( α , β , γ , t x , t y , t z ) = ( cos ⁡ β cos ⁡ γ + sin ⁡ α sin ⁡ β sin ⁡ γ cos ⁡ γ sin ⁡ α sin ⁡ β − cos ⁡ β sin ⁡ γ cos ⁡ α sin ⁡ β t x cos ⁡ α sin ⁡ γ cos ⁡ α cos ⁡ γ − sin ⁡ α t y cos ⁡ β sin ⁡ α sin ⁡ γ − cos ⁡ γ sin ⁡ β sin ⁡ β sin ⁡ γ + cos ⁡ β cos ⁡ γ sin ⁡ α cos ⁡ α cos ⁡ β t z 0 0 0 1 )
G 1 ( α , β , γ ) k 2 + G 2 ( α , β , γ ) k 2 + G 3 ( α , β , γ ) k 2 − l k = 0
{ G 1 ( α , β , γ ) k = T 11 M k x + T 12 M k y + T 13 M k z + T 14 − C k x G 2 ( α , β , γ ) k = T 21 M k x + T 22 M k y + T 23 M k z + T 24 − C k y G 3 ( α , β , γ ) k = T 31 M k x + T 32 M k y + T 33 M k z + T 34 − C k z
F k ( X , l k ) = 0 ( k = 1 , 2 , … , 6 )
X ( N + 1 ) = X ( N ) − F ( x n ) F ′ ( x n )
X ( N + 1 ) = X ( N ) − F ′ ( x n ) − 1 ( F ( x n ) + F ( x ∗ n + 1 ) )
x ∗ n + 1 = x n − F ′ ( x n ) − 1 F ( x n )
X ( N + 1 ) = X ( N ) − J ( N ) − 1 ( F ( X ( N ) , l k ) + F ( X ( N + 1 ) ∗ , l k ) )
X ( N + 1 ) ∗ = X ( N ) − J ( N ) − 1 F ( X ( N ) , l k )
J i j = ∂ F i ∂ X j
M a x | △ X | ≤ ε
0.9 R B > R A
{ σ α r = ∑ s = 1 M ( α r s − α r ) 2 M Δ α r = ∑ s = 1 M α r s M − α r
{ σ α = ∑ r = 1 P ( σ α r − σ ¯ α ) 2 P − 1 e α = ∑ r = 1 P ( Δ α r − Δ ¯ α ) 2 P − 1
δ lim α = ± t e α 2 + σ α 2
δ lim β = ± t e β 2 + σ β 2 δ lim γ = ± t e γ 2 + σ γ 2
f ( G i ) = 3 ( δ α i ) 2 + ( δ β i ) 2 + ( δ γ i ) 2
F ( G i ) = 1 f ( G i )
δ α i = ∑ r = 1 P ( α i r ∗ − α i r ) P
R M S E α = ∑ i = 1 M ( ν α i ) 2 M R M S E β = ∑ i = 1 M ( ν β i ) 2 M
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