Pose Selection Based on a Hybrid Observation Index for Robotic Accuracy Improvement

Abstract

The problem of the insufficient accuracy performance of industrial robots in high-precision manufacturing is addressed in this paper. Firstly, a kinematic error model based on an M-DH model was presented. Secondly, a hybrid observability index O₆ was proposed to select the optimal poses for parameter identification. O₆ is the combination of O₁ and O₃. The optimal poses were obtained by using the IOOPS algorithm. Thirdly, the fitness function for parameter identification was established, and the Levenberg–Marquardt (LM) algorithm was applied for the accurate identification of kinematic parameter errors. Finally, several experiments were conducted to evaluate the performance of the proposed hybrid observability index O₆. The average position error and average attitude error of Staubli TX60 robot were reduced by 89% and 49%. The results show that the proposed hybrid observability index O₆ has great stability and effectiveness for robot calibration.

1. Introduction

Industrial robots have been widely applied in plenty of fields [1,2,3]. The repeat positioning accuracy of industrial robots can reach from 0.01 to 0.1 mm. However, the absolute positioning accuracy is only achieved at a millimeter level. Research shows that the error factors leading to poor accuracy performance of industrial robots include the joint encoder errors, the reducer transmission errors, the kinematic parameter errors, the joint and linkage flexibility errors, and stiffness error caused by robot own weight and external forces, etc. [4,5,6,7,8,9]. The kinematic parameter error is the major error factor that affects the absolute positioning accuracy of industrial robots. Robot calibration technology can improve the accuracy performance of industrial robots [10,11,12].

Robot calibration methods are classified into error-model-based calibration methods and non-model-based calibration methods [13]. The non-model-based calibration methods consider various error factors in an integrative manner. Min K. et al. proposed a stable and highly accurate model-free calibration method by improving the existing kriging error compensation method [14]. However, non-model-based calibration methods cannot compensate errors in real time. Error-model-based calibration methods can be divided into four basic steps: error model establishment, robot error measurement, parameter identification, and error compensation [15]. The error model is established based on several kinematic models, such as the DH model [16], POE model [17], ZRM model [18], etc. Due to the non-singular property and completeness, the M-DH model is the most widely used model in robot calibration technology [19]. There are plenty of measurement devices used for obtaining positioning errors, such as the laser tracker [20,21], the ball bar [22], the stereo vision measurement system [23], and the IMU [24]. The laser tracker is the most commonly used measurement device in robot calibration. In the parameter identification steps, a multivariate optimization objective function is constructed based on the error model. The parameter errors can be identified by the traditional optimization algorithms and intelligent optimization algorithms [25]. Ma L. et al. applied the maximum likelihood method to achieve parameter identification. The proposed method significantly improved the robot’s error [26]. Chen X. et al. proposed an improved multi-objective PSO algorithm for kinematic parameter identification [27]. Except for the properties of the optimization algorithms, the selected measured poses also affect the identification accuracy. Yu Sun et al. studied the factors affecting the accuracy of robot calibration [28]. They proved that an optimal pose selection method can effectively improve the accuracy of robot calibration. In order to evaluate the goodness of the measured poses, five observability indexes—O₁, O₂, O₃, O₄, and O₅—were proposed [29,30,31,32]. The measured poses are selected based on the criterion of maximizing the observability indexes. The selected measured poses can improve the accuracy performance of robot calibration [33]. The impacts of each observability index on robot calibration were studied in [34,35]. Yu Z. et al. [36] compared the performance of different observability indexes for a 6 DOF serial robot. The results show that O₁ has the best performance among the five indexes. Horne A. et al. [37] investigated the performance of different observability indexes by simulating a 2 DOF robotic arm. The results show that O₅ and O₃ have good and poor performance, respectively. Wang W. et al. [38] proposed a universal observability index to calibrate serial robots. An improved PSO algorithm was used to select the optimal poses. The accuracy performance of surgical robot was effectively improved. However, there are no uniform conclusions about the performance of each observational index. Each observational index has its advantages and disadvantages in different robot configuration. At present, the 6 DOF industrial robots are the most commonly applied in research. Meanwhile, fewer studies have been conducted to investigate the calibration performance of different observation indexes for the 6 DOF industrial robots.

In this paper, a robot calibration method based on the hybrid observability index is proposed. The M-DH kinematic model and the error model are presented in Section 2. Section 3 introduces the different observability indexes and IOOPS algorithm. The IOOPS algorithm is applied to select the optimal measured poses. The performance of different observability indexes has been compared. Based on the analysis results, a hybrid observability index O₆ is proposed for robot calibration. Section 4 introduces the LM algorithm to identify the kinematic parameter errors. The experimental results are presented in Section 5. The conclusion of this study and future research are summarized in Section 6.

2. Kinematic Error Model

2.1. M-DH Kinematic Model

As shown in Figure 1A, the industrial robot to be calibrated is the Staubli TX60 robot. The joint coordinate system of the Staubli TX60 robot is illustrated in Figure 1B. The M-DH kinematic model is used to describe the mathematical model of the robot. Based on the definition of the M-DH kinematic model, the transformation matrix A_i between the adjacent joints is given as Equation (1).

$$\begin{array}{cc}{\mathit{A}}_i& =Rot(Z_i,{\theta }_i)Trans(Z_i,d_i)Trans(X_i,a_i)Rot(X_i,{\alpha }_i)Rot(Y_i,{\beta }_i)\hfill \\ & =\left[\begin{array}{cccc}-s{\theta }_{i}s{\alpha }_{i}s{\beta }_i+c{\theta }_{i}c{\beta }_i& -s{\theta }_{i}c{\alpha }_i& s{\theta }_{i}s{\alpha }_{i}c{\beta }_i+c{\theta }_{i}s{\beta }_i& a_{i}c{\theta }_i\\ c{\theta }_{i}s{\alpha }_{i}s{\beta }_i+s{\theta }_{i}c{\beta }_i& c{\theta }_{i}c{\alpha }_i& -c{\theta }_{i}s{\alpha }_{i}c{\beta }_i+s{\theta }_{i}s{\beta }_i& a_{i}s{\theta }_i\\ -c{\alpha }_{i}s{\beta }_i& s{\alpha }_i& c{\alpha }_{i}c{\beta }_i& d_i\\ 0& 0& 0& 1\end{array}\right]\hfill \end{array}$$

(1)

where a_i, d_i, α_i, θ_i, and β_i (i = 1, 2, …, 6) are the nominal values of the link length, the link offset, the link twist angle, the joint angle, and the joint torsion angle of the i-th joint, respectively. s and c are the abbreviations of sin() and cos(), respectively.

Therefore, the nominal pose matrix Tⁿ of the industrial robot can be obtained by Equation (2).

$${\mathit{T}}^n={\mathit{A}}_1^n{\mathit{A}}_2^n{\mathit{A}}_3^n{\mathit{A}}_4^n{\mathit{A}}_5^n{\mathit{A}}_6^n=\left[\begin{array}{cc}{\mathit{R}}_n& {\mathit{P}}_n\\ 0& 1\end{array}\right]$$

(2)

where R_n and P_n are the nominal rotation matrix and the nominal displacement vector, respectively. n denotes the nominal kinematic parameters.

The nominal M-DH parameters of the Staubli TX60 robot are shown in

Table 1. Nominal values of the M-DH kinematic model of the Staubli TX60 robot.

i	θi/rad	di/mm	ai/mm	αi/rad	βi/rad
1	π	0	0	π/2	0
2	π/2	0	290	0	0
3	π/2	20	0	π/2	0
4	π	310	0	π/2	0
5	π	0	0	π/2	0
6	0	70	0	0	0

. However, due to errors introduced by manufacturing and assembly, the kinematic parameters of the industrial robot are inconsistent with the nominal values. The errors of the kinematic parameters a_i, d_i, α_i, θ_i, and β_i are defined as Δa_i, Δd_i, Δα_i, Δθ_i, and Δβ_i. The actual pose matrix T^r of the industrial robot can be obtained by Equation (3).

$${\mathit{T}}^r={\mathit{A}}_1^r{\mathit{A}}_2^r{\mathit{A}}_3^r{\mathit{A}}_4^r{\mathit{A}}_5^r{\mathit{A}}_6^r=\left[\begin{array}{cc}{\mathit{R}}_r& {\mathit{P}}_r\\ 0& 1\end{array}\right]$$

(3)

where R_r and P_r are the actual rotation matrix and the actual displacement vector, respectively. r denotes the actual kinematic parameters.

2.2. Pose Error Model

The pose error of the industrial robot is defined as ∆T, which is the difference between the actual pose T^r and the nominal pose Tⁿ.

$$\Delta \mathit{T}={\mathit{T}}^r-{\mathit{T}}^n$$

(4)

With respect to the kinematic parameters, the partial differentiation is performed on each column vector of the pose matrix T. The high-order differential terms are ignored. Therefore, the linearized kinematic error model is given as (5).

$$\Delta \mathit{E}=\left[\begin{array}{c}{\mathit{P}}^r-{\mathit{P}}^n\\ {\mathit{n}}^r-{\mathit{n}}^n\\ {\mathit{o}}^r-{\mathit{o}}^n\\ {\mathit{a}}^r-{\mathit{a}}^n\end{array}\right]=\left[\begin{array}{c}{\mathit{H}}_{\mathit{p}}\\ {\mathit{H}}_{\mathit{n}}\\ {\mathit{H}}_{\mathit{o}}\\ {\mathit{H}}_{\mathit{a}}\end{array}\right]\Delta \mathsf{\eta }=\mathit{H}\Delta \mathsf{\eta }$$

(5)

where H is the Jacobi matrix of kinematic parameters, $\mathit{H}={\left[\begin{array}{cccc}\begin{array}{cc}{\displaystyle \frac{\partial \cdot }{\partial \mathsf{\theta }}}& {\displaystyle \frac{\partial \cdot }{\partial d}}\end{array}& {\displaystyle \frac{\partial \cdot }{\partial a}}& {\displaystyle \frac{\partial \cdot }{\partial \mathsf{\alpha }}}& {\displaystyle \frac{\partial \cdot }{\partial \mathsf{\beta }}}\end{array}\right]}_{(12\times N)\times 24}$. N is the number of measured poses. $\Delta \mathsf{\eta }={\left[\begin{array}{ccc}\dots & \begin{array}{cccc}\begin{array}{cc}\Delta \mathsf{\theta }& \Delta d\end{array}& \Delta a& \Delta \mathsf{\alpha }& \Delta \mathsf{\beta }\end{array}& \dots \end{array}\right]}_{24\times 1}^T$, $\Delta \mathsf{\eta }$ is the error vector of the kinematic parameters to be identified.

3. Pose Selection Based on a Hybrid Observability Index

3.1. Overview of Observability Indexes

The measured poses used for parameter identification largely influence the calibration results. In order to optimize the effect of robot calibration, five observational indexes were proposed to evaluate the quality of the measured poses. The observational indexes are calculated based on the none-zero singular values of the Jacobi matrix H. The singular value decomposition (SVD) of the Jacobi matrix H is defined as

$$\mathit{U}\sum {\mathit{V}}^T=SVD\left(\mathit{H}\right)$$

(6)

where U and V are all orthogonal matrices. $\mathit{U}={\left[\begin{array}{cccc}{\mathit{u}}_1& {\mathit{u}}_2& \dots & {\mathit{u}}_{(12\times N)}\end{array}\right]}_{(12\times N)\times (12\times N)}$. ∑ is a diagonal matrix composed of singular values. $\mathit{V}={\left[\begin{array}{cccc}{\mathit{v}}_1& {\mathit{v}}_2& \dots & {\mathit{v}}_{24}\end{array}\right]}_{24\times 24}$.

$$\mathsf{\Sigma }={\left[\begin{array}{ccccc}{\sigma }_1& 0& \cdots & 0& 0\\ 0& {\sigma }_2& & 0& 0\\ ⋮& ⋮& & ⋮& ⋮\\ 0& 0& \cdots & 0& {\sigma }_{24}\\ 0& 0& \cdots & 0& 0\\ ⋮& ⋮& & ⋮& ⋮\\ 0& 0& \cdots & 0& 0\end{array}\right]}_{(12\times N)\times 24}$$

(7)

where the value σ_i (i = 1, 2, …, 24) is the none-zero singular value of the Jacobi matrix H. σ₁ ≥ σ₂ ≥ σ₃ ≥ … ≥ σ₂₄.

By substituting the above Equation (7) into (5), the expression is given as follows:

$$\Delta \mathit{E}=\mathit{U}\sum {\mathit{V}}^T\Delta \mathsf{\eta }$$

(8)

From Equations (7) and (8), the singular values σ_i of the Jacobi matrix H correspond to the axes of the hyperellipsoid. It represents the sensitivity of each kinematical parameter to the pose. Menq C. [29] proposed the observability index O₁, defined as (9). O₁ is associated with hyperellipsoid volume. The larger O₁ is, the larger the hyperellipsoid volume is.

$$O_1={\displaystyle \frac{\sqrt[L]{{\sigma }_1{\sigma }_2\cdots {\sigma }_L}}{\sqrt{n}}}$$

(9)

where n is the number of calibration poses. L is the number of kinematic parameters to be identified.

Driels M. [30] proposed the observability index O₂, defined as (10). O₂ represents roundness of the hyperellipsoid.

$$O_2={\displaystyle \frac{{\sigma }_L}{{\sigma }_1}}$$

(10)

Nahvi A. [31] proposed an observability index O₃. O₃ is the smallest singular value. Nahvi A also proposed a hybrid observability index O₄.

$$O_3={\sigma }_L$$

(11)

$$O_4=O_2\times O_3={\displaystyle \frac{{\sigma }_L^2}{{\sigma }_1}}$$

(12)

Sun Y. [32] proposed the observability index O₅. Maximizing the value of O₅ ensures that the hyperellipsoid volume is maximized.

$$O_5={\displaystyle \frac{1}{\frac{1}{{\sigma }_1}+\frac{1}{{\sigma }_2}+\cdots +\frac{1}{{\sigma }_L}}}$$

(13)

3.2. Selecting Optimal Poses for Identification

The iterative one-by-one pose search (IOOPS) algorithm is derived from the DETMAX algorithm [39]. The IOOPS algorithm can search for a given number of optimal poses within an infinite pose set. The flowchart of the IOOPS algorithm is shown in Figure 2. Firstly, an initial pose set U is randomly determined. The pose set U is divided into the selected pose set γ^m and alternative pose set γ^R. Secondly, the add algorithm and remove algorithm are used to select the optimal poses. The parameter descriptions of the IOOPS algorithm are shown in

Table 2. Parameter description of the IOOPS algorithm.

Parameters	Clarification
U	Initial pose set, a randomly generated pose set in the measured poses
γm	The selected pose set
γm+1	The pose set γm is added by 1, and the set of selected pose set is γm+1
γR	The alternative pose set, γR = U − γm
O(γm)	Observability index of the selected pose set
γmbest	Current best pose set
ζi	Pose (i = 1, 2, …, n)
ζ+	The pose satisfying the condition max {O6(γm + ζi)}
ζ−	The pose satisfying the condition max {O6(γm+1 − ζi)}
O(best)	Observability index of the current best pose set
num	O(best) Number of times constant or approximate
K	Randomly exchanged the number of poses between γm and γR
a	The growth threshold of num
b	When num does not reach a, K = b
c	When num reaches a, K = c
d	The growth threshold 2 of num, algorithm terminates when num reaches d

. Details of the add algorithm and remove algorithm are shown as follows:

In the add algorithm, the pose ζ_i(i = 1, 2,…, n) is taken sequentially from the alternative pose set γ^R. If the max{O₆(γ^m + ζⁱ)} is maximum, the ζⁱ is denoted as ζ⁺. The selected pose set γ^m adds one pose, becoming γ^m+1. Thus, γ^m+1 = γ^m + ζ⁺.

In the remove algorithm, the pose ζⁱ(i = 1, 2,…, n) is removed sequentially from the selected pose γ^m+1. If the max{O₆(γ^m+1 − ζⁱ)} is maximum, the ζⁱ is denoted as ζ⁻. The selected pose set γ^m+1 deletes one pose. The selected pose becomes γ^m. γ^m = γ^m+1 − ζ⁻.

3.3. A Novel Hybrid Observability Index

In order to analyze the performance of different observability indexes, the four independent observability indexes (O₁, O₂, O₃, and O₅) are chosen for comparison. As given in Equation (12), the observability index O₄ is a combination of O₂ and O₃. Therefore, the observability index O₄ is not discussed. In the analysis process, 50 optimal poses are first selected from 200 measured poses with the IOOPS algorithm based on the four different observability indexes. The remaining 150 measured poses are used for validating the identified kinematic models. Secondly, the kinematic parameter errors are identified by the LM algorithm. The attitude errors and the positioning errors before and after robot calibration are shown in Figure 3 and Figure 4. In this paper, the average comprehensive position error (E_ACPE) and the average comprehensive attitude error (E_ACAE) are defined to evaluate the accuracy performance of the industrial robot.

$${\mathit{E}}_{ACPE}={\displaystyle \frac{1}{M}}{\displaystyle \sum _{j=1}^M\sqrt{(\Delta x_{\mathit{j}}^2+\Delta y_{\mathit{j}}^2+\Delta z_{\mathit{j}}^2)}}$$

(14)

$${\mathit{E}}_{ACAE}={\displaystyle \frac{1}{M}}{\displaystyle \sum _{j=1}^M\sqrt{(\Delta {\varphi }_{xj}^2+\Delta {\varphi }_{yj}^2+\Delta {\varphi }_{zj}^2)}}$$

(15)

where M denotes the number of measured poses,$\Delta x$,$\Delta y$, and $\Delta z$ are the position errors in the x-axis, y-axis, and z-axis. $\Delta {\varphi }_x$, $\Delta {\varphi }_y$, and $\Delta {\varphi }_z$ are the attitude errors in the x-axis, y-axis, and z-axis.

Table 3. Iimprovement ratio of Robot positioning and attitude accuracy for different observability index.

Improvement Ratio	Different Set	Observability Index
		O1	O2	O3	O5
Attitude accuracy	Identification set	45.0%	47.4%	52.6%	50.0%
	Validation set	44.4%	50.0%	50.0%	−31.6%
Positioning accuracy	Identification set	88.2%	85.9%	88.1%	79.1%
	Validation set	87.3%	86.8%	89.6%	89.2%

presents the improvement ratio of the attitude accuracy and positioning accuracy in the identification and validation sets. As shown in Table 3, O₅ is worst in terms of E_ACAE in validation set. O₁ and O₃ are better in terms of E_ACPE in both identification and validation sets. Moreover, O₃ is better in terms of E_ACAE in both identification and validation sets. Experimental results show that O₁ and O₃ have better performance. The poses selection is an optimized procedure which maximizes the observability index value. To ensure the maximum values of O₁ and O₃ simultaneously, a hybrid observability index O₆ is proposed. The expression of O₆ is given as (16).

$$O_6=O_1+{\mathit{kO}}_3$$

(16)

where k is the coefficient. The coefficient is determined, as described in Section 5.1.

4. Parameters Identification

There are plenty of algorithms for kinematic parameter identification, of which the most commonly employed is the least squares (LS) optimization algorithm. The formula of the LS optimization algorithm is given based on Equation (17) as follows:

$$\Delta \mathsf{\eta }={\mathit{H}}^{+}\cdot \Delta E={({\mathit{H}}^T\mathit{H})}^{-1}\cdot {\mathit{H}}^T\cdot \Delta E$$

(17)

When the matrix H^TH is invertible or a pathological matrix, the LS optimization algorithm cannot find the correct solution. The Levenberg–Marquardt (LM) optimization algorithm is an improvement on the LS optimization algorithm. The LM optimization algorithm can avoid this problem through adding a unit matrix. In this paper, the LM algorithm is used to identify the kinematic parameter errors. The formula of the LM optimization algorithm is given as follows:

$$\Delta \mathsf{\eta }={ {\mathit{H}}^{\prime }}^{+}\cdot \Delta E={({\mathit{H}}^T\mathit{H}+\mu \mathit{I})}^{-1}\cdot {\mathit{H}}^T\cdot \Delta E$$

(18)

where μ is a coefficient with an initial value of 0.01. I is the unit matrix.

Based on the M-DH kinematic error model obtained above, d₂, β₁, β₃, β₄, β₅, and β₆ are no need to be identified. Therefore, only 24 kinematic parameter errors need to be identified. As is well known, the position error and attitude error have a great difference in dimension and magnitude. To simplify the optimization process, the position error fitness function and the attitude error fitness function are fused in (19).

$$f=\min \left({\displaystyle \sum _{j=1}^N\sqrt{L({\mathit{n}}_j\cdot {\mathit{n}}_j+{\mathit{o}}_j\cdot {\mathit{o}}_j+{\mathit{a}}_j\cdot {\mathit{a}}_j)+{\mathit{p}}_j\cdot {\mathit{p}}_j}}\right)$$

(19)

where N is the number of measured poses, N = 50. L is the adjustment factor, which is used to balance position error and attitude vector in the objective function, L = 100.

5. Experiment Results

5.1. Robot Calibration Experimental Platform

As shown in Figure 5, the industrial robot calibration system is established in this paper. The laser tracker Leica AT960 is used to measure the poses of the Staubli TX60 robot. The T-Mac tool of the laser tracker is installed on the industrial robot flange. Spatial Analyzer and Matlab are used for processing the measured data. The positioning measurement uncertainty of the laser tracker is 15 μm + 6 μm/m × Dm. D indicates the distance between the laser tracker and the T-Mac tool. The measured distance in the experiment is nearly 1.5 m. The Staubli TX60 robot to be calibrated in this paper has a repetitive positioning accuracy of ±0.02 mm and a rated load of 3 Kg. The transformation relation between the measurement coordinate frame and the base coordinate frame can be obtained through the method presented in [40]. The experimental measurement procedure is as follows:

(1)

The transformation between the base coordinate frame of the Staubli TX60 robot and the measurement coordinate frame of the laser tracker Leica AT960.

(2)

The transformation between the frame of T-Mac tool and the tool frame of the Staubli TX60 robot.

(3)

The base coordinate frame of the Staubli TX60 robot was used as the reference coordinate system.

The measurement processes involved in this paper are in accordance with the ISO-9283 performance specification for industrial robots and its test method standard [41]. There are 1200 measured poses selected randomly in the front motion space of the robot. The motion space is a cube of which length is 1 m. The measured poses are selected by the RoboDyn software. The measured poses are distributed in the measured motion space as much as possible.

5.2. Robot Calibration Based on the Proposed Hybrid Observability Index

When the number of poses increases to a certain number, the identification accuracy does not increase. Therefore, the optimal number of poses should firstly be determined by the selection algorithm. In the algorithm, two random poses are selected among 200 measured poses initially. The six observability indexes of the selected poses are calculated. The above process is repeated T times. The number of randomly selected poses gradually increases. As shown in Figure 6a,b, it can be seen that the observability indexes O₁ and O₂ steadily increase when the number of poses reaches 50. From Figure 6c,f, it is shown that the observability indexes O₃ to O₆ increase monotonically. Therefore, the optimal number of poses is set as 50 for cross-comparison and fast calibration.

As shown in Figure 6a,c, the ratio of O₁ and O₃ stable values is closed to 10. Therefore, the coefficient k in Equation (16) is initially set as 10. To compare the performance and the stability of the proposed hybrid observability index O₆, 1200 measured poses are divided into six groups. Five groups are selected to verify the stability of the proposed hybrid observability index O₆, and the other one group is used to verify its performance.

Firstly, the IOOPS algorithm is applied to select 50 optimal poses from 200 measured poses in each group based on the proposed O₆ index. The optimal poses are used for parameter identification, and the remaining 150 poses are used for verification. Secondly, the parameter identification is conducted through the LM algorithm. The E_ACPE of the industrial robot after calibration was used to evaluate the comprehensive error of the method. Moreover, the maximum comprehensive position error (E_MCPE) and the maximum comprehensive attitude error (E_MCAE) are defined to evaluate the accuracy performance of the industrial robot.

$${\mathit{E}}_{MCPE}=Max(\sqrt{(\Delta x_{\mathit{j}}^2+\Delta y_{\mathit{j}}^2+\Delta z_{\mathit{j}}^2)})$$

(20)

$${\mathit{E}}_{MCAE}=Max(\sqrt{(\Delta {\varphi }_{xj}^2+\Delta {\varphi }_{yj}^2+\Delta {\varphi }_{zj}^2)})$$

(21)

where Max denotes the maximum value in the results.

The results of robot calibration are shown in Figure 7 and Figure 8. E_ACPE of the O₁, O₃, and O₆ are almost identical. The E_MCPE of the proposed O₆ is better than O₃. As summarized in

Table 4. Robot positioning and attitude enhancement accuracy for different observability indexes.

EMCPE after Calibration in Five Groups	O1	O2	O3	O4	O5	O6
Maximum value	0.2504	0.2502	0.3817	0.2510	0.7526	0.2509
Minimum value	0.1923	0.2077	0.1894	0.2096	0.3047	0.2201
Absolute deviation	0.0581	0.0425	0.1923	0.0414	0.4479	0.0308

, it can be seen that the calibration performance of the proposed hybrid observability index O₆ is more stable than other observability indexes.

To verify the accuracy performance of the proposed hybrid observability index O₆, 50 optimal poses were firstly selected from the sixth group. The identification results of the kinematic parameter errors of the Staubli TX60 robot are given in

Table 5. The identified parameter errors of the calibrated robot.

i	θi/rad	di/mm	ai/mm	αi/rad	βi/rad
1	−3.2792 × 10−4	−0.2226	−0.2389	5.3341 × 10−5	-
2	7.8894 × 10−4	-	0.3223	−5.4318 × 10−5	3.8910 × 10−5
3	2.3603 × 10−4	0.0826	0.0484	3.7122 × 10−4	-
4	0.0014	0.2044	−0.0812	1.1155 × 10−4	-
5	5.1444 × 10−4	7.4762 × 10−5	0.0205	3.9926 × 10−5	-
6	4.0758 × 10−4	−0.0624	−0.1351	0.0033	-

Note: “-” indicates that the parameter is ignored.

. The E_ACPE and E_ACAE of the Staubli TX60 robot is reduced from (0.716 mm, 0.167°) to (0.082 mm, 0.077°). The position and attitude accuracy performances of the Staubli TX60 robot are improved by 89% and 49%. The E_MCPE and E_MCAE of the Staubli TX60 robot is reduced from (1.187 mm, 0.281°) to (0.22 mm, 0.177°) by the LM algorithm. The position and attitude accuracy performances of the Staubli TX60 robot are improved by 81% and 37%. It can be seen that highly accurate parameter identification can be achieved based on the proposed hybrid observability index O₆.

5.3. Comparison of Robot Calibration Algorithms

The calibration performance of the LM, PSO, and DE algorithms are compared. The results are illustrated in Figure 9 and Figure 10. The E_ACPE and E_ACAE of the Staubli TX60 robot are reduced from (0.716 mm, 0.167°) to (0.2396 mm, 0.080°) by the PSO algorithm. Meanwhile, the E_ACPE and E_ACAE of the Staubli TX60 robot are reduced from (0.716 mm, 0.167°) to (0.1935 mm, 0.097°) by the DE algorithm. In summary, the accuracy of the parameters identified using the LM algorithm is better than the PSO and DE algorithms.

6. Discussion

The observability index is actually a loss function for optimizing the selected poses used for identification. Five observability indexes were investigated for the calibration of various robots. The calibration effectiveness of the typical observability indexes for six DOF serial industrial robots is compared through the experiments. Studies also show that O₁ has have better performance. The experimental results in this paper show that the selected poses based on O₁ and O₃ are relatively effective in positioning accuracy; meanwhile, O₃ is more effective in attitude accuracy. Therefore, both O₁ and O₃ were used as criteria for selecting measured poses, and the positioning and attitude accuracy of the calibration performance both improved. Based on the handling of objective functions in multi-objective optimization methods, a hybrid observability index O₆ is expressed as (16). Firstly, the target number of the selected poses is determined. The 50 selected poses can ensure the calibration efficiency and cross-comparison. Secondly, the coefficient k in Equation (16) is determined as 10. The performance of each observability index is compared with six different groups of measured poses. As shown in Figure 7 and Figure 8, the positioning accuracy E_ACPE and E_MCPE of O₆ is consistent with O₁. Moreover, the attitude accuracy E_ACAE and E_MCAE of O₆ is also consistent with O₁. As is well known, the coefficient k in Equation (16) can balance the effects of the observability indexes O₁ and O₃. The coefficient k can be adjusted to obtain a better performance in both positioning accuracy and attitude accuracy. However, it is hard to find the best solution of the coefficient k.

The selected measured poses are applied to identify the kinematic parameters with different optimization algorithms. The comparison results show that the LM algorithm is better than the PSO and DE algorithms.

7. Conclusions

This paper proposes a hybrid observability index for selecting the optimal poses to improve the accuracy performance of the industrial robot. Firstly, the kinematic error model of the robot is constructed based on the M-DH model. Secondly, the performance of each observability index is evaluated, and the proposed hybrid observability index O₆ is defined. O₆ is the linear combination of O₁ and O₃. The optimal poses is obtained by using the IOOPS algorithm. Thirdly, the fitness function and identification algorithm are presented based on the kinematic error model. Finally, several experiments have been conducted to evaluate the performance of the proposed hybrid observability index O₆. Based on the proposed hybrid observability index O₆, the ACPE and ACAE of the Staubli TX60 robot is reduced from (0.716 mm, 0.167°) to (0.082 mm, 0.077°). The average position error and average attitude error of the Staubli TX60 robot were reduced by 89% and 49%. The results show that the proposed hybrid observability index O₆ has great stability and effectiveness for robot calibration.

Future research will focus on the multi-objective optimization algorithm for selecting measured poses based on the two observability indexes O₁ and O₃. Moreover, the stiffness error compensation of the self-weight and external loads can make the robot suitable for complex working conditions. It will also be the primary focus of research in the future.